1_what determines the nature of gravity_a phenomenological approach

8/13/2019 1_What Determines the Nature of Gravity_A Phenomenological Approach

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Space Sci Rev (2009) 148: 501522

DOI 10.1007/s11214-009-9559-9

What Determines the Nature of Gravity?

A Phenomenological Approach

Claus Lmmerzahl

Received: 27 March 2009 / Accepted: 24 June 2009 / Published online: 16 July 2009

Springer Science+Business Media B.V. 2009

Abstract The gravitational field can only be explored through the motion of test objects. To

achieve this one first has to set up the correct equations of motion. Initially these equations

are based on Newtons laws. Corresponding experiments that support Newtons laws are

described. Furthermore, the basic characteristics of the motion of test objects in gravitational

fields are described. This leads to the notion of Einsteins Equivalence Principle which has

as consequence a metric theory of gravity. One particular metric theory is General Relativity

based on Einsteins field equations with its particular predictions for effects like periastron

advance, light deflection, etc. An overview over the experimental confirmation of General

Relativity, in particular those presented at this workshop, is given. This workshop summary

ends with open problems. We also describe some of the strategies for the experimental search

for a quantum gravity theory.

Keywords General relativity Special relativity Newtons axioms Experimental

relativity Equivalence principle Solar system tests Quantum gravity phenomenology

1 Introduction

In this article we present a general frame of how to define and explore the nature of gravity as

well as the mathematical formalism and the equations that represent gravitational phenom-

ena. In principle there are two ways to state physical equations: The first waya topdown

schemeis to postulate the equations. For General Relativity (GR), for example, one may

postulate a Lorentzian spacetime manifold, the Einstein field equations and the geodesic

equations for point-like masses and light, as well as the observables of the theory. All ob-

servable consequences will follow from these statements. A second waycorresponding to

a bottomup approachis to base the physical laws on a few basic observations and to build

up the theory in a constructive manner. Here we like to proceed along the second way as faras possible.

C. Lmmerzahl ()

ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany

e-mail:[email protected]
mailto:[email protected]:[email protected]


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502 C. Lmmerzahl

The advantages of the second way are: (i) a physical understanding of mathematical

schemes, (ii) each mathematical structure is directly related to an observable phenomenon

and, thus, is immediately physically interpretable, and (iii) if generalizations are necessary

(like those expected from quantum gravity) natural generalizations are offered through this

way.In what follows we can only give a very rough, short and incomplete description of the

scheme. However, we hope to show all the ingredients and what in principle has to be taken

into account in order to reach a certain stage of complete mathematical description of the

nature of gravity. At many instances we refer to other contributions to this workshop which

expand several short remarks given here.

2 How to Explore Gravity

The gravitational field and its properties canonlybe explored through the observation of the

dynamics of test objects: point particles, light rays or quantum fields (Ehlers 2006). Their

dynamics is governed by equations of motion. The simplest such equations are Newtons

laws. Much more complicated laws are feasible: one may consider dynamical equations

with higher order time derivative (Lmmerzahl and Rademaker2009), for example.

After we setting the main structure of equations of motion for test objects we ask for

characteristic features of the interaction of these objects with the gravitational field. These

are governed by the Einstein Equivalence Principle. From that we conclude that gravity is

a metric theory. Each of these theories shows the typical effects like perihelion shift, light

deflection, time delay, LenseThirring and Schiff effects, etc. Only for certain values of

these effects gravity is described by the Einstein field equations. These field equations are

then extrapolated to the strong field regime and can be confronted with observations of

binary systems and black holes.

We always consider fundamental equations only. Effective equations for the motion of

test objects with largely different features may come out from complicated calculations that

take, for example, radiation reaction into account.

3 The Structure of Dynamics

The notion of an inertial system, of the inertial law and the law of reciprocal actions

(actio = reactio) is assumed in all equations of motion, either non-relativistic or relativistic.

Any test or exploration of the gravitational interaction has to account for the structure of the

equations of motion which are used to measure gravitational effects.

In many instances we have equations of motion which are more general than those related

to Newtons axioms. Examples are the equations taking into account radiation reaction, or

dynamics with memory. However, these are effective equations of motion. In our bottom

up approach we are interested in the fundamental equations of motion only.

3.1 Existence of Inertial Frames

A condition for the existence of inertial frames is content of the first of Newtons laws. In an

intuitive sense an inertial frame is a local reference frame where all freely falling particles


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What Determines the Nature of Gravity? A Phenomenological Approach 503

move uniformly along straight lines. Here we already have to put in an intuitive understand-

ing of free motion. Though it is not clear of how to characterize a force-free motion (non-

gravitational forces) uniquely in an experimental way, Finsler spacetime (Lmmerzahl and

Perlick2009) provides a model for the non-existence of inertial frames.1

An indefinite Finslerian geometry is given by the line element

ds2 = F(x,dx) with F(x,dx) = 2F(x,dx), (1)

for all R and whereFis a function homogeneous of degree two. Then

ds2 = g (x,dx)dxdx with g (x,y) =

2F(x,y)

y y , (2)

where g (x,dx) is a Finslerian metric which, however, depends on the vector it is acting

on. The motion of light rays and point particles is given by the action principle 0 =

ds2

and leads to the equation of motion

d2x

ds2 =

(x,x)

dx

ds

dx

ds. (3)

Since the FinslerChristoffel connection

(x, x) =1

2g (x,x) g (x,x) + g (x,x) g(x,x) (4)

(hereg (x,x) is the inverse ofg (x,x) defined throughg (x,x)g(x,x) =

) depends

on the velocity x, it cannot be transformed away. As a consequence, there is no frame in

which all particles move uniformly along straight lines. We always have accelerated parti-

cles. This provides a model for that gravity cannot be transformed away and, thus, for the

non-existence of inertial systems. This is true for all equations of motion with a non-linear

connection of the form (3). Another consequence of a Finslerian metric is the anisotropy of,

e.g., light propagation violating Lorentz invariance (see below and Lmmerzahl et al.2009).

Free fall experiments and orbits of planets and satellites yield that the order of magnitude

of any hypothetical Finslerian deviation from ordinary Riemannian spacetimes should be

smaller than 109 m/s2 (Lmmerzahl and Perlick2009).

3.2 The Inertial Law

The inertial law

p = mx= F (5)

is characterized by its order of differentiation and the linear relation between force and

acceleration. We highlight both properties. Any change in these characteristic features ofthe inertial law dramatically influences the interpretation of, e.g., orbits of satellites, planets

or stars.

1We leave out BerwaldFinsler spacetimes where the spacetime metric depends on the connecting vector

while the equation of motion still is the ordinary Riemannian geodesics.


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3.2.1 Order of Equations of Motion

Newtons second law (5) withp = miv wheremi is the inertial mass implies an equation of

motion of second order for the position. The relation between momentum and velocity has

the structure of a constitutive law and, thus, may be generalized to p= p(m, v, v, v, . . . ).This implies higher-order equations of motion. Higher-order equations of motion may also

come from metrical fluctuations with a certain time correlation. Here we are concerned only

with fundamental equations: In the contrary effective equations of motion in general contain

higher order derivatives from radiation reaction.

The most simple model for a higher order dynamics is based on a second order La-

grangian L = L(t,x,x,x). This gives an equation of motion of fourth order. With a small

additional term of appropriate sign (see Lmmerzahl and Rademaker2009for more details)

the solution of this fourth order equation gives the standard solution of the usual second

order equation together with a kind of zitterbewegung. Therefore, the usual second order

equations of motion seem to be rather robust against small higher-order additions. In orderto be consistent we introduce interactions with external fields through a gauge principle.

Such higher order gauge principles result in novel gauge fields.

As we expect from this approach only a small zitterbewegung, an experimental detection

of such a phenomenon is rather difficult. One potentially feasible idea is to look for funda-

mental noise in electronic devices with characteristics differing from the standard Nyquist

or 1/fnoise. Corresponding proposals will be worked out.

3.2.2 Linearity

Newtons inertial law (5) is a definition of the force F. Measuring the path of a test object

and knowing its characteristic parameters determines the force.

Theories modifying this relation by introducing a function f (a) on the left hand side,

mf(a)= F(x), as MOND does, (Milgrom 2002), are equivalent to a theory of modified

gravity provided the functionf possesses an inverse. Thenma= mf1(F(x)/m), and this

is a mere redefinition of the force equivalent to a modification of the gravitational influence.

This modified Newtonian dynamics or modified gravity is rather successful in modeling

galactic rotations curves. The function f(a)is mainly determined by a characteristic accel-

eration scalea0 of the order 1010 m s2.

Though the inertial law defines the force, there is one aspect which may be subject toexperimental proof: If the force acting on a body is given by a gravitating mass, F= mU

with U = G

(x )/|x x |dV (G is Newtons graviational constant and the mass

density), then one may ask the question whether the acceleration decreases linearly with

decreasing gravitating mass which can be measured through its weight. If the gravitating

mass M is spherically symmetric, U =GM/r , then the question is whether x x for

M M, in particular in the case of small M. This is an operationally well defined ques-

tion which is worth to be explored experimentally.

A recent laboratory experiment performed tests of the linearity between force and accel-

eration in the extremely weak force regime, (Gundlach et al.2007). No deviation from New-

tons inertial law has been found for accelerations down to 5 1014 m s2. This experiment,

however, does not test MOND. Within MOND it is required that the full acceleration has

to be smaller than approx 1010 m s2 while in the above experiment only two components

of the acceleration were small while the acceleration due to the Earth attraction was still

present. Therefore such tests of MOND have to be performed in space (for the constraints of

tests on Earth, see Ignatiev2007). An earlier test (Abramovici and Vager1986) went down


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to accelerations of 3 1011 m s2. In both cases the applied force was nongravitational. It

might be speculated whether the MOND ansatz applies to all forces or to the gravitational

force only.

It has also been speculated whether the MOND ansatz can describe the Pioneer anomaly

(Milgrom2002; Anderson et al.2002) but this has not been convincingly confirmed. In anycase, it is a very remarkable coincidence that the Pioneer acceleration, the MOND charac-

teristic accelerationa0 as well as the cosmological acceleration are all of the same order of

magnitude,aPioneer a0 cH, whereHis the Hubble constant.

3.3 Law of Reciprocal Action

A key model for the violation of the law of reciprocal action is a difference in active and

passive gravitational masses. The notion of active and passive masses and their possible non-

equality has first been introduced and discussed by (Bondi 1957). Theactive massmais the

source of the gravitational field (here we restrict to the Newtonian case with the gravitationalpotentialU)U= 4 ma(x), whereas thepassive massmp reacts to it

mix= mpU (x). (6)

Here, mi is the inertial mass and x the position of the particle. The equations of motion for

a gravitationally bound twobody system then are

m1ix1= Gm1pm2ax2 x1

|x2 x1|3, m2ix2= Gm2pm1a

x1 x2

|x1 x2|3, (7)

where 1, 2 refer to the two particles and G is the gravitational constant. For the equation of

motion of the center of mass X , we find

X = Gm1pm2p

MiC21

x

|x|3 with C21=

m2a

m2p

m1a

m1p(8)

whereMi= m1i + m2iand x is the relative coordinate. Thus, ifC21= 0 then active and pas-

sive masses are different and the center of mass shows a self-acceleration along the direction

ofx . This is a violation of Newtonsactioequalsreactio. A limit has been derived by Lunar

Laser Ranging (LLR): no self-acceleration of the moon has been observed yielding a limit

of| CAlFe| 7 1013 (Bartlett and van Buren1986).

The dynamics of the relative coordinate

x= Gm1pm2p

m1im2i

m1

m1a

m1p+ m2

m2a

m2p

x

|x|3 (9)

has been probed in the laboratory by (Kreuzer1968) with the result |C21| 5 105.

Similar considerations have been made for active and passive charges or for magnetic

moments (Lmmerzahl et al.2007a).

4 The Structure of Gravity

After having set up the fundamental equations of motion for test objects one can start to

explore the structure of interactions. The gravitational interaction is first characterized by a

number ofuniversality principlesput together in the Einstein Equivalence Principle (EEP).


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506 C. Lmmerzahl

It consists of (i) the Universality of Free Fall (UFF), (ii) the Universality of the Gravitational

Redshift (UGR), and (iii) Local Lorentz Invariance (LLI), see Will (1993). These principles

further constrain the structure of the equations of motion of test objects: The EEP leaves

only freedom for a symmetric second rank tensor field to couple to the equation of motion

of test objects. As a consequence one arrives at a metric theory of gravity.Any metric theory of gravity shows the standard Solar system effects like perihelion

shift, light deflection, gravitational time delay, and the LenseThirring and Schiff effect.

Only for GR given by the Einstein field equations these effects attain certain values. There

is no constructive way to derive these Einstein equations. However, the Parametrized Post

Newtonian formalism (PPN), Will (1993), with approximate 10 undetermined parameters

(the number of parameters depends on the chosen version) provides a very powerful method

of parameterizing deviations from GR. For GR these parameters attain certain values. The

PPN formalism gives a theoretical frame within which by means of a finite number of ob-

servations and experiments it is possible to single out GR from other theories of gravity.

In the following two sections we describe the experiments and observations first leadingto a metric theory of gravity and second singling out GR from all other metric theories (see

also the contribution of C. Will).

5 The Foundations of Metric Gravity

5.1 Universality of Free Fall

The UFF states that all neutral pointlike particles move in a gravitational field in the sameway: The path of these bodies is independent of the composition of the body. The corre-

sponding tests are described in terms of the acceleration of these particles in the reference

frame of the gravitating body: the Etvs factor compares the normalized accelerations of

two bodies = a2 a112

(a2+a1)in the same gravitational field. In the frame of Newtons theory this

can be expressed as = 2 112

(2+1), where =mg/mi is the ratio of the (passive) gravita-

tional and inertial mass.

There are two principal schemes to perform tests of UFF. The first scheme uses the free

fall of bodies. In this case the full gravitational attraction towards the Earth can be exploited.

However, these experiments suffer from the fact that the time-of-flight is limited to roughly1 s and that a repetition needs new adjustment. The other scheme uses a restricted motion

confined to one dimension only, namely a pendulum or a torsion balance. The big advan-

tage is the periodicity of the motion which by far outweights the disadvantage that only a

fraction of the gravitational attraction is used. In fact, the best test today of the UFF uses

a torsion pendulum and confirms it to the order of 2 1013. Altogether we then have the

amazing equality mi= mg= ma= mp. New proposed tests in space, the approved mission

MICROSCOPE, (Touboul2001and the contribution of P. Touboul), and the proposal STEP,

(Lockerbie et al. 2001), will combine the advantages of free fall and periodicity (see also

the contributions by J. Mester and T. Sumner).

There are hints from quantum gravity inspired scenarios that the UFF might be violated

below the 1013 level, Damour et al. (2002a,2002b) and the contribution of T. Damour. Also

from cosmology with a dynamical vacuum energy (quintessence) one can derive a violation

of UFF at the 1014 level (Wetterich2003). The validity of the UFF has also been used for

setting bounds on the time variability of various constants such as the fine structure constant

and the electron-to-proton mass ratio (Dent2006).


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According to GR, spinning particles couple to the spacetime curvature (Hehl 1971;

Audretsch1981) and, thus, violate the UFF. However, the effect is far beyond any experi-

mental detectability. Therefore testing the UFF for spinning matter amounts to a search for

an anomalous coupling of spin to gravity. Motivations for anomalous spin couplings came

from the search for the axion, a candidate for the dark matter in the universe which also canresolve the strong PC puzzle in Quantum Chromodynamics (Moody and Wilczek1984). In

these models spin may couple to the gradient of the gravitational potential or to gravitational

fields generated by the spin of the gravitating body. The first case can easily be tested by

weighting polarized bodies what showed that for polarized matter the UFF is valid up to the

order of 108 (Hsieh et al.1989).

Also charged particles do couple to the spacetime curvature (DeWitt and Brehme1960)

but this effect is again too small to be detectable. Further, it is possible to introduce a charge-

dependent violation of the UFF by assuming a charge-dependent anomalous inertial and/or

gravitational mass. It is also possible to choose the model such that for a neutral atom UFF

is fulfilled exactly while it is violated for isolated charges (Dittus et al. 2004). It has beensuggested to carry out a corresponding experiment in space (Dittus et al. 2004).

5.2 Universality of Gravitational Redshift

A test of the universal influence of the gravitational field on clocks based on different

physical principles requires clock comparison during their common transport through dif-

ferent gravitational potentials. There is a large variety of clocks which can be compared:

(i) light clocks (optical resonators), (ii) various atomic clocks, (iii) various molecular clocks,

(iv) gravitational clocks based on the revolution of planets or binary systems, (v) the rotationof the Earth, (vi) pulsar clocks based on the spin of stars, and (vii) clocks based on particle

decay.

On a phenomenological level the comparison of two collocated clocks is given by

clock1(x1)

clock2(x1)=

1 (clock2 clock1)

U (x1) U (x0)

c2

clock1(x0)

clock2(x0)(10)

where clocki are clock-dependent parameters. If this frequency ratio does not depend on

the gravitational potential then the gravitational redshift is universal. This is a null-test of

clock 2 clock 1. It is obviously preferable to use large differences in the gravitational poten-tial which clearly shows the need for space experiments. In experiments today the variation

of the gravitational field is induced by the motion of the Earth around the Sun.

The best test up to date has been performed by comparing the frequency ratio of the 282

nm 199Hg+ optical clock transition to the ground state hyperfine splitting in 133Cs over 6

years. The result is |Hg Cs| 5 106 (Ashby et al. 2007; Fortier et al. 2007). Other

tests compare Cs clocks with the hydrogen maser, Cs or electronic transitions in I2 with op-

tical resonators. We are looking forward to ultrastable clocks on the ISS and on satellites in

Earth orbit or even in deep space as proposed by SPACETIME (Maleki2001), OPTIS (Lm-

merzahl et al.2004) and SAGAS (Wolf et al.2008), which should considerably improve the

scientific results (see also the contribution by S. Reynaud).

So far there are no tests using anti clocks, that is, clocks made of anti-matter. However,

since the production of anti-hydrogen is a working technique today, there are attempts to

perform high-precision spectroscopy of anti-hydrogen. These measurements first should test

special relativistic CPT invariance but, as a long-term goal, they could also be used to test

the Universality of the Gravitational Redshift for a clock based on anti-hydrogen.


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In many scenarios it is assumed that constants vary with (cosmological) time. Since dif-

ferent atomic or molecular states depend differently on these constants the question of th

constancy of constants is related to the UGR, cf. the contributions by J.-P. Uzan, N. Ko-

lachevsky, P. Petitjean, and E. Fischbach.

5.3 Local Lorentz Invariance

Lorentz invariance, the symmetry of Special Relativity (SR) which also holds locally in

GR, is based on the constancy of the speed of light and the relativity principle. For a recent

review, see Amelino-Camelia et al. (2005).

5.3.1 The Constancy of the Speed of Light

The constancy of the speed of light has many aspects:

1. The speed of light does not depend on the velocity of the source. Using the model

c = c + v, where v is the velocity of the source and some parameter one gets from

astrophysical observations 1011 (Brecher1977).

2. The speed of light does not depend on the frequency and polarization. The best results

come from astrophysics. From radiation at frequencies 7.1 1018 Hz and 4.8 1019 Hz of

Gamma Ray Burst GRB930229 one obtains c/c 6.3 1021 (Schaefer1999). Analy-

sis of the polarization of light from distant galaxies yielded an estimate c/c 1032

(Kostelecky and Mewes2002).

3. The speed of light is universal. This means that the velocity of all other massless parti-

cles as well as the limiting maximum velocity of all massive particles coincides withc. The maximum speed of electrons, neutrinos and muons has been shown in vari-

ous laboratory experiments to coincide with the velocity of light at a level (cparticle

c)/c 106 (Brown et al. 1973; Guiragossian et al. 1975; Alspector et al. 1976;

Kalbfleisch et al. 1979). Astrophysical observations of radiation from the supernova

SN1987A yield for the comparison of photons and neutrinos an estimate which is two

orders of magnitude better (Stodolsky1988; Longo1987).

4. The speed of light does not depend on the velocity of the laboratory. This can be tested in

KennedyThorndike experiments which is a clockclock comparison experiment where

the laboratory moves with varying speed (e.g. a laboratory on the surface of the Earth

moves with a velocity consisting of the rotation around its own axis and its revolution

around the Sun). The two clocks can be either two light clocks (different resonators or a

Michelson-type interferometer with different arm lengths) or a light clock and an atomic

clock. The best comparison yieldsc/c 1016 (Mller et al.2007).

5. The speed of light depends not on the direction of propagation. This has been confirmed

by modern MichelsonMorley experiments using optical resonators to a relative accuracy

ofc/c 1016 (Mller et al.2007).

6. A bit more involved is the combination of a finite velocity of signal propagation with

quantum systems and quantum measurements involving entanglement (spooky action

at a distance). Though quantum systems may be entangled over long distances and ameasurement of one part of the system has some influence on the properties of the other

part of the quantum system it is not possible to communicate with velocities larger than

the velocity of light.

This altogether means that the velocity of light is a universal structure and, thus, can be

interpreted as part of a spacetime geometry.


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5.3.2 The Relativity Principle

The relativity principle states that the outcome of all experiments when performed iden-

tically within a laboratory without reference to the external word, is independent of the

orientation and the velocity of the laboratory. For the photon sector this can be tested withMichelsonMorley and KennedyThorndike type experiments already discussed above. Re-

garding the matter sector the corresponding tests are HughesDrever type experiments. In

general, these are nuclear or electronic spectroscopy experiments. Such effects can be mod-

eled by an anomalous inertial mass tensor (Haugan1979) of the corresponding particle. For

nuclei one then gets estimates of the order m/m 1030 (Chupp et al. 1989). Also an

anomalous coupling of the spin to some given cosmological vector or tensor fields destroys

the Lorentz invariance. All anomalous spin couplings are absent to the order of 10 31 GeV,

see Walsworth (2006) for a review. Also higher-order derivatives in the Dirac and Maxwell

equations in general lead to anisotropy effects (Lorek and Lmmerzahl 2008).

A further aspect of anisotropy is that there might be some anisotropies in the Coulombor Newtonian potential (Kostelecky and Mewes 2002; Kostelecky2004). Anisotropies in

the Coulomb potential may affect the length of, e.g., optical cavities which may influence

the frequency of light in the cavity. However, it has been shown that the influence of the

anisotropies of the Coulomb potential are smaller than the corresponding anisotropies in

the velocity of light (Mller et al. 2003). Anisotropies in the Newtonian potential of the

Earth has recently been looked for by means of atomic interferometry; these measurements

constrain the anisotropies to the 108 level (Mller et al.2008).

Future spectroscopy of anti-hydrogen may yield further information about the validity of

the PCT symmetry.

5.4 The Consequence

The consequence of the validity of the EEP is that gravity is described by a Riemannian

metricg , a symmetric second rank tensor defined on a differentiable manifold being the

collection of all possible physical events. The purpose of this metric is twofold: First, it

governs the rate of clocks, that is,

s= ds, ds= g dxdx (11)is the time shown by clocks where the integration is along the worldline of these clocks.

Second, the metric gives the equation of motion for massive point particles as well as for

light rays,

0 =d2x

ds2 +

d xds

dx

ds(12)

where{

} = 12

g ( g+ g g)is the Christoffel symbol. Herex = x(s)is the

worldline of the particle parametrized by its proper time. It can be shown that the metric

also describes the propagation of, e.g, the spin vector, Dv

S= 0, whereSis a particle spin.

6 Motivating Einsteins Field Equations

There is no derivation of Einsteins field equations from a few key observations. However,

a PPN formalism (Will 1993), makes it possible to parametrize in terms of ten or more


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510 C. Lmmerzahl

parameters deviations of the metric from a metric following from Einsteins field equations

R1

2Rg= T (13)

where R andR are the Ricci tensor and scalar, respectively, T is the energy-momentumtensor of the matter creating the gravitational field, and the relativistic coupling constant.

In the case of the validity of Einsteins field equations these parameters take specific values.

As a consequence the precise measurement of the effects described in the next section will

also give a justification of the validity of Einsteins field equations.

While all this is going in the weakfield and low-velocity regime, one extrapolates the

field equation (13) to the strong field and large velocity regime. This extrapolation then can

be examined by observations of binary systems and effects near black holes where higher

order terms are needed for their correct description (Blanchet 2006).

One largely discussed generic deviation from GR is a modification of the Newtonian

1/r potential. Such deviations described by V (r) = Mr

(1 + er/) are parametrized by the

strength and range . Various experiments yield estimates for for a given range . The

high precision of LLR and ephemerides give very tight restrictions on for interplanetary

ranges. Higher-dimensional models predict deviations from the 1/r potential at short dis-

tances which motivated big experimental efforts in that direction, see e.g. the contribution

by R. Newman.

7 Proving Consequences of General Relativity

Gravity can be explored only through its action on test particles (or test fields). Accordingly

the gravitational interaction has been studied through the motion of stars, planets, satellites

and of light. There are only very few experiments which demonstrate the effects of gravity

on quantum fields.

There are two classes of tests: Weak gravity effects, mostly observed within the Solar

system, and strong gravity effects present in binary systems and near black holes.

7.1 Solar System Effects

For the calculation of the effects to be described one needs a solution of Einsteins fieldequations or an approximate solution in the frame of the PPN formalism.

7.1.1 The Gravitational Redshift

In a stationary gravitational field the gravitational redshift between two positions with radial

coordinatesr1 andr2 is given by

2

1

= gt t(r1)

gt t(r2)

1 GM

c

2 1

r1

1

r2, (14)

where r1 and r2 are the radial positions of the two observers. The right hand side of the

equation comes out if we assume the validity of Einstein theory of gravity. This effect has

best been observed in a space experiment where the time of a hydrogen maser in a rocket

has been compared with the time of an identical hydrogen maser on ground yielding a con-

formation of GR at the level of 1 part in 104 (Vessot et al.1980).


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7.1.2 Light Deflection

The deflection of light was the first prediction of Einsteins GR; it has been confirmed by ob-

servation four years after the theory has been completed. In the frame of the PPN formalism

we obtain

=1

2(1 + )

M

b, (15)

where Mis the mass of the Sun and b the impact parameter. Todays observations use Very

Long Baseline Interferometry (VLBI), and this has led to | 1| 104 (Shapiro et al.

2004).

7.1.3 Perihelion/Periastron Shift

Within the PPN formalism we obtain the perihelion shift

=1

3(2 + 2 )

6 M

a2(1 e2), (16)

wherea is the semimajor axis ande the eccentricity of the orbit. Today this post-Newtonian

perihelion shift has been determined as 4298 per century with an error of the order 104

(Pitjeva2005). Recently a huge periastron shift of a candidate binary black hole in the quasar

OJ287 has been observed where one black hole is small compared to the other (Valtonen

et al.2008). The observed perihelion shift is approximately 39 per revolution, which takes

12 years.

7.1.4 Gravitational Time Delay

In the vicinity of masses, electromagnetic signals move slower than in empty space, when

compared in a coordinate system attached to spatial infinity. This is the gravitational time

delay. There are two ways to confirm this effect: (i) direct observation, that is, by comparing

the time of flight of light signals in two situations for fixed sender and receiver, and (ii) by

observing the change in the frequency induced by this gravitational time delay.

Direct measurement The gravitational time delay for signals which pass through the vicin-

ity of a body of massMis given by

t= 2(1 + )GM

c3 ln

4xSatxEarth

b2 , (17)

where xSat andxEarth are the distances of the satellite and the Earth, respectively, from the

gravitating mass. If the gravitating body is the Sun and if we take b to be the radius of

the Sun then the effect would be of the order 104 s which is clearly measurable. This has

been measured using Mars ranging data of the Viking Mars mission giving | 1| 104

(Reasenberg et al.1979).

Measurement of frequency change Though the time delay is comparatively small, the in-

duced modification of the received frequency can indeed be measured with higher precision.

The reason is that clocks are very precise and, thus, can resolve frequencies also very pre-

cisely.


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512 C. Lmmerzahl

The corresponding change in the frequency is

y(t) = 0

0= 2(1 + )

GM

c31

b(t)

db(t)

dt, (18)

where 0 is the emitted frequency. It is the time dependence of the impact parameter whichis responsible for the effect. This effect has been measured by the Cassini mission. One im-

portant issue in the actual measurement was that three different wavelengths for the signals

have been used. This made it possible to eliminate dispersion effects near the Sun and to

verify with this time delay GR with an accuracy of| 1| 2.5 105 (Bertotti et al.2003).

7.1.5 LenseThirring Effect

For the Einstein field equation as well as within the PPN formalism a rotating gravitating

body gives metric componentsJidt dx

i

, where J is the angular momentum of the rotatingbody. On the level of the equations of motion this results in a Lorentz type gravitational

force acting on bodies called gravitomagnetism, see also the contribution of G. Schfer. The

influence of this field on the trajectory of satellites results in a motion of the nodes, which

has been measured by observing the LAGEOS satellites via laser ranging. Together with

new data of the Earths gravitational field obtained from the CHAMP and GRACE satellites

the confirmation recently reached the 10% level (Ciufolini2004, see also the contribution of

I. Ciufolini for the LAGEOS results and the contribution of L. Stella for the LenseThirring

effect in astrophysics). In the meantime the LARES mission has been approved. This is

another satellite of the same tye as LAGEOS which orbit will have a different inclination

than LAGEOS. This makes is possible to eliminate multipole moments of the Earth fromthe joint LAGEOS and LARES data. The launch is scheduled for early 2011.

This gravitomagnetic field also influences the proper time and, thus, the rate of clocks.

It can be shown that the difference of the proper time of two counterpropagating clocks is

s+ s= 4J/M. It should be remarked that this quantity does not depend on G andr .

This effect for clocks in satellites orbiting the Earth can be as large as 10 7 s per revolution

(Mashhoon et al.2001).

7.1.6 Schiff Effect

The gravitational field of a rotating gravitating body also influences the rotation of gyro-

scopes. This effect is right now under consideration by the data analysis group of the GP-B

mission flown in 2004. Data analysis is expected to be completed early 2010. Though the

mission met all requirements and, thus, was a big technological success it turned out after

the mission that contrary to all expectations and requirements the gyroscopes lost more en-

ergy than calculated. This requires the determination of further constants characterizing this

spinning down effect which effects the overall accuracy of the measurement of the Schiff ef-

fect which was expected to be of the order of 0.5%. Nevertheless, recent reports of the GP-B

data analysis group indicate that finally the error may go down to 1% (see the contribution

of F. Everitt and the GP-B team). For updates of the data analysis one may contact GP-Bswebsite.2

It should be noted that though both effects within GR are related to the gravitomagnetic

field of a rotating gravitational source, the LenseThirring effect and the Schiff effect are

2Seehttp://einstein.stanford.edu/.
http://einstein.stanford.edu/http://einstein.stanford.edu/


13/22


conceptually different and measure different quantities and, thus, should be regarded as in-

dependent tests of GR. In a generalized theory of gravity spinning objects may couple to

different gravitational fields (like torsion) than the trajectory of orbiting satellites. Further-

more, the LenseThirring effect is a global effect related to the whole orbit while the Schiff

effect observes the Fermi-propagation, a characterization of a torque-free dynamics, of thespin of the gyroscope.

7.1.7 The Strong Equivalence Principle

The gravitational field of a body contains energy which adds to the rest mass of the grav-

itating body. The strong equivalence principle now states that EEP is valid also for self

gravitating systems, that is, that UFF is valid for the gravitational energy, too. This has been

confirmed by LLR with an accuracy of 103 (Will1993), where the validity of UFF had to

be assumed. The latter has been tested separately for artificial bodies of a composition simi-

lar to that of the Earth and the Moon yielding a confirmation with an accuracy of 1.4 1013

(Baeler et al.1999).

7.2 Strong Gravity Effects

While most of the observations and tests of gravity are being performed in weak fields: Solar

system tests, galaxies, galaxy clusters, recently it became possible to observe phenomena in

strong gravitational fields: in binary systems and in the vicinity of black holes.

The observation of stars in the vicinity of black holes (Schoedel et al. 2007) may in one

or two decades give new improved measurements of the perihelion shift or of the Lense

Thirring effect. Binary systems present an even better laboratory for observing strong field

effects. See, e.g., the binary black hole candidate observed by (Valtonen et al.2008).

The inspiral of binary systems which has been observed with very high precision can be

completely explained by the loss of energy through the radiation of gravitational waves as

calculated within GR (Blanchet2006). The various data from such systems can be used to

constrain hypothetical deviations from GR. As an example, it can be used for a test of the

strong equivalence principle (Damour and Schfer1991) and of preferred frame effects and

conservation laws Bell and Damour (1996) in the strong field regime.

Recently, double pulsars have been detected and studied. These binary systems offer the

new possibility to analyze spin effects and, thus, open up a new domain of exploration ofgravity in the strong field regime (Kramer et al.2006a,2006b). Accordingly, the dynamics

of spinning binary objects has been intensively analyzed recently (Faye et al.2006; Blanchet

et al.2006; Steinhoff et al.2008).

A consequence of strong gravity is the emission of gravitational waves. At present ground

experiments are reaching their projected sensitivity and collect data. The space mission

LISA is sensitive to a lower frequency range more adapted to the long inspiral period of

binary systems and is a cornerstone mission of ESA/NASA. LISA is presently prepared

through the technologytesting LISAPathfinder mission (see the contribution of S. Vitale).

8 Open ProblemsUnexplained Observations

There are several observations which have not yet found a convincing explanation. In most

cases there is no doubt concerning the data. The main problem is the interpretation of the

observations and measurements.


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514 C. Lmmerzahl

8.1 Dark Matter

Dark matter is needed to describe the motion of galaxy clusters, as has been first spec-

ulated by F. Zwicky (1933), and for stars in galaxies, and has been also confirmed with

gravitational lensing, see e.g. Sumner (2002). Also structure formation needs this darkmatter. However, until now there is no single observational hint at particles which could

make up this dark matter. As a consequence, there are attempts to describe the same ef-

fects by a modification of the gravitational field equations, e.g., of Yukawa form (Sanders

1984), or nonlocal gravity (Hehl and Mashhoon 2008), or by a modification of the dy-

namics of particles, like the MOND ansatz (Milgrom 2002; Sanders and McGough2002),

recently formulated in a relativistic frame (Bekenstein 2004). Due to the lack of direct

detection of Dark Matter particles, all those attempts are on the same footing. There

are suggestions that at least a considerable part of the observations which usually are

explained by dark matter can be related to a stronger gravitational field which comeout while taking the full Einstein equations into account (Cooperstock and Tieu 2005;

Balasin and Grumiller2006).

8.2 Dark Energy

Observations of type Ia supernovae, (Riess et al. 1998; Perlmutter et al. 1999), WMAP

measurements of the cosmic microwave background (Spergel et al.2007), the galaxy power

spectrum and the Lymanalpha forest data lines (van de Bruck and Priester 1998; Overduin

and Priester2001; Tegmark et al.2004), indicate an accelerating expansion of the universeand that 75% of the total energy density consist of a dark energy component with negative

pressure (Peebles and Ratra2003).

Buchert and Ehlers (1997) have shown first in a Newtonian framework that within a

spatial averaging of matter and the gravitational field, rotation and shear of matter can

influence the properties of the averaged gravitational field which are described in effec-

tive Friedman equations. This also holds in the relativistic case (Buchert 2008). Therefore

it is an open question whether dark energy is just a result of a correct averaging proce-

dure. An influence of the averaging has been found in existing data (Li and Schwarz 2007;

Li et al.2008). These topics are illuminated in more detail in the contributions by Zakharov,Lasenby, Caldwell, and Goobar.

8.3 Pioneer Anomaly

The Pioneer anomaly, an unexplained anomalous acceleration of the Pioneer 10 and 11

spacecraft ofaPioneer= (8.74 1.33) 1010 m/s2 toward the Sun, is discussed in Anderson

et al. (2002) and the contribution of S. Turyshev. This acceleration seemed to turn on after

the last flyby at Jupiter and Saturn and stayed constant within a 3% range. Until now no con-

vincing explanation has been found. An anisotropy of the thermal radiation might explain

the acceleration. However, while the power provided by the plutonium decay decreases ex-

ponentially, the acceleration stays constant. Nevertheless, further work on a good thermal

modeling of the spacecraft is going on at ZARM (Rievers et al.2008). Moreover, an analysis

of the early tracking data is on the way. Improvements of ephemerides also helps to rule out

various suggested explanations and theories (Standish2008).


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8.4 Flyby Anomaly

It has been observed at various occasions that satellites having been subjected to an Earth

swing-by possess a significant unexplained velocity increase by a few mm/s. This unex-

pected and still unexplained velocity increase is called theflyby anomaly

. For a summary ofrecent investigations of this phenomenon, see Lmmerzahl et al. (2007b). Anderson et al.

(2008) have proposed the heuristic formula

v = vR

c2 (cos in cos out) (19)

which describes all flybys. Here Rand are the radius and the angular velocity, respectively,

of the Earth, and in andout are the inclinations of the incoming and outgoing trajectory.

However, the recent observation of a Rosetta flyby could not verify this empirical formula. 3

Until now no explanation has been found but, currently, it is expected that it is a mismod-

eling of either (i) the thermal influence of the Earth and the Suns radiation on the satellite,(ii) of reference systems (this is supported by the fact that all the flybys can be modeled

by (19), which contains geometrical terms only), (iii) of the flyby since this takes place at an

accelerated body, or (vi) of the satellites body being described by a point mass. There was

an ISSI workshop on this topic in March 2009 (cf. footnote 3).

8.5 Increase of Astronomical Unit

From the analysis of radiometric measurements of distances between the Earth and the ma-

jor planets including observations from Martian orbiters and landers from 1961 to 2003

a secular increase of the Astronomical Unit of approximately 10 m per century has been

reported (Krasinsky and Brumberg2004) (see also the article Standish2005and the discus-

sion therein). This increase cannot be explained by a time-dependent gravitational constant

Gbecause the G/G that would be needed is larger than the restrictions obtained from LLR.

Such an increase might be mimicked, e.g., by a long-term increase of the density of the Sun

plasma.

8.6 Quadrupole and Octupole Anomaly

Recently an anomalous behavior of the low-l contributions to the cosmic microwave back-ground has been reported. It has been shown that (i) there exists an alignment between the

quadrupole and octupole with >99.87% C.L. (de Oliveira-Costa et al. 2005), and (ii) that

the quadrupole and octupole are aligned to Solar system ecliptic to >99% C.L. (Schwarz

et al.2004). No correlation with the galactic plane has been found.

The reason for this is totally unclear. One may speculate that an unknown gravitational

field within the Solar system slightly redirects the incoming cosmic microwave radiation

(in the similar way as a motion with a certain velocity with respect to the rest frame of the

cosmological background redirects the cosmic background radiation and leads to modifica-

tions of the dipole and quadrupole parts). Such a redirection should be more pronounced for

low-l components of the radiation. It should be possible to calculate the gravitational fieldneeded for such a redirection and then to compare that with the observational data of the

Solar system and the other observed anomalies.

3Team meeting Investigation of the flyby anomaly, ISSI, Bern, March 26, 2009; http://www.issibern.ch/

teams/investflyby/.
http://www.issibern.ch/teams/investflyby/http://www.issibern.ch/teams/investflyby/http://www.issibern.ch/teams/investflyby/http://www.issibern.ch/teams/investflyby/


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516 C. Lmmerzahl

9 The Search for Signals of Quantum Gravity

There are many experiments proving that matter has to be quantized and, in fact, all ex-

periments in the quantum domain are in full agreement with quantum theory with all its

somehow strange postulates and consequences. Consistency of the theory also requires thatthe fields to which quantized matter field couple has to be quantized, too. Therefore, also

the gravitational interaction has to be quantized. In particular, there is no meaning of the

Einstein equation if the right hand side consists of quantized matter while the left hand side

is purely classical. Also the semiclassical Einstein equation with an expectation value on the

right hand side has been shown to lead to unwanted effects like faster than light propagation.

However, though gravity is an interaction between particles it also deforms the underlying

geometry. This double-role of gravity seems to prevent all quantization schemes from being

successful in the gravitational domain.

The incompatibility of quantum mechanics and GR also shows up in the role of time

which plays a different role in quantum mechanics and in GR. Furthermore, it is expectedthat a quantum theory of gravity would solve the problem of the singularities appearing

within GR. As a last issue, it is the wish that such a new theory also would lead to a true

unification of all interactions and, thus, to a better understanding of the physical world.

Any theory is characterized by their own set of constants. It is believed that the Planck

energy EPl 1028 eV sets the scale of quantum gravity effects. As a consequence, all ex-

pected effects scale with this energy or the corresponding Planck length, Planck time, etc.

In string theory other scales influence the modifications (as is explained in the contributions

by T. Damour and by B. Schutz). The implications of deviations from the standard model of

cosmology is the subject of the article by S. Sarkar.

9.1 Theoretical Approaches

The low energy limit of string theory, a quasiclassical limit of loop quantum gravity as well

as results from noncommutative geometry suggest that many of the standard laws of physics

will suffer modifications. At a basic level these modifications show up in the equations of

the standard model and in Einsteins field equations. These modifications then result in

violation of Lorentz invariance

different limiting velocities of different particles

modified dispersion relation leading to birefringence in vacuum

modified dispersion relation leading to frequency-dependent velocity of light in vac-

uum

orientation and velocity dependence of effects

time and position-dependence of constants (varying ,G, etc.)

modified Newton potential at short and large distances.

In recent years there have been increased activities to search for these possible effects. How-

ever, until now nothing has been found.

9.2 Experimental Approaches

The experimental search for signals of a new theory requires to measure effects which have

never been measured before. A strategy to find new things is (i) to explore new parameter

regions in extreme situations, (ii) to use more precise devices, (iii) to use high-precision

methods for new tests, or (iv) to test or measure exotic things.


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9.2.1 Extreme Situations

Rather often some kind of new physics has been discovered when exploring new situa-

tions. We discuss various situations of this kind.

Extremely high energy One possibility to explore new physics is to probe the physical

processes at very high energies. One example is the LHC where in future energies of the

order of 1013 eV should be achievable. It is the hope to find signals of the Higgs particle and

of supersymmetry. However, this energy range is still far away from the quantum gravity

scale. The best what one can do is to observe high energy cosmic rays which have energies

of up to 1021 eV. In fact, it has been speculated that the observations of high energy cosmic

rays which according to standard theories are forbidden owing to the GreisenZatsepin

Kuzmincutoff (Greisen1966; Zatsepin and Kuzmin1966) could indicate a modified dis-

persion relation.

Extremely low energy The other extreme, very low temperatures, might also be a tool to

investigate possible signals of quantum gravity. One may speculate that the influence of

possible spacetime fluctuations on the dynamics of quantum systems is more pronounced

for very low temperatures. One may even speculate that such spacetime fluctuations may

give rise to a temperature threshold above the absolute zero.

Very low temperatures may be achievable in BoseEinstein condensates (BEC) during

long time of free evolution. Recently, at the Bremen drop tower, within the DLR-funded

QUANTUS project a freely falling BEC has been created and its free expansion of longer

than 1 s has been observed. These BECs then may be used for novel investigations, including

a search for deviations from standard predictions. The contributions of W. Schleich and M.

Kasevich explore the importance and use of ultracold atoms.

Large distances The unexplained phenomena, dark matter, dark energy, and the Pioneer

anomaly are related to large distances. This could indicate that the laws of gravity have to

be modified at large distances. Recently, some suggestions have been made:

It has been discussed whether a Yukawa modification of the Newtonian potential may

account for galactic rotations curves (Sanders1984). In the context of higher dimensional braneworld theories deviations from Newtons poten-

tial also occurs (Dvali et al. 2000). At large distances the potential behaves like 1/r 2, as

one would expect from the Poisson equation in 5 dimensions. A comparison with cosmo-

logical and astrophysical observations has been reviewed by (Lue2005).

From considering a running coupling constant it has been suggested that the spatial parts

of the spacetime metric posses a part which grows linearly with distance (Jaekel and

Reynaud2005). This approach is in agreement with present solar system tests and also

describes the Pioneer anomaly (Jaekel and Reynaud2007).

Weak accelerations An acceleration a, being of physical dimension m s2 can be related

to a length scalel0= c2/a. Now, the largest length scale in our universe is the Hubble length

LH= c/H, where H is the Hubble constant. The corresponding acceleration is cH whose

order of magnitude remarkably coincides with the Pioneer acceleration and the MOND ac-

celeration scale. As a consequence, it seems mandatory to perform experiments to explore

the physics for such small accelerations, as discussed on page504.


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518 C. Lmmerzahl

Strong accelerations Analogously, since the smallest length scale is the Planck lengthlPl,

the corresponding acceleration is a = 2 1051 m s2 which, however, is far out of experi-

mental reach. For smaller acceleration which might be reached by electrons in the fields of

strong lasers one might be able to detect Unruh radiation or to probe the physics near black

holes (Schfer and Sauerbrey1998; Schtzhold et al.2006, respectively).

Strong gravitational fields See discussion in Sect.7.2.

9.2.2 Better Accuracy and Sensitivity

It is clear that for a search of tiny effects a better accuracy always is a good strategy. In fact, it

is amazing how the accuracy for testing Lorentz invariance, for example, has increased over

the last years. It took more than 20 years to improve the results of the experiment by (Brillet

and Hall1979) and within a few years the accuracy improved by two orders of magnitude

better and is still improving further.Similar developments can be observed in other areas of quantum optics. There are im-

provements in the performance of atomic interferometry which are expected to give an im-

provement of measuring the fine structure constant by one or two orders of magnitude.

New optical clocks will show an improvement in accuracy by three orders of magnitude.

A further improvement of experiments make use of atoms from ultracold BECs, which can

be created in free fall. A particular effort in this direction is made by the Center of Excellence

QUEST where new quantumoptical devices are being developed for novel spacetime re-

search.4

9.2.3 New Tests Misusing High Precision Devices

It might also be of interest to identify devices which have, at least in principle, the sensitivity

to find quantum gravity effects. One example for that are gravitational wave interferometers,

(Amelino-Camelia and Lmmerzahl2004). Todays already running gravitational wave in-

terferometers have a strain sensitivity of 1021. With the advanced LIGO the sensitivity

will become 1024. Thus, for a continuous gravitational wave with a frequency in the max-

imum sensitivity range between 10 and 1000 Hz a continuous observation over one year

would reach a sensitivity of a bit less than 1028. This is the sensitivity needed for observing

Planck scale effects (1028

eV) by optical laboratory devices (which have an energy scale of 1 eV). This sensitivity is just the sensitivity needed to detect Planck-scale modifications

in the dispersion relation for photons (Amelino-Camelia and Lmmerzahl 2004).

Another example for that is to misuse stable devices in order to search for a fundamental

noise. Such fundamental noise scenarios with a power spectral density for the strain L/L,

the relative length uncertainty, of the form

S() =L

c

LPl

L

c/L

, (20)

have first been discussed by Amelino-Camelia (2000) in relation to gravitational wave in-terferometers like GEO600. Here LPl is the Planck length, L a characteristic length of the

device, the frequency of the radiation involved (laser frequency in the gravitational wave

interferometer or in an optical cavity), and and are arbitrary exponents related to the

4Seehttp://www.questhannover.de.
http://www.questhannover.de/http://www.questhannover.de/


19/22


noise scenario (Amelino-Camelia2000; Ng2003). Perhaps more suited for the search for

such fundamental noise are ultrastable cavities. A first experimental search using such de-

vices has been carried out by Schiller et al. (2004). Noise scenarios of this kind may also

influence the dynamics of massive particles leading to an apparent violation of the UFF

measurable by atomic interferometry (Gkl and Lmmerzahl 2008).A particular noise with the exponents = 12

and = 0 is considered as holographic noise

related to the information stored on the surface of black holes. This case has been discussed

in Hogan (2008a,2008b) with respect to its detectability in GEO600. The same scenario

may give a violation of the UFF of up to an order of 1010 (Gkl and Lmmerzahl2008).

10 The Need for Improved Tests

One may think that improved tests of the foundations and the predictions of SR and GR are

just the wish of some esoteric very specialized physicists. However, there are many aspectsand reasons for trying to improve experiments:

Physical reason. Fundamental theories always have to be tested as much and as far as pos-

sible.

Practical reason. Metrology is the definition, preparation and dissemination of physical

units like the second, the meter, the kilogram, etc. with the highest possible precision.

The definition of units in most cases depends on fundamental symmetries. The definition

of the meter, for example, depends on the constancy of the speed of light. The definition

of the international atomic time (TAI) depends on the special relativistic time dilation as

well as on the gravitational redshift. Therefore each high precision test also contributes tometrology. This is also the reason why so many fundamental tests are carried through at

the national bureaus of standard like BIPM, NIST or PTB. It is also well known that the

Global Positioning Systems relies on SR and GR.

Theoretical reason. Since Quantum Gravity should show effectively some measurable de-

viations from standard physics, each high precision test of SR and GR can in principle, be

interpreted as a search for Quantum Gravity.

Acknowledgements I thank H. Dittus and V. Perlick for discussions. Financial support from the German

aerospace Center DLR and the German Research Foundation DFG is acknowledged.

References

A. Abramovici, Z. Vager, Phys. Rev. D34, 3240 (1986)

J. Alspector, G. Kalbfleisch, N. Baggett, E. Fowler, B. Barish, A. Bodek, D. Buchholz, F. Sciulli, E. Siskind,

L. Stutte, H. Fisk, G. Krafczyk, D. Nease, O. Fackler, Phys. Rev. Lett. 36, 837 (1976)

G. Amelino-Camelia, Phys. Rev. D62, 0240151 (2000)

G. Amelino-Camelia, C. Lmmerzahl, Class. Quantum Grav. 21, 899 (2004)

G. Amelino-Camelia, C. Lmmerzahl, A. Macias, H. Mller, In Gravitation and Cosmology, ed. by A. Ma-

cias, C. Lmmerzahl, D. Nunez. AIP Conference Proceedings, vol. 758 (Melville, New York, 2005),

p. 30J. Anderson, P. Laing, E. Lau, A. Liu, M. Nieto, S. Turyshev, Phys. Rev. D 65, 082004 (2002)

J. Anderson, J. Campbell, J. Ekelund, J. Ellis, J. Jordan, Phys. Rev. Lett. 100, 091102 (2008)

N. Ashby, T. Heavner, T. Parker, A. Radnaev, Y. Dudin, Phys. Rev. Lett. 98, 070802 (2007)

J. Audretsch, J. Phys. A14, 411 (1981)

S. Baeler, B. Heckel, E. Adelberger, J. Gundlach, U. Schmidt, H. Swanson, Phys. Rev. Lett. 83, 3585 (1999)

H. Balasin, D. Grumiller, Significant reduction of galactic dark matter by general relativity, 2006.

arXiv:astro-ph/0602519
http://arxiv.org/abs/astro-ph/0602519http://arxiv.org/abs/astro-ph/0602519


20/22

520 C. Lmmerzahl

D. Bartlett, D. vanBuren, Phys. Rev. Lett.57, 21 (1986)

J. Bekenstein, Phys. Rev. D70, 083509 (2004)

J. Bell, T. Damour, Class. Quantum Grav. 13, 3121 (1996)

B. Bertotti, L. Iess, P. Tortora, Nature 425, 374 (2003)

L. Blanchet, Living Rev. Relativ.9(4) (2006). http://www.livingreviews.org/lrr-2006-4

L. Blanchet, A. Buonanno, G. Faye, Higher-order spin effects in the dynamics of compact binaries II. Radia-tion field, 2006. arXiv:gr-qc/0605140

H. Bondi, Rev. Mod. Phys. 29, 423 (1957)

K. Brecher, Phys. Rev. Lett. 39, 1051 (1977)

A. Brillet, J. Hall, Phys. Rev. Lett. 42, 549 (1979)

B. Brown, G. Masek, T. Maung, E. Miller, H. Ruderman, W. Vernon, Phys. Rev. 30, 763 (1973)

T. Buchert, Gen. Relativ. Grav. 40, 467 (2008)

T. Buchert, J. Ehlers, Astron. Astrophys.320, 1 (1997)

T. Chupp, R. Hoara, R. Loveman, E. Oteiza, J. Richardson, M. Wagshul, Phys. Rev. Lett. 63, 1541 (1989)

I. Ciufolini, Gen. Relativ. Grav. 36, 2257 (2004)

F. Cooperstock, S. Tieu, General relativity resolves galactic rotation without exotic dark matter, 2005.

arXiv:astro-ph/0507619

T. Damour, G. Schfer, Phys. Rev. Lett. 66, 2549 (1991)T. Damour, F. Piazza, G. Veneziano, Phys. Rev. Lett. 89, 081601 (2002a)

T. Damour, F. Piazza, G. Veneziano, Phys. Rev. D66, 046007 (2002b)

A. de Oliveira-Costa, M. Tegmark, M. Devlin, L. Page, A. Miller, C. Netterfield, Y. Xu, Phys. Rev. D 71,

043004 (2005)

T. Dent, Varying constants in astrophysics and cosmology and ... in Proceedings of SUSY06 (2006, to

appear). arXiv:hep-ph/0610376

B. DeWitt, R. Brehme, Ann. Phys. (NY) 9, 220 (1960)

H. Dittus, C. Lmmerzahl, H. Selig, Gen. Relativ. Grav. 36, 571 (2004)

D. Dvali, G. Gabadadze, M. Porrati, Phys. Lett. B 485, 208 (2000)

J. Ehlers, Gen. Relativ. Grav.38, 1059 (2006)

G. Faye, L. Blanchet, A. Buonanno, Phys. Rev. D 74, 104033 (2006)

T.M. Fortier, N. Ashby, J. Bergquist, M. Delaney, S. Diddams, T. Heavner, L. Hollberg, W. Itano, S. Jefferts,K. Kim, F. Levi, L. Lorini, W. Oskay, T. Parker, J. Shirley, J. Stalnaker, Phys. Rev. Lett. 98, 070801

(2007)

E. Gkl, C. Lmmerzahl, Class. Quantum Grav.25, 105012 (2008)

Z. Guiragossian, G. Rothbart, M. Yearian, R. Gearhart, J. Murray, Phys. Rev. Lett. 34, 335 (1975)

J. Gundlach, S. Schlamminger, C. Spitzer, K.Y. Choi, B. Woodahl, J. Coy, E. Fischbach, Phys. Rev. Lett. 98,

150801 (2007)

K. Greisen, End of the cosmic ray spectrum? Phys. Rev. Lett. 16, 748 (1966)

M. Haugan, Ann. Phys.118, 156 (1979)

F. Hehl, Phys. Lett. A 36, 225 (1971)

F. Hehl, B. Mashhoon, (2008). arXiv:0812.1059v3[gr-qc]

C. Hogan, Phys. Rev. D 78, 087501 (2008a)

C. Hogan, Phys. Rev. D 77, 104031 (2008b)

C.H. Hsieh, P.Y. Jen, K.L. Ko, K.Y. Li, W.T. Ni, S.S. Pan, Y.H. Shih, R.J. Tyan, Mod. Phys. Lett. 4, 1597

(1989)

A. Ignatiev, Phys. Rev. Lett.98, 101101 (2007)

M.T. Jaekel, S. Reynaud, Mod. Phys. Lett. A 20, 1047 (2005)

M.T. Jaekel, S. Reynaud, In Lasers, Clocks and Drag-Free, ed. by H. Dittus, C. Lmmerzahl, S. Turyshev

(Springer, Berlin, 2007), p. 193

G. Kalbfleisch, N. Baggett, E. Fowler, J. Alspector, Phys. Rev. Lett. 43, 1361 (1979)

V. Kostelecky, Phys. Rev.69, 105009 (2004)

A. Kostelecky, M. Mewes, Phys. Rev. D 66, 056005 (2002)

M. Kramer, I. Stairs, R. Manchester, M. MacLaughlin, A. Lyre, R. Ferdman, M. Burgag, D. Lorimer, A.

Possenti, N. DAmico, J. Sarkission, B. Joshi, P. Freire, F. Camilo, Ann. Phys. (Leipzig)15, 34 (2006a)M. Kramer, I. Stairs, R. Manchester, M. MacLaughlin, A. Lyre, R. Ferdman, M. Burgag, D. Lorimer, A. Pos-

senti, N. DAmico, J. Sarkission, G. Hobbs, J. Reynolds, P. Freire, F. Camilo, Science 314, 97 (2006b)

G. Krasinsky, V. Brumberg, Celest. Mech. Dyn. Astron. 90, 267 (2004)

L. Kreuzer, Phys. Rev.169, 1007 (1968)

C. Lmmerzahl, V. Perlick, Confronting Finsler gravity with experiment, 2009. Preprint Univ. Bremen

C. Lmmerzahl, P. Rademaker, Gravity, equivalence principle and clocks, 2009. arXiv:0904.4779 [gr-gc].

Preprint, University of Bremen
http://www.livingreviews.org/lrr-2006-4http://arxiv.org/abs/gr-qc/0605140http://arxiv.org/abs/astro-ph/0507619http://arxiv.org/abs/hep-ph/0610376http://arxiv.org/abs/0812.1059v3http://arxiv.org/abs/0904.4779http://arxiv.org/abs/0904.4779http://arxiv.org/abs/0812.1059v3http://arxiv.org/abs/hep-ph/0610376http://arxiv.org/abs/astro-ph/0507619http://arxiv.org/abs/gr-qc/0605140http://www.livingreviews.org/lrr-2006-4


21/22


C. Lmmerzahl, C. Ciufolini, H. Dittus, L. Iorio, H. Mller, A. Peters, E. Samain, S. Scheithauer, S. Schiller,

Gen. Relativ. Grav.36, 2373 (2004)

C. Lmmerzahl, A. Macias, H. Mller, Phys. Rev. A 75, 052104 (2007a)

C. Lmmerzahl, O. Preuss, H. Dittus, inLasers, Clocks, and Drag-Free Exploration of Relativistic Gravity in

Space, ed. by H. Dittus, C. Lmmerzahl, S. Turyshev (Springer, Berlin, 2007b), p. 75

C. Lmmerzahl, D. Lorek, H. Dittus, Gen. Relativ. Grav. 41 (2009)N. Li, D. Schwarz (2007). arXiv:gr-gc/0702043v3[gr-gc]

N. Li, M. Seikel, D.J. Schwarz, Is dark energy an effect of averaging? 2008. arXiv.org: 0801.3420

N. Lockerbie, J. Mester, R. Torii, S. Vitale, P. Worden, In Gyros, Clocks, and Interferometers: Testing Rela-

tivistic Gravity in Space, ed. by C. Lmmerzahl, C. Everitt, F. Hehl (Springer, Berlin, 2001), p. 213

M. Longo, Phys. Rev. D 36, 3276 (1987)

D. Lorek, C. Lmmerzahl, In Proceedings of the 11th Marcel Grossmann Meeting, ed. by R. Jantzen, H.

Kleinert, R. Ruffini (World Scientific, Singapore, 2008), p. 2618

A. Lue, Phys. Rep.423, 1 (2005)

L. Maleki, SPACETIMEa midex proposal, 2001. JPL

B. Mashhoon, F. Gronwald, H. Lichtenegger, In Gyroscopes, Clock, Interferometers, ...: Testing Relativistic

Gravity in Space, ed. by C. Lmmerzahl, C. Everitt, F. Hehl. LNP, vol. 562 (Springer, Berlin, 2001), p.

83M. Milgrom, New Astron. Rev.46, 741 (2002)

J. Moody, F. Wilczek, Phys. Rev. D 30, 130 (1984)

H. Mller, C. Braxmaier, S. Herrmann, A. Peters, C. Lmmerzahl, Phys. Rev. D67, 056006 (2003)

H. Mller, P. Stanwix, M. Tobar, E. Ivanov, P. Wolf, S. Herrmann, A. Senger, E. Kovalchik, A. Peters, Phys.

Rev. Lett.99, 050401 (2007)

H. Mller, S.W. Chiow, S. Herrmann, S. Chu, K.Y. Chung, Phys. Rev. Lett. 100, 031101 (2008)

Y. Ng, Mod. Phys. Lett. A18, 1073 (2003)

J. Overduin, W. Priester, Naturwiss88, 229 (2001)

P. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 (2003)

S. Perlmutter, G. Aldering, G. Goldhaber et al., Astrophys. J. 517, 565 (1999)

E. Pitjeva, Astron. Lett.31, 340 (2005)

R. Reasenberg, I. Shapiro, P. MacNeil, R. Goldstein, J. Breidenthal, J. Brenkle, D. Cain, T. Kaufman, T.Komarek, A. Zygielbaum, Astrophys. J. Lett. 234, 219 (1979)

A. Riess, A. Filippenko, P. Challis, et al., Astron. J. 116, 1009 (1998)

B. Rievers, C. Lmmerzahl, M. List, S. Bremer (2008). Preprint, Univercity of Bremen

R. Sanders, Astron. Astrophys.136, 21 (1984)

R. Sanders, S. McGough, Ann. Rev. Astron. Astrophys. 40, 263 (2002)

B. Schaefer, Phys. Rev. Lett. 82, 4964 (1999)

G. Schfer, R. Sauerbrey, Probing black-hole physics in the laboratory using high intensity femtosecond

lasers, 1998. arXiv:astro-ph/9805106

S. Schiller, C. Lmmerzahl, H. Mller, C. Braxmaier, S. Herrmann, A. Peters, Phys. Rev. D 69, 027504

(2004)

R. Schoedel, A. Eckart, T. Alexander, D. Merritt, R. Genzel, A. Sternberg, L. Meyer, F. Kul, J. Moultaka, T.

Ott, C. Straubmeier, Astron. Astroph.469, 125 (2007)R. Schtzhold, G. Schaller, D. Habs, Phys. Rev. Lett.97, 121302 (2006)

D. Schwarz, G. Starkman, D. Huterer, C. Copi, Phys. Rev. Lett.93, 221301 (2004)

S. Shapiro, J. Davis, D. Lebach, J. Gregory, Phys. Rev. Lett.92, 121101 (2004)

D.N. Spergel, R. Bean, O. Dore, M.R. Nolta, C.L. Bennett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu,

L. Page, H.V. Peiris, L. Verde, M. Halpern, R.S. Hill, A. Kogut, M. Limon, S.S. Meyer, N. Odegard,

G.S. Tucker, J.L. Weiland, E. Wollack, E.L. Wright, Astroph. J. 170, 377 (2007)

E. Standish, inTransits of Venus: New Views of the Solar System and Galaxy, Proceedings IAU Colloquium

No. 196, ed. by D. Kurtz (Cambridge University Press, Cambridge, 2005), p. 163

E. Standish, inGravitation and Cosmology, ed. by A. Macias, C. Lmmerzahl, A. Camacho. AIP Conference

Proceedings, vol. 977 (Melville, New York, 2008), p. 254

J. Steinhoff, G. Schfer, S. Hergt, Phys. Rev. D 77, 104018 (2008)

L. Stodolsky, Phys. Lett. B201, 353 (1988)T. Sumner, Living Rev. Relativ. 5, 2002420112005 (2002)

M. Tegmark, et al., Phys. Rev. D69, 103501 (2004)

P. Touboul, Comptes Rendus de lAcad. Sci. Srie IV: Phys. Astrophys. 2, 1271 (2001)

M.J. Valtonen, H.J. Lehto, K. Nilsson, J. Heidt, L. Takalo, A. Sillanp, C. Villforth, M. Kidger, G. Poyner,

T. Pursimo, S. Zola, J.H. Wu, X. Zhou, K. Sadakane, M. Drozdz, D. Koziel, D. Marchev, W. Ogloza, C.

Porowski, M. Siwak, G. Stachowski, M. Winiarski, V.P. Hentunen, M. Nissinen, A. Liakos, S. Dogru,

Nature452, 851 (2008)
http://arxiv.org/abs/gr-gc/0702043v3http://arxiv.org/abs/0801.3420http://arxiv.org/abs/astro-ph/9805106http://arxiv.org/abs/astro-ph/9805106http://arxiv.org/abs/0801.3420http://arxiv.org/abs/gr-gc/0702043v3


22/22

522 C. Lmmerzahl

C. van de Bruck, W. Priester, in Dark Matter in Astrophysics and Particle Physics 1998: Proceedings of

the Second International Conference on Dark Matter in Astrophysics and Particle, ed. by H. Klapdor-

Kleingrothaus (Inst. of Physics, London, 1998)

R. Vessot, M. Levine, E. Mattison, E. Blomberg, T. Hoffmann, G. Nystrom, B. Farrel, R. Decher, P. Eby, C.

Baughter, J. Watts, D. Teuber, F. Wills, Phys. Rev. Lett.45, 2081 (1980)

R. Walsworth, inSpecial Relativity, ed. by J. Ehlers, C. Lmmerzahl (Springer, Berlin, 2006), p. 493C. Wetterich, Phys. Lett. B561, 10 (2003)

C. Will, Theory and Experiment in Gravitational Physics (revised edition) (Cambridge University Press,

Cambridge, 1993)

P. Wolf, C.J. Borde, A. Clairon, L. Duchayne, A. Landragin, P. Lemonde, G. Santarelli, W. Ertmer, E. Rasel,

F.S. Cataliotti, M. Inguscio, G.M. Tino, P. Gill, H. Klein, S. Reynaud, C. Salomon, E. Peik, O. Bertolami,

P. Gil, J. Paramos, C. Jentsch, U. Johann, A. Rathke, P. Bouyer, L. Cacciapuoti, D. Izzo, P. de Natale,

B. Christophe, P. Touboul, S.G. Turyshev, J.D. Anderson, M.E. Tobar, F. Schmidt-Kaler, J. Vigue, A.

Madej, L. Marmet, M.C. Angonin, P. Delva, P. Tourrenc, G. Metris, H. Muller, R. Walsworth, Z.H. Lu,

L. Wang, K. Bongs, A. Toncelli, M. Tonelli, H. Dittus, C. Lmmerzahl, G. Galzerano, P. Laporta, J.

Laskar, A. Fienga, F. Roques, K. Sengstock, Exp. Astron.23 (2008)

G.T. Zatsepin, V.A. Kuzmin, Upper limit of the spectrum of cosmic rays. JETP Lett. 4, 78 (1966)

F. Zwicky, Helv. Phys. Acta6, 110 (1933)

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