cosmology at the turning point of relativity revolution. the debates

UNIVERSITA DEGLI STUDI DI PADOVA

Dipartimento di Astronomia

Scuola di Dottorato di Ricerca in Astronomia

XXI Ciclo (2006 - 2008)

Tesi di Dottorato

Cosmology at the turning point

of relativity revolution.

The debates during the 1920’s

on the “de Sitter Effect”

Direttore della Scuola: Prof. Giampaolo PIOTTO

Dipartimento di Astronomia - Universita di Padova

Supervisore: Prof. Giulio PERUZZI

Dipartimento di Fisica - Universita di Padova

Co-supervisore: Prof. Luigi SECCO

Dipartimento di Astronomia - Universita di Padova

Dottorando: Matteo REALDI

UNIVERSITY OF PADOVA

Department of Astronomy

Ph.D. in Astronomy

XXI Cycle (2006 - 2008)

Dissertation Thesis

Cosmology at the turning point

of relativity revolution.

The debates during the 1920’s

on the “de Sitter Effect”

Ph.D. Coordinator: Prof. Giampaolo PIOTTO

Department of Astronomy - University of Padova

Supervisor: Prof. Giulio PERUZZI

Department of Physics - University of Padova

Co-supervisor: Prof. Luigi SECCO

Department of Astronomy - University of Padova

Candidate: Matteo REALDI

A mia mamma e mio papa.

A Gio, Anna, Luca,

Vale, Albe, Matteino, Bettax.

Alla zia Gisa e alla nonna Laura.

A Giuliano e Marinella.

Soprattutto, a Michela.

La chair est triste, helas! Et j’ai lu tous les livres.

(Mallarme, Brise marine)

Abstract

This thesis is devoted to a critical analysis of the cosmological debates

which took place during the 1920’s about the so-called “de Sitter effect”,

which represents the linking thread between the 1917 beginning of the-

oretical relativistic cosmology and the 1930 first diffusion and general

acceptance of the model of the expanding universe.

The de Sitter effect is a theoretical redshift-distance relation which

can be derived from the cosmological solution of field equations proposed

by de Sitter. This solution and the solution proposed by Einstein repre-

sent the first theoretical relativistic cosmological models. They appeared

in 1917, when stars, not yet galaxies, where considered the fundamental

pieces filling the universe, and the expanding universe still had to enter

modern cosmology. During the 1920’s it was just the de Sitter effect

which played a fundamental role in the first pioneering attempts to re-

late the theoretical relativistic description of the universe to astronomical

observations.

The models of the universe proposed by Einstein and de Sitter, both

based on general relativity, soon appeared as revolutionary tools in order

to investigate the properties of the universe as a whole and the connec-

tion among space, time and gravitation. In his own spherical model,

Einstein proposed a static, finite and unbounded universe. In this uni-

verse, according to Machian inspiration, inertia was fully determined by

all masses. Dealing with the universe as a whole and its properties,

Einstein took into account a hypothetical density of matter which was

uniformly and homogeneously distributed through space, foreshadowing

VII

VIII Abstract

what became later known as the Cosmological Principle. Einstein modi-

fied his field equations and introduced a new term with the cosmological

constant λ, which in Einstein’s intentions acted like an anti-gravity, in

order to express in general relativity his model of a static universe.

On the contrary, de Sitter found a suitable solution of field equations

which corresponded to a completely empty and static world. In de Sit-

ter’s static universe, a spectral displacement was expected from a mass

test for the form of the metric and the geodesic equations. This property

of de Sitter’s universe became known as the de Sitter effect. Already

in 1917 de Sitter related spectral shifts to velocity and distance of as-

tronomical objects through his own relativistic solution. He proposed

that spectral displacements which were observed in some stars and neb-

ulæ could be interpreted in his static and empty world as an apparent

(spurious) velocity of test particles due to the peculiar line element, su-

perimposed to a relative (Doppler) velocity which resulted from geodesic

equations. The first contribution led to a quadratic redshift-distance (or

equivalently velocity-distance) relation, while the latter involved a linear

dependence.

This first suggestion did not pass unnoticed, and during the 1920’s

several scientists dealt with the properties of de Sitter’s universe and pro-

posed different formulations of the redshift-distance effect which resulted

by the metric of such a model. Despite its lack of matter, de Sitter’s

universe attracted the attention of scientists because it offered more ad-

vantages than Einstein’s one with regard to astronomical consequences

and observations.

According to Eddington, a general cosmic recession was expected in

de Sitter’s universe just because of the presence of the cosmological con-

stant. Such a tendency of particles to scatter, which Eddington pro-

posed in 1923, could roughly account for the astonishing radial velocities

measured by Slipher in spiral nebulæ, the most part of which revealed

receding motions from the observer.

Moreover, the geometry of de Sitter’s world-model was not uniquely

Abstract IX

determined, and non-static pictures emerged by appropriate coordinate

changes, as done in the 1920’s by Weyl, Lanczos, Lemaıtre and Robert-

son. Their contributions marked the actual departure from the metric

of a static universe, by using a stationary frame of de Sitter’s model.

All of them took into account the empirical evidence of relevant veloci-

ties in spiral nebulæ, and each of them proposed an own version of the

redshift-distance relation in de Sitter’s world.

In 1924 Wirtz realized that the universe of de Sitter represented a

suitable model accounting for redshift and apparent diameter measured

in spiral nebulæ. In the same year, on the contrary, Silberstein criticized

the possibility of a general cosmic recession, and considered the distances

of globular clusters in order to verify the de Sitter effect. However, the

correctness of the method and the results proposed by Silberstein were

shortly after denied by Lundmark and Stromberg.

The de Sitter effect could offer an answer to the question of relevant

redshift measurements in nebulæ, however there was an ambiguous for-

mulation of such a theoretical relation between velocities and distances.

Nevertheless, up to 1930 such an effect was the only possible, however

puzzling, explanation of the redshift problem. A suitable test of redshift

relations was possible only with reliable determinations of distance of

spiral nebulæ.

In this controversial picture, the contributions of Hubble marked a

turning point in the comprehension of the structure of the universe.

Thanks to the revolutionary observations which Hubble furnished dur-

ing the 1920’s, spiral nebulæ were finally accepted in 1925 as ‘island

universes’, i.e. as true extra-galactic stellar systems. In 1929 Hubble

confirmed that such systems receded relatively to one another, and that

their radial velocities linearly increased with distances.

The puzzling question of the interpretation of the de Sitter effect

and the meaning of redshift was solved in 1930, when the cosmology of

Lemaıtre was reconsidered in order to explain the cosmic recession of

galaxies revealed by Hubble. The model of a non-empty expanding uni-

X Abstract

verse which Lemaıtre had already proposed in 1927 provided the proper

cosmological interpretation of redshift: the displacement of spectral lines

was due to the expansion of the universe. The cosmological solution of

Lemaıtre corresponded to a “third way” between the Einstein’s model,

which had matter but not motion, and the de Sitter’s model, which had

motion but not matter.

Since 1930, also the cosmological consequences of similar solutions

proposed by Friedmann in 1922 and 1924 were fully acknowledged. In

their analysis, Friedmann and Lemaıtre took into account dynamical

models of the universe, i.e. they considered the possibility of a not

empty, homogeneous and isotropic universe which world-radius increased

in time. Static and stationary models were eventually seen as limiting

cases of solutions of field equations describing an expanding universe.

As from 1930, the de Sitter effect, which during the 1920’s represented

the first hint in the intersection between the new theory of gravitation

and observed facts, was seen as an effect of minor importance, and the

expanding universe inaugurated another chapter of modern cosmology.

The historical analysis which is below proposed is useful to highlight

the richness of contributions, attempts and controversies which appeared

in the early connections between astronomical observations and predic-

tions offered by relativistic cosmology. In particular, scientists involved in

the 1920’s debates about the de Sitter effect approached and thoroughly

analyzed some fundamental questions in the framework of relativistic

cosmology, such as the nature of redshift measurements, the geometry of

space, the assumption of a homogeneous and isotropic universe.

Several passages from primary literature, original manuscripts, un-

published sources and correspondence among scientists have often been

quoted, in order to highlight the very contributions by actors involved in

those debates.

From the present thesis it emerges the fundamental role played by

the de Sitter effect in the 1920’s debates. It was a very fruitful phase for

the introduction of new ideas, discoveries and changes, and the history

Abstract XI

of that period permits to understand how cosmology developed passing

from a sphere of theoretical speculations to a truly empirical science.

Riassunto

Ci si riferisce alla cosmologia moderna, o scientifica, come alla scienza

che studia l’origine e l’evoluzione dell’universo, interpretando il quadro

che ne risulta sulla base delle leggi della fisica.

In questa tesi viene proposta una ricostruzione storica del ruolo svolto

dal cosiddetto “effetto de Sitter” durante le prime fasi della cosmologia

moderna. Vengono ricostruiti in particolare i dibattiti cosmologici cen-

trati su tale effetto che ebbero luogo negli anni Venti, prima cioe che

il concetto di universo in espansione facesse la sua comparsa ufficiale

nella storia della cosmologia moderna. Lo studio e basato prima di tutto

sulla documentazione originale dell’epoca, sulle corrispondenze tra scien-

ziati e su manoscritti inediti. L’indagine storica permette di evidenziare

l’importanza fondamentale che l’effetto de Sitter ebbe in quei dibattiti.

Proprio attorno a tale effetto, infatti, ruotarono i primi confronti tra le

previsioni teoriche relativistiche e le osservazioni su scala cosmologica, fa-

vorendo quindi il passaggio della cosmologia da pura discussione teorica

a scienza empirica.

L’effetto de Sitter si riferisce ad una relazione teorica tra redshift e dis-

tanza, e deriva dalla scelta della metrica e dalle equazioni delle geodetiche

nell’universo vuoto proposto da de Sitter. Questo modello di universo e il

modello di Einstein rappresentano i primi due approcci teorici relativis-

tici al problema cosmologico, e furono entrambi proposti nel 1917. Tali

modelli sono, appunto, basati sulla relativita generale, la cui diffusione

implico una svolta nella comprensione dell’intreccio tra spazio, tempo e

gravitazione e, in particolare, nello studio scientifico dell’universo come

XIII

XIV Riassunto

un tutto.

All’inizio del secolo scorso si considerava ancora l’universo come statico

e costituito essenzialmente da stelle e nebulæ. Bisognera attendere la

meta degli anni Venti perche venga accettata l’idea di un universo cos-

tituito da galassie, ritenute da allora i “mattoni” fondamentali nella de-

scrizione del cosmo.

Nel 1917 Einstein propose un modello di universo sferico in cui la ma-

teria era distribuita in maniera uniforme ed omogenea. La gravitazione

veniva interpretata come curvatura dello spazio e la materia era intera-

mente responsabile dell’origine dell’inerzia, soddisfacendo quello che Ein-

stein introdusse piu tardi come “principio di Mach”. Per rendere coerenti

i risultati della relativita generale con la supposta staticita dell’universo,

Einstein aveva introdotto nelle equazioni di campo un termine aggiuntivo

contenente la costante cosmologica λ. Questa nuova costante era inter-

pretabile come una sorta di repulsione cosmica, capace di controbilanciare

a grandi distanze l’effetto della gravita e di rendere statico l’universo.

Sempre nel 1917 apparve, ad opera di de Sitter, un altro lavoro sulla

cosmologia, impostato anch’esso sulla teoria della relativita generale. De

Sitter mostrava che se la densita media della materia nell’universo poteva

essere considerata nulla, la metrica dello spazio-tempo che ne risultava

comportava un mondo statico con condizioni fisiche coerenti.

A partire dal 1917 e nel corso degli anni Venti la discussione cosmo-

logica fu centrata essenzialmente su quale di questi due modelli potesse

rappresentare al meglio il cosmo, se il modello statico di Einstein o quello

“vuoto” di de Sitter. Tra i due modelli, quello di de Sitter forniva la pos-

sibilita di interpretare gli spostamenti osservati nelle righe spettrali di

stelle e nebulæ. Gia de Sitter aveva infatti notato che nel suo modello

di universo una particella di prova non poteva rimanere in uno stato di

quiete, ma avrebbe mostrato un moto rispetto all’osservatore. Tale re-

lazione teorica tra velocita (redshift) e distanza divenne in seguito nota

come effetto de Sitter. La causa di questo spostamento spettrale predetto

dall’effetto de Sitter poteva essere ricondotta sia ad un contributo grav-

Riassunto XV

itazionale che ad un effettivo moto relativo tra l’osservatore e l’oggetto

osservato (effetto Doppler).

Numerosi scienziati presero in considerazione le proprieta dell’universo

di de Sitter, e proposero differenti interpretazioni e formulazioni dell’effetto

de Sitter.

Nella sua analisi del 1923 sui modelli cosmologici relativistici, Ed-

dington riteneva che dalle proprieta dell’universo di de Sitter si potesse

dedurre una recessione cosmica estendibile a tutto l’universo. Questa

tendenza delle particelle ad allontanarsi le une dalle altre era dovuta,

secondo Eddington, alla presenza della costante cosmologica, e poteva

rendere conto delle sorprendenti velocita radiali di alcune nebulæ mis-

urate proprio in quegli anni da Slipher.

La geometria dell’universo di de Sitter non era univocamente deter-

minata, e alcuni protagonisti del dibattito cosmologico degli anni Venti

proposero modelli stazionari dell’universo di de Sitter. In questo ambito,

i contributi di Weyl, Lanczos, Lemaıtre e Robertson segnarono effettiva-

mente il distacco dallo studio di elementi di linea di universi puramente

statici. Ognuno di loro tento quindi di interpretare l’evidenza osservativa

delle elevate velocita radiali misurate nelle nebulæ, proponendo versioni

differenti della relazione redshift-distanza nell’universo di de Sitter.

Nel 1924 Wirtz propose una relazione tra la velocita di allontana-

mento e il diametro apparente misurato nelle nebulæ. Secondo Wirtz,

dunque, l’universo di de Sitter risultava idoneo per spiegare una reces-

sione cosmica suggerita dalle osservazioni. Nello stesso anno, al con-

trario, Silberstein critico tale tendenza a recedere, e tento di confermare

l’effetto de Sitter attraverso le distanze degli ammassi globulari. Di lı a

poco, Lundmark e Stromberg mostrarono che il metodo e i risultati di

Silberstein non erano corretti.

Le controversie legate all’effetto de Sitter e all’interpretazione del red-

shift osservato nelle nebulæ potevano trovare una soluzione solo tramite

una corretta stima delle distanze di tali oggetti. A tal proposito i con-

tributi di Hubble segnarono una svolta nella comprensione del cosmo.

XVI Riassunto

Grazie alle fondamentali osservazioni di Hubble, a partire dal 1925 le

nebulose a spirale vennero finalmente considerate come veri e propri sis-

temi extragalattici del tutto simili alla nostra galassia, la Via Lattea. Nel

1929, inoltre, Hubble confermo che esisteva una recessione cosmica, e che

ogni galassia si allontanava dalle altre con una dipendenza lineare tra la

velocita e la distanza.

Nel 1930 l’enigma legato all’effettiva causa del redshift trovo final-

mente una soluzione. In quell’anno, infatti, Eddington e de Sitter col-

legarono le osservazioni della recessione cosmica contenuta nelle osser-

vazioni di Hubble con le predizioni teoriche di nuovi modelli d’universo

basati sulla relativita generale. Essi interpretarono correttamente l’espan-

sione del cosmo evidenziata dalle osservazioni rivalutando un modello di

universo proposto ancora nel 1927 da Lemaıtre, ma passato inosservato.

L’idea di Lemaıtre era basata sulla formulazione di un modello interme-

dio tra quello di Einstein, che conteneva materia ma non movimento, e

quello di de Sitter, che era vuoto ma al contrario prediceva un movimento

delle particelle di prova.

Nella proposta di Lemaıtre era contenuta la spiegazione cosmolog-

ica del redshift: la causa dello spostamento delle righe spettrali andava

ricondotta proprio all’espansione dell’universo.

Solo a partire da allora venne interamente apprezzato anche il la-

voro svolto gia nel 1922 e nel 1924 da Friedmann. Questi aveva ottenuto

soluzioni coerenti delle equazioni di campo ammettendo una densita me-

dia di materia non nulla e una dipendenza della metrica dal tempo. Le

soluzioni di Friedmann prevedevano un’evoluzione dell’universo indipen-

dentemente dalla presenza o meno della costante cosmologica.

A partire dal 1930, dunque, il concetto dell’universo in espansione

inauguro un nuovo capitolo della cosmologia moderna. L’effetto de Sit-

ter venne da allora visto come un effetto di importanza secondaria, e

l’interesse in questa relazione tra distanza e redshift ando via via dimin-

uendo.

Come evidenziato nella presente tesi, fu proprio l’effetto de Sitter che

Riassunto XVII

caratterizzo il tortuoso percorso della cosmologia moderna relativistica

dalla concezione di un universo statico a quella di un universo in espan-

sione. I dibattiti cosmologici degli anni Venti, come avviene in tutte le

fasi di scoperta, furono quanto mai fecondi per la produzione di argomen-

tazioni e di idee innovative. I protagonisti di quei dibattiti affrontarono

in quegli anni questioni fondamentali quali il significato del redshift, la

metrica dell’universo, il confronto tra predizioni teoriche ed evidenze os-

servative, inaugurando in questa maniera l’approccio moderno alla com-

prensione dell’universo come un tutto. Anche se oggi pochi conoscono

quel periodo, e pur certo, tanto piu alla luce degli attuali sviluppi, il ruolo

fondamentale che questi dibattiti svolsero. La storia di quel periodo e un

significativo esempio di come procede l’impresa scientifica.

Contents

Abstract VII

Riassunto XIII

1 Introduction 1

1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Some issues of present cosmology 7

2.1 The Cosmological Principle . . . . . . . . . . . . . . . . 9

2.2 The Robertson-Walker metric . . . . . . . . . . . . . . . 11

2.3 Redshifts in cosmology . . . . . . . . . . . . . . . . . . . 16

2.3.1 Velocities and distances . . . . . . . . . . . . . . 18

2.4 Dark matter and dark energy . . . . . . . . . . . . . . . 22

3 Cosmology at the beginning of XX Century 27

3.1 The sidereal universe and the nebulæ . . . . . . . . . . . 27

3.2 Cosmological difficulties with Newtonian theory . . . . . 35

3.2.1 Olbers’s paradox . . . . . . . . . . . . . . . . . . 37

4 1917: the universes of general relativity 39

4.1 Einstein, the universe and the relativity of inertia . . . . 39

4.1.1 The debate with Willem de Sitter . . . . . . . . . 42

4.1.2 Towards the solution . . . . . . . . . . . . . . . . 49

4.1.3 “A ‘finite’ and yet ‘unbounded’ universe” . . . . . 52

4.1.4 The cosmological constant . . . . . . . . . . . . . 55

XIX

XX CONTENTS

4.2 The universe of de Sitter . . . . . . . . . . . . . . . . . . 58

4.2.1 The “mathematical postulate of relativity of inertia” 59

4.2.2 A universe without “world matter” . . . . . . . . 61

4.2.3 Einstein’s criticism . . . . . . . . . . . . . . . . . 63

5 The “de Sitter Effect” 71

5.1 De Sitter’s first suggestion . . . . . . . . . . . . . . . . . 74

5.1.1 Redshifts in de Sitter’s universe . . . . . . . . . . 79

5.2 Matter or motion? Eddington’s analysis . . . . . . . . . 90

5.3 Weyl, Lanczos and the redshift-distance law . . . . . . . 97

5.4 Silberstein’s contributions . . . . . . . . . . . . . . . . . 105

5.5 Lemaıtre’s 1925 notes . . . . . . . . . . . . . . . . . . . . 112

5.6 Shifts in de Sitter’s universe according to Robertson and

Tolman . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6 Observational investigations of redshift relations 129

6.1 The nature of the nebulæ around 1920 . . . . . . . . . . 129

6.2 Slipher and the radial velocities of spirals . . . . . . . . . 134

6.3 The K term and the solar motion . . . . . . . . . . . . . 136

6.3.1 Wirtz and de Sitter’s cosmology . . . . . . . . . . 139

6.4 Astronomers at work: Lundmark and Stromberg . . . . . 142

6.5 Hubble and the universe of galaxies . . . . . . . . . . . . 150

6.5.1 The contributions by Humason . . . . . . . . . . 153

6.5.2 Hubble’s 1929 relation . . . . . . . . . . . . . . . 154

7 The “third way” between Einstein’s and de Sitter’s solu-

tions 159

7.1 1930: Eddington, de Sitter and the expanding universe . 160

7.2 The importance of Lemaıtre’s 1927 proposal . . . . . . . 163

7.2.1 Rediscovering the models of Friedmann . . . . . . 169

7.3 The decline of the interest in the de Sitter effect . . . . . 172

Conclusion 177

CONTENTS XXI

Bibliography 183

Primary literature . . . . . . . . . . . . . . . . . . . . . . . . 183

Secondary literature . . . . . . . . . . . . . . . . . . . . . . . 201

Acknowledgements 211

Chapter 1

Introduction

Cosmology is the study of the origin, structure and evolution of the

universe as a whole, based on interpretations of astronomical observations

at different wave-lengths through laws of physics.

This thesis is devoted to a historical reconstruction of some debates

which happened less than a century ago, during the early developments

of modern scientific cosmology. Emphasis is placed on the first descrip-

tions of the universe as a whole at the turning point of general relativity

revolution, focusing on the pioneering attempts to relate theoretical pre-

dictions of relativistic cosmology with astronomical data.

Particular attention is given to the debates which took place during

the 1920’s about the so-called “de Sitter effect”, a redshift-distance rela-

tion which resulted from the model of the universe proposed by Willem

de Sitter (1872-1934). This cosmological solution and the solution pro-

posed by Albert Einstein (1879-1955), both based on relativistic theory

of gravitation, represent the first models of modern cosmology. They

were proposed by Einstein and de Sitter in 1917, when stars, and not

yet galaxies, were considered the fundamental pieces filling the universe

and the concept of the expanding universe still had to enter the history

of modern cosmology1.

1The formulation of the first two relativistic models of the universe represents a

milestone in modern cosmology. Its history has been faced by several authors; see for

1

2 Introduction

These models inaugurated a new phase in cosmological researches:

they marked a revolutionary step beyond Newtonian cosmology in the

investigation of connection among space, time and gravitation. The the-

ory of general relativity soon appeared as a suitable tool to study the

physics and the geometry of the universe. Since 1917, when both Einstein

and de Sitter proposed a spatially finite world2, appropriate theoretical

extrapolations of laws of physics and gravitational properties of matter

to the universe as a whole entered cosmology just through “the remark-

able rationality and inner logicality of the theory of general relativity”

[Tolman 1934, p. 331].

It was during the 1920’s that the two alternative solutions proposed

by Einstein and de Sitter were related to observational cosmology. In

this perspective, the history of the de Sitter effect and its related de-

bates represents the linking thread in order to understand how cosmol-

ogy developed from a sphere of theoretical speculations into an empirical

science.

In Einstein’s spherical world, according to Machian inspiration, the

whole of matter was responsible for the origin of inertia. Einstein intro-

duced in his field equations a new term, with the cosmological constant λ,

example the Editorial Note about the Einstein - de Sitter - Weyl - Klein debate in

the Collected Papers of Albert Einstein [CPAE 1998 Editorial ], and also [Kerszberg

1989, North 1965, Ellis 1989 ]. The author of present thesis dealt with this topic in his

Laurea degree thesis about the renewal of cosmology in XX Century (“La rinascita

della cosmologia nel XX secolo”, in Italian; University of Padova) and subsequently

in [Realdi-Peruzzi 2009 ].2As we shall see in Chapter 4, both Einstein’s and de Sitter’s universes were closed

respect to their spatial dimensions. According to de Sitter “infinity is not a physical

but a mathematical concept, introduced to make our equations more symmetrical and

elegant” [de Sitter 1933, p. 154]. It is worth noting that, mutatis mutandis, in origin

the fact that infinity is a foreign notion in physical realm was pointed out by Galileo

Galilei (1564-1642): “essendo il moto retto di sua natura infinito, perche infinita

e indeterminata e la linea retta, e impossibile che mobile alcuno abbia da natura

principio di muoversi per linea retta, cioe verso dove vi e impossibile di arrivare, non

vi essendo termine prefinito” [Galilei 1632, p. 43].

Introduction 3

in order to express in general relativity the observed evidence of a static

equilibrium of the universe. De Sitter proposed an alternative solution,

i.e. a model of an empty world. In de Sitter’s universe a mass test would

have escaped far away from an observer because of the presence of the

cosmological constant, so that de Sitter’s universe appeared not really

static. This property was later called “de Sitter effect”, and during the

1920’s some discussions arose about its form and its actual astronomical

consequences. Indeed the de Sitter effect related distances and velocities,

even better redshifts, and for this reason it seemed to be connected by

some means with the first relevant redshift measurements in extragalac-

tic nebulæ. Thus de Sitter’s model began to appear in a favorable light

compared to Einstein’s solution.

The interest in the de Sitter effect survived until 1930, when “truly”

expanding models of the universe, i.e. solutions already discovered in

1922 by Alexander Friedmann (1888-1925) and independently in 1927

by Georges Lemaıtre (1894-1966), were proposed in order to properly

explain the evidences of a general cosmic recession confirmed in 1929

by Edwin Hubble (1889-1953). It was Arthur Eddington (1882-1944),

together with de Sitter, who clarified in 1930 the relation between the

non-static and non-empty Friedmann-Lemaıtre models and observational

discoveries of receding galaxies.

Before the first diffusion and general acceptance of the model of the

expanding universe, the solution of de Sitter played a relevant role in the

early phases of modern cosmology: just because of its mentioned prop-

erties, it was the precursor of the other non-static solutions to which

attention was directed since the 1930’s. “Statique sans l’etre, vide mais

non neutre, virtuellement actif sur toute matiere qu’on voudrait y mettre,

resultat d’une symetrie en trompe-l’œil, solution batarde d’une equation

batarde, l’univers de de Sitter - Merleau Ponty wrote in 1965 - etait

donc un curieux complexe d’equivoques, qui cependant portait l’avenir

de la pensee cosmologiques” [Merleau Ponty 1965, p. 61]. In particu-

lar, the de Sitter effect, which was obtained through the metric of de

4 Introduction

Sitter’s universe, played a fundamental influence in contributions that

several scientists offered during the 1920’s both in theoretical and in

observational cosmology. Predictions and confirmations of an appropri-

ate redshift-distance relation marked the tortuous process towards the

change of viewpoint from 1917 paradigm of a static universe to 1930

picture of a universe evolving both in space and in time.

In this perspective, such a historical research centered on the rise and

decline of the interest in the de Sitter effect aims to highlight the va-

riety of ideas and contributions addressed to the cosmological question

during the 1920’s debates, which inaugurated the modern approach of

cosmologists to the universe as a whole and its properties. Indeed, deal-

ing with the geometry of curved spaces in astronomical context, some

important topics, which are still present in cosmological debates, entered

in that period the description of the universe through applications of

general relativity theory. In particular, concepts as the metric of the uni-

verse, the homogeneity and isotropy of space, as well the interpretation

of extragalactic redshift, were first faced in those debates.

In this framework, such a history of the debates about the de Sitter

effect reveals its utility. Individual attempts to explain the de Sitter

effect and interpretations of redshift measurements through de Sitter’s

line element can be viewed as the leading questions in the history of early

intersections between observations of distant astronomical objects and a

serious and coherent physical theory of the universe given by cosmological

solutions of relativistic field equations.

Therefore, from this point of view, the issue faced in this thesis rep-

resents a remarkable passage in the broader history of the encounter

between astronomical observations and the laws of physics, inaugurated

through the revolutionary scientific contributions by Galilei, who first

“did not separate the two sciences by an impassable barrier” [Drake 1993,

p. 237].

The convergence of astronomy and physics into cosmology has greatly

developed towards the completely new present cosmological picture. In

Summary 5

this picture, both general relativity and quantum physics today share

a fundamental role investigating the nature of space, time and gravita-

tion. Indeed at present the knowledge of outer space, i.e. of cosmological

scales, is deeply connected to that of inner space, i.e. sub-atomic scales,

looking for the formulation of a quantum theory of gravitation by inves-

tigating unification models of fundamental interactions.

1.1 Summary

An overview of some topics of present cosmology, as the Robertson-

Walker metric and the meanings of redshift in cosmology, is reported in

Chapter 2. Then the state of cosmological knowledge at the beginning

of XX Century, when the universe was still considered static and filled

by stars and nebulæ, is illustrated in Chapter 3.

Chapter 4 is devoted to the beginning of relativistic cosmology. A

critical analysis is furnished about the ideas and attempts which led

Einstein and de Sitter to the formulation of the first two rival models of

the universe as a whole through general relativity.

The main topic of the thesis is then developed in Chapters 5 and

6. The former, Chapter 5, is focused on the de Sitter effect, its math-

ematical formulation, and the several formulations of such a theoreti-

cal redshift-distance relation which were proposed by the actors of the

cosmological debates during the 1920’s: de Sitter, Eddington, Lemaıtre,

Hermann Weyl (1885-1955), Kornel Lanczos (1893-1974), Ludwik Silber-

stein (1872-1948), Howard Robertson (1903-1961), and Richard Tolman

(1881-1948). The following chapter, Chapter 6, unfolds the various at-

tempts to relate the de Sitter effect to observational data, as done by Carl

Wirtz (1874-1939), Knut Lundmark (1889-1958), and Gustav Stromberg

(1882-1962), who based their conclusions on the astonishing radial ve-

locities of nebulæ measured by Vesto Slipher (1875-1969). During the

early phases of observational cosmology, the attempts to deduce a re-

liable empirical redshift relation successfully culminated in 1929, when

6 Introduction

Hubble confirmed that a linear redshift-distance relation among galaxies

existed, and therefore that a systematic recession was actually revealed

by observations.

In Chapter 7, the 1930 first diffusion of the model of the expanding

universe is discussed, paying attention to Lemaıtre’s cosmology as the

“third way” between Einstein’s and de Sitter’s cosmological models which

solved the puzzle generated by the de Sitter effect.

Chapter 2

Some issues of present

cosmology

In this chapter an overview of some issues of present standard cos-

mology is given.

The starting points on which the following topics are based are rela-

tivistic solutions of field equations for an expanding universe. However,

it is important to note that the first hints and discussions about some

of these issues can be found in the interpretation of properties of static

and stationary cosmological models since 1917, i.e. before the general

acceptance of the expanding universe. In particular, the assumption of

homogeneity and isotropy of space entered modern cosmology by Ein-

stein’s model, and the meaning, even better the meanings, of redshift

(and also the existence of visual horizons) were faced in 1920’s debates

dealing with de Sitter’s cosmological solution, as we shall see in next

chapters.

Astronomy and cosmology are based on observations at several wave-

lengths, which are limited to regions of the universe accessible to tele-

scopes. The causal connection between observers and observed objects is

given by light: the fact that such a “sidereal messenger” travels through

space with a finite speed involves that the farthest we observe in space,

the more backward we observe in time.

7

8 Some issues of present cosmology

Among the four fundamental forces in nature, which are gravity, elec-

tromagnetic force, strong interaction and weak interaction, only gravity

and electromagnetism act on long range. Assuming a neutral charge

state of the universe, the fundamental role in the evolution of structures

in the universe is thus played only by gravitation. The fundamental

theory describing gravity is the general theory of relativity, published

by Einstein in its final form in 19161. In Einstein’s theory, space-time

can be described as a 4-dimensional Riemannian manifold. Through

non-Euclidean geometry, Einstein introduced the concept of curvature of

space-time, meaning that metrical properties of space-time are entirely

described by tensor quantities gµν ’s, which represent the gravitational

field.

The space-time interval, often called the metric, is given by:

ds2 = gµνdxµdxν , (2.1)

where µ, ν = 1, 2, 3, 4. Following initial notation used by Einstein, the

first three indexes refer to spatial coordinates, while dx4 refers to time co-

ordinate; the signature is (–,–,–,+). The summation convention is used,

i.e. identical upper and lower indices are implicitly summed.

In general relativity an important aspect is that gravity, which is rep-

resented through space-time curvature, is included in the line element.

Thus a particle in a gravitational field can be considered as moving along

geodesics of space-time.

Field equations, or “Einstein’s equations”, correspond to the relativis-

1The questions of a general covariant formulation of laws of physics, in particular

gravitation, and the “defect” of the existence of privileged observers also in special

relativity, i.e. the existence of inertial reference frames, led Einstein to the formulation

of his new theory of gravitation, based upon the principle of relativity and the principle

of equivalence. See [Janssen 2005 ] for a historical reconstruction of how Einstein

formulated general relativity. See also [Renn-Schemmel 2007 ] for a historical analysis

of different approaches of the problem of gravitation around the turn of the last

century.

The Cosmological Principle 9

tic generalization of Poisson’s equation2, and relate space-time geometry

(through Riemann curvature tensor Rµνσρ) to energy-momentum tensor

Tµν , which describes the matter and energy contribution:

Rµν − 1

2gµνR = −κTµν (2.2)

or equivalently

Rµν = −κ

(Tµν − gµν

1

2T

). (2.3)

Here Rµν is the Ricci tensor, i.e. the contracted Riemann tensor, R is

the Ricci scalar, i.e. a scalar curvature obtained from gµνRµν , and T is

obtained from gµνTµν ; κ is a constant equal to 8πGc4

, where c is the speed

of light and G the gravitational constant. The solution of field equations

with regard to the universe as a whole permits to determine the metric

of the universe, i.e. all information about the geometry of space-time.

2.1 The Cosmological Principle

From astronomical observations on large scale and from theoretical

models interpretation, the present picture is that of an expanding uni-

verse, which evolved from an initial singularity, i.e. from an extremely

dense and hot phase: this is the standard model of the universe, the so-

called “hot Big-Bang model”. There are some fundamental results which

2Such a scalar equation was formulated in 1813 by Simeon Denis Poisson (1781-

1840) and describes how gravitational potential φ (determined by density of matter

ρ) behaves:

∇2φ ≡ ∆φ =∂2φ

∂x2+

∂2φ

∂y2+

∂2φ

∂z2= 4πGρ.

The gravitational potential φ is defined as

φ(r) = −G

∫ρ

r′dV,

where G is the universal gravitational constant. Thus gravitational force for a point

mass at a distance r is F = −∇φ. Laplace’s equation is obtained from Poisson’s

equation when zero density of matter is considered: ∇2φ = 0.


support this picture, coming both from pure astronomical evidences and

from data comparison with the Standard Model of particle physics:

• the recession of galaxies, which was interpreted since the 1930’s as

a feature of an expanding universe

• the light elements abundance, which observed values agree with

predicted abundances from the primordial nucleosynthesis during

the early phases after the Big-Bang

• the cosmic background radiation, which is at present observed at

microwave lengths (CMB), and is interpreted as the relic of radia-

tion originated at the last scattering surface from proton-electron

ionized plasma at the time of decoupling, at a temperature of about

T ' 3000 K, when the universe was 380’000 years old, about

13.7 · 109 years ago. Such a radiation was first observed in 1965

by Arno Penzias (1933- ) and Robert Wilson (1936- )3, giving a

very proof that the universe expanded from a hot early phase. The

measured radiation, according to latest results4, corresponds to the

radiation of a black-body at a temperature of T ' 2.725 K.

Moreover, present results from cosmological surveys permit to assume

that the universe we observe is homogeneous and isotropic on scales larger

than approximately5 200 h−1 Mpc. Matter and radiation are assumed to

be uniformly distributed through space on the very largest scales, with

3Actually, Penzias and Wilson measured a background radiation since 1963, then

in 1965 Robert Dicke (1916-1997) and his colleagues at Princeton rightly interpreted

the meaning of such a radiation as a cosmic radiation relic. See [Penzias-Wilson 1965 ]

and [Dicke et al. 1965 ].4See [Komatsu et al. 2008 ] for a present cosmological interpretation of observations

of CMB and its anisotropies.5Here h correspond to H

100 , where H is the Hubble parameter defined in Section

2.2. Mpc means Megaparsec, one of the astronomical quantities which are used for

distances. It corresponds to 106 pc, i.e. to 3.09 · 1022 m. Other useful units of length

are the light year, equal to 9.46· 1015 m (1 pc ' 3.26 light years), and the astronomical

unit (AU), corresponding to 1.49 · 1011 m.

The Robertson-Walker metric 11

neither privileged directions nor privileged positions. As already men-

tioned, it was Einstein who first assumed in his 1917 static cosmological

model the homogeneity and isotropy of matter (see Chapter 4). Such

a fundamental condition is referred in the literature as the “Cosmolog-

ical Principle”. In particular, in the picture of an expanding universe,

the cosmos exhibits the same properties at a fixed time t, i.e. physical

properties of space depend only on time coordinate, which is called for

this reason “cosmic time”. The cosmological principle permits to identify

3-dimensional spatial surfaces at constant time which are homogeneous

and isotropic, thus maximally symmetric.

Since the mentioned 1965 observation of the cosmic microwave back-

ground radiation, Big-Bang model was clearly preferred to a rival theory,

the Steady State cosmology. Such a theory was proposed in 1948 by

Thomas Gold (1920-2004), Hermann Bondi (1919-2005) and Fred Hoyle

(1915-2001), and does not predict an initial singularity of the universe6.

The steady state model is based on a continuous creation of matter, in

order to satisfy the “Perfect Cosmological Principle”, meaning that the

universe looks the same both in every direction and at every time.

2.2 The Robertson-Walker metric

The only solution which satisfies the cosmological principle for an

expanding universe is the Robertson-Walker (RW) metric7:

ds2 = −a2(t)

[dr2

1− kr2+ r2dθ2 + r2 sin2 θ dφ2

]+ c2dt2. (2.4)

Here the signature is (–,–,–,+). The coordinate t is the cosmic time of

comoving objects. Polar coordinates (r, θ, φ) refer to a comoving reference

frame, in the sense that they are constant for each particle of the “perfect

fluid” (to which the content of the universe is approximated) at rest

6See [Bondi-Gold 1948 ], [Hoyle 1948 ].7Such a line element was independently proposed in 1935 by Robertson in [Robert-

son 1935 ] and in 1936 by Arthur Walker (1909-2001) in [Walker 1936 ].


with respect to this reference frame. The function a(t), historically first

denoted as R(t), is the cosmic scale factor, or expansion parameter, which

depends only on time and varies according to expansion8. The parameter

k determines the constant curvature of spatial sections. It can be negative

(k = −1), null (k = 0), or positive (k = +1), yielding respectively an

open universe (3-dimensional hyperbolical space, or 3-hyperboloid), a flat

universe (Euclidean space) or a closed universe (3-dimensional spherical

space). In fact, the curvature parameter can be scaled in such a way to

assume only values k = (1, 0, -1). The expansion parameter a(t) is related

to the curvature of space. Indeed closed and open spaces have positive

and negative Gaussian curvature CG = ka2 , respectively. The parameter

a(t) thus represents the radius of spatial curvature, which in cosmology

describes the modulus of Gaussian curvature radius RG = C−1/2G = a√

|k|[Coles-Lucchin 2002, pp. 9-13].

In the next chapters of present thesis, following notation in primary

sources of the period 1917-1930, the symbol R will be used in order to

represent the radius of the universe9. Indeed, before the 1930 diffusion

of the model of the expanding universe, such a notation was used to

represent the constant world-radius in early relativistic static and finite

models of the universe proposed by Einstein and de Sitter. It was Ed-

dington who introduced in 1930 the symbol a(t) for the world-radius in

an expanding universe, R(t) ≡ a(t) [Eddington 1930a]. Friedmann and

Lemaıtre, as we shall see, were the first who independently considered a

world-radius depending on time for a not empty universe.

8Following conventions in [Coles-Lucchin 2002 ], scale factor a(t) has the dimen-

sions of a length, and the comoving coordinate r is dimensionless.9The mentioned world-radius R is not the Ricci curvature scalar obtained from

gµνRµν , i.e. from the contraction of Ricci tensor which appears in Einstein’s field

equations. The Ricci scalar is related to the expansion parameter a(t) and the spatial

curvature k through [Carroll 1997, p. 220]:

R =6a2

(aa + a2 + kc2). (2.5)

.


During the 1920’s debates, several authors dealt with the properties

of de Sitter’s universe, and proposed non-static interpretations of the

line element of such an empty world. In this framework, the theoretical

contributions by Robertson, Lemaıtre, Weyl and Lanczos, which will be

analyzed in Chapter 5, marked the departure from the static picture of

the universe. In particular, they considered the universe of de Sitter

in a stationary frame. As we will see, each of the mentioned authors

interpreted in a different way the notion of a stationary world for their

own versions of de Sitter’s universe. In the retrospect, following [Ellis

1990, pp. 100-101], their contributions can be actually viewed as true

expanding versions of the universe of de Sitter.

It is useful to mention the present meaning in general relativity of the

concepts of static and stationary gravitational fields. Following [Landau-

Lifsitz 1960, Ita. tr. p. 325], a static gravitational field corresponds

to the constant field which is generated by a unique body, where in the

metric the gµν terms are not dependent on time coordinate x4 (i.e. the

body is at rest and both two time directions are equivalent), and all gα 4

are equal to zero10. On the contrary, in a constant gravitational field

which is generated by a body which has axial symmetry, as for example

by a body rotating along one of its symmetry axis, the two directions

of time are not equivalent. Such a gravitational field, where in general

gα 4 6= 0, is denoted as stationary. In other words, a stationary space-

time admits a time-like Killing vector field. Such a space-time is also

static if this Killing vector field is orthogonal to space-like hyper-surfaces

at constant time.

By using another radial comoving coordinate, r, an equivalent expres-

sion of the RW metric is:

ds2 = −a2(t)[dr2 + f(r)2(dθ2 + sin2 θ dφ2)

]+ c2dt2. (2.6)

Here the geometry of spatial sections is determined by the function f(r),

10The symbol α (and, in following pages, also the symbol β) refers to the spatial

coordinates of the metric: α, β = 1, 2, 3.


the comoving angular diameter distance, which is equivalent to r in the

metric 2.4, and is equal to (sin r, r, sinh r), for k = ( 1, 0,−1), respec-

tively.

The expansion rate of the universe is measured through the Hubble

parameter H(t):

H(t) ≡ da

dt

1

a(t)=

a(t)

a(t). (2.7)

In particular, H0 ≡ H(t0) is the value of Hubble parameter at present

time t0. Dimensionally, the Hubble parameter is the inverse of a time

and its present value is H0 ' 70.1 ± 1.3 Km/sec Mpc−1 [Komatsu et

al. 2008 ]. The rate of change of the expansion rate is measured by the

deceleration parameter q:

q(t) = −a a

a2. (2.8)

The geometry of space depends on its content: the spatial sections of the

universe can be closed, flat or open provided that the density parameter

Ω(t) ≡ ρ(t)

ρc(t)= 1 +

k

a2(2.9)

is respectively greater, equal or less than 1. Critical density ρc is the

density of a flat universe:

ρc(t) =3

8π G

(a

a

)2

=3

8π GH2(t). (2.10)

Since the material content of the universe is diluted during the expansion

over an increasing volume, matter (and also radiation) density decreases

with time. As already mentioned, the matter content is compared to

a perfect fluid with neither viscosity nor heat flow. Thus the energy-

momentum tensor is

Tµν = −p gµν + (p + ρ c2)UµUν , (2.11)

where U is the 4-velocity of the perfect fluid. The 4-velocity vector

represents the average motion of matter, and in comoving coordinates is


written as Uµ = (0, 0, 0, 1).

With these conditions and by considering the Robertson-Walker metric,

Einstein’s equations reduce to 2 equations, the well known Friedmann-

Lemaıtre (FL) solutions11, which describe the evolution of an expanding

universe:

a = −4

3π G

(ρ + 3

p

c2

)a, (2.12)

a2 + kc2 =8

3π Gρ a2. (2.13)

The first equation corresponds to the time-time component of Einstein’s

equations. The second equation can be obtained from the first one, by

considering Birkhoff theorem12 and the postulate of adiabatic expansion

of the universe:

d(ρ c2a3) = −p da3. (2.15)

The behavior of scale factor, a(t), depends thus on the total density ρ,

which is related to the pressure by the state equation of a perfect fluid,

p = ω ρ c2. (2.16)

If 0 ≤ ω ≤ 1 (the so-called Zel’dovich interval), models of a homogeneous

and isotropic universe have an initial singularity at a = 0. There is an

unavoidable singularity even for −13

< ω ≤ 0 [Coles-Lucchin 2002, p.

11Friedmann’s and Lemaıtre’s proposals will be described in Chapter 7.12Such a theorem was proposed in 1923 by George Birkhoff (1884-1944). It states

that a spherically symmetric gravitational field in an empty space is static and can

be described through the Schwarzschild metric, i.e. by the solution of Einstein’s

field equations considering a mass-energy source in empty space. Such a metric was

discovered in 1916 by Karl Schwarzschild (1873-1916):

ds2 = − dr2

(1− 2G M

c2r

) − r2(dθ2 + sen2θdφ2

)+

(1− 2GM

c2r

)c2dt2. (2.14)

Birkhoff’s relativistic result is analogous to the classic result obtained by Newton,

based on Gauss theorem for a gravitational field, which states that the gravitational

field outside a spherical object is the same as the whole mass of such an object is

concentrated at its center [Coles-Lucchin 2002, p. 24].


36]. The cases ω = 0, ω = 13, ω = 1 correspond to a dust, radiation, stiff

matter universe, respectively.

2.3 Redshifts in cosmology

Redshift or blueshift measurements and their interpretations play

a fundamental role in astronomy and in cosmology. The redshift (or

blueshift) z is due to an increasing (or decreasing) of wavelength λ (not

to confuse with the cosmological constant!), and corresponds to:

z =λ0 − λe

λe

⇒ 1 + z =λ0

λe

, (2.17)

where λ0 and λe denote, respectively, the wavelength measured by the

observer and the original emitted wavelength.

There are three kinds of shift to take into account in astronomical obser-

vations [Harrison 1981, p. 235]:

• the gravitational shift, which can be either red or blue, and is due

to light traveling close to massive bodies:

zG =1√

(1− Rs

R)− 1. (2.18)

Here Rs = 2GMc2

is the Schwarzschild radius of the source of gravi-

tational shift approximated to a sphere of radius R and mass M .

• the Doppler shift, either red or blue, originated by relative motions

between observer and observed object through space13:

zD =v

c, (2.19)

13The interpretation of the change in the frequency of sound-waves which is heard

from a moving source of sound was proposed in 1842 by Christian Doppler (1803-

1853). It was Armand Fizeau (1819-1896) who in 1848 rightly predicted the displace-

ment of lines for light coming from stars. For this reason such an effect is also known

as the Fizeau-Doppler effect.

Redshifts in cosmology 17

zD =

(c + v

c− v

)1/2

− 1. (2.20)

The first is the classic Doppler formula, to be used when the velocity

of the object v is small when compared to speed of light, and the

second is the special relativistic Doppler formula. This latter gives:

v(z) = c(z2 + 2z)

(z2 + 2z + 2). (2.21)

• the expansion (or cosmological) redshift, which formulation was

proposed by Lemaıtre in 1927, as we shall see later. Such an effect

is due to waves stretched by the expansion of the universe propor-

tionally to the scale factor:

zC =a0

ae

− 1, (2.22)

where a0 ≡ a(t0) is the scale factor at reception time (i.e. the

present scale factor), and ae ≡ a(te) is the scale factor at signal

emission.

Following [Ellis 1989, p. 374], several contributions to redshift mentioned

above can be generally resumed as:

(1 + ztot) = (1 + zDS)(1 + zGS

)(1 + zC + zGC)(1 + zDO

)(1 + zGO). (2.23)

The terms zGS, zGO

, zGCcorrespond to gravitational shift contributions

due to inhomogeneity of matter distribution at the source, near the ob-

server, and on large scale, respectively. The shifts zDSand zDO

are origi-

nated by the relative (Doppler) motion of the source and of the observer

respectively. Finally, zC is the expansion redshift due to increasing scale

factor in the universe. Such a summary will be useful in Chapter 5 de-

scribing the de Sitter effect and several interpretations of redshifts in de

Sitter’s universe.


2.3.1 Velocities and distances

Cosmological redshift, by equation 2.22, gives informations about how

much the universe increased in size (i.e. expanded) from the emission

time of an observed object. However a direct information about distances

and recession velocities can not be obtained. It is indeed necessary to

know also the geometry of the universe and the expansion rate of scale

factor.

Equation 2.21 can be applied only in (local) inertial frames. Neglect-

ing contributions by gravitational and Doppler redshifts, the general

relativistic relation between recession velocity and cosmological redshift

[Davis-Lineweaver 2004, p. 99] is given by:

vrec(t, z) = ca(t)

a0

∫ z

0

dz′

H(z′). (2.24)

Setting t = t0, we obtain the recession velocity that the object with the

measured redshift has today:

vrec(t0, z) = cH0

∫ z

0

dz′

H(z′). (2.25)

When we speak about distances, we have to distinguish between the-

oretical concepts of proper and comoving distances. The proper distance

is the radial distance at a fixed time; from equation 2.6, being dt = 0

because of the fixed time and dφ = 0, dθ = 0 because of the radial

direction, the integral of ds gives:

Dpr(t) = a(t)r. (2.26)

Thus proper distance is obtained from the comoving distance r multiplied

to scale factor. Differentiating this formula with respect to time we obtain

the theoretical velocity-distance law [Harrison 1993, p. 30]:

dDpr(t)

dt=

da(t)

dtr ⇒ vrec(t) = H(t)Dpr(t). (2.27)

Such a linear relation is valid for all distances in an expanding RW uni-

verse, and for example predicts also recession velocities exceeding speed


of light. In particular, the so-called Hubble radius measured at present

time

DH0 =c

H0

(2.28)

is the distance at which the recession velocity is equal to the speed of

light, and beyond the sphere of that radius, the Hubble sphere, recession

velocities exceed speed of light.

As we will see in Chapter 6, Hubble in 1929 confirmed that a linear

relation existed between distances and radial velocities, thus relation 2.27

is usually referred to as the Hubble law14. However, as pointed out in

[Harrison 1993, p. 31], the true Hubble law should be considered the

empirical redshift-distance relation

zc = H(t)DL. (2.29)

Here DL is the luminosity distance, an observable quantity which will

be described later. This relation is due to the direct estimate of velocity

from redshift through equation 2.19, as a habit in early observational

cosmology15. Such a relation is linear, i.e. coincides with the general

theoretical velocity-distance law (equation 2.27), only for small redshifts

and small distances compared to Hubble’s radius.

The comoving distance of an object can be calculated through the

fact that photons travel along null geodesic, ds = 0. In particular, at

present time t0, the comoving distance of an object which emitted light

at te is:

r(te) = c

∫ t0

te

dt′

a(t′), (2.30)

14Hereafter it comes also the notation of H for the “constant” of proportionality

between velocities and distances.15It is important to note that also Lemaıtre proposed such a relation already in 1927,

as we shall see. Two years later, Hubble, unaware of Lemaıtre’s result, confirmed this

linear relation from new estimates of distance and radial velocity of galaxies which

Hubble himself and Humason had obtained by using the most powerful telescope

operating at that time, the 100-inch Hooker reflector at Mt. Wilson.


The proper distance to such a comoving object is thus obtained (from

the above equation and equation 2.26) by multiplying to a(t0).

As mentioned, an important observable quantity is the luminosity

distance DL, which is useful to approximately estimate cosmological pa-

rameters through measurements of redshift. It is defined as:

D2L =

L

4π l, (2.31)

where l and L are the apparent and absolute luminosity of a distant

source, respectively. The luminosity distance varies with redshift, be-

cause the flux of photons is diluted traveling in an expanding RW uni-

verse [Coles-Lucchin 2002, p. 20]. It can be written as [Davis-Lineweaver

2004, p. 105]:

DL(z) = a(t)f(r)(1 + z). (2.32)

Thus, for small redshift and by a Taylor series expansion of scale fac-

tor, measurements of luminosity distances allow to determine H0 and q0

through the approximate formula [Coles-Lucchin 2002, p. 20]:

DL(z) =c

H0

[z +

1

2(1− q0)z

2 + ...

]. (2.33)

The luminosity distance of a source is related to observable quantities

(with which astronomers usually work) as the absolute magnitude M and

apparent magnitude m through the distance modulus equation. Such an

equation, neglecting absorptions, takes the form:

m−M = −5 + 5 log DL. (2.34)

Therefore, the luminosity distance can be obtained for some astronom-

ical objects which absolute magnitude is known. For example, Cepheid

variable stars are important distance indicators, since they have a firm

relation between their period of variation and their absolute luminos-

ity. As we shall see in next chapters, such standard candles played a

fundamental role in the rise of observational cosmology.


Another useful observable quantity is the angular diameter distance

DA(z) = yθ, i.e. the distance of an object of physical size y observed with

apparent angular size θ. In an expanding universe it can be expressed as

[Davis-Lineweaver 2004, p. 105]:

DA = a(t)f(r)(1 + z)−1, (2.35)

the meaning of f(r) being described in equation 2.6. The angular diam-

eter distance is related to the luminosity distance through:

DL = (1 + z)2 DA. (2.36)

The observable part of the universe is defined through the particle

horizon. Our particle horizon is the frontier which divides, at the instant

of observation t0, world-lines which intersect our past light cone, i.e.

world-lines which can be observed, from world-lines which lie outside our

past light cone [Harrison 1991, p. 61]:

rph(t0) = c

∫ t0

0

dt′

a(t′). (2.37)

From equation 2.26, the corresponding proper distance to the particle

horizon is:

Dph(t0) = a(t0)rph(t0). (2.38)

Our particle horizon is thus the distance light we receive or can receive

has traveled from t = 0 to t0.

Another important horizon, which is exemplified by de Sitter’s uni-

verse16, is the event horizon:

reh(t) = c

∫ +∞

t

dt′

a(t′). (2.39)

16Actually, through the “exponential” representation of de Sitter’s line element

(where a(t) ∝ eH t), which is at present used to describe a vacuum energy dominated

universe, it can be shown that de Sitter’s universe has an event horizon at distance

Deh = cH , i.e. at the constant proper Hubble radius [Rindler 1956, p. 139].


The event horizon, following [Rindler 1956, p. 134], is the frontier which

divides events which have been, are, will be observable by us, from eter-

nally unobservable events. The superior limit in the above integral is

t → +∞ by supposing an eternally expanding universe17.

2.4 Dark matter and dark energy

With regard to the material and energy content of the universe, cur-

rently the relevant contributions are considered to be given not by ordi-

nary matter, but from dark matter and dark energy.

Measurements of rotation curves of galaxies permit to estimate the

mass of these structure. However, a general discrepancy exists between

observed and predicted mass. A possible solution seems that to consider

17The existence of cosmological horizons is related to the problem of causal con-

nection among different regions of the universe. For example, from considerations on

the CMB, it can be shown that regions which now are separated by more than ∼ 2

are causally disconnected. Thus, the fair isotropy of CMB can not be explained: this

is the so-called horizon problem. A possible solution to such a problem is given by

the mechanism of inflation, first proposed by Alan Guth (1947- ) in 1981. The infla-

tionary theory is based on a spontaneous broken symmetry as predicted in Standard

Model of particle physics. Such a theory takes into account an accelerated phase of

the universe right after the Big-Bang, which took place between 10−34 sec and 10−32

sec, due to the repulsion effect of vacuum energy with negative pressure. During such

a inflationary phase the expansion is accelerated: a > 0 (ω < − 13 ). Thus the size

of the comoving effective horizon, which during inflation is not the particle horizon

(rph), but is the Hubble radius (rH), decreases with time: ˙rH ∝ −a. During inflation

the proper distance to the Hubble sphere remains constant and is coincident with the

event horizon [Davis-Lineweaver 2004, p. 100]. Therefore a region of the universe

which entered such a “horizon” before the beginning of inflation, i.e. which became

causally connected with regions within the Hubble sphere, can “escape” such an effec-

tive horizon during inflation (because the comoving effective horizon decreases with

time during such a phase) and eventually reenter the horizon once more, after the end

of inflation [Coles-Lucchin 2002, pp. 149-151]. We refer to [Harrison 2000, Chap-

ters 21-22] for further readings about the existence and description of cosmological

horizons, which represent an interesting and complex issue.

Dark matter and dark energy 23

the existence of a relevant fraction of non-luminous and non-ordinary

matter (thus dark matter), which is dominant in the total matter contri-

bution in the universe over the baryonic (ordinary) matter18. Even if a

small fraction of baryonic dark matter may also exist, the most fraction

of dark matter is characterized by a non-baryonic nature (the candidate

particles of dark matter are considered among exotic elementary parti-

cles predicted in supersymmetry theories). Its interaction with ordinary

matter occurs only by gravity. In the common accepted cosmological sce-

nario dark matter seems to be not relativistic (cold dark matter, CDM).

This kind of matter forms a (dark) halo around galaxies, so that the

rotational curves are altered by its presence.

In recent years, some fundamental observations of luminosity of dis-

tance indicators, type Ia supernovæ, revealed that the universe is accel-

erating, i.e. its rate of expansion is increasing [Riess et al. 1998, Perl-

mutter et al. 1999 ]. Accounting for this newly observed acceleration, an

unknown kind of energy, called dark energy, has been postulated. Such

a dark energy permeates the universe: its contribution is now dominant

and determines the accelerated expansion acting as a negative pressure.

One of the possible explanations of dark energy is vacuum energy.

The simplest form of a cosmic repulsion which accelerates the expansion

of the universe is a cosmological term in field equations. As we shall

see in Chapter 4, in 1917 Einstein introduced a new term, λ gµν , in field

equations 2.2 in order to obtain a suitable relativistic solution describing

a static universe. This new term acted like an anti-gravity term, and, in

Einstein’s intentions, could balance gravity effects on large scales. In the

current picture, accounting for an accelerated universe, a similar cosmo-

logical term (now denoted with Λ) can be inserted in field equations:

Rµν − 1

2gµνR− Λgµν = −κTµν . (2.40)

Equivalently, a new contribution can be considered in the right-hand side

18Already in 1933 Fritz Zwicky (1898-1974) suggested the existence of some kind of

invisible matter studying rotational curves of galaxies in Coma cluster [Zwicky 1933 ].


of field equations, among the energy-matter contributions which curve

space-time:

Rµν − 1

2gµνR = −κ Tµν = −κ(Tµν + ρΛ gµν). (2.41)

The present role of the cosmological constant is inspired by quantum

mechanics. It can indeed be interpreted not only as an intrinsic property

of space, but also as a kind of energy, in particular vacuum energy:

Λ =8π G

c4ρΛ. (2.42)

Vacuum energy has negative pressure (i.e. ω = −1 in equation 2.16),

and its contribution has to be taken into account in the total density

parameter Ω(t) as defined in 2.9. It is supposed to be constant on time, so

that its accelerating effect became dominant over time-decreasing matter

density about 5 · 109 years ago.

From measurements of anisotropies of CMB (which are of order of

10−5 and correspond to the “seeds” of structures in the universe), the cur-

rent picture is that of a flat universe (ktot ' 0 or equivalently Ωtot(t0) '1). Furthermore, investigations of clusters abundances and results from

gravitational lensing permit to estimate the present total matter (dark

and baryonic) contributions at about 30% in the density parameter. Thus

the remaining contribution is supposed to be done by dark energy. Com-

bining the results from type Ia supenovæ, CMB anisotropies, and acoustic

oscillations of baryons, the total energy content is assumed to be com-

posed by baryons ' 4.6%, dark matter ' 23%, dark energy ' 72%

[Komatsu et al 2008 ].

A ΛCDM model seems consistent with present observations. How-

ever, this issue has still to be clarified19, and other possible explanations

19For example, some problems are the present value of vacuum energy density

compared to the expected value from theoretical results of supersymmetry theories

and its fine-tuned coincident value with present matter density. See [Carroll 2001,

Peebles-Ratra 2003 ] for further readings on this subject.

Dark matter and dark energy 25

for the accelerated expansion have been proposed, as a quintessence dy-

namical field, or modified gravity. A suitable explanation of the accel-

erated universe represents one of the important challenges in cosmology

and theoretical physics, as well in search of quantum gravity theory.

Chapter 3

Cosmology at the beginning

of XX Century

In order to better understand the rise and early convergence of theo-

retical and observational cosmology, it is worthwhile to briefly illustrate

in the following pages the general scientific knowledge on the universe at

about a century ago.

3.1 The sidereal universe and the nebulæ

The structure and duration of the universe, quoting Simon Newcomb

(1835-1909), “is the most far-reaching problem with which the mind has

to deal” [Newcomb 1902, p. 226]. According to Newcomb, at the very

beginning of last Century first steps were made to attack such a problem

by scientific methods, not merely from a speculative point of view. Some

questions were involved, as the extent of the universe of stars through an

infinite or finite space, as well the arrangement of stars in space and the

duration of the universe in time. However, no certain answers could yet

be furnished by observations.

Moreover, as pointed out in 1911 by French mathematician Henri

Poincare (1854-1912) in the introduction to the “Lecons sur les Hy-

27

28 Cosmology at the beginning of XX Century

potheses Cosmogoniques”, the cosmogonic problem of the origin of the

universe was still far to be clarified. Science could not yet explain some

questions about the evolution of the cosmos already inspired by works of

Pierre-Simon Laplace (1749-1827). How the cosmos had evolved towards

the present and ordered state, starting from a hypothetical primeval

distribution of matter uniformly spread across space? According to

Poincare, “nous ne pouvons donc terminer que par un point d’interrogati-

on” [Poincare 1911, p. XXV]. Observations could be helpful to eventu-

ally interpret cosmogony as an experimental science, however the relation

between the Milky Way and other celestial objects, in particular spiral

nebulæ, still represented to Poincare a challenge without definite results.

Scientific investigations of heavens were firmly linked up with Galilean

experimental method and theoretical synthesis. From this point of view,

astronomy succeeded with positive results by applications of Newtonian

gravitation law and classical mechanics on scales referred to the solar

system. Some problems of celestial mechanics were solved, so to confirm

the law of motion proposed times before by Isaac Newton (1642-1727). A

long-standing question was the precession of the perihelion of Mercury.

The amount of the measured precession of the most inner planet con-

siderably differed from expected values of planetary perturbation, rep-

resenting an exception with regard to predictions of Newton’s theory.

Works by Urbain Le Verrier (1811-1877) and subsequently by Newcomb

settled, in 1882, a value for such a discrepancy to 43” per century. It

is well known that general relativity offers a solution for this dilemma:

the difference between the value of the precession of Mercury perihelion

predicted by general relativity and the value measured by observations

is negligible, even better, such a theoretical prediction represents one of

the crucial tests giving validity to Einstein’s theory of gravitation.

Astronomical observations were all the more detailed and careful:

astronomy quickly developed into astrophysics, thanks to advantages of

photometry and spectroscopy. However, despite technical progresses and

advancements of data interpretations through theoretical models, the

The sidereal universe and the nebulæ 29

source of stellar evolution was still unknown, and, moreover, no large-

scale systematic velocity fields were pointed out in stellar motions by

observed data. Thus at the beginning of last Century, during the rise

of special and general relativity, it was well accepted by astronomical

community the paradigm of a static universe, which fundamental filling

pieces were considered stars, not yet galaxies.

In fact, during the 1850’s there was a controversy about the concept

of time from the point of view of thermodynamics laws1. As postulated

by William Thomson (Lord Kelvin, 1824-1907) and Rudolph Clausius

(1822-1888), the second law of thermodynamics involved the irreversibil-

ity of physical processes and the concept of entropy. Such a state func-

tion tends to increase in isolated systems. Extrapolating the property

of increasing entropy to the universe as a whole, now considered as an

isolated system, the result of continuous energy dissipation, quoting Lord

Kelvin, “would inevitably be a state of universal rest and death, if the

universe were finite and left to obey existing laws” [Thomson 1862, p.

289]. Thus the universe would have ended in a “heat death”, correspond-

ing to the end of physical transformations and to an unchanging thermal

state of the universe. Therefore, according to Clausius, it could be pos-

sible to postulate in these terms both a beginning of the universe, i.e.

an initial state of the universe as a whole with zero entropy, and a final

state characterized by the maximum entropy. It was through the in-

terpretation of laws of thermodynamics proposed by Ludwig Boltzmann

(1844-1906) that the question about time duration of the universe was

clarified. Indeed, according to Boltzmann’s view, the direction of time

1It is interesting to note that, during the early phases of relativistic cosmology,

considerations on the universe as a whole from the thermodynamic point of view were

not essentially taken into account. Except for Tolman, who investigated in late 1920’s

how structures could evolve in a static universe [Tolman 1934 ], scientists involved in

cosmological debates mainly dealt with the geometry of the universe and with the

relation between the curvature of space-time and astronomical observations [Ellis

1993, p. 316].


could be appreciated in local, not global, thermodynamic phenomena.

Thus a definite time direction could not be given to the universe as a

whole, but only to some of its parts, as for example to the observable

portion of the universe [Boltzmann 1897 ].

This controversy apart, the universe was in general considered to

be unbounded in space and infinite in time, and, recalling the ancient

Greek conception, basically filled by “fixed” stars, meaning that stars

were aimed both by small velocities (compared to speed of light) and

small proper motions.

Astronomical investigations during those years were directed at the

comprehension of distances, positions and displacements of celestial bod-

ies, as a sort of continuation of pioneering works inaugurated by William

Herschel (1738-1822). Over a century earlier indeed Herschel tried to de-

termine through stars counts the structure of our Galaxy, the Milky Way.

Those observations were continued by Herschel’s son, John (1792-1871),

and then resumed by Jacob Cornelius Kapteyn (1851-1922). Dealing with

the problem of sidereal distances by a statistical approach, Kapteyn pro-

posed a model of the stellar system as a flat and static structure, the

so-called “Kapteyn universe”. In 1920 he assumed the Sun to be at the

center of such a system [Kapteyn 1920 ]. However, in order to better ex-

plain from a dynamical point of view his own discovery of star-streaming,

Kapteyn shortly after proposed that the Sun should have been at about

650 pc from the center of the Galaxy, then considered as a flat rotating

disk [Kapteyn 1922 ].

Detailed investigations of globular clusters marked a turning point

in the comprehension of the structure of the Milky Way. It was Harlow

Shapley (1885-1972) who gave a fundamental contribution in this field.

Shapley used statistical parallaxes in order to determine absolute mag-

nitude of RR Lyrae stars, i.e. pulsating variable stars (like Cepheids)

which change in brightness with a regular period. In this way Shapley

could estimate distances of these objects in globular clusters, by using

the period-luminosity relation discovered in 1912 by Hernrietta Leavitt


(1868-1921). In 1918 Shapley suggested a picture of our Galaxy as a flat

rotating disk with a diameter which he set in 1919 of about 300’000 light

years, filled by stars and nebulæ and surrounded by a spherical halo

of globular clusters. The center of the Galaxy was thousands of light

years far from the Sun (65’000 light years), in the direction of Sagittarius

[Shapley 1918, Shapley 1919, Paul 1993 ].

Doubts remained about the real extent of the Galaxy, and mainly

about the nature and distances of spiral nebulæ. During the 1880’s, J. L.

Emil Dreyer (1852-1926) compiled a vast catalogue of stars clusters and

nebulæ, the “New General Catalogue” (NGC), which contained, together

with following supplements, nearly 8’000 objects. Spectroscopy revealed

for some of these objects the same features of stellar systems: were spiral

nebulæ thus to be regarded as extragalactic systems? Could them be

compared to the Milky Way? On the contrary, did the whole of stars

and nebulæ belong to the Galaxy, considering it as a unique universal

system? It seemed to reappear, now supported by observations, Kantian

suggestion of “island universes”, one of the speculative cosmologies of

XVIII Century.

Indeed, some insights on philosophical grounds into the universe as

a whole date back to XVIII Century. Emanuel Swedenborg (1688-1772),

Thomas Wright (1711-1786), Johann Lambert (1728-1777) and Immanuel

Kant (1724-1804) tried to understand how the cosmos was arranged.

However, their remarks did not much involve scientific community. In-

deed, up to Herschel’s investigations, astronomers were usually interested

in celestial mechanics referred to the solar system. Kant put forward the

idea that nebulæ formed part of an infinite cosmos: in his 1755 “All-

gemeine Naturgeschichte und Theorie des Himmels” (Universal Natural

History and the Theory of Heavens), the philosopher from Konigsberg

suggested the existence of an unlimited number of flat shaped island

universes spread over an infinite space. On the contrary, Lambert hy-

pothesized that celestial bodies formed part of a hierarchical universe,

culminating with a unique structure.


Recalling Lambert’s idea, in 1908 and subsequently in 1922 Swedish

astronomer Carl Charlier (1862-1934) proposed a static and hierarchical

model of the universe. Charlier discarded the hypothesis of a uniform

distribution of matter through space, which, on the contrary, played an

important role in the formulation of first relativistic models of the uni-

verse. He supposed that celestial bodies formed gradually increasing

spherical systems, which Charlier explicitly called galaxies, of order 0, 1,

2, ... and radii R0, R1, R2, ... , respectively. According to Charlier:

• N1 stars were arranged as system G1, of order 1 and radius R1

• N2 systems G1 formed system G2, of order 2 and radius R2

• N3 systems G2 formed system G3, of order 3 and radius R3...

And so forth. Charlier proposed that this scheme could also be inverted.

Indeed a hierarchical structure of spherical systems could be found both

in the infinitely great and in the infinitely small2:

2Charlier’s speculative attempt was not an isolated hint in the history of connec-

tions between macro-systems and micro-systems during those years. For example,

Eddington attempted to relate the cosmological problem to the atomic one. In 1931

Eddington suggested that, starting from the solution of wave-equation, it could be

possible to obtain a value of the cosmological constant in agreement with expected

value of λ measured from galaxies recession. Indeed, according to Eddington, the cos-

mological constant, which Einstein introduced in 1917 and “abandoned” just in 1931

(see later, Section 7.2.1), had a non-zero value and gave the expansion of the universe

[Eddington 1931 ]. Eddington considered λ as one of the fundamental entities in na-

ture, together with α (which is the fine structure constant), the number of particles

expected in an expanding universe, and the ratio of electrostatic and gravitational

forces. He firmly attempted to develop a fundamental theory which could relate cos-

mology to laboratory physics [Eddington 1946 ]. However, his considerations did not

much appeal to physics community (see [Longair 2004 ] for further readings on Ed-

dington’s approach). Also Paul A. M. Dirac (1902-1984) pointed out the importance

of numerical coincidences between physical and cosmological quantities. Accounting

for identifications of large numbers, Dirac proposed in 1938 that the gravitational

constant could vary with time [Dirac 1938 ].


• N0 meteorites formed star G0

• N1 molecules formed meteorite G1

• N2 electrons formed molecule G2

• N3 sub-electrons formed electron G3

According to stars and nebulæ counts, and measuring their apparent

dimensions, Charlier proposed that all spherical systems satisfied the

relation:Ri

Ri−1

>√

Ni. (3.1)

This relation was useful, for example, to estimate extent of the second

order system G2 which contained our Galaxy:

R2

R1

>√

N2. (3.2)

By supposing that the galaxy G2 contained N2 ≈ 106 nebulæ, Charlier

proposed a radius R2 > 1000 R1, being R2 the “metagalaxy” radius and

R1 the Galaxy radius respectively. In this detailed description of an

infinite world, Charlier supported the hypothesis that collisions among

nebulæ caused the formation of spiral structures in each system [Charlier

1925 ].

The status of spiral nebulæ, together with the dimension of the Milky

Way and other astronomical issues as the role of distance indicators,

the interpretation of star counts and stellar evolution theory, were faced

in a famous discussion, “the Scale of the universe”, between Shapley

and Heber D. Curtis (1872-1942). It took place on April 26, 1920 in

Washington and passed into the literature as “the Great Debate”. During

that meeting, Shapley and Curtis presented opposite proposals about the

nature of celestial systems3.

3The great importance of the discussion between Shapley and Curtis was to stim-

ulate further researches in that topic. From this point of view, the history of modern

cosmology is marked by several debates highlighting important astronomical issues,


Shapley asserted that spiral nebulæ were not comparable in size and

in constitution with our Galaxy: “I prefer to believe - Shapley said -

that they are not composed of stars at all, but are truly nebulous ob-

jects” [quoted in Hoskin 1976, p. 177]. On the contrary, Curtis, holding

the same point of view of William W. Campbell (1862-1938), supported

theory of spirals as island universes of the order of size of our Galaxy, as

suggested long before by Kant. In Curtis’ picture, the Milky Way was

counted as a spiral-arms system, as already given in 1900 by Cornelius

Easton (1864-1929). In the retrospect, each of the two protagonists pre-

sented right arguments with regard to different topics they faced: Shap-

ley was correct in his estimates of globular clusters distance, and Curtis

rightly interpreted extragalactic position of spirals4.

The controversy on the existence of extragalactic systems was shortly

after solved, thanks to high resolution images produced by 100-inch (2.5

m) Hooker reflector telescope at Mt. Wilson. The 1917 “first light” of

this telescope represents a milestone in the history of the comprehension

of galactic structures. As we will see later, just through Mt. Wilson

observations, Hubble, whose name is closely connected to the rise of

observational cosmology, located Cepheid stars in M31 and M33 nebulæ.

Thus Hubble could estimate the distance of these objects. They were

placed, according to Hubble, at a distance of about 930’000 light years

which are also useful to illustrate how definite interpretations of astronomical ob-

servations are not easy to be established. One of the most important controversy in

cosmology focused on the rival theories of Big-Bang model and steady state cosmology

(we refer to [Kragh 1996 ] for further readings about it). With regard to other topics,

in 1972 Arp and Bahcall discussed about the conventional interpretation of redshift

measurements [Field 1973 ]. During the 1990’s other debates took place about the

distance scale to gamma ray bursts and the nature and scale of the universe [Debates

1998 ].4See [Hoskin 1976 ] and [Trimble 1995 ] for some review articles on the Great De-

bate. The nature of the nebulæ and relevant topics related to the astronomical obser-

vations of spirals which were faced during the 1920’s will be reconsidered in Chapter

6 of present thesis.

Cosmological difficulties with Newtonian theory 35

(285’000 pc), undoubtedly outside boundaries proposed by Shapley for

galactic globular clusters [Hubble 1925b]. Since 1925 spiral nebulæ were

eventually considered as true extragalactic systems similar to our Galaxy:

the realm of the stars was replaced in 1925 by the realm of the galaxies.

3.2 Cosmological difficulties with Newto-

nian theory

As seen, at the beginning of last century the most accepted view was

that of a sidereal and infinite universe. In fact the cosmological issue

was not truly faced: there were debates about the nature and position of

nebulæ, and questions about the finiteness or infiniteness of space were

investigated only from a speculative point of view. Spectroscopical mea-

surements permitted to determine radial velocities and proper motions

of many stars. Thanks to observations by Vesto M. Slipher (1875-1969),

first relevant redshifts were measured in nebulæ and interpreted as ra-

dial velocities5 (through 2.19 formula). However, any departure from the

picture of a static universe was not debated.

Newtonian theory of gravitation, based on usual Euclidean geometry,

involved some difficulties and contradictions when extrapolated to an

infinite universe. If gravitational force is admitted to be universally valid,

how the gravitational effect of an infinite number of stars spread over an

unbounded space can be balanced towards the observed equilibrium?

In other words, the integral of the contributions by all masses to the

gravitational force of a mass test does not converge.

Among several attempts to solve such cosmological difficulty in New-

tonian theory6, at the end of XIX Century both Hugo von Seeliger (1849-

1924) and Carl Neumann (1832-1925) proposed a modification in the

gravitational potential φ. In 1885 Seeliger highlighted that both gravita-

5See Chapter 6 for a description of early measurements of radial velocities in spirals.6See [Norton 1999 ] and [North 1965, Chapter 2], for further readings on this issue.


tional force F and potential φ diverged when a uniform density of matter

ρ was spread over an infinite volume V . Neumann, dealing with these

issues which he already faced in 1874, suggested that Poisson’s equa-

tion should have been modified in order to obtain a uniform and static

distribution of matter through space.

As a possible solution, both of them independently proposed in 1895

(von Seeliger) and 1896 (Neumann) a similar modification in gravita-

tional law. They introduced an exponential term in the expression of φ.

Their modifications can be resumed as:

φ(r) = −G

∫ρ

r′e−λ r′ dV. (3.3)

This new exponential term was crucial at great distances. In such a way

the gravitational force diminished on large scale more rapidly than usual

r−2 in Newton’s law. The new term balanced gravitational attraction

of matter and could be interpreted as a sort of “cosmic repulsion”, so

that an average density of matter that was constant everywhere was well

allowed.

In 1917, as we will see, Einstein used the same analogy and modi-

fied his relativistic field equations by the so-called “cosmological term”

to obtain a static model of the universe (which, however, Eddington

demonstrated in 1930 to be unstable).

In order to suitably deal with gravitation on large scale, another possi-

bility was to renounce Euclidean geometry and to consider curved spaces.

It is well known that Einstein followed this road towards his cosmolog-

ical solution of field equations. However, between the end of XIX Cen-

tury and the beginning of XX Century, just a few astronomers ventured

this path. Among them, Newcomb in 1877 gave attention to the con-

sequences of elliptical curved spaces in parallax measurements. In 1900

Karl Schwarzschild (1873-1916) investigated the effects of hyperbolical

and elliptical geometries of space in astronomical context7 [Schwarzschild

1900 ].

7This pioneering pre-relativistic work by Schwarzschild, as we will see in next chap-

Cosmological difficulties with Newtonian theory 37

3.2.1 Olbers’s paradox

Besides contradictions related to gravity effects on large scale, there

were also objections to Newtonian cosmology related to luminosity ef-

fects. Recalling the suggestion by William Stukeley (1687-1765), already

in XVIII Century Jean Philippe de Cheseaux (1718-1751) dealt with the

problem of the darkness of night sky. Indeed in an infinite and Euclidean

universe, accounting for an infinite number of uniformly distributed stars

with the same luminosity, it is expected an equal contribution to the to-

tal luminosity both from stars closest to the Earth and from stars placed

on a spherical surface at a greater distance than the formers. Indeed the

number of stars in latter shell increases by the square of the distance,

while the luminosity flux diminishes by the inverse square of the dis-

tance. Thus, considering the equal contribution from an infinite number

of shells, one could expect to see always a bright sky. In other words, the

total luminosity flux diverges. Indeed apparent luminosity l at distance

dL = r is related to absolute luminosity L by l = L4π r2 (see equation

2.31). Being n the constant stars density and 4π nr2dr the amount of

stars between r and r + dr, the total flux luminosity U is

U =

∫ ∞

0

(L

4πr2

)4πnr2 dr = Ln

∫ ∞

0

dr.

Thus such an integral diverges.

In order to explain the absence of such a bright sky, de Cheseaux

postulated a loss of light traveling across space, so that the contribution

by farthest stars was negligible.

In 1823 Heinrich Olbers (1758-1840) took into account the same issue.

Following de Cheseaux’s possible explanation, Olbers pointed out that

a loss of the order of 1/800 in stars luminosity was sufficient to explain

the evidence of dark sky. Such a cosmological enigma passed into the

literature as the “Olber’s paradox”, and could not be suitably solved by

ter, played a significant role in Einstein’s and de Sitter’s approaches to the question

of the geometry of the universe.


admitting an Euclidean universe which was infinite in space and time.

The concept of an initial singularity, which is a feature of Big-Bang

standard cosmology, offers a solution to this problem8. In an expanding

universe, being a(t) the parameter expansion (scale factor), the apparent

luminosity l is:

l =L a2(te)

4π a4(t0) r2e

, (3.4)

where re is star comoving distance, t0 is time of received signal and te is

time of emission signal. The amount of stars with luminosity between L

and L + dL, which light is emitted between te − dte and te and received

at time t0, is

dN = 4π a2(te) r2e n(te, L)dtedL, (3.5)

where n(te, L)dL is stars density at te. Thus the total density of stars

observed at t0 is

U0 =

∫ ∫l dN =

∫ t0

−∞

∫n(te, L)

[a(te)

a(t0)

]4

dLdte. (3.6)

If a initial singularity is postulated, as in Big-Bang models (but not for

example in steady-state cosmology), such a integral does not diverge,

being the inferior limit of integration t = 0, not t → −∞.

8The solution here presented is taken from [Weinberg 1972, p. 612].

Chapter 4

1917: the universes of general

relativity

This chapter is devoted to the historical reconstruction of significant

steps that Einstein and de Sitter took towards the formulation of the first

two relativistic models of the universe1. Such solutions of field equations

were proposed during the famous 1916-1918 debate between Einstein and

de Sitter. In this debate the origin of inertia represents one of the main

issues, and can be viewed as the leading thread which led to the first

world models.

4.1 Einstein, the universe

and the relativity of inertia

Einstein proposed his cosmological solution in the fundamental pa-

per “Kosmologische Betrachtungen zur allgemeinen Relativitatstheorie”

(Cosmological Considerations in the General Theory of Relativity), first

published in February 1917. In this paper Einstein illustrated the “rough

1Relevant parts of this chapter are taken from [Realdi-Peruzzi 2009 ]. With re-

gard to Einstein’s papers and correspondence, references are made to the English

translation of the Collected Papers of Albert Einstein, hereafter [CPAE ].

39

40 1917: the universes of general relativity

and winding road” [Einstein 1917b, p. 423] that he had to follow to obtain

the whole material origin of inertia in the framework of General Rela-

tivity. Einstein considered his cosmological result as the achievement of

what he called the “relativity of inertia” [Einstein 1913b, p. 197]. Indeed,

accounting for Mach’s ideas, Einstein pointed out that “inertia is simply

an interaction between masses, not an effect in which ‘space’ of itself were

involved, separate from the observed mass” [CPAE 1998E doc. 181, p.

176]. The interest which Einstein addressed to the universe as a whole

came from the necessity of a global explanation of the relations among

space, time and gravitation through the principle of relativity. In partic-

ular, the universe as a whole represented to Einstein an ideal setting in

which the relativity of inertia could be verified. This idea can be inferred

from a letter to Michele Besso (1873-1955) that Einstein wrote in May

1916. In this letter Einstein illustrated his interest in the implications of

a finite universe, “that is, a world of naturally measured finite extension,

in which all inertia is truly relative” [CPAE 1998E doc. 219, p. 213].

It is well known that some ideas of the Austrian scientist and philoso-

pher Ernst Mach (1838-1916) had a fundamental influence on Einstein’s

concept of inertia and on the formulation of general relativity2. In 1916,

a few days before the publication of his review paper containing the

new theory of gravitation [Einstein 1916b], Einstein wrote an obituary

of Mach [Einstein 1916a]. In these pages Einstein acknowledged the

philosopher as a true precursor of General Relativity, for having “clearly

recognized the weak points of classical mechanics, and thus came close

to demand a general theory of relativity” [Einstein 1916a, p. 144].

In Mach’s critical analysis, the Newtonian absolute space and absolute

time were considered as pure imaginary entities. By way of experiment

it was not possible to know anything about them. Mechanics should be

founded, according to Mach, on experimental knowledge about the rela-

tive motions and positions of bodies. No absolute space was necessary to

2See [Renn 1994 ] for a comprehensive study about Mach’s influence on Einstein’s

formulation of general relativity.

Einstein, the universe and the relativity of inertia 41

define inertial frames. Only the relative motion existed, and there was

not any difference between rotation and translation. The very existence

of relative motions was the basis of Mach’s interpretation of physical

measurements. Quoting Mach, “when we say that a body preserves un-

changed its direction and velocity in space, our assertion is nothing more

or less than an abbreviated reference to the entire universe” [Mach 1883,

Engl. tr. p. 286]. Motions had to be referred directly to all masses in the

universe, and not to an absolute space. In this way the average motions

referred to closest celestial bodies could be considered null, and in this

approximation the fixed stars became the reference frame.

These criticisms played an important role in Einstein’s concept of in-

ertia, which led Einstein to cosmology. Inertia could not be interpreted

as an absolute intrinsic property: according to Einstein it had its origin

in the interactions among bodies. Since 1912 Einstein dealt with this

statement in some papers3. After publishing in 1916 the general the-

ory of relativity in its final form, the considerations on the universe as

a whole, i.e. on the whole of matter, were for Einstein the natural and

yet necessary investigation both in the relationships among space, time

and gravitation, and in the derivation of local Dynamics by the effects

of the total world content [Barbour 1990, p. 57]. General relativity,

indeed, should have expressed that there was neither locally nor glob-

ally any independent property of space. Thus the relativity of inertia,

i.e. the requirement that the metric should be fully determined by mat-

ter, was a fundamental question. Einstein finally achieved this result in

1917, through a suitable relativistic model of the universe. According

to Einstein, this model, which he proposed in his 1917 “Cosmological

considerations”, was the very proof that his new theory of gravitation

could “lead to a system free of contradictions” [CPAE 1998E, doc. 306,

p. 293].

3See [Einstein 1995 ] for some selected quotations revealing Mach’s influence on

Einstein’s view of inertia.


4.1.1 The debate with Willem de Sitter

The road to Cosmology passed through The Netherlands. The initial

investigations about different effects of the masses of the universe can

be inferred in a few letters between Einstein and Hendrik A. Lorentz

(1853-1928) [CPAE 1993 E, doc. 467; CPAE 1998E, doc. 43, doc. 47,

doc. 225, doc. 226]. They discussed in particular about the relativity

of rotation, and about the relationships among fixed stars, centrifugal

forces and Coriolis forces4.

However, it was Willem de Sitter who gave important contributions

to the cosmological consequences of general relativity5. Einstein and de

Sitter met in Leiden in the fall of 19166. They held a fruitful corre-

spondence focused in particular on the boundary conditions at spatial

infinity.

According to Machian view, “in a consistent theory of relativity -

Einstein wrote - there can be no inertia relatively to ‘space,’ but only

an inertia of masses relatively to one another” [Einstein 1917b, p. 424].

This statement required that at very large distances from all masses the

sources of inertia could not influence a mass test: inertia of this body

had to be zero. Such a condition was represented by a space-time that

at infinity was pseudo-Euclidean.

In the absolute reference frame of Newtonian theory this condition

could be expressed, in relativistic notation, by requiring that at infin-

ity the potentials (gµν ’s) assumed the diagonal values of Minkowski flat

4See [Janssen 1999 ] for a detailed study about the rotation in Einstein’s theory.5It is important to note that just through the fundamental articles by de Sitter

about the astronomical consequences of general relativity, which were published in

the Monthly Notices of the Royal Astronomical Society in 1916 and 1917 [de Sitter

1916a, de Sitter 1916c, de Sitter 1916d, de Sitter 1917b], the scientific community in

Great Britain (thus in particular Eddington) and in United States became aware of

Einstein’s new theory of gravitation.6During a stay in Leiden, from September 27 to October 12, as inferred from

[CPAE 1998E, doc. 260, doc. 263].


space-time:

−1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 +1

The presence of some material sources did not influence these values,

except for the g44 (time-dependent) term, which became in this case:

g44 = 1 +2φ

c2. (4.1)

The gravitational potential φ was produced by these sources, and could

be calculated by Poisson’s equation.

How to formulate this condition inside the framework of the new the-

ory of gravitation soon became for Einstein a “fundamentally important

question” [Einstein 1917b, p. 421]. In de Sitter’s words, general relativ-

ity “has no room for anything whatever that would be independent of

the system of reference” [de Sitter 1916b, p. 527]. There was not any

absolute property, and relativistic field equations were the “fundamental

ones” [de Sitter 1916b, p. 529]. The energy-momentum tensor appeared

as the source of the potentials, i.e. as the source of the geometry of

space-time. However, this statement was not sufficient to assert that the

whole of the gµν ’s was of material origin. It was necessary to assign the

constants of integration, namely the boundary conditions. As de Sitter

wrote, “we must be prepared to have different constants of integration

in different systems of reference” [de Sitter 1916b, p. 531]. Above all, in

order to preserve the principle of relativity, the values for the potentials

at infinity had to be invariant for all transformations [de Sitter 1916c, p.

181].

The original manuscripts of de Sitter which are stored at Leiden Ob-

servatory are helpful to reconstruct the first attempts to solve the ques-

tion of boundary conditions at infinity. De Sitter reported in a notebook

some topics faced about the relativity of rotation during conversations

in Leiden among Einstein, de Sitter himself, Paul Ehrenfest (1830-1933)


and Gunnar Nordstrom (1881-1923) on September 28-29, 1916. “Ein-

stein - de Sitter wrote - wants the hypothesis of the closeness of the

world. He means by that that he makes the hypothesis (conscious that it

is a hypothesis which cannot be proven) that at infinity (that is at very

large, mathematically finite, distance, but further than any observable

material object (...)) there are such masses (...) that the gµν assume

certain degenerate values (these have not to be 0, that is a priori not to

be said), the same in all systems. (...) He is even prepared to give up

the complete freedom of transformation (...). If it is possible to find a set

of degenerate values of the gµν that is invariant for a not too restricted

group of transformations, is a question that can be solved mathemati-

cally. Is the answer no (what Ehrenfest and I expect), then Einstein’s

hypothesis of the closeness is untrue. Is the answer yes, then the hypoth-

esis is not incompatible with the relativity theory. However, I even then

maintain my opinion that it is incompatible with the spirit of the princi-

ple of relativity. And Einstein admits that I have the right to do so. Also

the rejection of the hypothesis is completely admissible in the relativity

theory” [de Sitter Archive, Box S12. Engl. tr. by Jan Guichelaar].

Einstein proposed a set of degenerate values for the gµν ’s which were

invariant at infinity [de Sitter Archive, Box S12; de Sitter 1916b, p. 532;

de Sitter 1916c, p. 181]:

0 0 0 ∞0 0 0 ∞0 0 0 ∞∞ ∞ ∞ ∞2

Einstein thought to achieve in this way the relativity of inertia, since these

values could satisfy the condition of vanishing inertia of a test mass at

infinity. He considered a reference frame in which the gravitational field


Figure 4.1: Detail from the notebook of de Sitter. The Dutch astronomer

reported the set of degenerate values which Einstein suggested during conver-

sations in Leiden with de Sitter himself, Ehrenfest and Nordstrom, September

28-29, 1916. According to Einstein, such values could satisfy the requirement

of the relativity of inertia [from de Sitter Archive, Box S12].

was spatially isotropic. In this approximation the space-time interval7

ds2 = gµνdxµdxν (4.2)

assumed the simpler form:

ds2 = −A[(dx1)2 + (dx2)

2 + (dx3)2] + B(dx4)

2. (4.3)

In a Galilean reference frame, by using the Minkowski metric (√−g =

1 =√

A3B), the momentum terms

m√−g gµα

dxα

ds(4.4)

became (to a first approximation for small velocities):

mA√B

dxα

dx4

, (4.5)

7Hereafter the notation which appears in equation 2.1, i.e. the upper-lower indices

summation, is used. However, in Einstein’s and de Sitter’s original papers the metric

was written as ds2 = gµνdxµdxν .


where α, β = 1, 2, 3. In this static case the potential energy was equal

to m√

B. By assuming that far from heavenly bodies space-time was

pseudo-Euclidean, the condition of vanishing inertia could be mathemat-

ically obtained with A diminishing to zero or equivalently with B tending

to infinity. The degenerate values for gµν ’s proposed by Einstein could

satisfy this requirement. Thereby the potential energy became infinite

at very great distances, so that any material point could not leave the

system [Einstein 1917b, p. 425].

According to de Sitter, these values for the gµν ’s were the “natural”

[de Sitter 1916c, p. 182] ones. Any other set of different values, i.e. any

variation from them, should be determined by some material sources.

The meaning of this set of values was that “at infinity the 4-dimensional

time-space is dissolved into a 3-dimensional space and a one-dimensional

time” [de Sitter 1916c, p. 182]: de Sitter pointed out that, by this ref-

erence frame, the time coordinate recalled some properties of Newtonian

absolute time.

Some portions of the observable universe, placed at very great dis-

tances from material sources, could be well approximated by diagonal

gµν ’s of Minkowskian flat space. Since these values were different from the

“natural” ones, an explanation of this variation was thus necessary. Ein-

stein considered the existence of some hypothetical and unknown masses.

These “supernatural masses” [de Sitter 1916c, p. 183] were placed at fi-

nite but very large distances from all observable heavenly bodies, and

were the source of degenerate values at infinity. Quoting de Sitter, this

hypothesis “implies the finiteness of the physical world, it assigns to it

a priori a limit, however large, beyond which there’s nothing but the

field of gµν ’s which at infinity degenerated into the natural values” [de

Sitter1916b, p. 532].

De Sitter criticized this attempt to explain the origin of inertia. He

did not accept this “envelope” as a physical reality, because it “will al-

ways remain hypothetical and will never be observed” [CPAE 1998E,

doc. 272, p. 260]. Observable stars and nebulæ were not part of this


boundary. Light received from those bodies had approximately the same

wave-length as the light from terrestrial bodies, i.e. there was roughly

the same deviation from the diagonal Minkowskian values for the former

as for the latter. Thereby those stars and nebulæ should have been in-

side the envelope [de Sitter 1916c, p. 182]. According to de Sitter, such

unknown and “invisible” masses took the same role of absolute space.

This explanation of the origin of inertia was not more satisfactory than

Newtonian explanation, and moreover was “practically equivalent to no

explanation at all, or to a confession of our ignorance” [de Sitter 1916c,

p. 183]. De Sitter acknowledged general relativity to represent “an enor-

mous progress over the Physics of yesterday” [de Sitter 1916c, p. 183].

He found this ad hoc envelope very hard to accept. “It is not possible -

de Sitter wrote to Einstein - that, in the end, the explanation for inertia

must be sought in the infinitely small rather than in the infinitely large?”

[CPAE 1998E, doc. 272, p. 261]. De Sitter preferred to doubt about the

origin of inertia rather than accept this solution8.

De Sitter’s remarks persuaded Einstein to modify his own hypothesis.

Since the boundary conditions problem was “a purely matter of taste,

8It is worthwhile to note that G. F. Bernhard Riemann (1826-1866) used the same

words in 1854, with reference to the relations between geometrical assumptions and

empirical determinations. In his fundamental lecture “On the hypotheses which lie

at the bases of geometry” Riemann said: “The questions about the infinitely great

are for the interpretation of nature useless questions. But this is not the case with

the questions about the infinitely small. It is upon the exactness with which we fol-

low phenomena into the infinitely small that our knowledge of their causal relations

essentially depends. The progress of recent centuries in the knowledge of Mechan-

ics depends almost entirely on the exactness of the construction which has become

possible through the invention of the infinitesimal calculus, and through the sim-

ple principles discovered by Archimedes, Galileo, and Newton, and used by modern

Physics. But in the natural sciences which are still in want of simple principles for

such constructions, we seek to discover the causal relations by following the phe-

nomena into great minuteness, so far as the microscope permits. Questions about

the measure-relations of space in the infinitely small are not therefore superfluous

questions” [Riemann 1854, Engl. tr. p. 150].


which will never gain scientific significance” [CPAE 1998E, doc. 273,

p. 261], Einstein abandoned the idea of the distant masses, and gave

a different interpretation. Observable regions of the universe contained

a small mass when compared to the total mass of the universe. Inside

those portions of space “the inertia is determined by the masses there,

and only by these masses” [CPAE 1998E, doc. 273, p. 261]. In such a

way any envelope was not necessary. In different regions of the universe,

the gµν ’s and so the inertia were determined both by masses inside those

regions and by boundary conditions at infinity. According to Einstein,

“no motive remains to place such great weight on the total relativity of

inertia” [CPAE 1998E, doc. 273, p. 262].

However, also this new interpretation soon became “intolerable” [CPAE

1998E, doc. 308, p. 296] for Einstein. The problem of the gµν ’s at in-

finity was still an open issue, and Einstein abandoned the hypothesis of

degenerate values. Velocities measured on stars were small when com-

pared with the speed of light. For this reason it was possible to consider

only the contribution of the√

g44 dx4 term in the expression of the space-

time interval. Also because of the same reason, the energy-momentum

tensor could be approximated to the T 44 term. However, these approx-

imations did not agree with degenerate boundary conditions. As Ein-

stein noted, “in the retrospect this result does not appear astonishing”

[Einstein 1917b, p. 426]. The small velocities of stars indeed allowed

to consider a quasi-static stellar system. The gravitational potential of

such a system could not assume extremely large (or infinite) values, nor

values much greater than those on Earth.

The possibility to assign values that were not invariant at infinity

appeared to Einstein as a renounce, and not as a true solution to the

problem of boundary conditions: “I must confess - Einstein wrote - that

such a complete resignation in this fundamental question is for me a

difficult thing” [Einstein 1917b, p. 426].

Another possibility was to assign at infinity the diagonal values of flat

space, i.e. to impose at very large distances the scheme of Special Rel-


ativity. Also this choice was not satisfactory for many reasons. Firstly,

such a defined reference frame contradicted the Principle of Relativity

[Einstein 1917b, p. 427]. Secondly, this hypothesis required null values

for the Riemann tensor Rµνσρ at infinity. This requirement corresponded

to assign 20 independent conditions, while only 10 terms of the curvature

tensor Rµν were considered in field equations. There was not any phys-

ical foundation for this remarkable statement [Einstein 1922a, p. 359].

Moreover, the hypothesis of a universe that was pseudo-Euclidean at in-

finity was not satisfactory at all because it involved a sort of spatial origin

of inertia. If the gµν ’s were constant at infinity, then on large scale the

physical properties of space would have not been dependent on matter

[Einstein 1922a, p. 359]. As Einstein pointed out, “inertia would indeed

be influenced, but not would be conditioned by matter (present in finite

space)” [Einstein 1917b, p. 427]. On the contrary, quoting Einstein,

“the essence of my theory is precisely that no independent properties are

attributed to space on its own” [CPAE 1998E, doc. 181, p. 176].

Einstein did not succeed in assigning suitable boundary conditions at

infinity, in order to satisfy both the Machian view on inertia, and the

Principle of Relativity. By observed data, the isotropy of space could be

roughly considered, in the sense that a uniform distribution of the stars

appeared as the “natural” [Einstein 1922a, p. 363] one.

4.1.2 Towards the solution

It was well known that classical celestial Mechanics involved some dif-

ficulties in the cosmological framework. According to Newton’s theory,

matter should have concentrated in a finite region of the infinite space,

the universe being in such a way “a finite island in the infinite ocean of

space” [Einstein 1917a, p. 362]. Considering at great distances the solu-

tion of Poisson’s equation, the density of stars should have diminished to

zero, and there the gravitational potential should have tended to a fixed

(or null) limiting value. This world was empty at infinity and was not


isotropic: it was neither possible to consider all points on average equiv-

alent, nor everywhere assign the same mean density of matter [CPAE

1998E, doc. 604, p. 630].

In Newtonian theory, as Einstein pointed out, it was possible to imag-

ine the universe gradually losing its content. As the radiation was able to

radially pass away, in the same way heavenly bodies with enough kinetic

energy could leave the system of stars and “lost in the infinite” [Ein-

stein 1917b, p. 422]. Furthermore, this world built on Newtonian laws

could not exist from the statistical point of view. Comparing stars to

gas molecules, the application of Boltzmann’s law required a finite den-

sity ratio among different points. This condition corresponded to a finite

difference of the gravitational potential between the center of the system

and the infinity. Thus a null value for the density at infinity would have

required a null value for the density at the center9 [Einstein 1917b, p.

422].

As seen in Chapter 3, at the end of XIX Century both von Seel-

iger and Neumann introduced an exponential term in the expression of

gravitational potential in order to solve the cosmological difficulties in

Newtonian theory. At the time of writing his “Cosmological consider-

ations”, Einstein was not aware of this attempt [Einstein 1919, p. 89],

and proposed a similar modification. Einstein pointed out that his own

method “does not in itself claim to be taken seriously” [Einstein 1917b, p.

423], but was useful to suggest a possible solution. According to Einstein,

also the solution proposed by von Seeliger, with which Einstein became

acquainted some months later, could “free ourselves from the distasteful

conception that the material universe ought to possess something of the

nature of a center”10 [Einstein 1917a, p. 362], but at the same time

was “a complication of Newton’s law which has neither empirical nor

theoretical foundation” [Einstein 1917a, p. 363].

9See [Kerszberg 1989, pp. 145-152], for some aspects on this subject.10This statement is taken from the “Part III: Considerations on the Universe as a

Whole” of [Einstein 1917a], which Einstein added in the third edition of 1918.


Einstein considered an extension in Poisson’s equation:

∇2φ− λφ = 4πGρ. (4.6)

The solution of such a modified expression was:

φ = −4πG

λρ0 (4.7)

This solution was dynamically correct. In this relation, λ and ρ0 re-

spectively denoted a universal constant and a uniform density of mat-

ter. Considering the universe as a whole, stars and planets represented

some local not homogeneous distributions of matter. Therefore ρ0 cor-

responded to the hypothetical mean density of fixed stars, if the latter

were assumed to be uniformly distributed through space. “In order to

learn something of the geometrical properties of the universe as a whole”

[Einstein 1922a, p. 363], it was indeed convenient to consider a continue

distribution of matter with a uniform and extremely small (but not zero)

density. Being ρ0 constant, φ was constant too, so the universe did not

have a center. There was not any preferred position nor any preferred

direction. Some local non-uniform distributions of matter were related

to some variations in the potential φ. In this case, the latter was roughly

equal to the Newtonian field, because λφ could be chosen very small

when compared with 4πGρ [Einstein 1917b, p. 423].

Einstein extended this modification in the framework of his new the-

ory of gravitation. He expressed in general relativity that both the po-

tential and the mean density of matter remained constant in space and in

time. In order to satisfy these conditions, Einstein respectively eliminated

spatial infinity and introduced in his field equations the so-called cosmo-

logical term, namely the fundamental tensor gµν multiplied by −λ, a uni-

versal but unknown constant. The condition of spatial closure ensured

that both the gravitational potential and the mean density of ponderable

matter remained constant in space. He introduced the cosmological con-

stant accounting for the supposed static nature of the universe, i.e. to

preserve the gravitational potential and the density of matter constant


in time. In this way field equations expressed the observational evidence

of the static equilibrium of the universe.

His fundamental paper “Cosmological considerations in the general

theory of relativity” was published in 1917, February 15. It represented

“the first serious proposal for a novel topology of the world as a large”

[Pais 1982, p. 286]. Einstein was aware that this solution could appear

“outlandish” [CPAE 1998E, doc. 293] or “adventurous” [CPAE 1998E,

doc. 298] to his colleagues, and thought “to have erected, from the

standpoint of astronomy, but a lofty castle in the air” [CPAE 1998E,

doc. 311, p. 301]. However, Einstein was satisfied because, by this

model of the universe, “nothing new can be found from general relativity

anymore: identity of inertia and gravity; the metric behavior of matter

is determined by the interaction of the bodies; independent properties of

space do not exist. With this, in principle, all is said” [CPAE 1998E,

doc. 453, p. 459].

4.1.3 “A ‘finite’ and yet ‘unbounded’ universe”

As already mentioned in Chapter 3, Schwarzschild studied in 1900

the astronomical consequences of an elliptic geometry of space. It is

interesting to note, as pointed out in [Schemmel 2005, p. 470], that

Schwarzschild dealt with the possibility of a closed universe with elliptic

geometry as a solution of field equations in some correspondence with

Einstein already in early 1916 [CPAE 1998E, doc. 188].

Einstein avoided the difficulty to obtain boundary conditions at spa-

tial infinity by using a very similar solution, i.e. by considering the uni-

verse “as a continuum which is finite (closed) with respect to its spatial

dimensions” [Einstein 1917b, p. 427].

There were two possibilities: there could be an infinite extension of

the universe or a finite one. Non-Euclidean geometry allowed to right-

fully consider the finiteness of space without contradictions respect to

experimental results [Einstein 1917a, p. 364]. A spatially infinite uni-


verse was possible only with a vanishing density of matter, in the sense

that “the ratio of the total mass of the stars to the volume of space (...)

approaches zero as greater and greater volumes are considered” [Einstein

1921a, p. 215]. Such a hypothesis of vanishing density was admissible

from a logical point of view, but was less probable than a finite aver-

age density of matter in the universe [Einstein 1922a, p. 368]. The

possibility to assign this non-zero density was useful to eliminate some

difficulties due to local non-homogeneous distributions of masses. This

method emulated the way of the geodesists, who, through an ellipsoid,

“approximate to the shape of the earth’s surface, which on a small scale

is extremely complicated” [Einstein 1917b, p. 428].

Einstein based his considerations both on the hypothetical density of

matter, and on the supposed static nature of the world-system, i.e. on

a “magnitude (‘radius’) of space independent of time” [Einstein 1917a,

p. 392]. At the beginning of XX Century, as seen, no large-scale sys-

tematic velocity fields were indeed pointed out in stellar motions by ob-

servations. Velocities measured on stars were very small when compared

with the speed of light, so a reference frame existed in which the mat-

ter approximately was permanently at rest. In this approximation the

energy-momentum tensor became:

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 ρ c2

The hypothesis of a constant gravitational field corresponded to the gµν ’s

independence of the time-coordinate x4 in the equation of motion of a

material point,d2xµ

ds2+ Γµ

νσ

dxν

ds

dxσ

ds= 0. (4.8)

Thus g44 = 1 for every xµ. Moreover, the static case corresponded to

g0α = 0. In this approximation the space-time interval resulted:

ds2 = dx24 − gαβdxαdxβ. (4.9)


The odd consequence of this approximation, as Einstein pointed out,

was that “now a quasi-absolute time and a preferred coordinate system

do reappear at the end, while fully complying all the requirements of

relativity” [CPAE 1998E, doc. 298]. According to Einstein, even if “the

‘spatial’ or ‘temporal’ nature is real, it is not ‘natural’ for one coordinate

to be temporal and the others spatial” [CPAE 1998E, doc. 270, p. 257].

It was necessary to give an expression for the gαβ’s, i.e. for the poten-

tial terms independent of time. Einstein considered a spherical spatial

continuum, as the geodesist way11. Quoting Einstein, “the curvature

of space is variable in time and place, according to the distribution of

matter, but we may roughly approximate to it by means of a spherical

space”12 [Einstein 1917b, p. 432]. The hypothetical uniform distribution

of matter allowed a constant curvature of space. On a two-dimensional

spherical surface, which is defined in a 3-dimensional Euclidean space, all

points are likewise equivalent. Einstein used the 3-dimensional analogy:

the universe could be well represented through a 3-dimensional spherical

space with constant curvature. As noted by de Sitter in March, 1917,

“the idea to make the 4-dimensional world spherical, in order to avoid the

necessity of assigning boundary conditions, was suggested several months

ago by Prof. Ehrenfest, in a conversation with the writer. It was, how-

ever, at that time not further developed” [de Sitter 1917a, p. 1219]. This

universe had no preferred points; it had a finite volume equal to 2πR3

(dependent on the radius R), and its surface was unbounded [Einstein

1917a, p. 367].

In this framework, the hypothesis of an isotropic and homogeneous

space can be seen as the first suggestion to what became some years later

11Einstein’s suggestion to assume global average property of matter when consid-

ering extragalactic scales (as the “geodesist way”) turned out to be one of the typical

features in modern approach to cosmology.12In order to better describe the universe as a whole, right after publishing the “Cos-

mological considerations”, Einstein acknowledged the elliptical geometry as “more

obvious” than the spherical one [CPAE 1998E, doc. 319, doc. 359].


the “Cosmological Principle”.

Introducing (as a mathematical tool) a 4-dimensional Euclidean space,

such a hyper-sphere was defined by:

R2 = ξ21 + ξ2

2 + ξ23 + ξ2

4 . (4.10)

All points of this hyper-surface filled the 3-dimensional continuum of

the spherical space with constant curvature R. The line element of the

4-dimensional hyper-surface was

dσ2 = dξ21 + dξ2

2 + dξ23 + dξ2

4 . (4.11)

The projection on the hyper-plane ξ4 = 0 could be used to obtain the

line element of the spherical space:

dσ2 = γαβdξαdξβ, (4.12)

γαβ = δαβ +ξαξβ

R2 − (ξ21 + ξ2

2 + ξ23)

, (4.13)

where α, β = 1, 2, 3 and δαβ = 1 if α = β; δαβ = 0 if α 6= β. In such

a way Einstein found for his static model of the universe this expression

for the spatial potentials:

gαβ = −[δαβ +

xαxβ

R2 − (x21 + x2

2 + x23)

]. (4.14)

This model fully achieved the relativity of inertia. The condition of

closure of the universe replaced boundary conditions at infinity. There

was not any independent property of space which claimed to the origin

of inertia, so the latter was entirely produced by masses in the universe.

4.1.4 The cosmological constant

As seen, the requirement of a gravitational field independent of time

led to the formulation of the metric of the universe. However, there was

not any agreement between these values and field equations. For this


Figure 4.2: Einstein’s static model of the universe. One of the three spatial

dimensions is shown, and Einstein’s spherical world corresponds to the surface

of the cylinder. The time coordinate is along the vertical axis [from Robertson

1933, p. 70].

reason Einstein modified the latter. In fact a spatially finite universe ap-

peared to be a good solution to the cosmological problem, i.e. to achieve

the relativity of inertia. “From the equations of the general theory of

relativity - Einstein wrote - it can be deduced that this total reduction of

inertia to interaction between masses, as demanded by E. Mach, for ex-

ample, is possible only if the universe is spatially finite” [Einstein 1921a,

p. 215]. It was thus necessary to find a suitable formulation of relativistic

field equations to express also in general relativity the observed evidence


of a static equilibrium of the universe13.

Einstein introduced a new term on the left-hand side of field equa-

tions:

Rµν − 1

2gµνR− λgµν = −κTµν . (4.15)

He added the fundamental tensor gµν multiplied by −λ, an unknown

constant. This constant was sufficiently small, so that the modified field

equations were “compatible with the facts of experience derived from the

solar system” [Einstein 1917b, p. 430]. Both the general covariance and

the laws of conservation of momentum and energy were still satisfied.

The solution of field equations for the simplest case of one of the

two world points with coordinates x1 = x2 = x3 = x4 = 0 led to the

connection among the new universal constant, the radius of the universe

and the mean density of matter:

λ =κρc2

2=

1

R2; (4.16)

13It is useful to note an interesting analogy about the idea of the quasi-static equilib-

rium of the universe. “I do not seriously consider believing - Einstein wrote to Besso -

that the universe is statistically and mechanically at equilibrium, even though I argue

as I do. The stars would all have to conglomerate, of course (if the available volume

is finite)” [CPAE 1998E, doc. 308, p. 296]. In 1897 Boltzmann illustrated some

analogue considerations on the equilibrium of the universe. According to Boltzmann,

“the universe as a whole is in thermal equilibrium, and therefore dead.”. However,

“there must be here and there relatively small regions of the size of our galaxy (which

we call worlds), which (...) deviate significantly from thermal equilibrium. Among

these worlds the state probability increases as often as it decreases. For the universe

as a whole the two direction of time are indistinguishable, just as in space there is no

up or down” [Boltzmann 1897, Engl. tr. p. 242]. The question of the arrow of time

has been faced in a famous debate between Einstein and Walter Ritz (1878-1909) in

1909. In Einstein words, the “irreversibility of electromagnetic elementary processes”,

as well the “irreversibility of the elementary processes of atomic motion” [Einstein

1909a, p. 359], “is exclusively due to reasons of probability” [Einstein 1909b]. Even if

we have not found explicit references to the paper of Boltzmann in Einstein’s papers

and correspondence, the Boltzmann’s proposal for the universe as a whole in thermal

equilibrium could have been of some importance for Einstein’s original idea of the

universe as a whole in static equilibrium.


M = ρ 2π2R3 =4π2

κc2R. (4.17)

By using the spatial density of matter from star counts (ρ0 ' 10−22

g/cm3), the previous relation led to a world-radius R = 107 light-years,

whereas the farthest visible stars were estimated at 104 light-years [CPAE

1998E, doc. 308, p. 297].

As Einstein pointed out, the new λ-term “is not justified by our ac-

tual knowledge of gravitation. (...) That term is necessary only for the

purpose of making possible a quasi-static distribution of matter” [Ein-

stein 1917b, p. 432]. The modified field equations were consistent with

the metric of the static and closed universe. According to Einstein, “the

burning question whether the relativity concept can be followed through

the finish or whether it leads to contradictions” was thus solved. Actu-

ally, through this extension of field equations, Einstein was “no longer

plagued with the problem, while previously it gave no peace” [CPAE

1998E, doc. 311, p. 301].

4.2 The universe of de Sitter

Right after Einstein’s model appeared, de Sitter proposed his own

solution of field equations. The Dutch astronomer admired Einstein’s

conception of the universe “as a contradiction-free chain of reasoning”

[CPAE 1998E, doc. 312], and gave a different solution also maintaining

the λ-term.

However, de Sitter preferred the original relativistic theory of gravi-

tation, “without the undeterminable λ, which is just philosophically and

not physically desirable” [CPAE 1998E, doc. 313, p. 305]. According

to de Sitter, indeed, the cosmological constant “is a name without any

meaning, which (...) appeared to have something to do with the consti-

tution of the universe; but it must not be inferred that, since we have

given it a name, we know what it means. (...) It is put in the equa-

tions in order to give them the greatest possible degree of mathematical

The universe of de Sitter 59

generality” [de Sitter 1932b, p. 121].

4.2.1 The “mathematical postulate of relativity of

inertia”

De Sitter approached the cosmological problem in a different way.

He proposed, as a kind of reply to Einstein’s model, a finite and empty

universe, carrying on a mathematical conception of inertia. The corre-

spondence of de Sitter reveals that, in this framework, some important

suggestions can be attributed to Ehrenfest, colleague of de Sitter in Lei-

den during those years [CPAE 1998E, doc. 321].

De Sitter was “sceptical” [CPAE 1998E, doc. 327] about Einstein’s

model, and about the assumption that the world was mechanically quasi-

stationary. According to de Sitter, “all extrapolation is uncertain. (...)

We only have a snapshot of the world, and we cannot and must not con-

clude (...) that everything will always remain as at that instant when

the picture was taken” [CPAE 1998E, doc. 321]. Otherwise “the extrap-

olation leads us - de Sitter wrote to Einstein - to assume that we have

solved a puzzle, when we have just clothed it in other words” [CPAE

1998E, doc. 321, p. 312]. Moreover, the hypothetical average density of

“world matter” ρ was objectionable, because the distribution of stars in

the observable portion of the universe was extremely not homogeneous.

De Sitter proposed a distinction between the “world matter” and the

“ordinary matter”. The former was hypothetically distributed through

space with density ρ0. The latter corresponded to observable objects as

planets and stars, i.e. to locally condensed world matter with density ρ1.

By this assumption, de Sitter pointed out that “inertia is produced by

the whole of world matter, and gravitation by its local deviations from

homogeneity” [de Sitter 1917b, p. 5].

Neglecting all pressures and internal forces, and supposing all matter

to be at rest, the energy-momentum tensor became:

T44 = (ρ0 + ρ1) c2 g44. (4.18)


De Sitter made the hypothesis to neglect gravitation on large-scale, and

to take ρ0 constant.

The 3-dimensional finite world proposed by Einstein satisfied what

de Sitter called the “material relativity requirement” [CPAE 1998E, doc.

321, p. 312], or equivalently the “material postulate of relativity of in-

ertia” [de Sitter 1917b, p. 5]. This requirement denied the possibility of

the existence of a world without matter: “if all matter - de Sitter wrote -

is supposed not to exist, with the exception of one material point which

is used as a test-body, has then this test-body inertia or not? The school

of Mach requires the answer No. (...) This world matter, however, serves

no other purpose than to enable us to suppose it not to exist” [de Sitter

1917a, p. 1222].

According to de Sitter, the postulate that at infinity all gµν ’s were

invariant for all transformations, i.e. the requirement that the metric

satisfied general covariance, was more important than the Machian pos-

tulate of relativity of inertia introduced by Einstein. Quoting de Sitter,

the relativity of inertia was only “a somewhat vague phrase to which var-

ious meaning were attached” [de Sitter 1933, p. 158]. This interpretation

is the main difference between Einstein’s and de Sitter’s approaches.

As de Sitter pointed out, in Einstein’s model, for the hypothetical

value R →∞, the whole of gµν ’s degenerated to

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 1

This set of values was invariant for all transformations for which, at

infinity, t′ = t. In other words, in Einstein’s world it was possible to

find systems of reference in which the gµν ’s only depended on space-

variables, and not on “time”. However, the “time” of such a systems

had “a separate position” [de Sitter 1917a, p. 1223], because it was

“the same always and everywhere” [de Sitter 1917b, p. 11]. For such a


reason, according to de Sitter, the time coordinate in Einstein’s model

was nothing else than an absolute time, and there the world matter took

“the place of the absolute space in Newton’s theory, or of the inertial

system” [de Sitter 1917b, p. 9].

De Sitter proposed that the potentials should have degenerated at

infinity to the values:

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

“If at infinity - de Sitter claimed - all gµν ’s were zero, then we could truly

say that the whole of inertia, as well as gravitation, is thus produced. This

is the reasoning which has led to the postulate that at infinity all gµν ’s

shall be zero” [de Sitter 1917b, p. 4]. De Sitter called this requirement

the “mathematical relativity condition” [CPAE 1998E, doc. 321, p. 312],

or the “mathematical postulate of relativity of inertia” [de Sitter 1917b,

p. 5]. Indeed, such a condition corresponded to the possibility that “the

world as a whole can perform random motions without us (within the

world) being able to observe it” [CPAE 1998E, doc. 321, p. 312]. “The

postulate of the invariance of the gµν ’s at infinity - de Sitter concluded

- has no real physical meaning. It is purely mathematical” [de Sitter

1917a, p. 1223].

4.2.2 A universe without “world matter”

In a letter to Einstein [CPAE 1998E, doc. 313] de Sitter proposed

his own solution for the metric of the universe as a whole, actually the

second relativistic model in modern Cosmology. The Dutch astronomer

considered field equations with the λ-term and without matter, i.e. he

assumed ρ0 = 0:

Rµν − 1

2gµνR− λgµν = 0. (4.19)


These equations were satisfied by the gµν ’s given by the space-time in-

terval:

ds2 =−dx2 − dy2 − dz2 + c2dt2

[1− λ12

(c2t2 − x2 − y2 − z2)]2. (4.20)

The coordinates (x, y, z, t) could have infinite values, provided that gµν ’s

were null at infinity. Such a condition was equivalent to the finiteness of

the world in natural (proper) measure. In fact the length of any semi-axis

in natural measure was:

Lα =

∫ ∞

0

√−gαα dxα. (4.21)

A finite world (i.e. a finite value of Lα) necessary implied gαα = 0 for

xα →∞, and vice versa [de Sitter 1917b, p. 5].

De Sitter pointed out that in his model no world matter was necessary,

and the insertion of the λ-term satisfied the mathematical postulate of

relativity of inertia. In this system there was not any universal time,

nor any difference between the “time” and the other coordinates: none

of these coordinates had any physical meaning [de Sitter 1917b, p. 11].

The cosmological constant determined the value of the curvature radius

R:

λ =3

R2. (4.22)

By using an imaginary “time”-coordinate ξ4 = ict, the geometry of de

Sitter’s world was that of a 4-dimensional hyper-sphere which could be

described in a 5-dimensional Euclidean space:

R2 = ξ21 + ξ2

2 + ξ23 + ξ2

4 + ξ25 . (4.23)

In hyper-spherical coordinates the line element of such a 4-dimensional

world resulted:

ds2 = −R2dω2 + sin2 ω[dζ2 + sin2 ζ(dψ2 + sin2 ψ dθ2)], (4.24)

where 0 ≤ θ ≤ 2π; 0 ≤ ψ, ζ, ω ≤ π. Equivalently, by replacing the

imaginary “time”-coordinate ξ4 with a real time-coordinate (ξ4 → iξ4),


the geometry of de Sitter’s world corresponded to a 4-dimensional hyper-

boloid in a 4+1-dimensional Minkowski space-time:

R2 = ξ21 + ξ2

2 + ξ23 − ξ2

4 + ξ25 . (4.25)

By pseudo-spherical coordinates (with iω′ = ω), the interval of space-

time resulted:

ds2 = R2dω′2 − sinh2 ω′[dζ2 + sin2 ζ(dψ2 + sin2 ψ dθ2)], (4.26)

where 0 ≤ θ ≤ 2π; 0 ≤ ψ, ζ ≤ π; −∞ < ω′ < +∞.

The potentials in the hyper-spherical coordinate system were:

gµν = −[δµν +

xµxν

R2 − (x21 + x2

2 + x23 + x2

4)

]. (4.27)

Thus the solution proposed by de Sitter,

ds2 =−dx2 − dy2 − dz2 + c2dt2

[1− λ12

(c2t2 − x2 − y2 − z2)]2, (4.28)

could be obtained by the stereographic projection of the 4-dimensional

hyper-sphere on the Euclidean space, or equivalently by the projection

of the hyperboloid on a 3+1-dimensional Minkowski space-time14 [CPAE

1998 Editorial, p. 353].

“If a single test particle - de Sitter wrote to Einstein - existed in the

world, that is, there were no sun and stars, etc., it would have inertia”

[CPAE 1998E, doc. 313, p. 303]. Actually in the universe proposed by

de Sitter a suitable metric was obtained without any physical masses.

Such forms of matter as stars and nebulæ were to be regarded as “test

particles” in a fixed background metric, which curvature was determined

by the cosmological constant [Bernstein-Feinberg 1986, p. 10].

4.2.3 Einstein’s criticism

Einstein acknowledged de Sitter’s solution to be “very interesting”

[CPAE 1998E, doc. 317, p. 308], but “must have been disappointed”

14See later, Chapter 5, for further descriptions of the geometry of de Sitter’s world.


Figure 4.3: Example of the stereographic projection of a spherical surface on

the Euclidean space [from Harrison 2000, p. 376].

[Pais 1982, p. 287], and tried to discard this anti-Machian solution: “I

cannot grant - Einstein wrote to de Sitter - your solution any physical

possibility” [CPAE 1998E, doc. 366]. Actually, the cosmological term

took a fundamental role in de Sitter’s model in order to involve a sort

of spatial (and not material) origin of inertia. “The gµν field - Einstein

replied to de Sitter - should be fully determined by matter, and not be

able to exist without the latter” [CPAE 1998E, doc. 317, p. 309].

First of all Einstein objected that the hyperboloid surface

1− λ

12(c2t2 − x2 − y2 − z2) = 0 (4.29)

was a singularity. On this surface there was a discontinuity, because

the g44 term “jumped” [CPAE 1998E, doc. 317, p. 309] from +∞ to

−∞, and gαα’s from −∞ to +∞. Such a surface lied in the physically

finite, but it was not possible to assume infinite values for the poten-

tials, because of the supposed static nature of the universe and the small

velocities measured on stars. Moreover, the 4-dimensional continuum

proposed by de Sitter did not have the property that all its points were

equivalent [CPAE 1998E, doc. 351]. It had indeed a preferred point, i.e.

the center of the conic section

1 + (c2t2 − x2 − y2 − z2) = 0. (4.30)


De Sitter replied that the hyper-surface involved a finite natural spatial

distance and an infinite natural temporal distance. Thus the disconti-

nuity was only apparent, and this problem was “not interesting” [CPAE

1998E, doc. 327]. Also the supposed preferred point was later shown

to be a geometrical consequence of that choice of coordinates, and not a

true physical aspect. “My 4-dimensional world - de Sitter remarked to

Einstein - also has the λ-term, but no world matter” [CPAE 1998E, doc.

363].

In order to better compare his own model with Einstein’s solution,

de Sitter proposed another expression of the metric [CPAE 1998E, doc.

355]. By using spherical polar coordinates, he represented the hyper-

boloid universe (system B) as the Einstein’s universe (system A), i.e. as

3-dimensional spheres embedded in a 4-dimensional Euclidean space:

ds2A = −dr2 −R2 sin2 r

R(dψ2 + sin2 ψdθ2) + c2dt2, (4.31)

ds2B = −dr2 −R2 sin2 r

R(dψ2 + sin2 ψdθ2) + cos2 r

Rc2dt2. (4.32)

De Sitter pointed out that, between the possible forms of space with

constant curvature, the elliptical space was more preferable than the

spherical one. He acknowledged the importance of the already mentioned

1900 work by Schwarzschild [CPAE 1998E, doc. 355]. In the elliptical

space, which also was closed respect to its dimensions, any two straight

lines could not have more than one point in common, so there were not

the “antipodal” points which on the contrary were present in spherical

space [de Sitter 1917b, p. 8]. Einstein agreed with de Sitter on the

choice of elliptical space [CPAE 1998E, doc. 359], but he noticed that

the spherical geometry he used in the “Cosmological considerations in

the general theory of relativity” was just an approximation. According to

Einstein, it was useful to show “through an idealization, that a spatially

closed (finite) system is possible. (...) The system could actually be

quite irregularly curved, also on a large scale, that is, it could relate to

the spherical world like a potato’s surface to a sphere’s surface” [CPAE

1998E, doc. 356, p. 346].


Figure 4.4: Example of the elliptical space, which does not have antipodal

points and covers half of the surface of a sphere [from Harrison 2000, p. 202].

By the new expression of the line element (the so-called “static form”)

it was clear that all the points in de Sitter’s world were equivalent. How-

ever, as Einstein pointed out, the g44 coefficient of the temporal term in

system B depended on position. Being g44 = cos2( rR), such a potential

changed its value from 1 (for r = 0) to 0 (for r = π2R). According to

Einstein, time clocks slowed down approaching r = π2R: this null value

of the potential involved that all masses had the tendency to aggregate

at this “equator” [CPAE 1998E, doc. 363]. “It seems - Einstein wrote in

1918 - that no choice of coordinates can remove this discontinuity. (...)

We have to assume that de Sitter solution has a genuine singularity on

the surface r = π2R in the finite domain. (...) The de Sitter system does

not look at all like a world free of matter, but rather like a world whose

matter is concentrated entirely on the surface r = π2R” [Einstein 1918b,

p. 37].

According to Einstein, a free of matter solution of field equations was

inconceivable. Through his critical comment to de Sitter’s solution, Ein-

stein advocated the belief that the cosmological constant did not involve

any sort of spatial origin of inertia. The idea of the material origin of

inertia inspired by Mach was so important that Einstein elevated the


relativity of inertia to one of the fundamental principles of his new the-

ory of gravitation. Einstein indeed at almost the same time proposed

both such a critical comment on de Sitter’s solution and a new paper

on the foundations of General Relativity. In the latter, Einstein wanted

to underline the importance of three independent aspects upon which

the theory was based. Together with the Principle of Relativity and the

Principle of Equivalence, he took into account a third aspect, which he

called “Mach’s Principle”15: “The G-field is completely determined by

the masses of the bodies. Since mass and energy (...) are the same, and

since energy is formally described by the symmetric energy tensor (Tµν),

it follows that the G-field is caused and determined by the energy tensor

of matter” [Einstein 1918a, p. 33].

De Sitter acknowledged Einstein’s remark about solution B to be

correct, but gave a different interpretation. According to the Dutch as-

tronomer, such a remark involved a philosophical, and not a physical

requirement [CPAE 1998E, doc. 501, p. 523]. In fact, the “equator” at

r = π2R was at a finite distance in space, but was physically inaccessible

[de Sitter 1918, p. 1309]. It was a sort of “mass-horizon”. The velocity

of a material particle became zero for r = π2R. Thus a material particle

which was on the polar line on the origin could have no velocity, nor

energy. “All these results - de Sitter stated - sound very strange and

paradoxical. They are, of course, all due to the fact that g44 becomes

zero for r = π2R. We can say that on the polar line the 4-dimensional

time-space is reduced to the 3-dimensional space: there is no time, and

consequently no motion” [de Sitter 1917b, p. 17]. The time needed by

a ray of light, or by a material particle, to travel by any point to the

equator was infinite. Thus the singularity at r = π2R could never affect

any physical experiment [de Sitter 1918, p. 1309].

The debate between Einstein and de Sitter finished with these differ-

15The Einstein-Mach question and its related topics are fundamental issues in the

history of physics. We refer to [Barbour 1995 ] for further readings on these still open

matters.


Figure 4.5: The static form of de Sitter’s universe in pseudo-spherical coor-

dinates. Such a universe corresponds to the surface of the hyperboloid (one

spatial dimension and the time coordinate). Static coordinates cover only the

part of the hyperboloid bounded by the generators. Lines at constant time

intersect each other at the ‘equator’ [adapted from Lord 1974, p. 121].

ent interpretations of such a property of de Sitter’s empty universe.

The issue about the correctness of de Sitter’s solution was solved by

Felix Klein (1849-1925). Einstein, indeed, faced this topic with the au-

thoritative mathematician and with Weyl. In some correspondence with

Einstein, Klein showed that the singularity at the equator in de Sitter’s

universe could be eliminated. Indeed, by using pseudo-spherical coordi-

nates of the line element of the hyperboloid form, it could be possible to

write the line element in the static form. Thus such a singularity could

“simply be transformed away” [CPAE 1998E, doc. 566, p. 593]: the pres-


ence of such a “mass-horizon” was only a geometrical consequence of the

choice of coordinates, i.e. it was a coordinate singularity, not an intrinsic

one16. The matter-free model proposed by de Sitter was free of singular-

ities, and its space-time points were all equivalent. Indeed, as shown in

[Schroedinger 1957, pp. 14-17] and in [Rindler 1977, p. 185], the appar-

ent singularity derives from the fact the coordinates of the static form of

de Sitter’s line element cover only a portion of de Sitter hyperboloid.

In front of the explanation proposed by Klein, Einstein admitted that

de Sitter’s solution existed. However, he still believed that this anti-

Machian universe was not a physical possibility [CPAE 1998E, doc. 567].

In particular, Einstein objected that, from Klein’s result, it emerged a

non-static interpretation of de Sitter’s model. “For in this world - Ein-

stein wrote - time t cannot be defined in such a way that the three-

dimensional slices t = const. do not intersect one another and so that

these slices are equal to one another (metrically)” [CPAE 1998E, doc.

567, p. 594]. In the hyperboloid representation, hyper-surfaces at differ-

ent times intersected each other at the equator, therefore time coordinate

could not be uniquely defined. According to Einstein, this singularity-

free solution of field equations without matter existed, however such a

solution was not static17.

Actually, the non-static character of de Sitter’s world which Einstein

pointed out from the geometrical interpretation of Klein can be con-

sidered as the very first hint in the departure from static cosmological

solutions of field equations in modern cosmology. Before the expanding

universe entered modern cosmology, however, it was the interest in the

16See [CPAE 1998 Editorial, Earman-Eisenstaedt 1999 ] for detailed reports on the

Einstein - Klein - Weyl discussions about such a singularity in de Sitter’s cosmological

model. A useful reading about Weyl’s contributions to cosmology, highlighting Weyl’s

interpretations of de Sitter’s solution, is [Goenner 2001 ].17As we shall see, before the 1930 general acceptance of the expanding universe,

Einstein objected both to the non-static universe which Friedmann proposed in 1922

and to the similar solution which Lemaıtre gave in 1927 that they did not correspond

to a physical possibility.


“de Sitter effect” which drew the attention in cosmological discussions to-

wards the connection between relativistic world models and astronomical

data by non-static interpretations of the line element.

Chapter 5

The “de Sitter Effect”

The so-called “de Sitter effect” is a theoretical redshift-distance rela-

tion which can be obtained through the metric of de Sitter’s universe1. In

this chapter the variety of theoretical predictions of such an effect which

appeared during the 1920’s is analyzed2.

Already in 1917 de Sitter related spectral shifts to velocity and dis-

tance of astronomical objects by his relativistic solution. He proposed

that spectral displacements which were observed in some stars and neb-

ulæ could be interpreted in his static and empty world as an appar-

ent (spurious) velocity of test particles due to the peculiar g44 term in

de Sitter’s line element, superimposed to a relative velocity which re-

1In present thesis we will generally refer to the de Sitter effect as a redshift-distance

relation. However, the strict interpretation would refer to a shift-distance relation,

since, as we shall see, the de Sitter effect predicted also approaching motions.2In Chapter 2 we mentioned different kinds of shift which can be related to astro-

nomical observations, i.e. the gravitational, Doppler, and cosmological (expansion)

shift. During the historical period which is considered in the present thesis, redshift

and blueshift measurements corresponded to velocity measurements, because of the

habit in the early phases of relativistic cosmology to directly estimate velocities from

spectral displacements by classic or special relativistic Doppler formula. In this sense,

the de Sitter effect was considered during the 1920’s, i.e. before the acceptance of

the expanding universe, as a redshift-distance relation or equivalently as a velocity-

distance relation.

71

72 The “de Sitter Effect”

sulted from geodesic equations. The first contribution led to a quadratic

velocity-distance relation, while the latter involved a linear dependence.

These suggestions did not pass unnoticed. During the 1920’s several

scientists dealt with the properties of de Sitter’s universe and proposed

different formulations of the redshift-distance effect which resulted by

the metric of such a model. Despite its lack of matter, de Sitter’s uni-

verse attracted the attention of scientists for several features, such as

the not univocal geometrical description, the presence of the singular

mass-horizon, and the possibility to explain spectral displacements in as-

tronomical objects. In particular, as we shall see, some analysis about

the de Sitter effect roughly admitted a linear relation between veloci-

ties and distances. Although there was an ambiguous formulation of the

theoretical relation between velocities and distances, the de Sitter effect

seemed to offer an answer to the question of relevant redshift measure-

ments in nebulæ, and up to 1930 such an effect was the only possible,

however puzzling, explanation of the redshift problem.

Moreover, it turned out that the empty universe of de Sitter could be

represented also by stationary frames (with a closed or a flat geometry

of spatial sections). Weyl, Lanczos, Lemaıtre and Robertson interpreted

de Sitter’s universe as a stationary world, by using different definitions

of a stationary space-time and different coordinates. Their contributions

marked the theoretical departure from a static picture of the universe3.

Because of the properties of this model, during the 1920’s de Sitter’s

solution was preferred to the “rival” static solution proposed by Einstein,

which did not allow to such an interpretation of redshift.

The interest in the de Sitter effect raised in 1917 and faded away in

1930, when the observed redshifts were eventually interpreted as due to

3As already mentioned (see Chapter 2), in the retrospect these authors actually

used an expanding FLRW frame. The possibility to write the line element of de

Sitter’s universe in many ways comes from the fact that such a space-time is a space-

time of constant curvature, and there is not a unique choice to specify the 4-velocity

which represents the average motion of matter [Ellis 1990, p. 100].

73

a motion of recession in an expanding universe. The history of the first

diffusion of the model of the expanding universe is strictly connected to

the history of the investigations of a suitable relation between velocity

and distance of spiral nebulæ. Even though de Sitter effect turned out to

be of minor importance and, compared to the genuine receding motion of

distant nebulæ, was regarded since 1930 as an “imitation recession” [Ed-

dington 1933, p. 2], several attempts to predict and to confirm through

observations the de Sitter effect foreshadowed the tortuous transition to

the expanding universe. After the 1925 acceptance that spiral nebulæ

were “island universes”, i.e. really extragalactic systems, in 1929 a linear

relation between velocity and distance of these objects was confirmed by

Hubble’s observations4. In 1930 the expanding models of the universe

independently proposed by Friedmann and Lemaıtre already in, respec-

tively, 1922 and 1927 were accepted by the scientific community in order

to describe the universe as a whole. Friedmann and Lemaıtre took into

account in their own dynamical models the possibility of a not empty, ho-

mogeneous and isotropic universe with a world-radius increasing in time.

Static and stationary models were eventually seen as limiting cases of

solutions of field equations describing an expanding universe.

Before this second renewal of cosmology, i.e. before the general accep-

tance of the expanding universe, investigations about the properties of

the model proposed in 1917 by de Sitter played a remarkable role in the

first connections between theoretical and observational cosmology which

took place during the 1920’s. Between the two rival relativistic models of

the universe, de Sitter’s solution offered more advantages than Einstein’s

one with regard to astronomical consequences and observations.

In this perspective, the history of the de Sitter effect which is below

4We recall that in the picture of present cosmology, the relation which Hubble pro-

posed in 1929, the so-called Hubble law, should be written as zc = Hr, i.e. strictly

as an empirical redshift-distance relation. Such a law coincides with the general the-

oretical velocity-distance relation v = Hr only for small distances and small redshifts

compared to Hubble radius.


reconstructed is useful to highlight the richness of contributions which

appeared in the early intersection of predictions offered by relativistic

cosmology and confirmations through astronomical observations. In par-

ticular, the actors during the 1920’s discussions about the de Sitter effect

approached and thoroughly analyzed some fundamental questions in the

framework of relativistic cosmology, such as the nature of redshift, the

geometry of space, the assumption of a homogeneous and isotropic uni-

verse.

5.1 De Sitter’s first suggestion

Einstein and de Sitter, whose insights into the relativity of inertia

inaugurated the rise of relativistic cosmology, followed different paths

comparing their own theoretical world models to astronomical measure-

ments.

As seen in Section 4.1.4, Einstein proposed in 1917 a possible value

of the radius of his own spherical universe by using star count estimates

for the density of matter5.

On the contrary, dealing with the features and the geometry of his

own empty world, de Sitter wrote the line element of this model in dif-

ferent ways, and studied properties of test particles and light rays in

it. Investigating the astronomical consequences of general relativity, de

Sitter compared in his papers results obtained through solutions A and

5Einstein, in collaboration with Erwin Freundlich (1885-1964), pursued a possible

confirmation that a non-zero cosmological constant could be revealed by observations.

In particular, in the framework of Newtonian approximation of general relativity, Ein-

stein investigated the application of Newtonian law of gravitation to globular clusters.

Being globular clusters in stationary equilibrium, through some assumptions on the

average masses and luminosity of stars the virial theorem could be used to obtain an

average theoretical velocity of stars. Therefore the comparison with observed veloci-

ties would have revealed the presence of a non-zero cosmological constant, which was

necessary to prevent the collapse of such stellar systems [Einstein 1921b].

De Sitter’s first suggestion 75

B6. It was just the attention de Sitter drew on redshift interpretation

which inaugurated the connection of observational astronomy on large

scale with theoretical predictions given by models of the universe based

on general relativity. In particular, as we shall see, de Sitter was the first

to take into account both redshifts measured on stars and relevant re-

cession velocities obtained from spectroscopic analysis of nebulæ, clearly

attempting to relate these observational evidences to the geometry of the

universe [Ellis 1989, p. 372].

De Sitter proposed his own cosmological solution of field equations

right after Einstein’s “Cosmological considerations” appeared in 1917

[CPAE 1998E, doc 313; de Sitter 1917a; de Sitter 1917b]. Despite this

model, i.e. solution B, was purely obtained from a “mathematical pos-

tulate” and corresponded to an empty world, de Sitter attached great

importance to its astronomical consequences.

In this framework, the correspondence with Kapteyn, whom de Sit-

ter asked for advices just during the debate with Einstein, reveals that

de Sitter was interested in the suitability of his own model with regard

to parallax measurements and to estimates of the mass of our sidereal

system. Most of all, de Sitter dealt with the possibility of a systematic

redshift from observations of stars and nebulæ. Actually, he was very

interested whether a general average redshift could be revealed by obser-

vational evidences. “These are hard nuts - Kapteyn replied to de Sitter

in June 1917 - you are giving me to crack” [van der Kruit-van Berkel

2000, p. 96].

System B was the 4-dimensional analogy of the 3-dimensional spher-

ical space of system A. The line element of system B, following de Sitter,

could be written as:

ds2 = −R2dω2 + sin2 ω[dζ2 + sin2 ζ(dψ2 + sin2 ψ dθ2)], (5.1)

6It is important to mention that, beside solutions A and B, de Sitter took into

account for the sake of comparison the solution of field equations without λ, i.e. the

line element of the special theory of relativity, which he denoted as solution C.


where ψ and θ were real angles, and ω and ζ were imaginary angles [de

Sitter 1917c, p. 229]. Equivalently, avoiding imaginary quantities by

ω = iω′ and ζ = iζ ′, the metric was:

ds2 = R2dω′2 − sinh2 ω′[dζ ′2 + sinh2 ζ ′(dψ2 + sin2 ψ dθ2)]. (5.2)

Furthermore, by the new real coordinates χ and η defined as

sin ω sin ζ = sin χ, (5.3)

r = R χ, (5.4)

tan ω cos ζ = tan iη, (5.5)

t = R η, (5.6)

the line element in the form 5.1 could be written in the “static” form:

ds2 = −dr2 −R2 sin2 r


Rc2dt2. (5.7)

According to de Sitter, elliptical geometry was more suitable than the

spherical one for the absence of antipodal points. In order to describe

spaces with constant positive curvature, elliptical space was “really the

simpler case, and - de Sitter wrote - it is preferable to adopt this for the

physical world” [de Sitter 1917b, p. 8]. Spherical space filled elliptical

space twice, and for small values of r compared to R these two spaces

could be approximated by the Euclidean space [de Sitter 1917c, p. 231].

Such an elliptical space, as the spherical one, could be projected on an

Euclidean space or on a hyperbolical space. In the first case, elliptical

space was projected on the whole of Euclidean space through the trans-

formation7:

r = R tan χ. (5.8)

The line element was:

ds2 = − d r2

(1 + r2

R2

)2 −r2[dψ2 + sin2 ψdθ2]

1 + r2

R2

+c2dt2

1 + r2

R2

. (5.9)

7The symbol r was introduced by de Sitter. It represents a spatial coordinate, not

a vector.


With regard to hyperbolical space, de Sitter considered a new coordinate

h, so that:

cos χdh = dr. (5.10)

The integral of this expression was:

sinhh

R= tan

r

R=

r

R. (5.11)

Thus spatial sections could be described also through hyperbolical space

(or space of Lobatschewsky) with constant negative curvature [de Sitter

1917b, p. 13]. The line element could be written:

ds2 =−dh2 − sinh2 h

R[dψ2 + sin2 ψdθ2] + c2dt2

cosh2 hR

. (5.12)

Also in this case, the elliptical space was projected on the whole of hyper-

bolical space. On the contrary, as de Sitter pointed out, the projection of

spherical space of system A, i.e. of Einstein’s model, filled both Euclidean

space and hyperbolical space twice [de Sitter 1917c, p. 233].

Therefore de Sitter showed that the spatial geometry of his static

and empty model of the universe, with positive curvature proportional

to the cosmological constant (ρ0 = 0; λ = 3R2 ), could be represented by

an elliptical geometry, and equivalently described through the projection

on Euclidean and hyperbolical spaces.

Hyperbolical geometry was useful to derive a formula for parallax.

Indeed, as de Sitter pointed out, the rays of light were not geodetic lines

in 3-dimensional space (r, ψ, θ), where the speed of light was equal to

v = c cos θ. They were not geodesics also in space (r, ψ, θ). Light rays

were straight (i.e. geodetic) lines in hyperbolical space (h, ψ, θ), where

speed of light was constant in all directions [de Sitter 1917b, p. 13]. The

parallax p of a star at a distance r from the Sun could be obtained in

such a space from:

tan p = sinha

Rcoth

h

R, (5.13)

being a the average distance between the Sun and the Earth. Thus:

p =a

Rcoth

h

R. (5.14)


From equation 5.11 it followed:

p =a

R sin χ=

a

r

√1 +

r2

R2. (5.15)

In system B, the parallax p of a star was never zero, and reached its

minimum value at h → ∞, i.e. at the “mass horizon” χ ≡ rR

= π2

[de

Sitter 1917b, p. 13]. On the contrary, in system A, being for spherical

space p ' ar, parallax p was zero for r = π

2R, i.e. at the largest distance

which was possible in elliptical space [de Sitter 1917c, p. 233].

A lower limit of the curvature radius, R > 4 · 106 AU, was found in

1900 by Schwarzschild by using hyperbolical spaces and parallaxes of the

order of 0′′.05. This value, according to de Sitter, could be applied in

system B, while measured parallaxes did not give a limit of R in system

A [de Sitter 1917c, p. 234].

De Sitter proposed different ways to estimate the curvature radius in

system A. First, he considered the angular diameter δ of an object of

linear diameter d at the distance r from the Earth. In elliptical space

the relation with R was given by:

δ =d

R sin χ. (5.16)

“It is very probable - de Sitter wrote in 1917 - that at least some of the

spiral nebulæ or globular clusters are galactic systems comparable with

our own in size” [de Sitter 1917b, p. 24]. Taking for example for one of

these systems d = 109 and δ = 5′, it followed R ≥ 1012 AU in system A.

Another estimate was obtained by considering for the star density of

the universe (ρ0) the same density at the center of the galactic system

(80 stars/1000 pc3); in this case, the curvature radius in system A was

R = 9 · 1011 AU.

In addition, being space finite and straight lines closed, one should

expect in system A to see the antipodal image of the Sun. Since this

was not the observed case, light should have been absorbed traveling

round the world. Taking an absorption of 40 magnitudes, which had


been already proposed in 1900 by Schwarzschild, de Sitter obtained a

value R > 14· 1012 AU of the radius in Einstein’s universe [de Sitter

1917c, p. 234]. However, de Sitter remarked that “all this of course is

very vague and hypothetical. Observation only gives us certainty about

the existence of our own galactic system, and probability about some

hundreds more. All beyond this is extrapolation” [de Sitter 1917c, p.

237].

Estimates of R in system B could not be obtained by using the fact

that the “back of the Sun” was not observed, since in such a system

“light requires an infinite time for the voyage round the world” [de Sitter

1917b, p. 26]. Also the relation between apparent and linear diameter of

spirals could not be applied. Such a relation was in system B:

δ =d

R sinh hR

=d

R tan χ. (5.17)

Thus δ was zero at r = π2R.

However, it was the g44 term which was useful to estimate the world

radius in system B, leading to the interpretation of spectral displacements

in de Sitter’s universe.

5.1.1 Redshifts in de Sitter’s universe

Since in the static form of system B the g44 term diminished with

increasing r, the frequency of light vibrations diminished with increasing

distances from the observer at rest at the origin of coordinate, i.e., as de

Sitter wrote, “the lines in the spectra of very distant objects must appear

displaced towards the red” [de Sitter 1917c, p. 235]. According to de

Sitter, this effect corresponded to a “spurious positive radial velocity” [de

Sitter 1917b, p. 26] of very distant stars and nebulæ. Such a displacement

was indeed produced by the inertial field, and was superposed on the

displacement due to the gravitational field of objects themselves. In

order to explain such a spurious velocity, de Sitter took into account the

gravitational contribution to redshift in spectral lines produced by stars,


i.e. one of the “crucial phenomena” [de Sitter 1933, p. 150] which could

be described through Einstein’s new theory of gravitation8. For a fixed

star in a general static field, i.e. for a fixed point in 3-dimensional space,

the line element could be written:

ds2 = f c2dt2, (5.18)

where, for small deviations from the diagonal values of Minkowski space-

time (denoted by de Sitter as Galilean values), f was obtained by the

Newtonian potential φ:

f = 1 + γ ' 1 +2φ

c2. (5.19)

Sincedt

ds=

1

c√

1 + γ, (5.20)

the measure of time was different at different places in the gravitational

field. Spectral lines originating in a strong gravitational field, for example

on the solar surface, would be displaced towards the red to an observer

in a weaker gravitational field. At the surface of the Sun:

dt

ds=

1.00000212

c. (5.21)

The ratio between the observed and emitted wavelengths was:

λ0

λe

=1√

1 + γ' 1− 1

2γ. (5.22)

Therefore, interpreting the observed shifts through classic Doppler’s for-

mula (z ≡ λ0−λe

λe= v

c), the displacement on the solar surface was the

same as produced by a radial velocity of 0.00000212c, or 0.634 km/sec

8It is worth noting that de Sitter considered also the blueshift contribution due to

the general gravitational field of a shell of fixed stars. Such a displacement towards

the violet, according to de Sitter, was smaller than about 13 km/sec, and its effect was

canceled by the displacement towards the red produced by the gravitational field of

each single star [de Sitter 1916c, pp. 175-177].


[de Sitter 1916a, p. 719]. For a star of mass M and density ρ (with solar

mass and density M¯ = ρ¯ = 1), such a gravitational shift K was:

K = 0.634 M23 ρ

13 . (5.23)

Referring to Campbell’s observations9, Helium stars (i.e. B stars), for

which K ' 1.4 km/sec, showed a systematic redshift corresponding to

a radial velocity of about v = +4.5 km/sec. De Sitter pointed out that

apparent (spurious) positive radial velocities were therefore produced by

the diminution of g44 in the line element of his own model. Indeed just

one third of this observed value could be associated to the gravitational

shift at the star surface. The remaining v = +3 km/sec corresponded

to the displacement by the inertial field in system B [de Sitter 1917c, p.

235].

De Sitter used such a displacement in order to calculate the radius

of his universe. Since the ratio of the observed and emitted wavelengths

was related to the velocity v through Doppler’s theory, it followed [de

Sitter 1917c, p. 235]:

f = g44 = 1 + γ ' 1− 2v

c= 1− 2 · 10−5. (5.24)

In the static form of system B, g44 = cos2 rR. Assuming the average

distance of Helium stars at about r = 3 · 107 AU, it resulted a curvature

radius of system B of R = 23· 1010 AU [de Sitter 1917b, p. 27].

De Sitter took into account the most relevant radial velocities, both

positive and negative, which were known in 1917, and attempted to re-

late these observational evidences to the geometry of his own universe.

This was the first suggestion which inaugurated the intersection between

astronomical observations on large scale and an appropriate description

of the universe as a whole given by relativistic solutions of field equations.

Referring to data from the 1917 Council of the Royal Astronomical

Society about spiral nebulæ [Eddington 1917 ], de Sitter pointed out that

9Campbell’s analysis of the K term from observations of velocities of B stars will

be described in next chapter.


the lesser (meaning the Small) Magellanic Cloud was estimated to be at

r > 106 AU, with a radial velocity v ' +150 km/sec. These values gave,

for system B, a curvature radius R > 2 ·1011 AU [de Sitter 1917b, p. 27].

By several independent observations, three objects (NGC 4594, NGC

1068 and the Andromeda nebula) showed very large radial velocities com-

pared with usual velocities of stars in Sun neighborhood [de Sitter 1917c,

p. 236]:

NGC 4594 Pease + 1180 km/sec

Slipher + 1190 ”

NGC 1068 Pease + 765 km/sec

Slipher + 1100 ”

Moore + 910 ”

Andromeda Wright – 304 km/sec

Pease – 329 ”

Slipher – 300 ”

Their average velocities were [de Sitter 1917b, p. 27]:

NGC 4594 + 1185 km/sec

NGC 1068 + 925 ”

Andromeda – 311 ”

Taking v = +600 km/sec as the mean of these velocities, and the curva-

ture radius R = 23· 1010 AU, the distance of these nebulæ was, at least,

r = 4 · 108 AU [de Sitter 1917c, p. 236]. Alternatively, by assuming for

these objects a distance of about r = 2 · 1010 AU, the curvature radius of

system B was found of the order of R = 3 · 1011 AU [de Sitter 1917b, p.

28].

As showed above, the displacement towards the red depending on the

g44 term for all spectral lines in de Sitter’s static universe was:

z ≡ λ0 − λe

λe

=1√g44

− 1 = secr

R− 1. (5.25)

By considering the spatial projection on Euclidean space, through which


the line element of solution B became:

ds2 = − d r2

(1 + r2

R2

)2 −r2[dψ2 + sin2 ψdθ2]

1 + r2

R2

+c2dt2

1 + r2

R2

, (5.26)

redshift, i.e. velocity, was related to distance by the quadratic form10:

λ0 − λe

λe

' 1

2

r2

R2. (5.28)

Superimposed to such a gravitational shift, which was responsible for

an apparent velocity, there was another effect in de Sitter’s universe,

which de Sitter interpreted as a sort of Doppler effect11 due to the veloc-

ity of particles in his empty world. From the geodesic equations, indeed,

a test particle in de Sitter’s universe showed a velocity contribution de-

pending on distances, which was a “velocity due to inertia” [de Sitter

1917b, p. 27].

With regard to the equations of motion of a material particle in the

field of pure inertia, by using coordinates (r, ψ, θ, ct), the differential

equations of the geodesic were [de Sitter 1917b, p. 15]:

d2r

c2dt2=

r

R2+ r

[(dψ

c dt

)2

+ sin2 ψ

(dθ

c dt

)2]

, (5.29)

10We recall a present summary of several contributions to shifts, which has been

mentioned in Chapter 2. Following [Ellis 1989, p. 374], the gravitational, kinematic

and cosmological shifts can be resumed as:

(1 + ztot) = (1 + zDS)(1 + zGS

)(1 + zC + zGC)(1 + zDO

)(1 + zGO). (5.27)

In the static frame, as in de Sitter’s original proposal, the term zC is equal to zero.

The redshift contribution 5.28, obtained by de Sitter as a gravitational shift from

the metric of solution B, corresponds to the term zGC , and reflects the space-time

curvature [Ellis 1990, p. 101].11Such a contribution is equivalent to an additional term zDS

in the shift summary

of the previous note. The same consideration can be applied to the analysis of the

contributions to redshift in de Sitter’s universe proposed by Eddington in 1923, as we

will see later. However, according to Eddington the amount of the velocity (Doppler)

effect was roughly the same as the quadratic distance (gravitational) effect.


d2θ

c2dt2= −2

r

(dr

c dt

) (dθ

c dt

)− 2 cot ψ

(dψ

c dt

)(dθ

c dt

), (5.30)

d2ψ

c2dt2= −2

r

(dr

c dt

)(dψ

c dt

)+ sin ψ cos ψ

(dθ

c dt

)2

. (5.31)

Taking ψ = 90, dψdt

= 0, the integration of geodesics gave:

r2

(dθ

dt

)= c = k1, (5.32)

(dr

dt

)2

+ r2

(dθ

dt

)2

=r2

R2+ k2. (5.33)

In order to find the equation of the orbit of a material particle, de Sitter

put k1 = (r0 v0) and k2 =(v2

0 − r20

R2

), where r0 was the minimum distance

from the origin, and v0 = r0

(dθdt

)0

was the particle velocity at that point.

By using a = r0 and b = (R v0), the equation of the orbit was [de Sitter

1933, p. 195]: (dr

dθ

)2

=r2(r2 − a2)(r2 + b2)

a2b2. (5.34)

The integration gave an hyperbola: in system B a material particle under

the influence of inertia alone did not describe a straight line with constant

velocity. The orbit was an hyperbola of which the real axis was r0 and

the imaginary axis was Rv0:

x2

a2− y2

b2= 1, (5.35)

with x = r cos(θ − θ0) and y = r sin(θ − θ0). For v0 = 1 the velocity

was equal to the speed of light, therefore in system B, in the reference

system (r, ψ, θ, ct), also the orbit of rays of light were hyperbolas whose

imaginary axis was R [de Sitter 1917b, p. 19].

The radial velocity was in this notation [de Sitter 1933, p. 195]:

(dr

dt

)2

=r2

R2

(1− a2

r2

)(1 +

b2

r2

). (5.36)

According to this relation, the velocities due to inertia had no preference

in sign.


The total shift in de Sitter’s universe, i.e. what de Sitter interpreted as

a real motion due to inertia together with a spurious redshift contribution

due to the diminution of g44, took the form12:

λ0 − λe

λe

' ± r

R+

1

2

( r

R

)2

. (5.38)

Strictly, it was the quadratic relation which became known as the de

Sitter effect [Ellis 1990, p.101].

In 1917, at the very beginning of relativistic cosmology, de Sitter re-

marked that, with regard to the relevant radial velocities observed in

nebulæ, “conclusions drawn from them are liable to be premature” [de

Sitter 1917b, p. 27]. If, however, future observations confirmed a system-

atical displacement towards the red, this evidence would have been an

indication in order to adopt system B in preference to system A. Since

these two systems differed in their physical consequences at large dis-

tance, according to de Sitter the study of systematic radial motions of

spirals was exactly the key method in order to decide between system

A and system B: “if continued observations should confirm the fact that

the spiral nebulæ have systematically positive radial velocities - de Sitter

remarked - this would be certainly an indication to adopt the hypothesis

B in preference to A” [de Sitter 1917b, p. 28].

“At the present time - de Sitter wrote in 1920 - the choice between

the systems A and B is purely a matter of taste. There is no physical

criterion as yet available to decide between them” [de Sitter 1920, p.

12It is useful to mention the general spectral shift in de Sitter’s universe which

North derived in his own book about the history of cosmology, “The measure of the

universe”. By using the metric of the static form of de Sitter’s universe, North gave

the general formula [North 1965, p. 95]:

1 + z = α[1 + v sec

( r

R

)]sec2

( r

R

), (5.37)

where α was a parameter fixing the orbit of the source. As we shall see in next

sections, during the 1920’s several scientists proposed different formulations of the de

Sitter effect. According to North, Weyl’s 1923 result is not confirmed through the

general formula proposed by North himself [North 1965, p. 101].


867]. Referring to data available in 1920, radial velocities of 25 spirals

were known. Among them, as de Sitter pointed out, only three were

negative, and the mean velocity was v ' +560 km/sec. Thus there was a

rough, not (yet) systematic, observed tendency of spirals to recede, which

could eventually lead to discriminate between Einstein’s and de Sitter’s

universes. However, quoting de Sitter, “the decision between these two

systems must, I fear, for a long time be left to personal predilection” [de

Sitter 1920, p. 868].

As seen above, first hints about the interpretation of relevant shift

displacements through relativistic solutions of field equations were pro-

posed in 1917 by de Sitter, investigating the features of his own model

of the universe. In the following years, as we shall see in next sections,

several authors dealt with different interpretations of the line element of

de Sitter’s universe and the connection of the de Sitter effect to shifts

measurements.

De Sitter directed Leiden Observatory from 1919 to 1934, and, after

1920, did not consider the cosmological question in other published pa-

pers up to the 1930 second renewal of cosmology. As Eddington wrote

in 1934 in an obituary for de Sitter, the Dutch astronomer was “the man

who discovered a universe and forgot about it” for at least ten years

[Eddington 1934, p. 925].

Some 1929 correspondence between de Sitter and Frank Schlesinger

(1871-1943), astronomer at Yale University Observatory, are useful to

emphasize de Sitter’s point of view on redshift in his own model. In these

letters de Sitter resumed the question about the velocity-distance effect

at the turning point of Hubble’s 1929 confirmation that a linear velocity-

distance relation existed. De Sitter considered the twofold contribution to

redshift predicted by his empty model, and tried to explain the absence of

observations of approaching objects which on the contrary were predicted

through the term corresponding to negative radial velocities. In a letter

to Schlesinger dated November 8, 1929, de Sitter pointed out that, being


the approximate formula of the velocity-distance effect

v

c= ± r

R+

1

2

( r

R

)2

, (5.39)

the first term represented a “real velocity due to repulsive force of the

origin of the coordinate, which is a consequence of the adopted form of

the gravitational potential” [de Sitter Archive, Box 17.7A]. The observed

velocities, de Sitter wrote in this letter, were all positive, and could not

be represented by a linear formula. “Why - de Sitter inferred - are all

the spirals found on the receding branches and none on the approaching?

The evident solution (evident once you think about it) is: some of them

(say one half) have originally been on the approaching branches, but

have long since past their nearest point, and are now receding” [de Sitter

Archive, Box 17.7A].

As de Sitter noted, this suggestion could be attributed to Eddington.

Indeed, some months earlier de Sitter and Eddington discussed about

spiral nebulæ during a trip to South-Africa, where the Meeting of the

British Association for the Advancement of Science took place. On July

2, 1929, de Sitter reported that “Eddington is of the opinion that two big

negative velocities are too much of an exception. They are however both

in Andromeda, in the direction of our movement by the rotation of the

galaxy. He had never noticed this, and thinks (...) that the explanation

by ‘de Sitter’s world’ has become much more probable. (...) The velocity

is ± at + bt2. The term with a must be preponderant for the clear (and

big) nebulæ. Why always +? (this is my question). Eddington seems to

want to answer: at the creation all the r’s were small” [de Sitter Archive,

Box 43.6. Engl. tr. by Jan Guichelaar].

At the end of December, 1929, de Sitter wrote in another letter to

Schlesinger that he had derived empirical formulas for distances of spirals

as a function of the diameter and the magnitude, which, quoting de

Sitter, “still represent what I think is the best we can do at present” [de

Sitter Archive, Box 17.7A]. De Sitter later published these results in [de

Sitter 1930a, de Sitter 1930b]. The application of such relations to radial


velocities led to a linear relation between velocity and distance:

∆ = 2000 v, (5.40)

∆ being the distance in units of 1024 cm, and v the velocity in that of

light as a unit. The factor 2000 was the radius of the universe when

solution B was adopted (R = 2 · 109 light-years) [de Sitter Archive, Box

17.7A].

Figure 5.1: Detail from a letter by de Sitter to Schlesinger (November 8,

1929). In this letter de Sitter, following Eddington’s suggestion, considered

the possibility that some spirals “have originally been on the approaching

branches, but have long since past their nearest point, and are now receding”

[from de Sitter Archive, Box 17.7A].

“We are confronted - de Sitter wrote to Schlesinger - with the math-

ematical problem: what becomes of the empty world B if you fill it with

matter. I have not yet been able to solve this problem completely, but I

have reasons to expect that the solution will be intermediate between the

solutions A and B” [de Sitter Archive, Box 17.7A]. As we shall see, also

Eddington considered (already in 1923) the possibility that the actual

world corresponded to an intermediate state between Einstein’s and de

Sitter’s universes.


In 1930, having learned about 1927 Lemaıtre’s contributions to the

cosmological question, de Sitter “accepted with enthusiasm” [Eddington

1934, p. 925] such a new development of the theory and the related

interpretation of redshift as due to a motion of recession in an expanding

universe.

In 1933, in the appendix of a report on the astronomical aspect of

relativity theory, de Sitter returned on the twofold source of redshift in his

own empty and static model of the universe. He explained that “in 1917

(...) it was not realized that the velocity drdt

would always be positive,

and it was thought that this Doppler effect would not be systematic,

the redshift 12

(rR

)2being the only systematic effect” [de Sitter 1933, p.

195]. From late 1920’s, velocities of very distant celestial objects turned

out to be all positive; therefore according to de Sitter such an evidence

could be explained in his own model by postulating, as already mentioned

in his 1929 letter to Schlesinger, that “all observable bodies are on the

receding branches of their respective hyperbolas, having passed the apex

long ago, so that none remain on the approaching branches” [de Sitter

1933, p. 195].

In such a 1933 report, de Sitter showed that, being the Doppler ef-

fect a first-order effect proportional ro rR, a linear relation was actually

predicted also by his static model, having corrected the Doppler formula

to the second order. For a general velocity q, the correct Doppler effect

formula was:λ0 − λe

λe

= q − 1

2q2. (5.41)

Taking

q =dr

dt=

r

R, (5.42)

the contribution by the quadratic term was canceled, leaving the linear

effect:λ0 − λe

λe

=r

R. (5.43)

This effect, de Sitter concluded, was “in exact agreement with the result

from the new theory of the expanding universe” [de Sitter 1933, p. 196].


5.2 Matter or motion? Eddington’s analy-

sis

Eddington and the British scientific community became aware of de-

tails of Einstein’s new theory of gravitation through de Sitter’s papers

which appeared in 1916 and 1917 in the Monthly Notices of the Royal

Astronomical Society13.

As from 1917, Eddington dealt with the general theory of relativity in

many papers and books, giving fundamental contributions in this field14.

With regard to the curvature of space and time and the considerations

on the universe as a whole, in front of the two rival possibilities proposed

by Einstein and de Sitter, Eddington was inclined to prefer the empty

solution of de Sitter.

Eddington found Einstein’s model objectionable for some aspects. “I

feel - Eddington wrote to de Sitter in August, 1917 - a strong objection

to system A. It seems to me that the world-matter in that is simply the

aether coming back again” [de Sitter Archive, Relativity Box, A2]. Re-

ferring to the presence of antipodal points in Einstein’s world, Eddington

regretted “being unable to recommend this rather picturesque theory of

anti-suns and anti-stars. It suggests that only a certain proportion of

the visible stars are material bodies; the remainder are ghosts of stars,

haunting the places where stars used to be in a far-off past” [Eddington

1918, p. 87].

Einstein’s hypothesis allowed to a direct relation between the world

radius R and the total amount of matter M in it. As Eddington wrote,

13In June 1916, Eddington had written to de Sitter: “Hitherto I had only heard

vague rumors of Einstein’s new work. I do not think anyone in England knows the

details of his paper” [de Sitter Archive, Relativity Box, A2].14We mention, for example, Eddington’s contribution to the fundamental 1919 ex-

periments devoted to the confirmation of the amount of light-bending as predicted

by general relativity, for which we refer to [Kennefick 2007 ]. A useful paper about

Eddington’s works in relativity and Einstein’s reaction to them is [Stachel 1986 ].

Biographical material about Eddington can be found in [Douglas 1956a].

Matter or motion? Eddington’s analysis 91

“there is something rather fascinating in a theory of space by which,

the more matter there is, the more room is provided” [Eddington 1918,

p. 90]. However, following de Sitter’s remarks, Eddington criticized the

introduction of such a vast quantity of undetected world matter15.

Also the cosmological constant was objectionable, being a “very artifi-

cial adjustment” [Eddington 1918, p. 87], which was “very hard to accept

at any rate without some plausible explanation of how the adjustment

is brought abroad” [Eddington 1920, p. 163]. However, as mentioned in

Chapter 3, in following years the role of the cosmological constant gained

great importance in Eddington’s search for a fundamental theory which

could relate constants in nature.

The alternative solution proposed by de Sitter seemed “much less

open to objection” [Eddington 1918, p. 87]. “If the real time is used -

Eddington wrote about solution B - the world is spherical in its space

dimensions, but open towards plus and minus infinity in its time dimen-

sion. (...) It might seem that this kind of fantastic world building can

15Already in 1916, i.e. before Einstein’s and de Sitter’s cosmological solutions

appeared, Eddington criticized Einstein’s initial assumption of unobservable super-

natural masses which influenced local observable phenomena. In a letter to de Sitter

dated October 13, 1916, Eddington criticized the supernatural masses proposed by

Einstein which were responsible for the degeneration of potentials at infinity in the

form

0 0 0 ∞0 0 0 ∞0 0 0 ∞∞ ∞ ∞ ∞2

Quoting Eddington, such hypothesis contradicted “the fundamental postulate that

observable phenomena are entirely conditioned by other observable phenomena. It is

no great advance - Eddington wrote - to be told that instead of being conditioned by a

framework of reference (which we can materialize as the aether), they are conditioned

by things equally outside observations at infinity. But then (one asks) where is infinity

according to the new conceptions? I do not know that is matters, because - Eddington

remarked - infinity is necessarily outside observation and that is the main point” [de

Sitter Archive, Relativity Box, A2].


have little to do with practical problems. But that is not quite certain”

[Eddington 1920, pp. 159-160]. The great advantages of this model were

the presence of the potentials gµν which at infinity were invariant for all

transformations, and the absence of any assumption of the existence of a

large amount of “not yet recognized” world matter [Eddington 1918, p.

88].

De Sitter’s model especially offered the possibility to explain the large

velocities of spirals, and permitted to estimate the value of the world

radius: the theory of de Sitter, according to Eddington, “is of course

very speculative, but is the only clue we possess as to the dimensions of

space” [Eddington 1929, p. 167]. In a letter to Shapley, December 1918,

Eddington explained that “de Sitter’s hypothesis does not attract me

very much, but he predicted this (spurious) systematic redshift before

it was discovered definitely; and if, as I gather, the more distant spirals

show a greater recession that is a further point in its favour” [quoted in

Smith 1982, p. 174].

In his 1920 book “Space, time and gravitation” Eddington mentioned

the two interpretations of spectral displacement proposed by de Sitter in

1917. With regard to measurements of motion in the line-of-sight of

nebulæ, Eddington pointed out that “the data are not so ample as we

should like; but there is no doubt that large receding motions greatly

preponderate” [Eddington 1920, p. 161]. Therefore this evidence could

be interpreted as a genuine phenomenon of recession; however, it could

not be discarded the possibility that such an effect was really due to the

slowing down of atomic vibrations, allowing to an apparent recession.

The de Sitter effect was faced in details in Eddington’s book “The

mathematical theory of relativity”. Such a book, which appeared in

1923, was later acknowledged by Einstein as “the finest presentation of

the subject in any language” [Douglas 1956b, p. 100]. In the chapter

which Eddington addressed to the curvature of space and time, solu-

tions A and B were considered by Eddington “as two limiting cases, the

circumstance of the actual world being intermediate between them. De


Sitter’s empty world is obviously intended as a limiting case; and the

presence of stars and nebulæ must modify it, if only slightly, in the di-

rection of Einstein’s solution” [Eddington 1923, p. 160]. This is a very

important remark highlighting Eddington’s approach to the search of a

suitable model of the universe, because it contained the suggestion to in-

vestigate intermediate solutions between a model which had matter but

not motion, i.e. Einstein’s world, and a model which was empty but

showed motion, i.e. de Sitter’s world.

As we shall see, the 1927 model of the universe proposed by Lemaıtre

would have represented such an intermediate solution. In 1930 Lemaıtre

drew the attention of Eddington to the solution of an expanding (and not

empty) universe which Lemaıtre had already discovered in 1927. There-

fore in 1930 Eddington acknowledged that such a dynamical solution

discovered by Lemaıtre was “a brilliant and remarkably complete solu-

tion of the various questions connected with the Einstein and de Sitter

cosmogonies” [Eddington 1930a, p. 668]. Through Lemaıtre’s solution

Eddington proved that Einstein’s world was unstable, claiming that “the

equilibrium having been disturbed, the universe will progress through a

continuous series of intermediate states towards the limit represented by

de Sitter’s universe” [Eddington 1930b, p. 850].

Nonetheless, in 1923, the possibility to discriminate whether de Sit-

ter’s or Einstein’s form was “the nearer approximation to the truth”

[Eddington 1923, p. 160] was strongly challenged by the preponderance

of positive velocities of spirals, which, according to Eddington, favoured

the model of de Sitter for its interpretation of redshift.

In order to describe the properties of de Sitter’s world, Eddington

considered the original line element in the form [Eddington 1923, p. 156]:

ds2 = −R2dχ2 −R2 sin2 χ(dθ2 + sin2 θ dφ2) + R2 cos2 χdt2. (5.44)

Thus, speed of light at the origin was equal to R. The coordinate χ

corresponded to ( rR), being r the same radial coordinate used by de


Sitter. Since, for a clock at rest, the metric was:

ds = R cos χdt, (5.45)

there was a displacement of spectral lines towards the red in distant

objects at rest “due to the slowing down of atomic vibrations, which

would be erroneously interpreted as a motion of recession” [Eddington

1923, p. 161].

With regard to the question of the mass horizon, Eddington pointed

out that, because of the symmetry of the original line element proposed

by de Sitter, “there can be no actual difference in the natural phenomena

at the horizon and at the origin” [Eddington 1923, p. 157]. The singu-

larity in ds2 could be removed, or equivalently introduced, by transfor-

mations of coordinates: “it is impossible - Eddington noted - to know

whether to blame the world-structure or the inappropriateness of the

coordinate-system. (...) I believe then that the mass-horizon is merely

an illusion of the observer at the origin, and that it continually recedes

as we move towards it” [Eddington 1923, pp. 165-166].

By the new coordinate r defined as:

r = R sin χ, (5.46)

the line element of system B became:

ds2 = − 1(1− r2

R2

)dr2 − r2dθ2 − r2 sin2 θ dφ2 +

(1− r2

R2

)dt2. (5.47)

Here the usual unit of t (with c = 1) was restored. From the geodesic

equations, and by considering a particle at rest (drds

= 0), it followed:

d2r

ds2=

1

3λ r, (5.48)

where λ = 3R2 . Therefore, as Eddington pointed out, “a particle at rest

will not remain at rest unless it is at the origin” [Eddington 1923, p. 161],

and was repelled away with an acceleration increasing with the distance

to the origin. Actually, such a tendency to scatter of particles revealed


that de Sitter’s world “becomes non-statical as soon as any matter is

inserted in it. But this property - Eddington claimed - is perhaps rather

in favour of de Sitter’s theory than against it” [Eddington 1923, p. 161].

Indeed, such an effect could explain the relevant velocities measured

in spiral nebulæ, which represented “one of the most perplexing problems

of cosmogony” [Eddington 1923, p. 161]. In his book Eddington reported

radial velocities measured in 41 nebulæ by Slipher. The table with these

observations [Eddington 1923, p. 162] was arranged just by Slipher, and

contained many unpublished data related to objects of the New General

Catalogue. In addition, Eddington remarked that, beside these data,

an additional nebula (NGC 1700) showed a large receding velocity by

the observations of Francis Pease (1881-1938). Among these 42 objects,

there was a strong preponderance of positive velocities, which suggested

a recession, however not entirely systematic.

Figure 5.2: List of radial velocities of spiral nebulæ measured by Slipher,

which appeared in Eddington’s 1923 book ‘The mathematical theory of rela-

tivity’ [from Eddington 1923, p. 162].

With regard to this motion of recession observed in spirals, Eddington


followed de Sitter’s double explanation: beside the motion due to inertia,

which for Eddington corresponded to a general tendency of particles to

scatter (according to equation 5.48), there was a spurious motion due to

the singular g44 term in de Sitter’s metric (equation 5.45).

In particular, Eddington pointed out that the acceleration 13λ r (equa-

tion 5.48), if continued for the time (in natural unit) Rχ = r, would have

caused a change of the velocity of a test particle of the order of:

1

3λ r2 =

r2

R2. (5.49)

Since (from the metric in the form 5.47) the shift due to g44 term resulted

z ' 1

2

(r

R

)2

, (5.50)

Eddington concluded that “the Doppler effect of this velocity would be

roughly the same as the shift to the red caused by the slowing down of

atomic vibrations” [Eddington 1923, p. 164].

Eddington interpreted redshift “as an anticipation of the motion of

recession which will have been attained before we receive light” [Edding-

ton 1923, p. 164]: nebulæ did not have the motion revealed by redshift at

the moment of emission of light, but they acquired such a motion during

the time light took to travel towards the observer.

However, in 1923 observations did not reveal a systematic shift to

the red: from Slipher’s data, some nebulæ (NGC 221, NGC 224, NGC

598) showed relevant negative velocities. This approaching motion (due

to blueshift measurements) towards the observer at the origin could not

be explained through geodesic equations. Therefore, quoting Eddington,

“the cosmogonical difficulty is perhaps not entirely removed by de Sitter’s

theory” [Eddington 1923, p. 162].

Eddington returned on such an interpretation of the effect foreseen by

de Sitter also in his 1933 book “The expanding universe”. He explained

that redshift of galaxies corresponded, now in the theory of the expanding

universe, to an actual motion of recession, and were no more to be inter-

preted as the “rather mysterious slowing down of time at great distances

Weyl, Lanczos and the redshift-distance law 97

from the observer” [Eddington 1933, p. 49]. With regard to his own

considerations about de Sitter effect which he had proposed in 1923, Ed-

dington remarked in 1933 that “it was a question of definition” whether

the slowing down of atomic vibrations was to be regarded as a genuine

or spurious motion. “During the time that its light is traveling to us -

Eddington wrote about de Sitter effect - the nebula is being accelerated

by the cosmical repulsion and acquires an additional outward velocity

exceeding the amount in dispute” [Eddington 1933, p. 49]. Therefore,

according to Eddington, in the de Sitter effect the velocity was spurious

at the time of emission, and became genuine at the time of reception.

The analysis of Friedmann’s and Lemaıtre’s expanding models, as Ed-

dington pointed out in 1933, showed that the slowing down of time had

to be regarded as a second-order term, which was small compared to the

effect due to the expansion.

5.3 Weyl, Lanczos and the redshift-distance

law

In 1923, the same year when Eddington’s book “The mathematical

theory of relativity” appeared, explicit formulations of redshift-distance

relations in de Sitter’s universe were independently proposed by Weyl

and Lanczos. These significant contributions are related to the depar-

ture from a static frame which both Weyl and Lanczos proposed in their

own considerations about de Sitter’s world. In particular, Weyl derived in

1923 a redshift-distance relation which was linear at small distances. In

1933, in a summary paper about relativistic cosmology in the framework

of the expanding universe, Robertson acknowledged that the “appropri-

ate linearity of velocity of recession with distance was first predicted by

Weyl in 1923, on the basis of the more restricted cosmology of de Sitter”

[Robertson 1933, pp. 68-69].

After the 1917 proposal by de Sitter of an empty universe as a so-


lution of field equations, Weyl participated in discussions with Einstein

and Klein about the mass-horizon singularity in de Sitter’s static form,

advocating that de Sitter’s universe actually had matter at its horizon

[CPAE 1998E, doc. 556]. By the spherical coordinate system in the static

form of de Sitter’s line element, system B corresponded to a really static

world “that - Weyl remarked - cannot exist without a mass-horizon”

[Weyl 1921, Engl. tr. p. 282]. However, as already mentioned at the end

of Chapter 4, Klein showed that the singularity at the mass-horizon was

due to the choice of coordinates, and was not an intrinsic singularity.

In the hyperboloid version, which was only partially covered by static

coordinates, such a world was not static on the whole, and, according to

Weyl, “was separated from any static solution of the same topology by

an abyss” [quoted in Earman-Eisenstaedt 1999, p. 198].

The de Sitter hyperboloid was “an homogeneous state of the world”

[Weyl 1930, p. 300]. Despite the static coordinates represented only

a sector of such a hyperboloid, Weyl shifted his own attention to this

version of the universe of de Sitter. Indeed such a world-description

seemed more satisfactory than Einstein’s one for the observed fact that

the further distances were considered, the more increasing velocities were

involved.

Both the analysis of de Sitter (1917) and Eddington (1923) were lack-

ing in a suitable assumption about the undisturbed state of stars when

the homogeneity and isotropy of space were taken into account. In this

framework, Weyl introduced the concept of causal connection of the el-

ements of proper time by postulating the existence of a system of null

geodesics diverging towards the future, in which all time moments re-

mained equivalent. Following [Bergia-Mazzoni 1999, p. 332], the as-

sumption that all null geodesics were oriented in one direction towards

the infinite future, and in the other direction towards the infinite past,

avoided the overlapping of the past and future light cones. “The different

points on the world line of a point-like source - Weyl wrote in 1923 - are

the origin of (3-dimensional) surfaces of constant phase that form null


cones opening towards the future” [Weyl 1923a, p. 322. Engl. tr. in Har-

vey et al., p. 1017]. These null cones opening to the future time-direction

filled a portion of the de Sitter’s universe which Weyl denoted as “the

range of the influence of a star. (...) There are ∞3 stars or geodesics

to which the same range of influence belongs as to the arbitrarily chosen

star” [Weyl 1930, p. 302]. De Sitter’s hyperboloid “is distinguished from

the well-known Einstein’s solution, which is based on a homogeneous dis-

tribution of mass, by the fact that the null cone of future belonging to

a world-point does not overlap with itself; in this causal sense, the de

Sitter space is open” [Weyl 1930, p. 301].

Such an assumption - Weyl claimed - “means that the stars of the

system are able to act upon one another to eternity” [Weyl 1924, p. 476],

and “is that, in the undisturbed state, the stars form such a system of

common origin” [Weyl 1930, p. 303]. The hypothesis suggested by Weyl

was thus that celestial objects could be assumed as uniformly distributed

and at rest respect to the spatial coordinates, whatever the origin of

coordinates.

Figure 5.3: Weyl’s postulate. World-lines belong to a pencil diverging towards

the future [from Harrison 2000, p. 294].

“Weyl 1923 postulate - Ellis notes - was intended to supply a way

of uniquely choosing the family of stationary expanding geodesic world


lines” [Ellis 1989, p. 375]. Weyl’s fundamental assumption became later

known as “Weyl Principle” and actually was the first proposal of causal

connection in relativistic cosmology16. By quoting Ehlers, “to specify a

cosmological model, he [Weyl] realized, it does not suffice to choose a

space-time, i.e. a Lorentz manifold. One must also specify a congruence

of time-like lines to represent the mean motion of the ‘stars’ (galaxies).

Moreover, if one wishes to express that all stars have a common ori-

gin, this congruence should be such that all its members share the same

domain of action” [Ehlers 1988, p. 95].

Through this hypothesis Weyl introduced a unique definition for spec-

tral displacement. According to Weyl, the Doppler effect and what Weyl

denoted as the Einstein effect (meaning the gravitational one) were in-

dissolubly related [Weyl 1923b, p. 375]. In de Sitter’s hyperboloid, ne-

glecting the matter contributions to gravity, world-lines belonged to a

∞3 pencil diverging towards the future. Such a divergence gave proof

of a universal tendency of matter to scatter, due to the presence of the

cosmological term [Weyl 1923a, p. 322]. By considering the world lines

of a source (s) and of an observer (σ), each point of the first corresponded

to a point on the latter, i.e. to the intersection of the observer world line

with the future null cone emanating from the source: σ = σ(s). Peri-

odic processes at the source would have resulted as periodic also by the

observer, however with the period increased according to the ratio:

α =dσ

ds. (5.51)

A spectral line measured by the observer with frequency ν corresponded

to a spectral line of the distant source with frequency να: the one appeared

displaced with respect to the other [Weyl 1923a, p. 322]. In terms of

redshift z, the general definition proposed by Weyl resulted [Weyl 1923b,

p. 375]:

z =λ0 − λe

λe

= α− 1, (5.52)

16We refer to [North 1965, Kerszberg 1986, Bergia-Mazzoni 1999, Goenner 2001 ]

for further readings on the role played by Weyl Principle and the reactions to it.


which according to Weyl assumed the form:

z = tand

R. (5.53)

Therefore, such a relation was linear at small distances. In this relation,

R was the constant curvature radius, and d was the “naturally measured

distance of the star in the static space at the same moment t at which

the observation takes place” [Weyl 1930, p. 306]. In 1923 Weyl did not

furnish explicit non-static coordinates, however in a 1930 paper about

redshift and relativistic cosmology he used the stationary (expanding)

frame which Robertson proposed in 192817: “the cosmology proposed by

Robertson - Weyl wrote in 1930 - is identical with the one proposed by

me” [Weyl 1930, p. 301].

The coordinates in the stationary frame which Weyl adopted were

[Weyl 1930, p. 305]:

r = ρ · eτ/R, (5.55)√

1−( r

R

)2

· et/R = eτ/R, (5.56)

and the relation between the observed frequency ν0 and the emitted fre-

quency νe was:

νe

ν0

=dτ0

dτe

= eτ0/R · e−τe/R = eτ0/R(e−τ0/R +

ρ

R

). (5.57)

At the moment of observation t = τ0 of a star (i.e. of the source) at

distance r, from the above equation it followed [Weyl 1930, p. 306]:

νe

ν0

= 1 +r√

R2 − r2= 1 + tan

d

R. (5.58)

17It is useful to note that, according to Weyl, a stationary gravitational field was

characterized by the metric:

ds2 = −dσ2 + f2dt2, (5.54)

where dσ was the spatial interval, f was the speed of light (c), g0 α=gα 0=0, and space

coordinates were not depending on time [Weyl 1921, Engl. tr. p. 241].


Observations revealed that some spiral nebulæ showed relevant displace-

ments towards the red, corresponding to a radial velocity of the order

of 1000 km/sec. Conclusions about the nature and the distance of these

objects were far to be drawn. However, Weyl remarked that it was nec-

essary to consider both the possibility of the extragalactic nature of the

nebulæ, and the interpretation of redshift which followed from his own

analysis [Weyl 1923a, p. 323].

The redshift-distance relation which, on the contrary, was proposed

in 1923 by Lanczos derived from a different formulation of de Sitter’s

line element suggested by Lanczos in 1922. In a footnote of a 1922 paper

about de Sitter’s universe, Lanczos wrote the line element of solution B

as [Lanczos 1922, p. 539]:

ds2 = −(et + e−t)2

4(dφ2 + cos2 φ dψ2 + cos2 φ cos2 ψ dχ2) + dt2. (5.59)

Spatial sections were proportional to the factor cosh2 t, revealing the non-

static character of such a metric. This solution corresponds to closed

spatial sections (k = +1) in an expanding frame [Ellis 1990, p. 100]. It

entirely covers the de Sitter hyperboloid and it is non-singular [Earman-

Eisenstaedt 1999, p. 203].

In his 1922 paper Lanczos dealt with the question of the aggregation of

matter around the equator of de Sitter’s world, i.e. at the so-called mass

horizon, which was advocated by Weyl when considering the static form

of de Sitter’s universe. In particular, Lanczos criticized Weyl’s attempt

to demonstrate such a presence of matter aggregation.

Lanczos pursued this point of view in a subsequent paper, which

appeared in 1923 and was devoted to the redshift in de Sitter’s universe.

In this paper Lanczos pointed out that a massive horizon was not needed

to be postulated in solution B, because it was just an apparent singularity,

depending on the choice of coordinates.

Therefore, also the interpretation of redshift as a distance (gravita-

tional) effect was objectionable. In order to deal with the possibility to

discriminate whether a redshift depended on the metric structure or was


a pure Doppler effect, according to Lanczos it was necessary to estab-

lish invariant geometrical relations. A static coordinate system was not

suitable for this purpose18 [Lanczos 1923, p. 169].

Lanczos acknowledged that, through Weyl’s 1923 hypothesis, it was

possible to uniquely determine spectral shifts. However, in his own anal-

ysis Lanczos argued that the ratio between the frequency of a signal

emitted by a source and the observed frequency was equal to the ratio

between the proper time of the source and the observer [Lanczos 1923,

p. 170]:νe

ν0

=tet0

. (5.60)

Therefore, according to Lanczos, the relation

ν0

νe

=ds0

dse

(5.61)

was useful to univocally determine redshift. However, any redshift de-

pended not only on the relative positions of the source and the observer,

but also on the angle between the two intervals. In his analysis Lanczos

derived a relation so that the ratio of observed and emitted wavelengths

depended only on the two angles that the line element of the source and

the line element of the observer formed with the line which connected

them [Lanczos 1923, p. 177]:

ν0

νe

=cos γe

cos γ0

. (5.62)

Such a relation, according to Lanczos, was valid both in special relativ-

ity and in general relativity. Since redshift was determined by the same

quantities in both theories, and since in special relativity a spectral dis-

placement was interpreted as a Doppler effect, Lanczos concluded that

18According to Lanczos, the static form of the universe of de Sitter was not sta-

tionary. Lancozs considered that a universe was stationary “if the coefficients of its

metric are independent of time in a coordinate system in which the masses are at

rest on average. (...) A necessary and sufficient condition for this - Lanczos asserted

- is that the time lines of our coordinate systems are geodesics. Therefore the static

solution given by de Sitter is not an example of a stationary world” [Lanczos 1924,

Engl. tr. p. 363].


any redshift had to be interpreted as a Doppler effect, and there was not

any shift which depended on the metric, thus in particular there were

not gravitational shifts [Lanczos 1923, p. 177].

Figure 5.4: The universe of de Sitter in the version proposed by Lanczos in

1922. Spatial sections at constant time are closed (k =+1). Such coordinates

cover the whole hyperboloid without singularities [adapted from Lord 1974, p.

123].

Lanczos then studied the Doppler shift predicted through this version

of the metric of de Sitter’s universe. He derived the relation [Lanczos

1923, p. 184]:

1 + z =νe

ν0

= cosa

R− sin

a

Rsinh τ0, (5.63)

where a was the geodesic distance between the source and the observer,

R the curvature radius, and τ0 the time of light reception. Therefore,

such a theoretical relation was approximately linear with regard to the

Silberstein’s contributions 105

distance, however with a time-dependent additional term19.

As seen, in 1923 the question of the interpretation of the prepon-

derance of redshift measurements through the line element of de Sitter’s

universe was far to be clarified. The de Sitter effect represented a pos-

sible solution, however puzzling for the arbitrariness of possible expla-

nations of redshift. The general tendency of particles to scatter, which

was suggested by Eddington and, mutatis mutandis, by Weyl, was not

systematically confirmed by observations. It was this general recession

which was strongly criticized by Silberstein, a Polish-American physicist.

5.4 Silberstein’s contributions

Silberstein dealt with shifts in de Sitter’s universe in many papers,

the majority of which appeared in 1924. He used the static form of the

line element and proposed a relation between velocity and distance. Such

a relation was obtained in a different way with respect to Weyl’s 1923

result and was approximately linear for very distant objects. However,

Silberstein proposed that such a relation was valid both for receding

and for approaching objects, discarding the assumption that there was

a general tendency of particles to scatter in de Sitter’s universe. In

his own analysis, Silberstein pointed out that globular clusters, and not

spiral nebulæ, were useful to confirm such a theoretical relation and to

determine the curvature radius of de Sitter’s world20.

19With regard to the summary of redshift contributions proposed by Ellis (see

Chapter 2), if an expanding frame is used, as by Weyl in 1930 and Lanczos in 1923,

and, as we shall see, also by Lemaıtre in 1925 and Robertson in 1928, one obtains

expansion redshifts (zC) [Ellis 1990, p. 101; Goenner 2001, p. 123].20It is worth noting that in several papers Silberstein referred to Shapley’s data.

Following [Flin-Duerbeck 2006, p. 1091], Shapley’s rejection of the existence of ex-

ternal galaxies could have played a significant role in Silberstein’s approach to the

interpretation of his own theoretical relation and the determination of the value of

the curvature radius. However, Shapley was disturbed by the polemical style which

Silberstein used in some of his own papers. See [Smith 1979, p. 144] for further


Already in 1922, in the first edition of his book “The theory of general

relativity and gravitation”, Silberstein mentioned de Sitter’s interpreta-

tion of redshift related to the g44 term. Disregarding the contribution of

gravitational shift due to the potential of objects themselves, the remain-

ing spectral displacement observed in B stars, as seen in section 5.1.1,

could be explained by the decrease of g44 term in de Sitter’s metric, lead-

ing to a possible determination of the curvature radius. According to

Silberstein, “there is, for the present, nothing cogent in the attribution

of the said remainder of spectrum shift to the dwindling of g44 with mere

distance, and it would certainly be premature either to reject or to ac-

cept the results of this attractive piece of reasoning” [Silberstein 1922,

pp. 136-137].

Two years later, Silberstein pointed out that, with regard to the ac-

tual contributions to this second-order effect due to mere distance, “the

distance-effect is insolubly amalgamated with the velocity - or usual

Doppler - effect” [Silberstein 1924a, p. 350]. The total shift was pro-

portional to the first power of rR, both for receding and for approaching

objects. “What concerns us here - Silberstein wrote in 1924 - is the actu-

ally interesting case of a star and an observer behaving as free particles,

i.e. describing world-geodesics, when they cannot be at rest relatively to

each other” [Silberstein 1924a, p. 350]. Silberstein acknowledged that

it was Weyl’s merit to have proposed a general treatment of redshift

in terms of world-lines. However, the universal tendency of matter to

scatter which Weyl had suggested was “an arbitrary hypothesis” [Silber-

stein 1924c, p. 909], and the “mythical assumption” [Silberstein 1924a,

p. 350] that the world-lines belonged to a pencil of geodesics diverging

towards the future was a “sublime guess” [Silberstein 1924c, p. 909].

Also Eddington’s suggestion of a universal scattering which appeared in

Eddington’s book “The mathematical theory of relativity” was consid-

ered by Silberstein as “a fallacy based upon a hasty analysis” [Silberstein

readings on Shapley’s reaction to Silberstein’s papers.


1924a, p. 350], and was “entirely undesirable” [Silberstein 1924c, p. 909].

The result which Weyl proposed in 1923, i.e. the redshift-distance

relation21

z = tanr

R, (5.64)

was contradicted both by measurements of the motion of the Andromeda

nebula, which showed a blueshift, i.e. an approaching velocity of the or-

der of v ' −316 km/sec, and by the approaching motion of other spirals.

Therefore, according to Silberstein, globular clusters represented suitable

objects to take into consideration more than spirals, being globular clus-

ters “equally interesting and probably as distant celestial objects (...).

Among them, negative radial velocities are by no means an exception.

Rather the contrary” [Silberstein 1924c, p. 909]. Moreover, referring

to data furnished by Shapley, Silberstein acknowledged that estimates

about the radial velocities and distances of globular clusters were known

with small errors.

Silberstein considered the static form of the line element of de Sitter’s

empty space-time:

ds2 = −dr2 −R2 sin2 r


Rc2dt2. (5.65)

From this metric, and by considering dψ = 0 and dθ = 0, the geodesic

equation which represented the most general radial motion of a free par-

ticle was [Silberstein 1924c, p. 910]:

R

c

dσ

dt= ± cos σ

√1− cos2 σ

γ2≡ ±v

c, (5.66)

where

γ = cos2 σc dt

ds=

(1− v2

0

c2

)− 12

. (5.67)

In these relations, σ corresponded to rR, and v0 was the particle velocity

at the instant of emission.

21In his papers, Silberstein used the symbol D = δλλ , which corresponds in our

notation to z = λo−λe

λe.


In order to compute the Doppler effect, Silberstein used the general

principle, formulated by Weyl, which related spectral displacements to

the ratio of the proper time of the observer and of the source:

z =ds

ds′− 1. (5.68)

For ds′ = c dα, being dα the interval of time between two light-signal

emissions from the source (initially at r = 0) to the observer, it followed:

c(dt− dα) = sec σ dr, (5.69)

cdt =sec σ dr√1− cos2 σ

γ2

. (5.70)

Therefore Silberstein found for what he called “the complete Doppler

effect” the form [Silberstein 1924c, p. 912]:

z = γ

[1±

√1− cos2 σ

γ2

]− 1 = γ

[1± v

csec σ

]− 1. (5.71)

The positive sign corresponded to receding objects, while the negative

sign to approaching ones. The distance from the observer and the source

at the moment of reception was r = Rσ. “The Doppler effect - Silberstein

noted - is in general by no means a universal function of distance alone”

[Silberstein 1924c, p. 913].

Actually, in the general formula of the Doppler effect there were two

terms: a term depending on the individual velocity v0, which was domi-

nant near the observer, and a second term depending upon rR, which, ac-

cording to Silberstein, was significant for very remote celestial objects22.

Silberstein also proposed a method to separate these two effects by ob-

serving the displacement of spectral lines at six months intervals [Silber-

stein 1924d, Silberstein 1924e, Silberstein 1924f ]. However, Eddington

criticized such an attempt. The velocities (V1 and V2) of the observer at

six months intervals were considered in the reference frame of the Earth

22These contributions are equivalent to a Doppler (zD) and a gravitational (zG)

effect, respectively, in the present summary of several contributions to redshifts.


with respect to the Sun. They could not be related to the velocity v0,

which was the velocity a star had at some epoch in the remote past or

future, “so that - Eddington wrote - (v0 − V1) has no obvious relevance

to the problem” [Eddington 1924, p. 747; see also Douglas 1924 ].

For near stars, the contribution of v0

cwas dominant with respect to

cos σ, thus in this approximation the Doppler effect reduced to the special

relativistic effect [Silberstein 1924a, p. 350]:

z =

√c± v0

c∓ v0

− 1. (5.72)

On the contrary, for the most distant celestial objects, the approxima-

tions cos σ ' 1 and γ ' 1 could be used, and the redshift was [Silberstein

1924a, p. 351; Silberstein 1924b, p. 363]:

z = ± sin σ = ±v

csec σ ' ± r

R. (5.73)

It was such a linear relation which Silberstein used in order to determine

the value of the curvature radius of de Sitter’s world by radial velocities,

both positive and negative, and distances of seven globular clusters. He

obtained a mean value of R = 6 · 1012 AU [Silberstein 1924a, p. 351],

which was almost confirmed also by (positive) velocities and distances of

the two Magellanic Clouds [Silberstein 1924b, p. 363]. In addition, being

the parallax p predicted through de Sitter’s model:

p ' tan p =a

R sin σ, (5.74)

it followed the relation:

zp = ± a

R, (5.75)

which, according to Silberstein, removed “every doubt as to the physical

meaning of r” [Silberstein 1924c, p. 914].

The most distant spiral which was known at that time, NGC 584,

showed a radial velocity of about v = +1800 km/sec. Therefore, by

using R = 6 · 1012 AU, such an object would have been at a distance

r = 3.6 · 1010 AU. “Huge as this may seen - Silberstein noted - it will


be remembered that Shapley’s latest estimate of the semi-diameter of

our galaxy is only four times smaller. (...) Whether these estimates

will or will not fit into the general scheme of modern galactic and extra-

galactic astronomy, is not known to me and must be left to the scrutiny

of specialists” [Silberstein 1924c, p. 917].

Weyl, in reply to [Silberstein 1924c], made a list of some objectionable

aspects of Silberstein’s analysis. First, the radial distance r used by

Silberstein was “very artificial”, because it was the distance of the star,

i.e. the source, from the observer at the moment of observation, “but -

Weyl noted - in the static space of the star”. On the contrary, in his own

analysis, Weyl had considered r “the distance of the star measured in the

static space of the observer at the moment of observation” [Weyl 1924,

p. 476]. With regard to Silberstein’s disapproval of Weyl’s hypothesis of

diverging world-lines, Weyl replied that “going further than de Sitter and

Eddington, I strongly emphasized the necessity of adding an assumption

regarding the undisturbed state of stars, if anything in the theoretical line

regarding the displacement to the red is to be formulated” [Weyl 1924, p.

476]. Finally, according to Weyl, Silberstein used the same assumption

introduced by Weyl for both future and past direction of time, a method

which appeared “quite abstruse” to Weyl [Weyl 1924, p. 477].

In subsequent papers, Silberstein showed that, from velocity and dis-

tance of ten objects (eight clusters and the Magellanic Clouds), a linear

relation was actually confirmed by plotting, as suggested by Henry N.

Russell (1877-1957), the modulus of the redshift: r = |z|R. Silberstein,

however, discarded data belonging to other three globular clusters (NGC

5904, 6626, 7089), which velocities were “suspiciously small’ [Silberstein

1924d, p. 602]. From data of these ten objects, the value of the curvature

radius, through the rigorous formula

z =r2

R2+

v20

c2, (5.76)

was of the order of R ≥ 9.1· 1012 AU [Silberstein 1924d, p. 602], while the

approximate linear formula led to a world radius of de Sitter’s universe


not exceeding R = 8 · 1012 AU [Silberstein 1924e, p. 819].

Figure 5.5: Linear relation between the modulus of spectral displacement

(velocity) and distance of ten globular clusters proposed by Silberstein in 1924.

Note the three objects which Silberstein discarded for their “suspicious small”

velocities [from Silberstein 1924d, p. 602].

Silberstein further developed the theoretical redshift relation for any

inertial motion, i.e. not exclusively for radial motions. He derived the

general relation [Silberstein 1924g, p. 623]:

z =

cos2 σγ

1∓√

1− cos2 σγ2

(1 + p2

R2 sin2 σ

) − 1, (5.77)

where p and γ were integration constants. In 1929, as we shall see, such

a relation was taken into account by Tolman in his own investigation of

redshifts in de Sitter’s universe.

Silberstein’s contributions marked a crucial passage to the consider-

ations on the universe as a whole through general relativity and the in-

terpretation of astronomical observations. Indeed several debates, some-

times at polemical level, arose about his systematic analysis to confirm a

suitable curvature radius through his own linear theoretical relation and

observations of astronomical objects. As seen, Eddington and Weyl inde-

pendently criticized Silberstein’s proposal for some theoretical features;


furthermore, as we will see in next chapter, Lundmark and Stromberg

denied the correctness of Silberstein’s result with regard to observational

aspects. Nevertheless, Silberstein’s proposal played an important role in

Lemaıtre’s analysis about de Sitter’s universe which Lemaıtre proposed

in 1925, and also Robertson and Tolman mentioned Silberstein’s result

in their own contributions about the de Sitter effect which appeared in

1928 and 1929, respectively.

In 1930 Silberstein collected his views concerning relativistic cosmol-

ogy in the book “The size of the universe”. In this book Silberstein

maintained the proposal of a shift relation, both for red and for blue

shifts, which was derived from the static metric of de Sitter’s universe.

He did not mention other works published at the end of the 1920’s, only

referring to the original papers by Einstein and de Sitter. Silberstein

also pursued the objection to the general tendency of particles to scatter

suggested by Weyl, and, moreover, criticized Hubble’s measurements of

distances in extra-galactic nebulæ23 [Silberstein 1930 ].

5.5 Lemaıtre’s 1925 notes

A possible relation between spectral displacement and distance in de

Sitter’s world was considered by Lemaıtre in a short note on the universe

of de Sitter which appeared in 1925 [Lemaıtre 1925a, Lemaıtre 1925b],

two years before the fundamental paper which Lemaıtre wrote about a

non-empty expanding universe.

In this 1925 note on de Sitter’s universe, which was presented to

the Society of Physics at Washington, Lemaıtre showed that, through

appropriate new coordinates, “the field is found homogeneous but not

statical. Furthermore the geometry is Euclidean. The singularity at de

Sitter’s horizon disappears” [Lemaıtre 1925b]. Indeed Lemaıtre proposed

a transformation in such a way that space in de Sitter’s universe had

23See [Robertson 1932 ] for a strong criticism of Silberstein’s book.

Lemaıtre’s 1925 notes 113

null curvature (k = 0), and spatial sections were proportional to an

exponentially time-dependent term. This result was independently found

also by Robertson in 1928, as a stationary form of de Sitter’s universe24.

In 1924-1925 Lemaıtre was Ph.D. candidate at the Massachusetts In-

stitute of Technology; during his stay in United States, he directed his

work in two lines: theoretical astrophysics and the theory of relativity.

Lemaıtre had previously been (in 1923) a student of Eddington at Cam-

bridge Observatory, just in the period when Eddington’s “Mathematical

Theory of Relativity” appeared. Therefore, the suggestion by Eddington

about the non-static character of de Sitter’s world played an important

role in the attention which Lemaıtre gave to the universe as a whole.

However, as noted in [Lambert 2000, pp. 70-81], it was a conference by

Silberstein which Lemaıtre attended in 1924, and subsequent discussions

between Lemaıtre and Silberstein about the Doppler effect and the mea-

sure of curvature radius in de Sitter’s universe, which most influenced

Lemaıtre’s early approach on relativistic cosmology25.

Following Lemaıtre’s notation (with c = 1), the model of the universe

24It is important to note that the result found by Lemaıtre and Robertson would

have corresponded years later to the exponentially expansion predicted in the steady

state model, and the exponential form of de Sitter’s universe is at present used in

order to describe the expansion of vacuum energy dominated universes.25With regard to a dynamical interpretation of de Sitter’s world, in the draft of

a 1963 obituary for Roberston stored at Lemaıtre Archive, Lemaıtre wrote: “I was

better prepared to accept it following an opinion expressed by Eddington. (...) The

errors by Silberstein have been very stimulating. I had myself had a long discussion

with him [Silberstein] in 1924 at a British Association Conference in Toronto and my

work, as possibly later on the work of Robertson, results as a large part as a reaction

against some unsound aspects of Silberstein’s theories” [Lemaıtre Archive, Box D32].

It seems very probable that, through Silberstein’s papers, Lemaıtre became aware of

Weyl’s assumption and results about redshifts. In some undated sheets which are

stored at Lemaıtre Archive, Lemaıtre repeated the calculations about the de Sitter

effect as in [Silberstein 1924c], referring also to the 1923 works of Weyl mentioned in

previous sections [Lemaıtre Archive, Box R3].


proposed by de Sitter was written with line element:

ds2 = R2[−dχ2 − sin2 χ(dθ2 − sin2 θ dφ2) + cos2 χdτ 2]. (5.78)

Such a description of de Sitter’s world was objectionable, since there

was a preferred point, the “center at χ = 0”: free particles or rays of

light did not move along geodesics of the space, except for those passing

through such a center. The purpose of the note on de Sitter’s universe

- as Lemaıtre wrote - was “to look for a separation of space and time

which is free from this objection” [Lemaıtre 1925a, p. 188].

By new coordinates, the line element could be written as:

ds2 = R2(− cosh2 τ ′[dχ′2 + sin2 χ′(dθ′2 + sin2 θ′dφ′2)] + dτ ′2

). (5.79)

This result was equivalent to the “expanding frame” obtained by Lanczos.

In this form the radius of space was constant at any place, but was

variable with time, since it was proportional to cosh τ ′. According to

Lemaıtre, by such a new metric “the central point in space is removed,

but now a central time has been introduced” [Lemaıtre 1925a, p. 189].

Indeed, at any instant τ ′ 6= 0, there were no geodesics of space which

were geodesics of the universe, i.e., quoting Lemaıtre, “the origin of time

becomes a time absolutely distinct from every other” [Lemaıtre 1925a,

p. 189]. By the transformation:

χ = arcsinr

t, (5.80)

τ =1

2ln(t2 − r2), (5.81)

and by changing polar coordinates to Cartesian ones, (r, θ, φ) → (x, y, z),

the line element was:

ds2 = R2−dx2 − dy2 − dz2 + dt2

t2. (5.82)

Therefore the geometry of spatial sections was Euclidean, and the cur-

vature was null. Time coordinate t was expressed by coordinate T :

T = ±∫

dt

t= ± ln t, (5.83)

Lemaıtre’s 1925 notes 115

and t = 0 represented the infinite past or future. The metric assumed

the form:

ds2 = R2[−e± 2T (dx2 + dy2 + dz2) + dT 2], (5.84)

where the coefficient −e−2T referred to parallel geodesics to the future,

while −e2T referred to parallel geodesics in the past direction. This choice

of coordinate, according to Lemaıtre, could free from the objection of

introducing the spurious asymmetry in space and time as in de Sitter’s

original metric26, which “is not simply the mathematical appearance of

center of an origin of coordinates, but really attributes distinct absolute

properties to a center” [Lemaıtre 1925a, p. 192].

Lemaıtre then considered the Doppler effect, and calculated the shift

predicted by such a metric. He found the same result obtained by Silber-

stein, except, as Lemaıtre pointed out, for the interpretation of the sign

in such an effect. However, Lemaıtre did not refer to observational data.

Lemaıtre considered two light sources, Me and M0, of the same proper

interval ds, which described geodesics of constant x, y, z:

ds

R=

dtete

=dt0t0

. (5.85)

The equation of the light ray from Me to M0 was t0 = te+r. By supposing

the light source Me at the origin, as done by Silberstein, Lemaıtre found

for the Doppler shift the linear form [Lemaıtre 1925a, p. 191]:

∆λ

λ0

=λe − λ0

λ0

=dtedt0

− 1 =tet0− 1 = − r

t0= − sin χ. (5.86)

The observer was supposed to describe geodesics passing neither through

the origin, nor at a minimum distance to it. This Doppler effect was ob-

tained in the case t = 0 in the past, meaning that lines of the universe

converged in the future. The same result, however with the opposite sign,

26With regard to the interpretations of the universe of de Sitter, Lemaıtre wrote

in 1929 that “Weyl a montre qu’il pouvait s’interpreter comme un espace Euclideen

ou la confirguration formee par les points materiels se dilate en restant semblable

a elle-meme. Lanczos a donne une interpretation analogue pour un espace ferme”

[Lemaıtre 1929, p. 31].


Figure 5.6: The ‘exponential form’ of the universe of de Sitter proposed by

Lemaıtre in 1925 and independently by Robertson in 1928. Spatial sections at

constant time have flat geometry (k = 0) [adapted from Lord 1974, p. 124].

was obtained by reversing the sign of t, and supposing geodesics converg-

ing in the past [Lemaıtre 1925a, p. 192]. Thus, the two solutions found

by Silberstein (with both the positive and the negative sign) could not

be compounded. Lemaıtre wrote about Silberstein’s result that “no way

is found of introducing his double sign without spoiling the homogeneity

of the field” [Lemaıtre 1925b].

“Our treatment - Lemaıtre remarked at the end of his note - evidences

the non-statical character of de Sitter’s world, which gives a possible

interpretation of the mean receding motion of spiral nebulæ” [Lemaıtre

1925a, p. 192]. According to Lemaıtre, the observed redshifts could be

physically interpreted as a feature of de Sitter’s universe. In particular,

redshift measurements were interpreted in such a version of de Sitter’s

Shifts in de Sitter’s universe according to Robertson and Tolman 117

universe as relative motions through space.

However, quoting Lemaıtre, “de Sitter’s solution has to be abandoned,

not because it is not static, but because it does not give a finite space

without introducing an impossible boundary” [Lemaıtre 1925a, p. 192].

Lemaıtre pointed out in a Ph.D. report of June 1925 that an infinite

space represented “a very unsatisfactory feature” in the consideration of

the universe as a whole [Lemaıtre Archive, Box 8]. In 1925, Lemaıtre at-

tended a talk by Hubble about Cepheid stars as distance indicators, and

in the same year he visited Lowell Observatory, where Slipher was dealing

with measurements of relevant spectral displacements in (extragalactic)

nebulæ. These important observational aspects and the fundamental re-

quirement of a finite space marked the connection between theoretical

cosmology and astronomical observations which Lemaıtre developed in

following years [Lambert 2000, p. 91]. As we shall see, Lemaıtre con-

sidered a finite spherical universe filled by matter which world radius

was depending on time. He investigated the homogeneous and isotropic

model of the universe proposed by Einstein, inserting a world radius in-

creasing with time. In such a truly expanding model of the universe,

which Lemaıtre discovered in 1927, redshifts were directly related to the

ratio of time-dependent world-radii, involving the actual cosmological in-

terpretation of redshift.

5.6 Shifts in de Sitter’s universe according

to Robertson and Tolman

The fact that de Sitter’s universe could be regarded as spatially open

was realized, as seen above, by Lemaıtre in a note published in 1925.

Three years later, also Robertson investigated the empty universe of

de Sitter. In a paper which appeared in 1928, Robertson, unaware of

Lemaıtre’s proposal, proposed a similar result, and, dealing with Doppler

shifts, suggested that a linear relation between velocities and distances


existed in de Sitter’s universe. Furthermore, Robertson appreciated that

such a theoretical law could account for available data furnished by Hub-

ble and Slipher.

By a suitable transformation of coordinates, Robertson proposed a

cosmological model which was equivalent to the original model of de Sit-

ter, but, as Robertson remarked, differed in its physical interpretations.

In such a version of de Sitter’s world, there were neither the paradoxes of

the mass-horizon, nor the arrest of time at the horizon. The geometry of

space proposed by Robertson had “the advantage of simplicity” because

it was Euclidean [Robertson 1928, p. 847]. However, the line element

explicitly depended on time: “consequently - Robertson wrote - natural

processes are not reversible, but space-time is isotropic in time, in the

sense that at any time the line element has the same form as at any

other” [Robertson 1928, p. 847]. The model of de Sitter’s universe in the

form proposed by Robertson corresponded to a stationary model, i.e.,

quoting Robertson, “it presents the same view to observers at different

times” [Robertson-Noonan 1968, p. 346]. This version of the universe

of de Sitter, according to Robertson, was the only non-static stationary

model: “the fundamental world lines - Robertson noticed - expand away

from each other, but they also present the same appearance at any cosmic

time” [Robertson-Noonan 1968, p. 365]

The empty model of de Sitter represented a suitable description of

the actual world, since the total matter in the world could be assumed

to have “little effect on its macroscopic properties” [Robertson 1928, p.

835]. The line element, following Robertson notation, was:

ds2 = − dρ2

1− k2ρ2− ρ2(dθ2 + sin2 dφ2) + (1− k2ρ2)c2dτ 2, (5.87)

where k =√

λ3

= 1R. The singularity at ρ = 1

k= R involved the presence

of the mass-horizon. Robertson proposed the transformation:

ρ = r ekct, (5.88)

τ = t− 1

2kclog(1− k2r2e2kct). (5.89)


The line element became:

ds2 = −e2kct(dr2 + r2dθ2 + r2 sin2 dφ2) + c2dt2, (5.90)

or equivalently, by Cartesian coordinates:

ds2 = −e2kct(dx2 + dy2 + dz2) + c2dt2. (5.91)

The coordinate r assumed in this frame infinite values, while in the origi-

nal form it was 0 < ρ ≤ R; through such a transformation the singularity

was removed from the finite region [Robertson 1928, p. 836]. The new

form of de Sitter’s line element was thus “dynamical, in that its coeffi-

cients depend on time t” [Robertson 1928, p. 837].

Although the mass horizon paradox did not appear in this new frame,

and space was there unlimited, Robertson noted that the observable

world was not unlimited: de Sitter’s universe presented a kind of cos-

mological horizon. “The closed character - Robertson pointed out - is

maintained in the sense that the only events of which we can be aware

must occur within a sphere of finite radius” [Robertson 1928, p. 837].

Investigating geodesic equations, it followed that R was the maximum

distance from which particles or light could enter in causal connection

with an observer at the origin: R represented “our observable universe”

[Robertson 1928, p. 839]. Following [Earman-Eisentaedt 1999, p. 202], in

fact, the Robertson form of de Sitter’s line element is non-singular, how-

ever it covers only a portion of de Sitter’s hyperboloid, and space-time is

not free of singularities for the choice of logarithmic coordinate.

Dealing with the properties of his own form of de Sitter’s line ele-

ment, Robertson found a linear relation between velocity and distance.

“The measurements of velocities - Robertson noted - can, in principle

at least, be reduced to the measurements of distances. (...) The mea-

surement of the radial component offers more difficulty; in practice it is

accomplished by means of the Doppler effect” [Robertson 1928, p. 843].

Therefore Robertson developed the theory of Doppler effect in de Sitter’s

universe. He considered the observer at the origin of coordinates, and he


supposed that gravitational shifts could be neglected. Robertson studied

the relation between time intervals of what he denoted as “equivalent”

observers, i.e. all observers at constant (r, θ, φ) [Robertson 1928, p. 837].

Being le = r ekcte the distance between a light source and the observer

at the origin27, light emitted from the source in the interval te, te + dte

arrived at the origin in the interval t0, t0 + dt0, and the relation between

time intervals was [Robertson 1928, p. 843]:

dt0 =dte

1− kle. (5.92)

Since the proper-time of atom vibrations was the same at the origin and

at the source, such a relation led to a shift ∆λ given by:

∆λ

λe

≡ λ0 − λe

λe

=dt0dte

− 1 =kle

1− kle. (5.93)

Such a shift “ would be attributed in practice” [Robertson 1928, p. 843]

to the Doppler effect due to a velocity of recession v:

zD =∆λ

λe

=

√1 + v

c

1− vc

− 1. (5.94)

Therefore it followed the relation:

v = ckle(1 + O[k]). (5.95)

Thus, through the application of Doppler effect for receding sources, the

observer at the origin would have measured a velocity v ' ckle of the

source [Robertson 1928, p. 843].

Robertson further developed such a relation to the general case of a

source which had a non-vanishing coordinate velocity δe ≡ (δr, δθ, δφ), i.e.

he considered the case when “the proper time of vibration of the moving

atom is not the same as the time interval measured by a stationary

27Note that in his 1928 paper Robertson used the notation l0, dt0, dt for, respec-

tively, the measured distance of the source from the origin, the time interval measured

by the source, and the time interval measured by the observer at the origin [Robertson

1928, p. 843]. In our notation these quantities become le, dte, dt0, respectively.


observer” [Robertson 1928, p. 843]. The radial component of the velocity

resulted:

vr ' ekcteδr + ckle. (5.96)

“If we assume - Robertson wrote - that there is no systematic correlation

of coordinate velocity with distance from the origin, we should expect

that the Doppler effect would indicate a residual positive radial velocity

of distant objects because of the term ckle” [Robertson 1928, p. 844].

The stationary form of the line element of de Sitter’s universe pre-

dicted the relation:

z =v

c' l

R. (5.97)

Essentially, the metric of de Sitter’s universe predicted a residual mo-

tion of recession through Doppler effect, which, by averaging proper

motions, was the cause of the “excess of recessional velocity of spiral

nebulæ” [Robertson 1928, p. 847]. Such a linear relation, as Robertson

acknowledged, was found also by Weyl in 1923. Comparing his own result

(equation 5.97) with Silberstein’s one, Robertson remarked that in his

own velocity-distance relation it was not possible to introduce a negative

sign, unless changing the sign of k [Robertson 1928, p. 844].

Robertson pointed out that such a theoretical linear relation was

nearly verified from distances of extragalactic nebulæ obtained by Hub-

ble in [Hubble 1926b], and from velocities proposed by Slipher which

appeared in [Eddington 1923 ], giving a radius of de Sitter’s universe

of about R = 2 · 1027 cm [Robertson 1928, p. 845]. Therefore, in his

1928 paper Robertson proposed that, neglecting gravitational shift and

eliminating proper motion, the effect of recession predicted by de Sitter’s

cosmology was linear, and was approximately confirmed by observational

data of spirals.

In a subsequent paper which appeared in 1929 Robertson acknowl-

edged that already in 1925 Lemaıtre had discovered the same coordinate

system Robertson independently proposed in 1928. In this paper Robert-

son stated that the only stationary cosmological models (in the sense seen


above) were those of Einstein and de Sitter. These models, according to

Robertson, arose “from particular cases of a class of solutions whose gen-

eral member defines a non-stationary cosmology” [Robertson 1929, p.

822]. The general form of the line element was [Robertson 1929, p. 826]:

ds2 = −e2f(t)

(dr2

1− r2

R2

+ r2 dθ2 + r2 sin2 θ dφ2

)+ c2dt2. (5.98)

In the retrospect, this form actually corresponds to the general line ele-

ment of an expanding FLRW frame, where the spatial curvature is arbi-

trary [Ellis 1989, p. 379].

With regard to the question of the Doppler effect, in the general case

the resulting Doppler shift was attributed to a velocity of recession:

v = c tanh [f(t0)− f(te)] . (5.99)

“Our choice of coordinates in the actual universe - Robertson concluded -

has been such that the above considerations apply to the residual Doppler

effect, after having averaging to eliminate the effect due to accidental

‘proper’ motions. Of the two stationary cosmologies, only that of de

Sitter will show such a residual effect, as in Einstein’s f is constant”

[Robertson 1929, p. 827].

Interestingly, Robertson mentioned in a note of this paper the solu-

tions proposed by Friedmann in 1922 and 1924 which described an ex-

panding universe. The cosmological consequences of Friedmann’s models

were fully acknowledged only in 1930. In 1929 Robertson found these so-

lutions objectionable, because they were non-stationary. These solutions,

quoting Robertson, introduced “untenable assumptions on the matter-

energy tensor, and require that Einstein’s field equations be satisfied

instead of making full use of the intrinsic uniformity of such a space”

[Robertson 1929, p. 828].

Just before the rise of the theory of the expanding universe, also Tol-

man took into account the redshift-distance relation predicted through

de Sitter’s static model. In 1929 a detailed paper by Tolman appeared


about the astronomical implications of the line element of de Sitter’s

universe.

In this paper, Tolman studied the metric of de Sitter’s universe in the

form which Eddington already proposed in 1923:

ds2 = − 1(1− r2

R2

)dr2 − r2dθ2 − r2 sin2 θ dφ2 +

(1− r2

R2

)dt2. (5.100)

According to Tolman, this equation was “the most suitable for our pur-

poses, since the variables occurring in it will later appear to be the most

natural ones to correlate with the results of astronomical measurements”

[Tolman 1929a, p. 248].

With regard to the geodesic equations, Tolman found the relation:

dr

dt= ±

√k2 − 1 +

r2

R2− h2

r2+

h2

R2, (5.101)

where h and k were constants of integration: h could assume positive

or negative values according to the direction of motion of the particle,

and k was a positive parameter, “if - Tolman wrote - we are going to

interpret t as the time and exclude the possibility that the proper time s

and coordinate time t could ever run backward” [Tolman 1929a, p. 249].

In his analysis of Doppler effect, Tolman considered both cases of the

observer and the emitting source at the origin of coordinates. In the

first case, with the observer at the origin, the source was at a distance r

and moved with a velocity drdt

at the time of light-signal emission. The

interval ds was proportional to the unshifted wavelength. The period dt0

of the light arriving at the origin was related to the period of emission dtε,

and the time dte which was taken by light to traverse the radial distance

dr = drdt

dtε:

dt0 = dtε ± dte, (5.102)

where

dte =k(

1− r2

R2

)2

dr

dtds, (5.103)


and

dtε =k

1− r2

R2

ds. (5.104)

Therefore, since dt0 was proportional to the wavelength observed at the

origin, the Doppler effect was [Tolman 1929a, p. 259]:

z =k ±

√k2 − 1 + r2

R2 − h2

r2 + h2

R2

1− r2

R2

− 1. (5.105)

In the second case, with the source at the origin and the observer at

distance r from the source, spectral displacement was:

z =1− r2

R2

k ∓√

k2 − 1 + r2

R2 − h2

r2 + h2

R2

− 1. (5.106)

Tolman remarked that the complete Doppler formula which Silberstein

had proposed in 1924 was correct only for the special case h = 0 and

k = 1 [Tolman 1929a, p. 261].

Furthermore, Tolman proposed a formulation of the Doppler effect

in the case drdt

= 0, i.e. when the particle was at the perihelion of his

hyperbolic orbit. In this case, being rm the minimum distance from the

origin to the perihelion, the Doppler effect resulted:

z =1− r2

m

R2 + h2

r2m− h2

R2

1− r2m

R2

− 1. (5.107)

According to this formula, the Doppler effect at the perihelion was pos-

itive and could be interpreted as due to a motion of recession, although

the source had no radial velocity at the time of emission. “The produc-

tion of a positive Doppler effect by a source which is at rest - Tolman

wrote - was the original reason which led investigators to hope that the

de Sitter universe might furnish an explanation for the preponderating

shift toward the red in the light of extragalactic nebulæ” [Tolman 1929a,

p. 262].

With regard to the astronomical implications, Tolman admitted that

there was an approximate uniform distribution of nebulæ in regions of


space accessible to telescopes. The average Doppler effect, moreover, ap-

peared to be approximately proportional to the first power of the distance

[Tolman 1929a, p. 266]:

zav ∝ r

R. (5.108)

The observed preponderance of shifts towards the red could be recon-

ciliated with theoretical Doppler effect only by some special assumptions

for the values of k and R. However, as Tolman pointed out, “we cannot

greatly increase k or decrease R without getting larger Doppler effects

than those actually observed” [Tolman 1929a, p. 268]. Alternatively, a

reconciliation was obtained through another special assumption, i.e. by

assuming that the number of nebulæ “which make perihelion at a given

radius increases very rapidly with the radius” [Tolman 1929a, p. 269].

Actually, the great preponderance of positive velocities involved strong

ad hoc requirements for the values of the parameters k and h. With

k = 1 and h = 0, the theoretical Doppler shift reduced to

z =r

R, (5.109)

only if the negative sign in the complete Doppler formula was neglected.

Moreover, the value of the theoretical average Doppler effect at any radius

r was found by Tolman of the order of [Tolman 1929a, p. 270]

zav ≥ 1

2

r2

R2, (5.110)

which “would not appear to account for the very rare occurrence of neg-

ative Doppler effects and for linear increase in average Doppler effect

with distance, without the imposition of restrictions on the parameters

determining the orbits of the nebulæ” [Tolman 1929a, p. 274].

Therefore, according to Tolman, “the de Sitter line element (...) does

not appear to afford a simple and unmistakably evident explanation of

our present knowledge of the distribution, distances, and Doppler effects

for the extra-galactic nebulæ” [Tolman 1929a, p. 273]. “Further obser-

vational material on the nebulæ - Tolman concluded - would be of great


importance. In particular it is desirable to be certain as to the form of

the relation between Doppler effect and distance, and also as to the rela-

tive frequency of negative and positive Doppler effects” [Tolman 1929a,

p. 274].

Tolman investigated in another paper the possible line elements of

the universe. The natural requirement was, according to Tolman, the

condition of homogeneity and isotropy, together with the possibility to

write the line element in a form which was static respect to time. “The

requirement of spherical symmetry - Tolman wrote - is an obvious one to

impose, since otherwise the universe regarded on large scale would have

different properties in different directions. The requirement of symmetry

with respect to past and future time means that the large scale behaviour

of the universe is reversible, and the static form of the line element means

that by and large the universe is in a steady state” [Tolman 1929b, p.

298]. A further requirement was that both the density of matter and the

pressure were constant throughout the universe, and the spatial velocity

had to be zero. Therefore, the solutions proposed by Einstein and de

Sitter (and the model of the special theory of relativity) were the only

models which satisfied the mentioned requirements.

However, at the end of his paper Tolman suggested that other models

could be investigated, by imposing different requirements. “In particu-

lar - Tolman concluded - it should be noted that our assumption of a

static line element takes no explicit recognition of any universal evolu-

tionary process which maybe going on. The investigation of non-static

line elements would be very interesting” [Tolman 1929b, p. 304].

As seen, at the end of the 1920’s the question of the explanation of red-

shift measured in galaxies through theoretical relativistic models of the

universe, which was foreshadowed in the discussions about the de Sitter

effect, was actually close to a solution. Investigations of non-static and

non-stationary models of the universe, in the light of the observational

evidence of a cosmic recession, inaugurated in 1930 a second renewal of

cosmology.


fram

ere

dsh

ift

(vel

oci

ty)-

dis

tance

rela

tion

de

Sit

ter,

1917

stat

icz

=v c'±

r R+

1 2

( r R

) 2

Eddin

gton

,19

23st

atic

z'

1 2

( r R

) 2

Wey

l,19

23+

1930

stat

ionar

y,k

=0

z=

tan

d R

Lan

czos

,19

23st

atio

nar

y,k

=+

1z

=co

sa R−

sin

a Rsi

nh

τ 0−

1

Silber

stei

n,19

24st

atic

z=±

sin

σ=±

v cse

cσ'±

r R

Lem

aıtr

e,19

25st

atio

nar

y,k

=0

z=

sin

χ

Rob

erts

on,19

28st

atio

nar

y,k

=0

z'

l R

Tol

man

,19

29st

atic

z=

( k±

√k

2−

1+

r2

R2−

h2

r2

+h2

R2

)/( 1−

r2

R2

) −1

Table 5.1: Summary of different interpretations of the model of de Sitter

and different formulations of the redshift-distance law related to the de

Sitter effect. Part of this summary is based on [Ellis 1990, p. 100].

Chapter 6

Observational investigations

of redshift relations

In this chapter the first observations on large scale during the 1920’s,

i.e. the beginning of observational cosmology, are presented. In par-

ticular, a critical reconstruction is here proposed of attempts to find a

suitable relation between velocities and observable quantities, such as the

apparent diameter and the distance of nebulæ, and therefore to possibly

confirm the de Sitter effect.

6.1 The nature of the nebulæ around 1920

In the book “Stellar movements and the structure of the universe”,

which appeared in 1914, Eddington devoted a chapter on the Milky Way,

star clusters and nebulæ. The general view which Eddington supported

was that there was firstly an inner and flattened star system, whose

density diminished from the center outwards, and secondly a certain

number of star clouds around the inner system, which “make up the

Milky Way. It is to the inner system - Eddington suggested - that our

knowledge of stellar motions and luminosity relates. Whether the other

clouds are continuous with the inner system or whether they are isolated,

129

130 Observational investigations of redshift relations

is a question at present without answer” [Eddington 1914, p. 233]. With

regard to spiral nebulæ, Eddington remarked that these systems were

not correlated to the planetary of irregular ones, which on the contrary

intimately belonged to our stellar system. Despite “direct evidence is

entirely lacking as to whether these bodies are within or without the

stellar system, (...) the island universe theory is much to be preferred as

a working hypothesis; and its consequences are so helpful as to suggest

a distinct probability of its truth ” [Eddington 1914, pp. 242-243].

The possibility to obtain a reliable distance of the nebulæ would have

permitted to discriminate between the galactic or extragalactic position

of these objects, and thus to understand their nature1. This challenge

was solved in 1925, thanks to the most powerful telescope which oper-

ated during the 1920’s, the Hooker 100-inch reflector at Mt. Wilson,

and to the man who directed such a telescope towards the deepest space

ever observed until then, namely Hubble, who inaugurated the cosmo-

logical observations, even better measurements, of objects at very large

distances2.

Hubble, as we shall see, determined the distance of some nebulæ by

1See [Fernie 1970 ] for the historical quest for the nature of the nebulæ.2It is important to note that in 1922 Ernst Opik (1893-1985) estimated the distance

of the Andromeda nebula at about 450’000 pc. He did not base his own calculations

on some empirical distance indicators. Indeed Opik determined the distance of M31

through a method based on the observed rotational movement. “Assuming that the

centripetal acceleration at a distance r from the center is equal to the gravitational

acceleration due to the mass inside the sphere of radius r - Opik wrote - an expression

is derived for the absolute distance” [Opik 1922, p. 406]:

D =E sin ρ

i

(v0

ω

)2

, (6.1)

where v0 was the velocity of motion along a circular orbit, ρ the angular distance from

the center, i the apparent luminosity, ω the orbital velocity of the Earth, and E was

the energy radiated per unit mass, which Opik assumed the same as for our Galaxy.

However, this result did not attract much interest, nor the method stimulated further

developments.

The nature of the nebulæ around 1920 131

using the period-luminosity relation typical of Cepheid variable stars3.

As mentioned in Chapter 3, such a fundamental relation was suggested

by Henrietta Leavitt in 1912. Leavitt used the catalogue of 1777 variable

stars in the two Magellanic Clouds which was given by studying plates

obtained at the Harvard Observatory at Arequipa (Peru). She took into

account 25 variables in the Small Magellanic Cloud, for which both peri-

ods and brightness at maxima and minima were available. Leavitt then

noted that “a straight line can readily be drawn among each of the two

series of points corresponding to maxima and minima, thus showing that

there is a simple relation between the brightness of the variables and their

periods” [Leavitt-Pickering 1912, p. 2]. It was the Danish astronomer

Ejnar Hertzsprung (1873-1967) who considered the result proposed by

Leavitt, and first calibrated such a relation in 1914, allowing to deter-

mine the absolute magnitude at the maximum of brightness [Hertzsprung

1914 ]. By using statistical parallaxes of 13 Cepheids, Hertzsprung ob-

tained the zero-point of such a calibration curve, and proposed a relation

between the absolute visual magnitude (MV ) and the period (P ). In

modern notation, the relation found by Hertzsprung was [Fernie 1969,

p. 708]:

< MV >= −0.6− 2.1 log P, (6.2)

with P measured in days. Therefore, the determination of apparent

magnitude could be used to obtain the distance of such objects through

the distance modulus equation.

Shapley used such a relation in 1918 to determine the extent of the

Milky Way, by assuming that globular clusters belonged to it and formed

a galactic halo. In his paper “On globular clusters and the structure of

the galactic system”, Shapley pointed out that, in order to determine the

distance of globulars, “the Cepheid variables (...) are of so much greater

weight, because of more definite knowledge of the dispersion of absolute

3We refer to [Fernie 1969 ] for a detailed reconstruction of the history of the period-

luminosity relation.


brightness, that the other types [B stars and red giants] can best be used

as checks or as secondary standards” [Shapley 1918, p. 43].

Figure 6.1: Magnitude-period curve of Cepheid stars which Shapley took into

account in order to study the structure of the Milky Way [from Shapley 1918,

p. 44].

Shapley obtained < M >= −0.4 and < M >= −4.0 for Cepheids in

the Small Magellanic Clouds and in the local system with, respectively,

periods less than a day and periods longer than a day. Furthermore, he

fixed the absolute photographic magnitude for the 25 brightest stars in

69 globular clusters at M = −1.5 [Shapley 1918, p. 44]. Having in this

way determined their distances, Shapley concluded that “the globular

clusters outline the extent and arrangement of the total galactic orga-

nization. Adopting this view of the stellar system, all known sidereal

objects become part of a single enormous unit, in which the globular clus-

ters and Magellanic Clouds (...) are clearly subordinate factors” [Shapley

1918, p. 50]. At the end of this paper, Shapley listed the observational

evidences unfavorable to the “island universe” hypothesis. He took up

these objections in a subsequent 1919 paper “On the existence of external

galaxies”. This paper contained the main issues that Shapley would have

The nature of the nebulæ around 1920 133

faced in 1920 in the famous discussion “The scale of the universe”, i.e.

in the so-called “Great Debate” with Curtis. According to Shapley, the

arguments against the stellar interpretation of spiral nebulæ were: the

relevant size of our Galaxy, the internal motions in spirals measured by

Adrian van Maanen (1884-1947) (which, however, Shapley acknowledged

to be still uncertain), and the absolute magnitude of Novæ, which “would

far transcend any luminosity with which we are acquainted” [Shapley

1919, p. 266]. Also the first relevant radial velocities measured in spirals

by Slipher (which will be described in next section) seemed to oppose

the extragalactic nature of nebulæ: “high speed - Shapley noted - is not

a condition impossible of production by the forces in our galactic sys-

tem” [Shapley 1919, p. 265]. Shapley concluded that “the evidence now

supporting the island universe interpretation appears unconvincing. (...)

We have, however, no evidence that somewhere in space there are not

other galaxies; we can only conclude that the most distant sidereal or-

ganizations now recognized (globular clusters, Magellanic Clouds, spiral

nebulæ) cannot successfully maintain their claims to galactic structure

and dimensions” [Shapley 1919, p. 268].

On the contrary, during the “Great Debate” Curtis advocated the

theory that spirals were island universes. According to him, the main

favorable points were “the tremendous space-velocities of spirals” [Curtis

1920, p. 326], which could not be related in any way with the thirty-

fold smaller velocities measured in stars, together with the evidences of

a spiral structure also in our system. Other favorable points were the

spectrum of the great part of spirals, which “is practically identical with

that given by a star cluster” [Curtis 1920, p. 326], and eventually the

presence of Novæ in those systems.

Curtis illustrated in a scheme that spirals were distributed in greatest

numbers around the poles of our Galaxy: nearly 400’000 and 300’000 spi-

ral nebulæ were placed, respectively, around the north and south galactic

pole4. “Our stellar system - Curtis wrote - is shaped like a thin lens, and

4It is worth noting that Curtis proposed such a scheme already in 1917 [Curtis


is perhaps 3’000 by 30’000 light years in extent. In this space occur

nearly all the stars, nearly all the new stars, nearly all the variable stars,

most of the diffuse and planetary nebulæ, but no spiral nebulæ” [Curtis

1920, p. 320].

Figure 6.2: Position of nebulæ with respect to the Milky Way according to

Curtis [from Curtis 1920, p. 320].

6.2 Slipher and the radial velocities of spi-

rals

Around 1920, questions about spiral nebulæ have as the main top-

ics their positions in space, i.e. their still unknown distances, and the

astonishing evidence that some of these objects showed large spectral

displacements, which were interpreted as large velocities for the habit to

directly relate spectral shift z to the corresponding velocity.

1917, p. 100]. However, in the 1917 scheme the number of spirals which, according

to Curtis, were placed around the poles was about 100’000, both on the north side

and on the south side of the Milky Way. The comparison between the 1917 and the

1920 schemes is therefore useful in order to have a snapshot of the new estimates of

the amount of spirals by observations accumulated during those years.

Slipher and the radial velocities of spirals 135

The first radial velocity of a nebula was determined in 1912 by Slipher.

From that year for over a decade Slipher supplied these fundamental ob-

servations. As seen in the previous chapter, radial shifts, and in partic-

ular redshifts, became during the 1920’s the most important empirical

evidence supporting the model of the universe proposed by de Sitter, be-

cause, unlike Einstein’s model, this solution of field equations predicted

spectral displacements for objects in it.

At Lowell Observatory, Slipher used a 24-inch refractor with a high

dispersion prism and a very short-focus (fast) camera. Indeed, “no choice

of the telescope - Slipher noted - as regards aperture, or focal-length, or

ratio of aperture to focus, will increase the brightness of the spectrum

of an extended source” [Slipher 1915, p. 22]. On the contrary, the

spectrograph and the camera were the most important factors to take

into account for the brightness.

Slipher acknowledged that “I have given more attention to the ve-

locity since the study of the spectra had been undertaken with marked

success by Fath at Lick and Mt. Wilson, and by Wolf at Heidelberg”

[Slipher 1915, p. 22]. The first object which Slipher observed was the

Andromeda nebula: “the early attempts - he noted - recorded well the

continuous spectrum crossed by a few Fraunhofer groups, and were par-

ticularly encouraging as regards the exposure time required” [Slipher

1913, p. 56]. By taking several spectrograms from September, 1912

to December, 1912, the observations revealed a mean radial velocity of

v = −300 km/sec. According to Slipher, “the magnitude of this velocity,

which is the greatest hitherto observed, raises the question whether the

velocity-like displacement might not be due to some other cause, but I

believe we have at the present no other interpretations” [Slipher 1913,

p. 56]. Therefore, the conclusion was that the Andromeda nebula was

approaching the solar system with this unexpected velocity.

By 1915 Slipher obtained spectral displacements of 15 nebulæ, the

most part of which showed redshift, i.e. positive velocity. Among them,

the objects NGC 1069 and NGC 4594 had about v = +1100 km/sec. “As


well as may be inferred - Slipher remarked - the average velocity of the

spirals is about 25 times the average stellar velocity” [Slipher 1915, p.

23]. In a subsequent paper, which appeared in 1917, Slipher proposed a

list of the radial velocity of 25 spiral nebulæ, and also compared his own

results with velocities measured by Pease, Wright and Moore [Slipher

1917 ].

It is important to recall the fact that in 1917 de Sitter related mea-

surements of radial velocity to the cosmological consequences of his own

world-model, however by referring only to three nebulæ. By quoting what

Hubble wrote in 1936, “Slipher’s list of 13 velocities, although published

in 1914, had not reached de Sitter, probably as a result of the disruption

of communications during the war” [Hubble 1936, p. 109].

6.3 The K term and the solar motion

The possibility to relate the relevant radial velocities of spiral nebulæ

to the stellar system was soon realized by some astronomers around 1916,

in the framework of the determination of the solar motion towards the

so-called Sun apex.

The components of the motion of the Sun, indeed, corresponded to

(−X,−Y,−Z) through the general formula:

v = X cos α cos δ + Y sin α cos δ + Z sin δ, (6.3)

where v was the observed radial velocity of a star (or of a group of stars),

and (α, δ) were, respectively, the right ascension and declination of such

a star.

Already at the end of 1915, O. H. Truman, at the Iowa State Ob-

servatory, suggested that “it seems likely that a distinct motion of our

own spiral nebula with respect to them [spiral nebulæ] would manifest

itself, and conversely, if we can find such a motion of our system with

respect to the nebula, it will be quite strongly urged upon us that they

are other sidereal system” [Truman 1916, p. 111]. Referring to Slipher’s

The K term and the solar motion 137

1915 measurements, Truman concluded that “our nebula is moving with

a velocity of +670 km/sec in the direction of R.A. 20 hrs, Dec −20. (...)

I think the determination of radial velocities of spiral nebulæ should be

rigorously pursued, in order to quickly find out whether the above results

are indeed real” [Truman 1916, p. 112].

In 1916, Reynold Young (1886-1977) and William Harper (1878-1940)

(Dominion Observatory, Canada), in front of the results proposed by

Slipher, noted that “it seems quite possible that our solar system and the

whole universe or spiral to which it belongs may be rushing through space

with a speed as yet undetermined, but of the same order of magnitude as

other spiral nebulæ” [Young-Harper 1916, p. 134]. Indeed observations

of such radial motions in spirals could be used to determine the direction

and speed of our system. By using data of 15 nebulæ, Young and Harper

found the velocity of v = −598 ± 234 km/sec for what they denoted as

the “velocity of the universe” [Young-Harper 1916, p. 135].

The relation of the stellar system to radial motions of spiral neb-

ulæ was faced in 1916 also by George Paddock (1879-1955), astronomer

at Lick Observatory, who further developed the equation of motion by

taking into account the so-called K term. The method which Paddock

suggested referred to what Campbell had already proposed in 1911 in

order to determine the solar motion from stellar velocities.

In 1911, indeed, Campbell found that a constant K term, which

Campbell interpreted as a “systematic error”, had to be added in the

equation of the solar motion [Campbell 1911, p. 93]:

v = v0 cos d + K, (6.4)

where

cos d = cos δ0 cos δ cos(α0 − α) + sin δ0 sin δ. (6.5)

Campbell proposed the above equation by taking into account 35 groups

of B stars; v was the mean observed radial velocity of each of the 35

groups, v0 was the velocity of the Sun with respect to the system of B


stars, d was the average angular distance of each group of B stars, and

(α0, δ0) were the right ascension and the declination of the Sun apex.

“An error of obscure source - Campbell concluded in 1911 - causes the

radial velocities of Class B stars to be observed too great by a quan-

tity, K, amounting to several kilometers” [Campbell 1911, p. 105]. In

1913, in his famous book “Stellar motions”, Campbell highlighted that

“the universe of Classes B to B9 stars is expanding, with reference to

the instantaneous position of the solar system as a center, at the rate

of 4.93 km/sec. (...) A personal equation in the measurements of the

spectrograms, systematically positive, amounting to 5 km/sec, cannot

be regarded as possible” [Campbell 1913, p. 203]. As seen in previous

chapter, in 1917 de Sitter took into account this puzzling systematic red-

shift of B stars which Campbell had pointed out. De Sitter proposed

that the great part of this shift could be interpreted as a spurious ve-

locity through the g44 potential in his own cosmological solution of field

equations.

In 1916, Paddock related such a K term to the velocities of spiral

nebulæ. He inaugurated, as we will see, a method which would have

been used by several astronomers “for the purpose of finding a possible

space motion of our system of stars, including the Sun, relative to a

possible system of spirals of which our stellar system may be a unit,

and the spirals each perhaps a system of stars” [Paddock 1916, p. 109].

Interestingly, Paddock, in front of the relevant average positive radial

velocities of groups of spirals at the north and south galactic poles, noted

that “accordingly a solution for the motion of the observer through space

should doubtless contain a constant term to represent the expanding

or systematic component whether there be actual expansion or a term

in the spectroscopic line displacements not due to velocities” [Paddock

1916, p. 113]. Such a required constant term was represented by the K

term. Paddock used the same form of the equation of motion with the

K term already found by Campbell. He analyzed the results proposed

by Young and Harper about nebulæ, and obtained a K term ranging


from + 248 to + 295 km/sec [Paddock 1916, p. 114]. In order to explain

such a large value of the K term, Paddock argued that “the algebraic

average numbers of the velocities of spirals will probably diminish with

increasing numbers of observed velocities. Probably, likewise, the value

of the apparent systematic term will diminish, so that it may therefore

be concluded that its appearance here is the result of insufficient data”

[Paddock 1916, p. 115].

6.3.1 Wirtz and de Sitter’s cosmology

Accounting for a large K term, a possible explanation of radial ve-

locities of nebulæ through the cosmology of de Sitter was proposed in

1924 by Wirtz, a German astronomer sometimes called “the European

Hubble without a telescope” [Sandage 2005, p. 500].

Wirtz started his astronomical activity at Strasbourg Observatory in

1902, where, by using a large refractor with small focal ratio, the main

researches were directed to the study of the nebulæ [Duerbeck-Seitter

2005, p. 168]. In 1918, now in Kiel, Wirtz considered the relation of the

K term to the motion of spirals, as done by Paddock in 1916. Wirtz

obtained a value of K = +656 km/sec: “if one gives this value a literal

interpretation - Wirtz wrote - the system of spiral nebulæ disperses with

the velocity 656 Km/sec relative to the momentary position of the solar

system as center” [Wirtz 1918, p. 115. Engl. tr. in Seitter-Duerbeck

1999, p. 238].

In 1922 Wirtz, in order to calculate the solar motion, took into ac-

count measurements related to 29 spirals. He found a notable value of

K = +840± 141 Km/sec [Wirtz 1922, p. 351]. Moreover, by investigat-

ing radial motions and observational properties of such a group of spirals,

Wirtz suggested that a linear relation existed between velocity and abso-

lute magnitude, in the sense that the nearest nebulæ showed a tendency

to approach, whether the distant ones receded from our galactic system

[Wirtz 1922, p. 352].


It was in 1924 that Wirtz clearly related his own statistical work on

radial velocity measurements to the model of the universe proposed by

de Sitter. In his 1924 paper “De Sitters Kosmologie und die Radialbe-

wegungen der Spiralnebel” (The de Sitter cosmology and radial motions

of spiral nebulæ), Wirtz appreciated that both Einstein’s and de Sit-

ter’s worlds corresponded to limiting cases, the actual world being an

intermediate state between them [Wirtz 1924, p. 21]. Referring to the

question of the mass horizon in the universe of de Sitter with which de

Sitter, Eddington and Weyl previously dealt, the slowing down of natural

clocks at increasing distances from the origin actually was, according to

Wirtz, a phenomenon accessible to astronomical observations. Spectral

displacements towards the red corresponded to such an effect, together

with the redshift effect suggested by Eddington due to the acceleration

of particles which increased with distances in de Sitter’s world. The fact

that, by inserting test particles, the empty world of de Sitter became

non-static was a remarkable feature rather than an objectionable one.

Through the theory of de Sitter, indeed, it could be predicted an increas-

ing redshift for increasing distances. However, as Wirtz pointed out, even

though Doppler radial velocities of many nebulæ were known, distance

measurements were not available. Nevertheless, the apparent diameter

of nebulæ (Dm), i.e. the apparent measure of major axes, could be taken

into account, by supposing that the linear diameter was nearly the same

for all spirals. Therefore radial motions, i.e. radial velocities, should

have increased with decreasing apparent diameters measured in spiral

nebulæ [Wirtz 1924, p. 23]. Wirtz used the list of 41 radial velocities ob-

tained by Slipher which Eddington had published in his 1923 book “The

mathematical theory of relativity”, together with the value v = +700

Km/sec for NGC 2681. With regard to the apparent diameter of these

42 objects, Wirtz referred to 1918 works of Curtis at Lick Observatory,

and 1917-1920 data by Pease at Mt. Wilson Observatory. The analysis

of the nebulæ which approximately had the same logarithm of apparent

diameter revealed that Doppler radial velocities decreased for groups of


spirals with increasing apparent diameters. Vice versa, by considering

groups of spirals with the same radial velocities, the largest velocities

corresponded to the smallest logarithms of apparent diameter. In the

first case Wirtz determined the average relation [Wirtz 1924, p. 24]:

v = 914− 479 · log (Dm). (6.6)

In the second case he determined the average relation:

log (Dm) = 0.96− 0.000432 · v. (6.7)

The average velocity was v = +574 km/sec. Thus, according to Wirtz,

it was clear that the radial motion of spiral nebulæ remarkably increased

with increasing distance. Actually, a diagram with values of velocities

and logarithmic diameters showed a V -shaped or triangular form. Wirtz

deduced from such a relation that apparently small nebulæ had either

very small or large velocities, while apparently large nebulæ showed only

small velocities: “the dispersion of the linear dimensions of the nebulæ

- Wirtz noted - fills the triangular plane in such a way that among the

near nebulæ absolutely small and large objects are visible, while in the

depth of space only the absolutely largest are subject to observing their

radial motions” [Wirtz 1924, p. 24. Engl. tr. in Seitter-Duerbeck 1990,

p. 376]. For the largest nebulae, Wirtz determined the relation between

velocity and logarithmic diameter in the form:

v = 2200− 1200 · log (Dm). (6.8)

According to Wirtz, such an empirical law, together with the velocity-

magnitude relation which he had proposed in 1922, could be interpreted

just through the properties of de Sitter’s world model.


6.4 Astronomers at work: Lundmark and

Stromberg

As seen, in 1924 the connection between the theoretical relativistic

description of the universe as a whole and observational cosmology was

analyzed and complicated through some important contributions. In

that very year, indeed, Silberstein took into account the static form of

de Sitter’s universe and observations of globular clusters. By consid-

ering the modulus of spectral shift, he proposed that a linear relation

existed between shift and distance. However, according to Silberstein,

the recession of astronomical objects at large distances was not system-

atic. Furthermore, as just seen, in 1924 Wirtz followed Eddington’s and

Weyl’s suggestions about a general tendency to scatter of test particles

in de Sitter’s world. Wirtz argued that the evidence of a relevant value

of the K term could be explained just through solution B (the de Sit-

ter’s model), and proposed that a linear relation existed between velocity

(redshift) and logarithm of apparent diameter.

Therefore, beside the still unsolved question of the nature of the neb-

ulæ, some puzzling questions marked the first steps in the rise of scien-

tific cosmology: did observations reveal a general systematic recession of

nebulæ? If so, which was the form of such a redshift-relation and the in-

terpretation of redshift? How to reconcile the de Sitter effect, predicted

by the metric of the truly empty de Sitter’s universe, with the actual

non-empty universe? Eventually, which were the distances of nebulæ

and which was the radius of the universe?

The relation of the spirals to the stellar system and the determination

of the curvature of space-time in de Sitter’s universe were faced by the

Swedish astronomer Knut Lundmark in 1924 and 1925. At the time of

the 1920 discussion between Shapley and Curtis about the nature of the

nebulæ, Lundmark agreed with Curtis on the large distances at which

spirals were placed, and seemed to prefer the extragalactic position of

Astronomers at work: Lundmark and Stromberg 143

them. However, by using Hubble’s words, Lundmark “was noncommit-

tal” [Hubble 1936, p. 89]. With regard to the motion of spirals, Lund-

mark had pointed out in 1920 that “it is obviously by no means out of

the question, that the great radial velocities found for the spirals are

not real, but that the measured displacements of spectral lines represent

other phenomenon than a Doppler-effect” [Lundmark 1920, p. 47].

In 1924, Lundmark discussed the results obtained by Silberstein, and

carefully examined the question of the nature of measured redshifts. By

using data of the Andromeda nebula obtained by Wright at Lick Ob-

servatory, Lundmark showed that the spectral displacement was nearly

constant for 16 lines ranging from 3970 A to 4860 A, with the average

value [Lundmark 1924, p. 748]:

zav =∆λ

λ= 0.00116± 0.00008. (6.9)

Therefore, according to Lundmark, “the shift is evidently a Doppler one.

The same applies to the velocities of the globular clusters. Another

question is, whether such a large Doppler shift represents motion in the

line of sight alone or is caused in other ways?” [Lundmark 1924, p. 748].

Lundmark was skeptical about Silberstein’s proposal that the motions of

spirals and globular clusters showed any effect of the curvature of space-

time. In order to compute the motion of the Sun, Lundmark used data of

18 globular clusters, and found the values A = 20h.4, D = +60, v0 = 305

km/sec, K = +31 km/sec, while with data of 43 spirals it resulted A =

20h.3, D = +75± 30, v0 = 651± 135 km/sec, K = +793± 88 km/sec

[Lundmark 1924, p. 748]. “The explanation may simply be - Lundmark

pointed out - that our local system as a whole has the motion found

above relatively to the huge systems of globulars and spirals” [Lundmark

1924, p. 749]. However the question remained open until 1929, when

more information was accumulated.

Furthermore Lundmark criticized in his 1924 analysis the determina-

tion of the curvature radius of the universe proposed by Silberstein, who,

quoting Lundmark, “has not given, and will probably not be able to give,


any justification for the use of the velocities of the globular clusters for

a determination of R” [Lundmark 1924, p. 750]. Since for these objects

the K term was small, Lundmark suggested that this result indicated

that globulars were both nearer than spirals, and “little if at all affected

by the slowing down of atomic vibrations in distant objects in de Sitter’s

world which might be erroneously interpreted as a motion of recession”

[Lundmark 1924, p. 750]. Moreover, Silberstein’s result was objection-

able because Silberstein had used in his own calculations selected values

of available radial velocities, excluding just objects which did not give a

constant value of the curvature radius R. By using data of 18 globular

clusters, Lundmark concluded that “there is little or no correlation be-

tween v and r, in contrary to what is to expected from the theory of Dr.

Silberstein” [Lundmark 1924, p. 752]. From such data, by assuming the

possibility to set a value for R, it followed a mean value R ' 19.7 · 1012

km, three times larger than the radius adopted by Silberstein.

Figure 6.3: Diagram with radial velocities and distances of globular clusters

proposed by Lundmark in 1924 [from Lundmark 1924, p. 753].


Since the value of R was uncertain, Lundmark suggested that other

classes of distant objects could be used in order to calculate the curvature

radius. He took into account radial velocities and distances of, respec-

tively, 30 Cepheid stars, 8 Novæ, 27 O stars, 29 R stars, 25 N stars

and 31 Eclipsing variables5. For these classes of objects any progression

seemed to be absent when the radial velocities were plotted according to

the corresponding distances. The average values of the curvature radius

were, respectively, 7.5, 41, 4.0, 6.7, 2.3, 2.7 · 1012 km [Lundmark 1924,

pp. 756-763].

At the end of this 1924 paper, Lundmark investigated the relation

between velocity and distance of 44 spiral nebulæ. He estimated the

distance of the Andromeda nebula at 200’000 pc by means of the Novæ

maximum brightness method, and he used such a value as the unit of dis-

tance scale. Lundmark hypothesized “that the apparent angular dimen-

sions and the total magnitudes of the spiral nebulæ are only dependent

on the distance” [Lundmark 1924, p. 767]. Lundmark then concluded

that “there may be a relation between the two quantities, although not

a very definite one” [Lundmark 1924, p. 768]. This issue was faced also

in a subsequent paper published in 19256, where the Swedish astronomer

remarked that “a rather definite correlation is shown between apparent

dimensions and radial velocity, in the sense that the smaller and presum-

ably distant spirals have the higher space-velocity” [Lundmark 1925, p.

5With regard to Cepheid distances, Lundmark followed the derivation proposed

by Shapley, who used data of proper motion together with the period-luminosity law.

Distances of Novæ were determined by assuming that the mean absolute maximum

magnitude had almost a constant value.6It is interesting to note that at the end of this 1925 paper about motions and

distances of spirals, Lundmark discussed the question of the extension of the universe,

recalling Charlier hierarchical model which has been mentioned in Chapter 3 of present

thesis. “Our present knowledge - Lundmark wrote - as to the space-distribution of

the stars and the spirals can be summed up in the statement: our stellar system and

the system of spiral nebulæ are constructed according to the conceptions expressed in

the Lambert-Charlier cosmogony” [Lundmark 1925, p. 893].


867]. In this paper Lundmark returned on the K term, and suggested

that its value was not constant in space. He proposed that the solar

motion could be expressed through7 [Lundmark 1925, p. 867]:

v = X cos α cos δ + Y sin α cos δ + Z sin δ + k + lr + mr2. (6.10)

Here k, l, m were constants, and r was the relative distance of a spiral

derived through the apparent diameter and the total absolute magnitude,

through the assumption of the same absolute dimension of spirals. By

using data of 44 spirals, and by taking the distance of the Andromeda

nebula as the distance unit, it followed:

K = +513 + 10.365r − 0.047r2 km/sec. (6.11)

“According to the above expression - Lundmark noted - the shift reaches

its maximum value, 2250 km/sec, at some 110 Andromeda units, which

(...) corresponds to a distance of 108 light years. (...) One would scarcely

expect to find any radial velocity larger than 3000 km/sec among spirals”

[Lundmark 1925, p. 867].

In 1925 another important analysis on the subject appeared by Strom-

berg, at Mt. Wilson Observatory. In this paper Stromberg summarized

all 63 measured radial velocities of globular clusters and “non-galactic

nebulæ” [Stromberg 1925, pp. 354-355]. According to Stromberg, be-

side the interesting fact that the large solar velocity which was obtained

from these data “indicated that a fundamental reference system could

be defined by them8”, such an analysis was useful to ascertain “whether

the velocities give any evidence of a curvature of space-time” [Stromberg

1925, p. 353].

With regard to the Sun’s apex, Stromberg obtained the values α =

315, δ = +62. The velocity was about 350 and 300 km/sec, which

7Note that Lundmark wrote sin δ instead of cos δ in the first term on the right-hand

side.8Stromberg had previously dealt with the existence of a universal world frame in

[Stromberg 1924 ].


was determined, respectively, by data of spirals and globulars [Stromberg

1925, p. 356]. The value of the K term resulted +616 km/sec, however,

as suggested also by Lundmark in the same year, Stromberg noted that

“the assumption of a constant K term for the nebulæ is probably only

approximately correct” [Stromberg 1925, p. 358].

Dealing with the curvature of space, Stromberg recalled the conse-

quence that from the static form of de Sitter’s universe a redshift in

spectral lines was expected, due to the apparent slowing down of atomic

vibrations and producing “a fictitious positive radial velocity” [Stromberg

1925, p. 359]. Therefore, radial velocities could be used in order to verify

their relation to distances. Stromberg pointed out in his analysis that Sil-

berstein’s conclusion about both red and blue shift “cannot be regarded

as conclusive. One thing is obvious, however. If the nebulæ studied are

at about the same distance (...) of globular clusters, we cannot regard the

large positive K term as a de Sitter effect, as the K term for the clusters

must be very small” [Stromberg 1925, p. 359]. Since distances of spirals

were not known, it was reasonable to have an indication of the distances

by assuming the same total brightness. However, no reliable correlation

between velocities and distances resulted from data. “De Sitter’s effect -

Stromberg pointed out - can be regarded as disproved by the clusters if

their distances are of the same order as those of the nebulæ. Silberstein’s

effect seems possible, but cannot be established by the data” [Stromberg

1925, p. 361]. Therefore Stromberg concluded that it was not possible

to confirm any dependence of radial motion on distance from the Sun.

As seen, through the authoritative contributions of Lundmark and

Stromberg, the explanation of relevant redshifts came to a standstill. In

particular, there was not a clear interpretation of the nature of redshift

which was shared by scientific community.

In 1925, dealing with the question of the finiteness of the universe,

Archibald Henderson (1877-1963) wrote that “if, as now appears prob-

able, the spirals are isolated systems, this recession must be explained,

it appears, either as a wholesale error or else as a relativistic effect. (...)


Figure 6.4: Summary of velocities measured in globular clusters and nebulæ

up to 1925 [from Stromberg 1925, pp. 354-355].


Much additional data will be required and many further researches made

before it will be possible categorically to decide between the infinite,

limiteless, Euclidean universe of Newton and the finite, unbounded, non-

Euclidean universe of Einstein or of de Sitter” [Henderson 1925, p. 223].

On the contrary, Campbell remarked in 1926 that “the motions of the

spirals seem to free them from the charge that they are retainers of our

stellar system. (...) Although other conditions than the radial velocity

of the light source as a whole are known to displace spectral lines from

their normal positions, there seems now to be no inclination to doubt

that the large displacements observed by Slipher are chiefly and perhaps

wholly Doppler-Fizeau effect” [Campbell 1926, p. 80].

In 1927 large radial velocities in (extragalactic) nebulæ were discussed

in the book “Astronomy” by Russell, Raymond Dugan (1878-1940) and

John Stewart (1894-1972). Such velocities indicated a solar motion of

about 400 km/sec, and the mean velocity of spirals was about v = +700

km/sec. Referring to de Sitter’s suggestion of spurious velocities, in front

of the observed recession these authors pointed out that “whether this

represents a real scattering of the nebulæ away from this region where

the Sun happens to be is very doubtful. It may arise from some other

cause” [Russell-Dugan-Stewart 1927, p. 850.]

The question of the nature of the nebulæ, as already mentioned, was

eventually solved by Hubble, who showed in 1925 that spirals were true

extragalactic systems. Again, as we shall see, it was just Hubble who

clarified the issue about the form of redshift relation. Hubble empirically

established in 1929 that such a relation between redshift (velocity) and

distance among distant nebulæ existed, was actually linear (at least from

available data), and could represent a possible confirmation of the de

Sitter effect.


6.5 Hubble and the universe of galaxies

The contributions given by Hubble represent a milestone in the his-

tory of astronomy and cosmology9. Quoting Osterbrock and Sandage,

Hubble “was a demon of energy, observing and making new discoveries

in quick succession” [Osterbrock 1990, p. 273], and “in only 12 years,

from 1924 to 1936, brought to an almost modern maturity the four foun-

dations of observational cosmology, even as its principles are practiced

today” [Sandage 1998, p. 2]. Indeed, venturing into the realm of the

nebulæ, Hubble clarified that spirals were really extragalactic systems,

i.e. galaxies (1925), then proposed a galaxy classification system shaped

like a tuning-fork (1926), established that the relation between velocities

and distances of nebulæ was linear (1929), and inaugurated a program of

galaxy counts as a function of magnitudes, N(m), attempting to measure

the curvature of space-time (1934).

As mentioned in Chapter 3, Hubble proposed a very large distance

of the Andromeda nebula (M31) by using 60-inch and 100-inch reflectors

at Mt. Wilson Observatory. Indeed, observing outer regions of M31 and

M33, “a survey of the plates made with the blink comparator - Hubble

noted - has revealed many variables among the stars, a large proportion

of which show the characteristic light-curve of the Cepheids” [Hubble

1925a, p. 252]. Among variables and Novæ discovered in such nebulæ,

Hubble determined period and magnitude of 22 Cepheids in M33 and 12

in M31, by taking, respectively, 65 plates and 130 plates. Photographic

magnitudes were obtained from 12 comparisons of selected areas with

the 100-inch telescope, with exposures from 30 to 40 minutes [Hubble

9A vast literature exists about the life and works of Hubble. For further readings

we refer to [Hetherington 1996, Christianson 1995, Christianson 2004, Sandage 1989,

Osterbrock-Brashear-Gwinn 1990 ]. Comprehensive studies about the developments

of redshift observations in nebulæ and the question of an empirical velocity-distance

relation can be found in [Smith 1979, Smith 1982 ]. Moreover, a great description of

measurements of distances and radial velocities in extragalactic nebulæ was proposed

by Hubble himself in his own famous book “The realm of the nebulæ” [Hubble 1936 ].

Hubble and the universe of galaxies 151

1925b, p. 140]. “The now familiar period-luminosity relation - Hubble

wrote - is conspicuously present” [Hubble 1925a, p. 254]. By assuming

that the variable stars were actually related to the spirals, the distance

modulus which resulted was m−M = 21.8 for M31, and m−M = 21.9

for M33. Having corrected such values by half the average ranges of the

Cepheids (because the original method by Shapley was based on median

magnitudes), the final value was about m −M = 22.3 for both objects.

This value corresponded to a distance of about 285’000 pc (930’000 light

years) [Hubble 1925b, p. 142].

Thus Hubble gave the fundamental confirmation that the “island uni-

verse” theory was correct. Hubble furnished through a reliable empirical

method, based on distance indicators, an astonishing distance for the An-

dromeda nebula, clearly outside the boundary of globular clusters which

Shapley had previously proposed for the Milky Way.

In the same year, Hubble proposed the distance of another object,

the irregular nebula NGC 6822. Through the application of the period-

luminosity law to 11 Cepheid variables in such a system, he found a

distance modulus of m − M = 21.65, corresponding to a distance of

about 214’000 pc. (700’000 light years) Therefore such an object “was

definitely assigned to a region outside the galactic system” [Hubble 1925c,

p. 409]. Moreover, Hubble remarked that “of especial importance is the

conclusion that the Cepheid criterion functions normally at this great

distance. (...) This criterion seems to offer the means of exploring extra-

galactic space” [Hubble 1925c, p. 432].

With regard to M33, by using data of 35 Cepheids, Hubble set the

distance modulus of such a nebula at m−M = 22.1, corresponding to a

distance of 263’000 pc (850’000 light years) [Hubble 1926a].

In 1926, in a paper devoted to a general classification of extragalac-

tic nebulæ, Hubble explained his own “working hypothesis” on which

he would have later based the effort to calculate the distances of other

galaxies. In such a paper, where Hubble furnished the distinction among

elliptical, normal spiral, barred spiral and irregular nebulæ, Hubble sug-


gested that “the various types are homogeneously distributed over the

sky, their spectra are similar, and the radial velocities are of the same

general order. These facts, together with the equality of the mean mag-

nitudes and the uniform frequency distribution of magnitudes, are con-

sistent with the hypothesis that the distance and absolute luminosity

as well are of the same order for the different types” [Hubble 1926b, p.

332]. The possibility to obtain the absolute magnitude was restricted

to a very few number of galaxies which distances were known. Hubble

adopted the mean value MT = −15.2 from data referred to 8 nebulæ: our

Galaxy, M31, M32, M33, M101, the Magellanic Clouds and NGC 6822.

Furthermore, the mean absolute magnitude which Hubble derived from

the brightest stars of such systems was MS = −6.3. [Hubble 1926b, p.

356]. Therefore, through this generalization, apparent magnitudes could

be used to determine distances.

Hubble confined at the end of this paper some considerations on the

theoretical cosmological consequences of general relativity and on the

possibility to calculate the radius and the mass of Einstein’s universe by

estimates of the density of matter [Hubble 1926b, p. 369]. However, fol-

lowing [Smith 1982, p. 181], these considerations were slightly confused,

revealing some misunderstandings in the difference between solution A

(Einstein’s model) and solution B (de Sitter’s model).

It was in 1928 that Hubble focused his own attention on the astro-

nomical consequences of de Sitter’s solution, in particular to the redshift

relation. In that year Hubble met de Sitter and Eddington in Leiden,

during the 1928 International Astronomical Union Meeting. Following

[Christianson 1995, p. 198], the idea to increase the set of radial velocities

obtained by Slipher was suggested to Hubble by de Sitter himself. With

regard to 1928, i.e. to such an important date for Hubble’s “cosmological”

plans, it is interesting to note that, in his reconstruction of the history

of Mt. Wilson Observatory, Sandage remarks that “Robertson, who was

my professor of mathematical physics at CalTech in 1951, told me in

1961 (...) that in 1928 he had discussed with Hubble his prediction and


partial verification of an expanding universe based on Friedmann’s 1922

solution. Robertson had used Slipher’s velocities and his own distance

estimates based on Hubble 1926 calibration of mean galaxy luminosity

to obtain observational results supporting the theory” [Sandage 2005, p.

501]. As seen in previous chapter, in 1928 Robertson derived a linear

relation between distance and velocity from the stationary (expanding)

metric of de Sitter’s universe, suggesting that such a relation was roughly

confirmed by comparing radial velocities measured by Slipher (1923) and

distances obtained by Hubble (1926).

In 1929, Milton Humason (1891-1972), the man who would have ex-

tended the radial velocity scale, wrote that “about a year ago Mr. Hub-

ble suggested that a selected list of fainter and more distant extragalactic

nebulæ (...) be observed to determine, if possible, whether the absorp-

tion lines in these objects show large displacements towards longer wave-

lengths, as might be expected on de Sitter’s theory of curved space-time”

[Humason 1929, p. 167]. This statement is therefore useful to understand

Hubble’s intention with regard to distant nebulæ, i.e. to obtain further

measurements of distances and velocities in order to establish the general

form of the relation between these observable quantities.

6.5.1 The contributions by Humason

While Hubble attempted to measure distances, Humason dealt with

radial velocities of galaxies, giving a sort of continuity in the monumental

work of Slipher about spectrographic measurements of relevant redshifts.

The first important result in such an exploration of deep space was

proposed by Humason in 192710. By using a two-prism spectrograph

(with a camera of 3-inch of focal length) at the Cassegrain focus of the

100-inch reflector at Mt. Wilson, Humason measured the radial velocities

of M101 and NGC 6822. For the former, Humason observed N1, N2, Hβ

10According to [Hetherington 1996, p. 121], such a 1927 paper by Humason was

probably written by Hubble himself.


and Hγ emission lines after exposures of 4 and 5 hours. The weighted

mean value from the two plates was about v = +216 km/sec [Humason

1927, p. 317]. For the latter, the mean velocity from the same lines

just mentioned, weighted from plates of exposures of 5 and 6 hours, was

v = +133 km/sec. “The radial velocity of both NGC 6822 and M101 -

Humason concluded - are unusually low for non-galactic objects. This

is consistent with the marked tendency already observed for the smaller

velocities to be associated with the larger (and hence probably closer)

nebulæ and those which are highly resolved” [Humason 1927, p. 318].

Therefore, an empirical progression of velocities with respect to distances

was suggested by Humason in this 1927 work.

Such a suggestion was clearly related by Humason to the de Sitter

effect in another paper, which appeared in 1929 and which was commu-

nicated at the same time of Hubble’s famous 1929 article. In this work

Humason reported the result about the large velocity of NGC 7619, which

was obtained with exposure times of 33 and 45 hours. Humason obtained

a mean value of v = +3779 km/sec, twice larger than the available largest

velocity of NGC 584, for which Slipher had obtained v = +1800 km/sec.

According to Humason, this result, together with what Hubble would

have shortly after communicated, suggested “an influence of distance

upon the observed line shift-such as would be produced, for example, on

de Sitter’s theory, both by the apparent slowing down of atomic vibra-

tions with distance and by a real tendency of material bodies to scatter

in space” [Humason 1929, p. 167].

6.5.2 Hubble’s 1929 relation

The 1929 paper about the velocities and distances among extragalac-

tic nebulæ, which marked a turning point in the rise of scientific cosmol-

ogy, was introduced by Hubble as “a re-examination of the question of

the K term” [Hubble 1929, p. 168].

Hubble based the determination of distance upon the criteria of abso-


lute magnitude and luminosity which he had already suggested in 1926.

He took into account 24 nebulæ which velocities were known, four of

them obtained by Humason. For seven of these stellar systems the dis-

tance was obtained by direct investigation of many stars in them. Then

Hubble calculated the distance of other 13 nebulæ by using the apparent

magnitude of their brightest stars and the absolute magnitude criteria

(MS = −6.3), i.e. “the criterion of a uniform upper limit of stellar lu-

minosity” [Hubble 1929, p. 170]. For other 4 objects, which were in the

Virgo Cluster, Hubble assigned the distance of 2 · 106 pc by using the

distribution of nebular luminosity. Since “the data (...) indicate a lin-

ear correlation between distances and velocities” [Hubble 1929, p. 170],

Hubble proposed a new expression of the equation of the solar motion, in-

troducing a direct proportionality between velocity and distance through

the K term, which became later known as H, the “Hubble constant”11:

v = X cos α cos δ + Y sin α cos δ + Z sin δ + Kr. (6.12)

By using the 24 nebulæ individually the result was:

K = +465± 50 km/sec, (6.13)

while the combination of these 24 objects in 9 groups according to their

distances implied:

K = +513± 60 km/sec. (6.14)

There were other 22 nebulæ for which the velocities were known, which

distances were, however, not available. In order to estimate the K cor-

relation term also for these nebulæ, Hubble derived their mean distance

from the mean absolute magnitude, and compared this value with the

mean radial velocity. It followed a value of K = +530 km/sec at a dis-

tance of 1.4 · 106 pc, which agreed with the relation obtained from the

first 24 objects.

“The results - Hubble pointed out - establish a roughly linear relation

between velocities and distances among nebulæ for which velocities have

11See [Trimble 1996 ] for a 1925-1975 history of such a “constant”.


been previously published, and the relation appears to dominate the

distribution of velocities” [Hubble 1929, p. 173].

Figure 6.5: The linear velocity-distance relation among extra-galactic nebulæ

which Hubble proposed in 1929 [from Hubble 1929, p. 172].

Concluding his own investigation, Hubble referred to the possibility

that such a relation could actually represent the de Sitter effect, “and

hence that numerical data may be introduced into discussions of the gen-

eral curvature of space” [Hubble 1929, p. 173]. Both the interpretations

of the de Sitter effect, namely the slowing down of atomic vibrations and

the scattering of particles, had to be taken into account. According to

Hubble, “the relative importance of these two effects should determine

the form of the relation (...), and in this connection it may be empha-

sized that the linear relation found in the present discussion is a first

approximation representing a restricted range in distance” [Hubble 1929,

p. 173].

Following [Hetherington 1996 pp. 124-128], Hubble introduced the

linearity of the velocity-distance relation by emphasizing the empirical as-

pect of his own work, based on the widely accepted method of the K term.

The refutation of Silberstein’s results by Lundmark and Stromberg, in-

deed, was not yet forgotten. Therefore, Hubble cautiously mentioned the

theoretical aspects of his work, i.e. the connection to the de Sitter effect,

just at the end of the paper, claiming that “new data to be expected in


the near future may modify the significance of the present investigation”

[Hubble 1929, p. 173].

The relation which Hubble found between velocity and redshift be-

came known as the “Hubble law”:

v = zc = Kr. (6.15)

Such a relation, as already mentioned, marked the turning point in the

connection between observations at large distance and theoretical rela-

tivistic cosmology. Indeed, despite the fact that, according to Hubble, the

linearity of the relation had to be confirmed by future observations12, it

was now evident that extragalactic nebulæ, few counterexamples apart13,

showed a systematic redshift. However, the cause of such a redshift was

still unknown.

A possible explanation of redshift was proposed by Zwicky in 1929.

Beside the theoretical possibilities which the model of de Sitter offered,

Zwicky added and advocated the idea that such spectral shifts could be

attributed to a “gravitational drag of light” analogue to the Compton

effect [Zwicky 1929, p. 775]. This interpretation became later known

as the “tired-light hypothesis”. “According to the relativity theory -

Zwicky wrote - a light quantum hν has an inertial and a gravitational

mass hνc2

. It should be expected, therefore, that a quantum hν passing

a mass M will not only be deflected but it will also transfer momentum

and energy to the mass M and make it recoil. During this process, the

light quantum will change its energy, and therefore its frequency” [Zwicky

1929, p. 775]. According to Zwicky, spectral displacements towards the

red were expected by considering light traveling at a distance L, which

12In addition, Shapley in 1929 criticized the actual linearity of Hubble’s relation

[Shapley 1929 ]. However, Shapley accepted such a result in 1930. See [Smith 1982,

pp. 183-185] for further readings on Shapley’s reaction to Hubble’s proposal.13See for example the objects with negative velocity, i.e. with an approaching

motion, in the summary proposed by Stromberg in 1925.


would have lost the momentum:

∆

(hν

c

)=

1.4 π GρDL

c· hν

c2, (6.16)

and therefore∆ν

ν=

1.4 π Gρ DL

c2. (6.17)

Here ρ was the density of matter in the universe, and D the distance at

which the perturbing effect faded out [Zwivky 1929, p. 778].

It was in 1930 that eventually the cosmological explanation of red-

shift, i.e. the explanation of spectral shifts as due to the expansion of

the universe, entered modern cosmology by revaluing the interpretation

which Lemaıtre had pointed out already in 1927.

velocity relation

Wirtz, 1924 v = a− b · log (Dm)

Lundmark, 1925 v = k + lr + mr2

Stromberg, 1925 no definite relation

Hubble, 1929 v = Kr

Table 6.1: Summary of different empirical relations proposed during the

1920’s. The observable quantities are the velocity (v), the angular diam-

eter (Dm) and the distance (r) of nebulæ (galaxies).

Chapter 7

The “third way” between

Einstein’s and de Sitter’s

solutions

In this chapter the second renewal of relativistic cosmology, i.e. the

general acceptance of the model of the expanding universe, is analyzed.

The rise of the expanding universe involved the decline of the interest in

the de Sitter effect. Indeed, in 1930 truly non-static and non-empty mod-

els of the universe, already proposed in 1922 by Friedmann and indepen-

dently in 1927 by Lemaıtre, were rediscovered by scientific community.

As from 1930, such solutions were taken into account in order to explain

the observational evidence of the systematic recession of extragalactic

stellar systems revealed by Hubble’s observations. The puzzling ques-

tions of the origin of extragalactic redshifts and of the redshift-distance

relation could be finally rightly interpreted by considering the metric of a

universe which world-radius increased in time, as in particular Lemaıtre

had suggested in his 1927 work.

159

160 The “third way” between Einstein’s and de Sitter’s solutions

7.1 1930: Eddington, de Sitter and the ex-

panding universe

During the 1930 January Meeting of the Royal Astronomical Society,

Eddington and de Sitter, in front of the sensational result by Hubble

that galaxies showed a systematic redshift, agreed on the conclusion that

non-static intermediary solutions between Einstein’s and de Sitter’s uni-

verses represented a suitable description of the actual universe. “The

question now arises - de Sitter pointed out during such a meeting - how

can we account for the linear connection between the velocities and the

distances?”. Eddington then remarked that ”one puzzling question is

why there should be only two solutions. I suppose the trouble is that

people look for static solutions. Solution A is such a static solution. So-

lution B is, on the contrary, non-static and expanding, but as there isn’t

any matter in it that does not matter” [R.A.S. Meeting 1930, pp. 38-39].

Eventually, it was the time to rediscover the expanding model which

Lemaıtre investigated in his 1927 paper “Un univers homogene de masse

constante et de rayon croissant, rendant compte de la vitesse radiale des

nebuleuses extra-galactiques” (A homogeneous universe of constant mass

and increasing radius, accounting for the radial velocity of extra-galactic

nebulæ1) [Lemaıtre 1927 ]. The original letters and manuscripts which

are stored at the de Sitter and Lemaıtre Archives permit to reconstruct

such an important discovery.

When the report of the mentioned R.A.S. meeting appeared in “The

Observatory” in February 1930, Lemaıtre recalled the attention of Ed-

dington on his own 1927 work. Together with copies of his 1927 paper,

1It is important to note that in the proofs of this 1927 paper, Lemaıtre had initially

written “variable radius” (“rayon variable”). Before the publication in the Annales de

la Societe Scientifique de Bruxelles, he had then changed this word with “increasing

radius” (“rayon croissant”) [Lemaıtre Archive, Box R10].

In the following pages of present thesis, quotations from the original 1927 paper refer

to the 1972 reprinted French version (see bibliography for details).

1930: Eddington, de Sitter and the expanding universe 161

Lemaıtre sent Eddington a letter in which he noted that “I just read the

February number of the Observatory and your suggestion of investigat-

ing non-statical intermediary solutions between those of Einstein and de

Sitter. I made this investigation two tears ago. I consider a universe

of curvature constant in space but increasing in time. And I emphasize

the existence of a solution in which the motion of the nebulæ is always a

receding one from time minus infinity to plus infinity” [Lemaıtre Archive,

Box D17].

Figure 7.1: Draft of the 1930 letter which Lemaıtre sent Eddington. In this

letter Lemaıtre recalled the attention of Eddington on the expanding model

of the universe which Lemaıtre had already discovered in 1927 [from Lemaıtre

Archive, Box D17].

Eddington immediately realized the importance of such a solution.

On March 19, 1930, Eddington sent de Sitter a copy of the French paper

by Lemaıtre, adding that “this seems a complete answer to the problem

we were discussing”. In a postcard that Eddington attached to the paper,

Eddington remarked that “by the way it was the report of your remarks

and mine at the R.A.S. which caused Lemaıtre to write to me about

it. (...) A research student McVittie and I had been worrying at the


problem and made considerable progress; so it was a blow to us to find

it done much more completely by Lemaıtre (a blow softened, as far as

I am concerned, by the fact that Lemaıtre was a student of mine)” [de

Sitter Archive, Relativity Box, A2]. Indeed around 1930 Eddington was

working in conjunction with George McVittie (1904-1988) on the stability

of Einstein’s spherical universe. With regard to the “brilliant solution”

by Lemaıtre, “although not expressly stated - Eddington remarked - it is

at once apparent from his formulæ that the Einstein world is unstable,

an important fact which, I think, has not hitherto been appreciated in

cosmological discussions” [Eddington 1930a, p. 668].

On March 25, 1930, de Sitter sent Lemaıtre an enthusiastic letter,

in which he showed his own satisfaction because the ds2 proposed by

Lemaıtre actually represented a suitable, simple and elegant solution to

the cosmological problem: “M. Eddington - de Sitter wrote to Lemaıtre

- m’a envoye il y a quelque jours un exemplaire de votre petit, mais

important, memoire du 1927 (...). J’avais moi meme (...) tache de trouver

une formule pour ds2 que comprendrait les deux solutions que j’ai appele

A et B (...). Votre solution, simple et elegante, me paraıt entierement

satisfaisante” [Lemaıtre Archive, D15]. Replying to this letter on April

5, 1930, Lemaıtre appreciated the interest which de Sitter gave to such

a work2: “je vous remercie beaucoup de l’interet que vous voulez bien

temoigner pour ma note sur l’univers de rayon variable (...). J’attends

avec grand interet les precisions que vous avez obtenues sur la relation

entre v and r et les diagrammes que vous avez montre aux membres de

la R.A.S.” [Lemaıtre Archive, D15].

Eddington and de Sitter further managed for an English translation

of Lemaıtre’s 1927 paper, which appeared in the Monthly Notices on

March, 1931 [Lemaıtre 1931a].

2In this letter, Lemaıtre also drew the attention of de Sitter on the 1922 work by

Friedmann and on Einstein’s reaction to it.

The importance of Lemaıtre’s 1927 proposal 163

Figure 7.2: Copy of the 1930 letter which de Sitter wrote to Lemaıtre, where

de Sitter acknowledged that the 1927 proposal by Lemaıtre was a solution to

the cosmological problem [from Lemaıtre Archive, Box D15].

7.2 The importance of Lemaıtre’s 1927 pro-

posal

As seen in Chapter 5, in 1925 Lemaıtre had faced some features of de

Sitter’s universe. By introducing new coordinates, Lemaıtre had shown

that the singularity of time-coordinate in the static form of de Sitter

universe was removed, however involving a non-static picture of the uni-

verse. According to Lemaıtre, such a representation of the universe as a

whole had to be discarded, because corresponded to an infinite universe

(the curvature of spatial sections was k = 0).


However, Lemaıtre realized that this change of coordinates repre-

sented an interesting suggestion for a world-radius changing with time.

As he wrote in a note of his 1927 paper (which was not translated in the

1931 English version), by using two coordinates, one spatial and the other

temporal, the de Sitter universe in the static form could be represented

as the surface of a sphere, where spatial lines corresponded to meridians,

whereas time lines corresponded to parallels on the sphere. The largest

parallel (the equator) of such a sphere corresponded to a geodesic, and

the “pole” of the sphere represented the singularity of time-coordinate.

On the contrary, by changing the coordinates, Lemaıtre showed that the

homogeneity was finally respected. Now the time lines corresponded to

meridians, and spatial lines to parallels: therefore, as Lemaıtre pointed

out, the world-radius changed with time3.

The empty universe of de Sitter, nevertheless, “is of extreme interest

as explaining quite naturally the observed receding velocities of extra-

galactic nebulæ” [Lemaıtre 1931a, p. 483]. On the contrary, as Lemaıtre

pointed out, Einstein’s truly static and finite universe was in agreement

“with the existence of matter, giving a satisfactory relation between the

radius and the mass of the universe” [Lemaıtre 1931a, p. 483]. A third

way between such solutions seemed desirable. Therefore, Lemaıtre con-

cluded that “in order to find a solution combining the advantages of

those of Einstein and de Sitter, we are led to consider an Einstein uni-

verse where the radius of space or of the universe is allowed to vary in

an arbitrary way” [Lemaıtre 1931a, p. 484].

Lemaıtre approximated the content of the universe to a rarefied gas

which was uniformly and homogeneously distributed through space, whose

molecules were the extragalactic nebulæ. As he soon recognized, “when

3“Les coordonnees respectant l’homogeneite reviennent a prendre pour lignes tem-

porelles un systeme de meridiens et pour lignes spatiales les paralleles correspondants,

alors le rayon de l’espace varie avec le temps” [Lemaıtre 1927, p. 90]. The author of

present thesis would like to thank Prof. Dominique Lambert for having pointed out

to him such an important passage in Lemaıtre’s analysis.


the radius of the universe varies in an arbitrary way, the density, uniform

in space, varies with time” [Lemaıtre 1931a, p. 484]. Notably, Lemaıtre

introduced in cosmology the concept of a time-dependent hypothetical

average density of matter in the universe, which Lemaıtre denoted with

δ. Moreover, he took into account also the contribution by the pressure.

Even though the pressure by matter could be considered negligible, the

radiation pressure, which Lemaıtre denoted with p, could not to be dis-

carded. The total energy density was thus:

ρ = δ + 3p. (7.1)

Lemaıtre then considered the line element of the universe (with c = 1)

with a time-dependent world radius, R ≡ R(t):

ds2 = −R2dσ2 + dt2, (7.2)

being dσ the spatial line element. Field equations reduced to two equa-

tions, which described the variation of the world radius with respect to

the world content, i.e. to radiation and matter [Lemaıtre 1927, p. 91]:

3R′2

R2+

3

R2= λ + κρ; (7.3)

R′2

R2+ 2

RR′′

R2+

1

R2= λ− κ p. (7.4)

In these relations λ was the cosmological constant, R′ ≡ dRdt

, and R′′ ≡dR′dt

. As mentioned in Chapter 2, these equations, together with the simi-

lar solutions (however without the contribution by pressure) which Fried-

mann proposed in 1922, became known in cosmology as the Friedmann-

Lemaıtre (FL) equations.

Lemaıtre, in his 1927 analysis, noted that the second of these equa-

tions could be replaced by introducing the condition of the adiabatic

expansion of the universe:

d(V ρ) + p dV = 0, (7.5)


being V = π2R3 the total volume. This condition, indeed, could be

written as:dρ

dt+

3R′

R(ρ + p) = 0, (7.6)

which expressed the energy conservation and was equivalent to the second

(FL) field equation. “The variation of total energy - Lemaıtre highlighted

- plus the work done by radiation-pressure in the dilatation of the universe

is equal to zero” [Lemaıtre 1931a, p. 485]. He then looked for the solution

of a universe of constant mass: M = V δ = constant. He introduced α

and β, where α was a constant related to the density of matter:

κδ =α

R3, (7.7)

and β was a constant of integration proportional to the pressure:

κ p =β

R3. (7.8)

In this way the first (FL) field equation reduced to [Lemaıtre 1927, p.

93]:

t =

∫dR√

[λ R2

3− 1 + α

3R+ β

R2 ]. (7.9)

The case of α = β = 0 corresponded to the de Sitter’s solution, and the

world-radius could be written in the form already found by Lanczos in

1922:

R =

√3

λcosh

√λ

3(t− t0). (7.10)

On the contrary, Einstein’s solution was obtained by making β = 0 and

by considering a constant radius [Lemaıtre 1927, p. 93]:

α = κ δ R3 =2√λ

. (7.11)

According to Lemaıtre, the cosmological constant was related to R0,

which was the asymptotic value at t = −∞ from which the radius of the

universe R increased without limit. The relation between λ and R0 was

expressed in the form:

λ =1

R20

. (7.12)


The radius of the actual world then resulted [Lemaıtre 1927, p. 94]:

R3 = R2ER0. (7.13)

RE was the radius of the finite universe of Einstein, which depended on

the density of matter and increased in time:

κδ =2

RE

. (7.14)

The value of R0 could be deduced from the radial velocities of nebulæ, as

Lemaıtre showed in a section of his 1927 paper devoted to the “Doppler

effect due to the variation of the radius of the universe”: already in the

title of this section, Lemaıtre pointed out which was the cause of the

redshift. From the form of the line element it followed for a ray of light

traveling from σ1 to σ2:

σ1 − σ2 =

∫ t2

t1

dt

R. (7.15)

“A ray of light emitted slightly later - Lemaıtre explained - starts from

σ1 at time t1 + δ t1 and reaches σ2 at time t2 + δ t2” [Lemaıtre 1931a, p.

487]. Therefore it followed:

δ t2R2

− δ t1R1

= 0, (7.16)

which involvedδ t2δ t1

− 1 =R2

R1

− 1 =v

c. (7.17)

R1 and R2 were the radius of the universe at the time of emission and

reception, respectively. It was thus Lemaıtre who offered in 1927 the

actual cosmological interpretation of redshift, since the ratio of these radii

minus 1 corresponded to “the apparent Doppler effect due to the variation

of the radius of the universe” [Lemaıtre 1931a, p. 487]. Redshifts which

were observed in nebulæ were not due to a relative motion between the

observer and the observed object, nor to the slowing down of atomic

vibrations. “The recession velocities of extragalactic nebulæ - Lemaıtre


pointed out - are a cosmical effect of the expansion of the universe”

[Lemaıtre 1931a, p. 489].

For sources which were near enough, it was useful to use the approx-

imate formula [Lemaıtre 1927, p. 96]:

v

c=

R2 −R1

R1

=dR

R=

R′

Rdt =

R′

Rr, (7.18)

which gave:R′

R=

v

c r. (7.19)

It is interesting to note that Lemaıtre actually found a linear relation

between velocities and distances. In present notation we have R′R≡ a

a≡

H(t), and the previous relation can be written as vc

= H r.

In 1927 Lemaıtre also derived a value of the proportionality constant

(575 km/sec Mpc−1) close to the value which Hubble proposed in his 1929

paper, which became later known as “Hubble constant”. However, such

passages in the 1927 French paper were not translated in the 1931 English

version. With regard to the radial velocity of nebulæ, Lemaıtre based his

calculations on data by [Stromberg 1925 ]. For distances, he referred to

[Hubble 1926 ], and to the Hubble’s suggestion to use an average absolute

magnitude (M = −15.2) for nebulæ. The distance was obtained from

the relation [Lemaıtre 1927, p. 96]:

log r = 0.2 m + 4.04, (7.20)

being m the apparent magnitude. For 42 nebulæ Lemaıtre found an

average distance of 0.95 Mpc, corresponding to an average radial velocity

of 600 km/sec. “On trouverait - Lemaıtre added in a note - 670 km/sec

a 1.16 · 106 pc, 575 km/sec a 106 pc. Certains auteurs ont cherche a

mettre en evidence la relation entre v et r et n’ont obtenu qu’une tres

faible correlation entre ces deux grandeurs. (...) Il semble donc que ces

resultats negatifs ne sont ni pour ni contre l’interpretation relativistique

de l’effet Doppler” [Lemaıtre 1927, p. 97]. From (7.19) he obtained:

R′

R=

v

c r=

625 · 105

106 · 3.08 · 108 · 1010= 0.68 · 10−27 cm−1. (7.21)


By using the approximate formula

R0 =c r

v√

3, (7.22)

Lemaıtre found for the world radius at t = −∞ the value R0 ' 2.7 · 108

pc [Lemaıtre 1931a, p. 487].

In order to understand the cause of the expansion of the universe,

Lemaıtre suggested at the end of his paper that “we have seen that

the pressure of radiation does work during the expansion. This seems

to suggest that the expansion has been set up by the radiation itself”

[Lemaıtre 1931a, p. 489].

7.2.1 Rediscovering the models of Friedmann

In 1930, also the cosmological equations proposed by Friedmann in

1922 and 1924 were eventually widely accepted by the scientific commu-

nity. Already in 1922, dealing with the question of the curvature of space,

Friedmann explained that his purpose was to show that Einstein’s and

de Sitter’s solutions “are special cases of more general assumptions, and

secondly to demonstrate the possibility of a world in which the curvature

of space in independent of the three spatial coordinates but does depend

on time” [Friedmann 1922, Engl. tr. p. 49]. In this paper Friedmann

took into account a positive spatial curvature of a universe which radius

depended on time. Subsequently, in 1924 he considered both a static and

a non-static case, now admitting a negative spatial curvature [Friedmann

1924 ].

In 1922, Friedmann proposed a model of an expanding universe, in

which he discarded the contribution by the pressure, and where stellar

velocities were small with respect to the speed of light. In the expression

of the metric, he considered R ≡ R(x4): “R is proportional to the radius

of curvature, that will be proportional to time also” [Friedmann 1922,

Engl. tr. p. 50]. Furthermore, by an appropriate choice of coordinates,

“space can be made orthogonal to time. We cannot offer - Friedmann


noted - any philosophical or physical justification for these assumption;

they simplify the calculations” [Friedmann 1922, Engl. tr. p. 51]. The

metric was written as:

ds2 = R2(dx21 + sin2 x1dx2

2 + sin2 x1 sin2 x2 dx23) + M2dx2

4. (7.23)

M was a function of all four coordinates. By putting R = −R2

c2, Ein-

stein’s metric arose if M = 1, whereas de Sitter’s metric corresponded to

M = cos x4. The general case corresponded to M as a function of time

coordinate x4. In the latter case, field equations for µ = ν = 4 yielded

to [Friedmann 1922, Engl. tr. p. 53]:

3R′2

R2+ 3

c2

R2− λ c2 = 8π Gρ. (7.24)

On the contrary, for spatial indexes (µ = ν = 1, 2, 3) it followed:

R′2

R2+ 2

RR′′

R2+

c2

R2− λ c2 = 0. (7.25)

In these equations it was R′ = dRdx4

, and R′′ = d2Rdx2

4. By replacing x4 with

t, the second (FL) equation reduced to:

R

c2

(dR

dt

)2

= A−R +

(λ

3

)R3, (7.26)

and thus:

t =1

c

∫ R

a

√√√√[

x

A− x + (λ3)x3

]dx + B, (7.27)

where a,B, A were arbitrary constants [Friedmann 1922, pp. 53-54]. The

density of matter resulted:

ρ =3A

c2κR3, (7.28)

and A was related to M , the total mass in the universe:

A =κ c2M

6π2. (7.29)

The quantity λ was not determined. Different world-models therefore

depended on the value which λ assumed with respect to λE = 4c2

9A2 , i.e.


to the value of the cosmological constant in Einstein’s static universe.

Notably, the case λ > λE involved a “creation of the world (...); the time

since the creation of the world - Friedmann suggested - might be infinite”

[Friedmann 1922, Engl. tr. p. 56].

However, in his 1922 and 1924 works Friedmann did not mention any

possible relation to astronomical consequences, in particular to redshift,

and at the beginning of the 1920’s such papers basically appeared as

a mathematical speculation [North 1965, p. 117]. Both the 1922 pa-

per by Friedmann and the 1927 paper by Lemaıtre were published in

lesser-known journals, which were not strictly focused on astronomical

topics. Therefore until 1930 they remained nearly unknown to most of

the scientists, nowadays called cosmologists, who were involved in the

first debates about relativistic cosmology. As seen, Robertson referred to

Friedmann’s paper in his own 1929 analysis. On the contrary, at the time

of writing his own 1927 paper, Lemaıtre was not aware of Friedmann’s

equations. It was Einstein who pointed out to Lemaıtre the existence

of Friedmann’s paper in 1927, as Lemaıtre acknowledged at the end of

a 1929 conference [Lemaıtre 1929, p. 32]. Indeed in 1927 Lemaıtre had

the chance to meet Einstein at the fifth Solvay Conference in Bruxelles

and became acquainted with Friedmann’s result. Lemaıtre could also

briefly talk with Einstein about his own analysis of the expanding uni-

verse. With regard to Einstein’s reaction on the contents of Lemaıtre’s

1927 paper, Lemaıtre referred some years later that “du point du vue

physique, cela lui parait tout a fait abominable” [Lemaıtre 1958, p. 129].

According to Einstein, such an expanding solution did not correspond

to a physical possibility. This was the same remark that Einstein had

already made in 1918 to the de Sitter’s universe, and in 1922 with respect

to the result found by Friedmann.

In 1922, indeed, Einstein objected in a brief note that Fridemann’s

result concerning a non-stationary world “seems suspect” to Einstein

himself [Einstein 1922b, Engl. tr. p. 66]. According to Einstein, “from

the field equations it follows necessarily that the divergence of the matter


tensor vanishes. This (..) leads to the condition ∂ρ∂ x4

= 0, which (...) im-

plies that the world-radius R is constant in time” [Einstein 1922b, Engl.

tr. p. 66]. However, thanks to Aleksander Krutkoff, Einstein could read

a letter where Friedmann explained the details of his own calculations.

Therefore in a subsequent paper Einstein accepted that Friedmann’s re-

sult was “both correct and clarifying” [Einstein 1923, Engl. tr. p. 67].

Nevertheless, despite the correctness of the mathematical method, Ein-

stein still believed that the physical interpretation of Friedmann’s result

was not conceivable. In 1931, in front of the empirical evidence that

galaxies were receding one to another, Einstein abandoned the cosmo-

logical constant which he had introduced in 1917 in order to express in

general relativity the static nature of the universe. Now this supposition

was contradicted by the observed recession of nebulæ which claimed to

the expanding universe4 [Einstein 1931 ].

7.3 The decline of the interest in the de

Sitter effect

The cosmology of Lemaıtre finally solved the problems related to

the models of Einstein and de Sitter, which were unsatisfying from the

point of view of a suitable comparison with astronomical observations.

The expanding universe gave a solution of the puzzling question of the

origin of redshift. Therefore, as from 1930, the de Sitter effect, i.e. the

interpretation of spectral displacements which the model of de Sitter

offered, faded away just because, quoting de Sitter himself, solution B

“must be rejected, (...) and the true solution represented in nature must

be a dynamical solution” [de Sitter 1930a, p. 482].

4It is interesting to note that in 1923, when both Eddington and Weyl proposed

their own interpretations of a cosmical recession in de Sitter’s world, Einstein wrote

to Weyl that “if there is no quasi-static world, then away with the cosmological term”

[quoted in Pais 1982, p. 288].

The decline of the interest in the de Sitter effect 173

On April 17, 1930, de Sitter wrote to Shapley that “I have been very

busy lately on spiral nebulæ and on the relativistic explanation of the

big velocities. I had come to the conclusion that my solution B could

not be accepted as an adequate explanation, as it supposes the universe

to be empty (...). Only very lately I have found the true solution, or at

least a possible solution, which must be somewhere near the truth, in

a paper published already in 1927 by Lemaıtre of Louvain, which had

escaped my notice at the time” [de Sitter Archive, Box 17.4 C]. De Sitter

acknowledged that non-static and non-empty world-models eventually

represented the actual universe. “The important point - the Dutch as-

tronomer wrote to Tolman on May 7, 1930 - is that we must look for

a dynamical solution of the field equations as the true interpretation of

nature” [de Sitter Archive, Relativity Box, A3]. As de Sitter noted, the

dynamical solution which Lemaıtre proposed “requires all observed ra-

dial velocities to have the same sign, while in the solution B the sign was

indeterminate for each individual body” [de Sitter 1930a, p. 487]. De

Sitter took into account observations of apparent magnitude and angular

diameter, in order to test by a method which differed from Hubble’s one

the linearity of the observed relation among galaxies. He then confirmed

through available data the existence of a linear redshift-distance relation,

adding proof to Hubble’s result [de Sitter 1930b].

With regard to the comparison between the de Sitter effect and the

explanation of redshift proposed by Lemaıtre, Eddington pointed out

in 1930 that de Sitter “introduced the slowing down of time at great

distances from the origin, which does not occur in the new formulæ. The

present description involves fewer paradoxes and is undoubtedly easier to

apprehend (...). It has, moreover, the advantage that we now approach

de Sitter’s world as the limit of a series of worlds of gradually diminishing

density; whereas formerly we had to start with a completely empty world,

and very cautiously put a few material bodies in it. (...) Of course it is

possible that the recession of the spirals is not the expansion theoretically

predicted; it might be some local peculiarity masking a much smaller


genuine expansion. But the temptation to identify the observed and the

predicted expansions is very strong” [Eddington 1930a, pp. 676-677].

In his 1933 report on the astronomical aspect of general relativity, de

Sitter remarked that “all that observations tell us is that light coming

from great distances, and which therefore has been a long time on the

way, is redder when it arrives than when it left its source. Light is red-

dened by age: traveling through space, it loses its energy and gets older.

Or, expressed mathematically: the wave-length of light is proportional

to a certain quantity R, which increases with the passing of time5” [de

Sitter 1933, p. 155].

A different interpretation of spectral shift was proposed in 1932 by

Edward Arthur Milne (1896-1950). Milne proposed a special relativistic

model of the universe. In this kinematical model, by using the cosmolog-

ical principle, the universe was bounded, and expanded into an infinite

flat space. Following [Harrison 2000, p. 374], such a universe can be

(mathematically) transformed into an infinite and unbounded universe

within the Robertson-Walker expanding frame. In such a version, Milne’s

universe is homogeneous and isotropic, its spatial curvature is negative,

and the radius of the universe is R = t, i.e. the universe expands at

constant rate. As Milne noted in 1934, in his own special relativistic

model “the Doppler shift s remains constant in time for any one patch;

it increases at any one epoch of observation as b [the surface brightness]

decreases” [Milne 1934, p. 26].

With regard to the cosmological (expansion) shift, also Hubble doubted

about its general relativistic interpretation. In 1937, in “The observa-

5It is worth noting that in the draft of such a 1933 report de Sitter used different

words. On p. 15 of this draft, de Sitter wrote: “the interpretation of Lemaıtre’s

non-static solution as an expanding universe, R being the ‘Radius’ of the universe, is

not imperative. Apart from any interpretation of the meaning of the quantity R or

of the observed redshift as a velocity, we can enounce the net result as follows: the

ratio between the observed and emitted wave-length is the same as the ratio of the

values of R at the times of observation and emission, respectively. Since R increases

with the time, light is reddened by age” [de Sitter Archive, Relativity Box, A8].

The decline of the interest in the de Sitter effect 175

tional approach to cosmology”, Hubble acknowledged that the interpre-

tation of redshift as radial motion was the only permissible explanation

that was known “until evidence to the contrary is forthcoming” [Hubble

1937, p. 26]. Beside such an interpretation of redshift as velocity-shift,

Hubble mentioned also the possibility of redshift as loss of energy in

transit. “Redshifts - he wrote - are produced either in the nebulæ, where

the light originates, or in the intervening space through which the light

travels. If the source is in the nebulæ, then the redshifts are probably

velocity-shifts and the nebulæ are receding. If the source lies in the inter-

vening space, the explanation of redshifts is unknown but the nebulæ are

sensibly stationary” [Hubble 1937, p. 31]. Within this alternative, which

corresponded to a universe which was not rapidly expanding, Hubble

pointed out that an exact linear relation between redshift and distance

had to be expected. On the contrary, redshifts as velocity-shifts involved

a departure from the linearity [Hubble 1937, p. 41]:

∆λ

λ= kr + lr2 + mr3 + ... (7.30)

“Theories may be revised - Hubble concluded - and new information may

alter the complexion of things, but meanwhile we face a rather serious

dilemma” [Hubble 1937, p. 44].

Alternative proposals apart, “it is now clear - Robertson wrote in his

1933 review on relativistic cosmology - that the existence of the so-called

velocity-distance relation formed no essential part of the deduction [of

Friedmann-Lemaıtre cosmological models], which was based entirely on

the evidence for the uniform distribution and state of motion of matter

in the large, and on the acceptance of the general theory of relativity

(...). If we consider the observed redshift as arising from the nature of

space-time, we find in it additional evidence for the theory” [Robertson

1933, p. 83].

New issues characterized the cosmological debates during the first

years of the 1930’s. Scientists involved in those discussions dealt, among

others, with the concept of the evolution of the universe and its time-


scale, the cause of the expansion of the universe, the value of the Hubble

constant. In 1931 the concept of the beginning of the world entered the

relativistic investigation of the universe. In that year Lemaıtre suggested

the hypothesis of the “primeval atom”, which corresponded to the ini-

tial state of the universe with minimum entropy [Lemaıtre 1931b]. In

1932 Einstein and de Sitter, whose 1917 rival models were the objects of

the cosmological question before the diffusion of the expanding universe,

proposed a metric which corresponded to an expanding model, where

the contribution by pressure was neglected and the curvature was zero

(flat spatial sections). In this model, Einstein and de Sitter did not take

into account the cosmological term in field equations. Indeed, through

the dynamical solutions rediscovered in 1930, the existence of a finite

mean density of matter could be now theoretically achieved without the

introduction of λ [Einstein-de Sitter 1932 ].

Whereas on the one hand general relativity offered the theoretical

base for a suitable description of the universe as a whole beyond New-

tonian cosmology, on the other hand observations on large scale now

revealed that the consequences and predictions of such a theory could be

now empirically verified or confuted. As seen, such a connection entered

modern cosmology just during the debates about the de Sitter effect,

which foreshadowed a non-static picture of the universe as a whole, and,

moreover, led to the development of cosmology from theoretical specula-

tion into an empirical science.

Conclusion

In present thesis we critically reconstructed the debates which took

place during the 1920’s about the de Sitter effect, i.e. the history of the

different predictions and possible confirmations of the redshift-distance

relation which was obtained through the metric of de Sitter’s universe.

From this analysis it emerges the fundamental role played by the de

Sitter effect in the rise of cosmology as an empirical science. Such a

redshift-distance relation was the leading thread in the early phases of

the modern scientific view of the universe, during the tortuous process

from the 1917 beginning of theoretical relativistic cosmology towards the

1930 diffusion of the model of the expanding universe which was claimed

by astronomical observations.

As shown in present thesis, the 1920’s debates were characterized by

a richness of ideas, attempts, controversies and failures related to the

cosmological question in the light of relativity revolution. Dealing with

the curvature of space-time in astronomical context, scientists involved

in cosmological debates from 1917 to 1930 approached and thoroughly

analyzed fundamental issues, some of them still present in modern cos-

mology. They addressed themselves to the question of which was the

most suitable model which represented the actual universe, and began

to verify relativistic models through observations. In this framework, it

was the interest in the de Sitter effect which led to link together theoret-

ical relativistic cosmology with astronomical observations on large scale,

inaugurating the modern approach of cosmologists in the comprehension

of global properties of the universe.

177

178 Conclusion

In 1917 Einstein showed that a new cosmology was allowed by his

new theory of gravitation, overcoming Newtonian difficulties at infinity.

In his picture of a static and finite universe, Einstein achieved the re-

quirement that, through a suitable metric and by introducing the cosmo-

logical constant, space-time was globally influenced by gravitation, and

inertia was entirely produced by all masses in the world. In his spherical

world, Einstein made the working hypothesis of a uniform and homoge-

neous density of matter in order to deal with properties of gravitation

and inertia on large scale, foreshadowing the fundamental assumption

which became known as the Cosmological Principle. In the same year de

Sitter demonstrated that, from a mathematical point of view, an empty

universe actually corresponded to another suitable solution of relativistic

field equations with the cosmological term.

The constant radii of both these finite universes depended on the value

of a new universal constant, λ. In Einstein’s intention such a constant

acted a a sort of anti-gravity in order to counterbalance gravitational

effects on large scale. Nevertheless, as de Sitter soon pointed out, a

redshift-distance relation could be derived from the metric of de Sitter’s

universe: a mass test in his own empty universe showed a displacement of

spectral lines, which de Sitter interpreted as due both to a gravitational

shift caused from the form of the metric, and to a Doppler effect from

the geodesic equations.

Therefore de Sitter related this feature, the so-called de Sitter effect,

to observations of shifts in spectral lines from stars and nebulæ, in order

to estimate the value of the world-radius of his own model. General

relativity offered the possibility to coherently extrapolate laws of physics

and properties of light and matter to the universe as a whole, and de

Sitter inaugurated the verification of the first two theoretical models

with respect to their astronomical consequences.

De Sitter’s pioneering attempts did not pass unnoticed. Despite its

lack of matter, the empty universe of de Sitter drew the attention of

several scientists looking for the best approximation of the actual world.

Conclusion 179

In this framework, the de Sitter effect was taken into account to rightly

interpret redshift measurements in nebulæ.

In his own 1923 analysis, Eddington pointed out that a general cosmic

recession was expected in de Sitter’s universe just because of the pres-

ence of the cosmological constant. Such a tendency of particles to scatter

could roughly account for the astonishing radial velocities measured by

Slipher in spiral nebulæ, the most part of which revealed receding mo-

tions from the observer. Remarkably, Eddington realized that a suitable

description of the actual world corresponded to an intermediate solution

between Einstein’s model, which had matter but not motion, and de

Sitter’s model, which had motion but not matter.

The geometry of de Sitter’s world-model was not uniquely deter-

mined, and non-static interpretations of the universe emerged by ap-

propriate coordinate changes, as done in the 1920’s by Weyl, Lanczos,

Lemaıtre and Robertson. Their contributions, which referred to a sta-

tionary frame of de Sitter’s universe, showed that theoretical cosmology

properly allowed to deal with non-static line elements, i.e. with a non-

static picture of the universe.

Beside the controversies around the interpretation of relativistic cos-

mological models, during the 1920’s first astronomical observations on

large scale inaugurated the beginning of the observational approach to

cosmology.

In 1924 Wirtz realized that the universe of de Sitter represented a

suitable model to account for redshift and apparent diameter of nebulæ.

In the same year, on the contrary, Silberstein criticized the possibility

of a general cosmic recession, and considered the distances of globular

clusters in order to verify the de Sitter effect. The correctness of the

method and of the result proposed by Silberstein was shortly after denied

by Lundmark and Stromberg. There was neither a general agreement on

the meaning of redshift, nor a clear and widely accepted understanding of

the properties of de Sitter’s universe. A suitable test of redshift relations

was possible only with a reliable determination of the distances of spiral

180 Conclusion

nebulæ.

In this controversial picture, the contributions of Hubble marked a

turning point in the comprehension of the structure of the universe.

Thanks to the revolutionary observations which Hubble furnished dur-

ing the 1920’s, spiral nebulæ were accepted as ‘island universes’, i.e. as

true extra-galactic stellar systems. Moreover, Hubble finally confirmed

in 1929 that such systems receded relatively to one another, and that

their radial velocities linearly increased with distances.

The puzzling question of the ambiguous interpretation of the de Sitter

effect and the meaning of redshift was solved in 1930, when the cosmol-

ogy of Lemaıtre was reconsidered in order to explain the cosmic recession

of galaxies revealed by Hubble. The model of a non-empty and expand-

ing universe which Lemaıtre had already proposed in 1927 provided the

proper cosmological interpretation of redshift: the displacement of spec-

tral lines was due to the expansion of the universe.

As from 1930, the de Sitter effect, which during the 1920’s represented

the first hint in the intersection between the new theory of gravitation and

observed facts, was seen as an effect of minor importance. The expand-

ing universe inaugurated another chapter of modern cosmology. Static

and stationary universes were eventually considered as limiting cases of

the general dynamical solutions of Friedmann-Lemaıtre equations, which

brought new questions and new challenges in the investigation of the

properties of the universe.

From 1917, when Einstein proposed his cosmological considerations in

general relativity, the knowledge of the universe as a whole greatly devel-

oped towards the new 1930 paradigm of the expanding universe, which

resulted the natural and more comprehensive and coherent theoretical

interpretation of the observational evidence of a cosmic recession.

The advancement of scientific cosmology was, and is nowadays, char-

acterized by new ideas, discoveries, changes. The history of the early

developments of modern cosmology during the 1920’s reveals how the is-

sues faced in that fruitful period represent a remarkable passage towards

Conclusion 181

the comprehension of the universe through the laws of physics.

“The theory of today - de Sitter wrote in 1932 - is not the theory

of tomorrow. (...) Science is developing so very rapidly nowadays, that

it would be preposterous to think that we had reached a final state in

any subject. The whole of physical science, including astronomy, is in a

state of transition and rapid evolution. Theories are continually being

improved and adapted to new observed facts. It would certainly not be

right to suppose at the present time that we had reached any state of

finality. We are, however, certainly on the right track” [de Sitter 1932a,

pp. 103-104].

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Acknowledgements

I want to express my deep gratitude to my thesis supervisor, Prof.

Giulio Peruzzi, for his patience, encouragement, and thoughtful guidance.

A very special thanks to my co-supervisor, Prof. Luigi Secco, for his

constant support and interesting suggestions. I am sincerely grateful

to my advisor, Prof. Jurgen Renn, for his kind attention and helpful

comments. I am deeply indebted to Dr. Jan Guichelaar, for his help and

very useful translations, and to Prof. Dominique Lambert, for interesting

discussions. I want to sincerely thank Prof. Frans van Lunteren and Dr.

David Baneke for their kind attention and collaboration, and for the

permission to reproduce original manuscripts from the de Sitter Archive.

My special thanks to Mark Hurn, Adam Green, Tatiana Turato, Claudia

Toniolo and Ruth Kessentini, for their kindness and patience. I gratefully

acknowledge Liliane Moens, for her great help and the permission to

reproduce original manuscripts from the Lemaıtre Archive. I am grateful

to Prof. Donald Lynden-Bell, Prof. Malcolm Longair and Dr. Matthias

Schemmel, for their useful suggestions. I also wish to thank Prof. Cesare

Chiosi, for his support, and Prof. Bepi Tormen, for helpful comments.

Thanks also to Luca and Monica, for their immediate help. Finally,

my sincere thanks to my grandmother, Laura, for her great help with

translations.

This work has been supported in part by the Italian Ministry of the

University and Research under the Research Projects of National Interest

(PRIN).

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