espacio, tiempo y materia en el cosmos - csic · a lobo, 14-ii-2007 espacio, tiempo y materia en el...

EspacioEspacio, Tiempo, Tiempo y materia en y materia en el Cosmosel Cosmos

J. Alberto LoboICE/CISC-IEEC

A Lobo, 14-ii-2007 Espacio, Tiempo y Materia en el Cosmos

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Summary

Part IPart I::GeneralGeneralRelativityRelativityTheoryTheory

• Concepts of motion: From Aristotle to Newton• Newton’s laws: Absolute Space and Time• Mach’s criticism and Mach’s Principle• Einstein’s Equivalence Principle• General Theory of Relativity:

– Curvature of space-time– Field equations: Space, Time and Matter link together

Part IIPart II::GeneralGeneralRelativisticRelativisticCosmologyCosmology

• Newtonian Cosmology, Olbers’ paradox• General Relativistic Cosmology, an evolutionary paradigm• The Cosmological Principle• Friedmann equations• Big Bang Cosmology: parameters of the Universe• The Accelerating Universe• Outlook and future talks


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Concepts of motion

Aristotle expressed in his treatise“Physica” that a force is necessaryto keep bodies in motion, no matterwhether accelerated or not.

Galileo Galilei, some 1800 years on,proved by experiment that uniformmotion does not require a force. Andthat forces cause accelerations.


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Newton’s laws of Mechanics

Isaac Newton wrote a fundamental book in1689: “Principia Mathematica PhilosophiaeNaturalis”. There, he developed his theoriesof motion and of Universal Gravitation.

Newton’s laws of mecahnics are explained in High School courses. Werecall them here now:

1. Under no forces, bodies move at constant speed, or stay at rest.2. Forces cause accelerations, according to the equation:

3. Every force action has a reaction counterpart, equal and opposite.

m=F a


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Let’s review the 1st law

Is it a consequence of the second? For, it may be argued,

, 0 0 constaif and nd tanm= = = =F a F a v⇒

Problem is: how can one possibly asceratain that F=0?This is in fact a logically circular question:

0 constant= =F v

The first law actually defines the class of reference systemsin which the second law applies. These are called Inertialreference systems. Newton inferred from here the existenceof Absolute Space (Scholium of his PM), and that it was thecause of inertia.


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The bucket of water experiment

Waterat rest

Rope wound

Waterrotates

Rope unwinds


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Inertia and Absolute Space

Newton derived important conclusions from his experiment:

Accelerated motion is an objective concept.

1. There is an absolute reference system whichenables the intrinsic definition of acceleration.

3. This is Absolute Space. It is therefore the causal agent of inertia.

4. It cannot however be individuated.

5. “Fixed stars” are (perhaps) the best approximation...


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For the next 200 years...

D. Papin J. Watt U. Leverrier P.S. Laplace I. Kant L. Boltzmann

Immense success of Newtonian Mechanics and its variety ofapplications somewhat left in stand by those almost philosophicaldiscussions:

• Mechanical machines of many types: industry, etc.• Celestial Mechanics• Mechanical theory of Thermodynamics• ...


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Ernst Mach’s criticism

Ernst Mach

What would happen if the rest of theUniverse should start revolving aroundthe bucket?

He conjectured the same would happen...

Mach went on to formulate what was tobecome known as Mach’s Principle:

–The inertia of the bodies is determinedby the distribution of matter throughout theUniverse.

Unfortunately, Mach’s principle is notany quantitative...


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General Relativity

The old riddle was solved by Einstein’s General Theory of Relativity,towards the end of 1915.

4

1 82

GR R g Tcµν µν µνπ

− = −

1950

Because that solution heavily relies on Gravitation –it actually constitutesthe new theory of gravitation–, it will be useful to review a few basic factsabout gravity...


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Bodies in free fall


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Port Aventura


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Simple theoretical analysis

Rope tension: T

M= ≠

= ≠

0

0⇓

a

a g

F

Rope tension: T

+ =

=

0

0⇓

a

T F

Gravity force: F Gravity force: F


14

A very contemporary example of Galileo’s experiment: The astronaut is in free fallat fhe same rate as the spacecraft, so he feels weightlessly floating. Nevertheless, gravity up there is not negligible: 8 m/sec2...


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“Hipotesis non fingo”

Let h = 5 m

h

L

v

212

L

h

tv

tg

=

=v = 700 m/sL = 700 m

v = 100 km/sL = 100 km

22 /hvL g⇒ = ⇒


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^^^^^^^^^

v = 100 km/sL = 100 km

But take a careful look at these numbers:

100 km>100 km !!!

The ocean’s surface actually bends…

If the Earth were flat then L=100 km

But the Earth is not flat!! Hence L > 100 km !!!


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Is it thinkable to shoot the bullet fast enough that it eventually hitsthe pirate from the back ???

Like this:

mmm...


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Perhaps the Moon is, after all, an example of that...:

Somebody must have shot it from where it revolves around the Earth...


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Newton’s Law of Universal GravitationNewton’s Law of Universal Gravitation

v = 0 v small v large

2

GMmFr

=


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Let’s pause for a moment...

...and write down the two main results seen so far:

1.1. The definition of Inertial Reference System is circular,therefore Newton’s laws –and Special Relativity laws,too— suffer from a severe logical inconsistency.

2.2. Gravitational fields impinge the same acceleration onall bodies, regardless of their mass and/or composition.

Inertial Reference Systems may not have a prvileged status,any Reference System should be equally appropriate to describePhysical laws. General, rather than Special Relativity must be theultimate concept in Physics.

It is not possible to tell whether a given Reference System isaccelerated by a gravitational field or by some other agent.


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Equivalence Principle

In 1915, Einstein came up with a far reaching conjecture whichsettled the basis for a new theory of gravity, and disposed of theepistemological inconsistencies of Newtonian Mechanics.

The Equivalence PrincipleThe Equivalence Principle

The laws of Physics are those of Special Relativity whenexpressed in a Reference System which is falling freely ina gravitational field.


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Equivalence Principle

-- The EP disposes of the servitudes of needing Inertial Systems.General Relativity is soundly established instead.

-- Inertial and gravitational masses must be equal.

-- Absolute space is a superfluous concept.

-- Freely falling particles (generally) follow curved trajectories.These are straight lines if no gravitational fields are present,and shortest distance curves (or geodesics) if they are.

-- Space-time is therefore not absolute, it has geometricproperties determined by the distribution of gravitatingmass and energy.

-- The old tenet of Euclidean Geometry thus gives way to therules of Riemannian Geometry.


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Field equations and geodesics

The EP sets the general framework. But a very important questionis still pending:

does matter distribution actually determine geometry?How

The answer is provided by Einstein’s field equations:

4

1 82

GR R g Tcµν µν µνπ

− = −

which is a complicated set of non-linear partial differential equationsfor the field unknowns gµν.

Once the gµν’s are known, the trajectories of test particles and lightrays are determined by the geodesic equation:

0x x xµ µ ρ σρσ+ Γ =


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Bending of light rays

A truly remarkable prediction of GR is the bending of light rays:just like material particles, light rays (photons) follow non-straightline trajectories as they come close to gravitating objects.

Take for example a ray which grazes the Sun surface:

2

4GMRc

δφ = 1.76 arc - secondsEinstein’s formula:


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Eddington’s 1919 observations

Night


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Eddington’s 1919 observations

Day


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SolarEclipse


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Gravitational lenses


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End of Part IEnd of Part I


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Summary

• Concepts of motion: From Aristotle to Newton• Newton’s laws: Absolute Space and Time• Mach’s criticism and Mach’s Principle• Einstein’s Equivalence Principle• General Theory of Relativity:

– Curvature of space-time– Field equations: Space, Time and Matter linked together

Part IPart I::GeneralGeneralRelativityRelativityTheoryTheory

• Newtonian Cosmology, Olbers’ paradox• General Relativistic Cosmology, an evolutionary paradigm• The Cosmological Principle• Friedmann equations• Big Bang Cosmology: parameters of the Universe• The Accelerating Universe• Outlook and future talks

Part IIPart II::GeneralGeneralRelativisticRelativisticCosmologyCosmology


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Cosmology: Preamble

Cosmology is concerned with the entire Universe –a big problemindeed, and a very old one, too...

The scientific approach to Cosmology is however much younger,about 300 years only.

Newtonian Cosmology faces a number of insurmountable problems:

• Space must be infinite: what would be there beyond its limitsotherwise?

• Likewise, time must be infinite: the Universe is thus eternal.• As a consequence, the gravitational potential is also infinite.

This creates a problem of stability versus infinitely long life.• Olbers’s paradox: the sky is dark at night...

General Relativistic Cosmology fixes these issues because itattributes dynamic properties to space and time.


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Contemporary Cosmology

Modern Cosmology –so called Standard, or Big Bang model– isbased upon a few major observational facts:

• The Universe is very highly isotropic –at large scales.• The Universe is expanding at the present epoch.• The Universe is filled with thermal radiation of ~2.7o Kelvin.• Abundance of light elements: H—73%, He—26%.

The Standard Model rests on two fundamental hypotheses:

• General Relativity theory• Cosmological Principle.

It also needs many other sources of scientific knowlwdge, such as HighEnergy Physics, Chemistry, Thermodynamics, Astrophysics, Statistics, etc.Cosmology is definitely a very pluri-disciplinar activity.

This talk addresses only the “GR connection”.


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The Cosmological Principle

The Cosmological Principle is the assumption that:

1. The Universe is large scale isotropic.2. Us, humans, are standard observers of the Universe

as regards its large scale properties.

Hence the Universe is: and⎧⎪⎨⎪⎩

isotropic

homogeneous


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Friedmann-Robertson-Walker

The CP has an immediate mathematical consequence as to which isthe large scale geometry of the Universe:

22 2 2 2 22 2 2

2 ( sin )1

( ) drds c dt rt dr

ak

dθ θ ϕ⎡ ⎤

= − + +⎢ ⎥−⎣ ⎦(FRW)

where a(t) is the scale factor of the Universe, and k is a trichotomicconstant:

101

k+⎧⎪= ⎨⎪−⎩

cl

hyperboli

osed

op

,

,

spherical

cenflat,open( ) 0a t > Universe expands

( ) 0a t < Universe shrinks


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The expanding Universe

Whether the Universe actuallyexpands or shrinks, i.e., whether

or( ) 0a t > ( ) 0a t <

is to be determined byobservation.

FRW geometry

It appears that, at the currentepoch, the Universe is in anexpansive phase.

This is determined by the fact that light from remote galaxies arrives red-shiftedto us.

Timeadvancesin thisdirection


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Cosmological red-shift

Cosmological redCosmological red--shiftshift

Here

Far away

Even farther

reception emission

emission

λ λλ

−=z , measure of redshift


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Cosmological red-shift

This was fisrst discovered in 1929 by Edwin Hubble, who interpretedredshift as an indication that galaxies move away from us.

Hubble also inferred that the expansion speedis proportional to distance:

0 ,v H d= Hubble’s law

where H0 is known as Hubble’s constant:

0 75 km / sec MpcH ⋅

In the light of FRW geometry, redshift is an evolutionary effect:

reception

emission

( )1

( )a ta t

= −z

Redshift is therefore a measure of age and size of the evolving Universe


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The geometry of space

k = +1

k = 0

k = −1


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Friedmann equations

The evolution of the Universe is described by Einstein’s equations:

4

1 82

g GR R g Tcµµν µν µννπ

− = −+ Λ

where Λ is the so called Cosmological Constant.

Tµν includes both ponderable matter and radiation in the Universe.Two magnitudes characterise their contribution: density and pressure.Expanding Einstein’s equations one finds Friedmann’s equations. Inthe present time, the Universe is matter-dominated, i.e., pressure isnegligible. The following obtains:

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2 2

83 3

a G kHa a

π ρ≡ = + +

Λ 2p cρ, Friedmann equationFriedmann equation


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Λ=0 Cosmological evolution

ρ0 > ρc

ρ0 = ρc

ρ0 < ρc


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Λ=0 Cosmological evolution

There is a critical density, which determines whether the Universeis spherical, hyperbolic or flat:

2 0 c0

c 0 c

0 c

1 if3 , 0 if8 1 if

kH kG k

ρ ρρ ρ ρ

π ρ ρ

= + >⎧⎪= = =⎨⎪ = − <⎩

flathyperboli

spher l

c

ica

But we do not know which of the three is true, as measuring ρ0 isa very difficult task. Indeed, under ρ0 go both visible and darkforms of matter...

To complicate things further, the Universe was recently seen tobe under accelerated expansion!!


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Parameters of the Universe

It is customary to define dimensionless parameters which characterisethe evolution of the Universe in terms of the critical density. They are:

0M 2

0

83

GHπ ρ

Ω =

matter

20

kk

HΩ =

Curvature

203HΛ

ΛΩ =

dark energy, or Λ

The Friedmann equation is then rewritten as:

( ) ( ) ( )3 22 20M 1 1kz z HzH Λ

⎡ ⎤= + + + +⎣ ⎦ΩΩ Ω

and gives rise to complicated casuistics, as we see next...


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Possible states of the Universe


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Composition of the Universe

We live in a Universe which is:

• Spatially flat• Therefore of infinite volume• In accelerated expansion• Dominated by dark energy• With only 4% of visible matter• 13 billion years old• Will expand forever

This model is certainly not free ofproblems... To discuss them is theobjective of this series of talks.


45LISA and the GW UniverseLISA and the GW Universe


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End of PresentationEnd of Presentation

espacio, tiempo y materia en el cosmos - csic · a lobo, 14-ii-2007 espacio, tiempo y materia en el...

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