THE CONFRONTATION BETWEEN GENERAL RELATIVITY ANDEXPERIMENT

CLIFFORD M. WILL�

Department of Physics and McDonnell Center for the Space Sciences, Washington University,St. Louis, MO 63130, USA; E-mail: cmw@wuphys.wustl.edu

Abstract. We review the experimental evidence for Einstein’s general relativity. Tests of the EinsteinEquivalence Principle support the postulates of curved spacetime and bound variations of funda-mental constants in space and time, while solar-system experiments strongly confirm weak-fieldgeneral relativity. The Binary Pulsar provides tests of gravitational-wave damping and of strong-field general relativity. Future experiments, such as the Gravity Probe B Gyroscope Experiment,a satellite test of the Equivalence principle, and tests of gravity at short distance to look for extraspatial dimensions could further constrain alternatives to general relativity. Laser interferometricgravitational-wave observatories on Earth and in space may provide new tests of scalar-tensor gravityand graviton-mass theories via the properties of gravitational waves.

1. Introduction

Because of its elegance and simplicity, and because of its empirical success, generalrelativity has become the foundation for our understanding of the gravitational in-teraction. Yet modern developments in particle theory suggest that it is probably notthe entire story, and that modification of the basic theory may be required at somelevel. String theory generally predicts a proliferation of scalar fields that couldresult in alterations of general relativity reminiscent of the Brans-Dicke theory ofthe 1960s. In the presence of extra dimensions, the gravity that we feel on our four-dimensional ‘brane’ of a higher dimensional world could be somewhat differentfrom a pure four-dimensional general relativity. Some of these ideas have motivatedthe possibility that fundamental constants may actually be dynamical variables, andhence may vary in time or in space. However, any theoretical speculation alongthese lines must abide by the best current empirical bounds. Decades of high-precision tests of general relativity have produced some very tight constraints.In this article we review some of these experimental constraints. We begin withan overview of experimental tests of general relativity. For further discussion oftopics in this section, and for references to the literature, the reader is referred toTheory and Experiment in Gravitational Physics (Will, 1993) and to the ‘living’review article (Will, 2001). We end with a discussion of possible future tests of

� Supported in part by the US National Science Foundation, Grant No. PHY 00-96522 and theNational Aeronautics and Space Administration, Grant No. NAG 5-10186

Astrophysics and Space Science 283: 543–552, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

[ 105 ]

544 C. WILL

gravitational theory that will use the direct detection of gravitational radiation usingground-based and space-based interferometers (Will, 1999).

2. Empirical status of general relativity

2.1. THE EINSTEIN EQUIVALENCE PRINCIPLE

The Einstein equivalence principle (EEP) is a powerful and far-reaching principle,which states that (i) test bodies fall with the same acceleration independently oftheir internal structure or composition (Weak Equivalence Principle, or WEP),(ii) the outcome of any local non-gravitational experiment is independent of the ve-locity of the freely-falling reference frame in which it is performed (Local LorentzInvariance, or LLI), and (iii) the outcome of any local non-gravitational experimentis independent of where and when in the universe it is performed (Local PositionInvariance, or LPI). The Einstein equivalence principle is the heart of gravitationaltheory, for it is possible to argue convincingly that if EEP is valid, then gravitationmust be described by ‘metric theories of gravity’, which state that (i) spacetimeis endowed with a symmetric metric, (ii) the trajectories of freely falling bod-ies are geodesics of that metric, and (iii) in local freely falling reference frames,the non-gravitational laws of physics are those written in the language of specialrelativity.

General relativity is a metric theory of gravity, but so are many others, in-cluding the Brans-Dicke theory. In this sense, superstring theory is not metric,because of residual coupling of external, gravitation-like fields, to matter. The-ories in which varying non-gravitational constants are associated with dynamicalfields that couple to matter directly are also not metric theories. A direct testof WEP is the comparison of the acceleration of two laboratory-sized bodies ofdifferent composition in an external gravitational field. A measurement or limiton the fractional difference in acceleration between two bodies yields a quantityη ≡ 2|a1 − a2|/|a1 + a2|, called the ‘Eötvös ratio’.

The best limit on η currently comes from the ‘Eöt-Wash’ experiments carriedout at the University of Washington, which used a sophisticated torsion balance tryto compare the accelerations of bodies of different composition toward the Earth,the Sun and the galaxy. Another strong bound comes from lunar laser ranging(LURE), which checks the equality of free fall of the Earth and Moon toward theSun. The results are:

ηEot−Wash < 4 × 10−13 , ηLURE < 5 × 10−13 . (1)

In fact, by using laboratory materials whose composition mimics that of the Earthand Moon, the Eöt-Wash experiments (Baessler et al., 1999) permit one to infer anunambiguous bound from lunar laser ranging on the universality of acceleration ofgravitational binding energy at the level of 1.3 × 10−3 (test of the Nordtvedt effect

[ 106 ]

CONFRONTATION BETWEEN GENERAL RELATIVITY AND EXPERIMENT 545

– see Sec. 2.2 and Table I.) Elsewhere in this volume, Damour and others discussfurther the implications of WEP experiments for theories of varying fundamentalconstants.

The best tests of Local Lorentz Invariance are the ‘mass anisotropy’ or ‘Hughes-Drever’ experiments. A simple way of interpreting these experiments is to supposethat a non-metric coupling to the electromagnetic interactions results in a change inthe speed of electromagnetic radiation c relative to the limiting speed of materialtest particles c0, in other words, c �= c0. Such a Lorentz-non-invariant electro-magnetic interaction would cause shifts in the energy levels of atoms and nucleithat depend on the orientation of the quantization axis of the state relative to ourvelocity in the rest-frame of the universe, and on the quantum numbers of the state,with sizes proportional to the parameter δ ≡ |(c0/c)

2−1|. Searches for such energyanisotropies using laser-cooled trapped atom techniques have placed the strongbound

δ < 10−21 . (2)

For further discussion, see Haugan and Will (1987).Local Position Invariance requires, among other things, that the internal binding

energies of atoms be independent of location in space and time, when measuredagainst some standard atom. This means that a comparison of the rates of twodifferent kinds of clocks should be independent of location or epoch, and thatthe frequency shift between two identical clocks at different locations is simplya consequence of the apparent Doppler shift between a pair of inertial framesmomentarily comoving with the clocks at the moments of emission and receptionrespectively. The relevant parameter is α ≡ ∂ ln EB/∂(U/c2), where EB is theatomic or nuclear binding energy, and U characterizes the external gravitationalpotential. The best bounds come from a 1976 rocket redshift experiment usingHydrogen masers, and a 1993 clock intercomparison experiment (a ‘null’ redshiftexperiment). The results are:

αMaser < 2 × 10−4 , αNull < 10−3 . (3)

Clock intercomparisons have also bounded a time variation of the fine structureconstant over cosmological timescales; from Rubidium fountain clocks comparedwith Cesium clocks, one finds the bound α/α < 8 × 10−15 yr−1 (Salamon et al.,1999). A better bound comes from analysis of fission yields of the Oklo naturalreactor, which occurred in Africa 2 billion years ago, namely (α/α)Oklo < 6 ×10−17 yr−1 (Damour and Dyson, 1996). These and other bounds on variations ofconstants, including reports of positive evidence for variations from quasar spectra,are discussed by several authors in this volume. For a review of these and other testsof EEP, see Will (2001).

[ 107 ]

546 C. WILL

TABLE I

Current limits on the PPN parameters

Parameter Effect Limit Remarks

γ − 1 (i) time delay 2 × 10−3 Viking ranging

(ii) light deflection 3 × 10−4 VLBI

β − 1 (i) perihelion shift 3 × 10−3 J2 = 10−7 from

helioseismology

(ii) Nordtvedt effect 10−3 η = 4β − γ − 3 assumed

ξ Earth tides 10−3 gravimeter data

α1 orbital polarization 10−4 lunar laser ranging

PSR J2317+1439

α2 solar spin 4 × 10−7 alignment of sun

precession and ecliptic

α3 pulsar acceleration 2 × 10−20 pulsar P statistics

η1 Nordtvedt effect 10−3 lunar laser ranging

ζ1 – 2 × 10−2 combined PPN bounds

ζ2 binary motion 4 × 10−5 Pp for PSR 1913+16

ζ3 Newton’s 3rd law 10−8 Lunar acceleration

ζ4 – – not independent

1Here η = 4β − γ − 3 − 10ξ/3 − α1 − 2α2/3 − 2ζ1/3 − ζ2/3

2.2. BOUNDS ON THE POST-NEWTONIAN PARAMETERS OF METRIC THEORIES

When we confine attention to metric theories of gravity, and further focus on theslow-motion, weak-field limit appropriate to the solar system and similar systems,it turns out that, in a broad class of metric theories, only the numerical valuesof a set of parameters vary from theory to theory. This framework, called theparametrized post-Newtonian (PPN) formalism is a convenient tool for classifyingalternative metric theories of gravity, for interpreting the results of experiments,and for suggesting new tests of metric gravity. The framework contains ten PPNparameters: γ , related to the amount of spatial curvature generated by mass; β,related to the degree of non-linearity in the gravitational field; ξ , α1, α2, and α3,which determine whether the theory violates local position invariance or localLorentz invariance in gravitational experiments (violations of the Strong Equi-valence Principle); and ζ1, ζ2, ζ3 and ζ4, which describe whether the theory hasappropriate momentum conservation laws. The parameter γ , or more precisely, thecombination (1 + γ )/2, governs the deflection of light and the Shapiro time delayin ranging. The combination (2 + 2γ − β)/3 governs the perihelion advance. Thecombination 4β−γ −3−10ξ/3−α1−2α2/3−2ζ1/3−ζ2/3 governs the Nordtvedt

[ 108 ]

CONFRONTATION BETWEEN GENERAL RELATIVITY AND EXPERIMENT 547

effect, a possible violation of WEP induced by gravitational self-energy of massivebodies. In general relativity, γ − 1, β − 1, and the remaining parameters all vanish.

Three decades of experiments, ranging from the standard light-deflection andperihelion-shift tests, to Lunar laser ranging, planetary and satellite tracking, andgeophysical and astronomical observations, have placed bounds on the PPN para-meters consistent with general relativity. The results are summarized in Table I.Scalar-tensor theories of gravity are characterized by a coupling function ω(φ)

whose size is inversely related to the ‘strength’ of the scalar field relative to the met-ric. In the solar system, the parameter |γ −1|, for example is equal to 1/(2+ω(φ0)),where φ0 is the value of the scalar field today outside the solar system. Solar-systemexperiments (primarily VLBI) constrain ω(φ0) > 3000. See Will (2001) for anup-to-date review.

2.3. THE BINARY PULSAR

The binary pulsar PSR 1913+16, discovered in 1974, provided important new testsof general relativity, specifically of gravitational radiation and of strong-field grav-ity. Through precise timing of the pulsar ‘clock’, the important orbital parametersof the system could be measured with exquisite precision. These included non-relativistic ‘Keplerian’ parameters, such as the eccentricity e, and the orbital period(at a chosen epoch) Pb, as well as relativistic ‘post-Keplerian’ parameters, such asthe mean advance of periastron, the effect of time-dilation and gravitational redshifton pulses, and the rate of decrease of the orbital period (Table II). The last effectis a result of gravitational radiation damping (apart from a small correction dueto galactic differential rotation). According to GR, all three post-Keplerian effectsdepend on e and Pb, which are known, and on the two stellar masses which areunknown. By combining the observations with the GR predictions, one obtainsboth a measurement of the two masses, and a test of GR, since the system isoverdetermined. The results are

m1 = 1.4411 ± 0.0007M� , m2 = 1.3873 ± 0.0007M� ,

P GRb /P OBS

b = 1.002 ± 0.003 . (4)

The results also test the strong-field aspects of GR in the following way: the neutronstars that comprise the system have very strong internal gravity, contributing asmuch as several tenths of the rest mass of the bodies (compared to the orbitalenergy, which is only 10−6 of the mass of the system). Yet in general relativity, theinternal structure is ‘effaced’ as a consequence of the Strong Equivalence Principle,and the orbital motion and gravitational radiation emission depend only on themasses m1 and m2. By contrast, in alternative metric theories, SEP is not valid ingeneral, and internal-structure effects can lead to significantly different behavior,such as the emission of dipole gravitational radiation. Unfortunately, in the case ofscalar-tensor theories of gravity, because the neutron stars are so similar in PSR1913+16 (and in other double-neutron star binary pulsar systems), dipole radiation

[ 109 ]

548 C. WILL

TABLE II

Parameters of the binary pulsar PSR 1913+16 (Numbers in parentheses denoteerrors in last digit.)

Parameter Symbol Value1

Keplerian Parameters

Eccentricity e 0.6171308(4)

Orbital Period Pb (day) 0.322997462736(7)

Post-Keplerian Parameters

Mean rate of periastron advance 〈ω〉 (oyr−1) 4.226621(11)

Gravitational redshift/time dilation γ ′ (ms) 4.295(2)

Orbital period derivative Pb (10−12) −2.422(6)

is suppressed by symmetry; the best bound on the coupling parameter ω(φ0) fromPSR 1913+16 is in the hundreds. For further discussion, see Will (2001).

2.4. FUTURE EXPERIMENTAL TESTS

There are a number of possibilities for future tests of alternative gravitation the-ories. (i) High-precision WEP experiments, can test super-string inspired modelsof scalar-tensor gravity, or theories with varying fundamental constants in whichweak violations of WEP can occur via non-metric couplings. The project MICRO-SCOPE, designed to test WEP to a part in 1015 has been approved by the Frenchspace agency CNES for a possible 2004 launch. A proposed NASA-ESA SatelliteTest of the Equivalence Principle (STEP) may reach the level of η < 10−18. Inaddition, laboratory tests of the inverse square law of Newtonian gravitation atmillimeter scales and below are underway to search for such additional couplingsor for the presence of large extra dimensions; the challenge of these experimentsis to distinguish gravitation-like interactions from electromagnetic and quantummechanical (Casimir) effects (Long et al., 1999; Hoyle et al., 2001). (ii) The NASARelativity Mission called Gravity Probe-B, will measure the precession of an arrayof gyroscopes in Earth orbit. Although its primary science goal is a one-percentmeasurement of the gravitomagnetic dragging of inertial frames caused by therotation of the Earth, it will also measure the larger geodetic precession causedby space curvature, and could thus measure γ to better than a part in 104, givinga bound on ω of 104 or higher. Launch of the mission is scheduled for April,2003. (iii) Orbiting optical interferometers to measure the deflection of light atthe micro-arcsecond level could bound ω at the level of 105 or greater.

[ 110 ]

CONFRONTATION BETWEEN GENERAL RELATIVITY AND EXPERIMENT 549

3. Gravitational-Wave Tests of Gravitation Theory

The detection of gravitational radiation by either laser interferometers or resonantcryogenic bars will, it is widely stated, usher in a new era of gravitational-waveastronomy (Thorne, 1987; Barish and Weiss, 1999). Furthermore, it will yield newand interesting tests of general relativity (GR) in its radiative regime (Will, 1999).

3.1. POLARIZATION OF GRAVITATIONAL WAVES

A laser-interferometric or resonant bar gravitational-wave detector measures thelocal components of a symmetric 3 × 3 tensor which is composed of the ‘electric’components of the Riemann tensor, R0i0j . These six independent components canbe expressed in terms of polarizations (modes with specific transformation proper-ties under null rotations). Three are transverse to the direction of propagation, withtwo representing quadrupolar deformations and one representing an axisymmetric‘breathing’ deformation. Three modes are longitudinal, with one an axially sym-metric stretching mode in the propagation direction, and one quadrupolar mode ineach of the two orthogonal planes containing the propagation direction. Generalrelativity predicts only the first two transverse quadrupolar modes, independentlyof the source, while scalar-tensor gravitational waves can in addition contain thetransverse breathing mode. More general metric theories predict up to the full com-plement of six modes. A suitable array of gravitational antennas could delineate orlimit the number of modes present in a given wave. If distinct evidence were foundof any mode other than the two transverse quadrupolar modes of GR, the resultwould be disastrous for GR. On the other hand, the absence of a breathing modewould not necessarily rule out scalar-tensor gravity, because the strength of thatmode depends on the nature of the source.

3.2. SPEED OF GRAVITATIONAL WAVES

According to GR, in the limit in which the wavelength of gravitational waves issmall compared to the radius of curvature of the background spacetime, the wavespropagate along null geodesics of the background spacetime, i.e. they have thesame speed, c, as light. In other theories, the speed could differ from c because ofcoupling of gravitation to ‘background’ gravitational fields. For example, in sometheories with a flat background metric η, gravitational waves follow null geodesicsof η, while light follows null geodesics of g (Will, 1993). Another way in which thespeed of gravitational waves could differ from c is if gravitation were propagatedby a massive field (a massive graviton), in which case vg would be given by, in alocal inertial frame,

v2g

c2= 1 − m2

gc4

E2≈ 1 − 1

2

c2

f 2λ2g

, (5)

[ 111 ]

550 C. WILL

where mg , E and f are the graviton rest mass, energy and frequency, respectively,and λg = h/mgc is the graviton Compton wavelength (λg � c/f assumed). Anexample of a theory with this property is the two-tensor massive graviton theory ofVisser (Visser, 1998).

The most obvious way to test for a massive graviton is to compare the arrivaltimes of a gravitational wave and an electromagnetic wave from the same event,e.g. a supernova. For a source at a distance D, the resulting bound on the difference|1 − vg/c| or on λg is

|1 − vg

c| < 5 × 10−17

(200Mpc

D

) ( t

1s

), (6)

λg > 3 × 1012 km

(D

200 Mpc

100 Hz

f

)1/2 (1

f t

)1/2

. (7)

where t ≡ ta − (1 + Z) te is the ‘time difference’, where ta and te are thedifferences in arrival time and emission time, respectively, of the two signals, andZ is the redshift of the source. In many cases, te is unknown, so that the best onecan do is employ an upper bound on te based on observation or modelling.

However, there is a situation in which a bound on the graviton mass can be setusing gravitational radiation alone (Will, 1998). That is the case of the inspirallingcompact binary. Because the frequency of the gravitational radiation sweeps fromlow frequency at the initial moment of observation to higher frequency at the finalmoment, the speed of the gravitons emitted will vary, from lower speeds initially tohigher speeds (closer to c) at the end. This will cause a distortion of the observedphasing of the waves and result in a shorter than expected overall time ta of pas-sage of a given number of cycles. Furthermore, through the technique of matchedfiltering, the parameters of the compact binary can be measured accurately (Cutleret al., 1993), and thereby the emission time te can be determined accurately.

A full noise analysis using proposed noise curves for the advanced LIGO ground-based detectors, and for the proposed space-based LISA antenna yields boundsthat are summarized in Table III. These potential bounds can be compared withthe solid bound λg > 2.8 × 1012 km, derived from solar system dynamics, whichlimit the presence of a Yukawa modification of Newtonian gravity of the formV (r) = (GM/r) exp(−r/λg) (Talmadge et al., 1988), and with the model-depend-ent bound λg > 6 × 1019 km from consideration of galactic and cluster dynamics(Visser, 1998).

3.3. TESTS OF SCALAR-TENSOR GRAVITY

Scalar-tensor theories generically predict dipole gravitational radiation, in additionto the standard quadrupole radiation, which results in modifications in gravitational-radiation back-reaction, and hence in the evolution of the phasing of gravitationalwaves from inspiralling sources. The effects are strongest for systems involvinga neutron star and a black hole. Double neutron star systems are less promising

[ 112 ]

CONFRONTATION BETWEEN GENERAL RELATIVITY AND EXPERIMENT 551

TABLE III

Bounds on λg from gravitational-wave observations of inspirallingcompact binaries

m1(M�) m2(M�) Distance (Mpc) Bound on λg (km)

Ground-based (LIGO/VIRGO)

1.4 1.4 300 4.6 × 1012

1.4 10 630 5.4 × 1012

10 10 1500 6.0 × 1012

Space-based (LISA)

107 107 3000 6.9 × 1016

106 106 3000 5.4 × 1016

105 105 3000 2.3 × 1016

104 104 3000 0.7 × 1016

TABLE IV

Bounds on ω for NS-BH sys-tems using LISA data ana-lysis. Signal-to-noise ratio =10, integration time is oneyear prior to the innermoststable circular orbit

mBH (M�) Bound on ω

1000 244549

5000 56326

10000 29906

50000 4780

100000 1458

because the small range of masses near 1.4 M� with which they seem to occurresults in suppression of dipole radiation by symmetry. Double black-hole systemsturn out to be observationally identical in the two theories, because black holes bythemselves cannot support scalar ‘hair’ of the kind present in these theories. Dipoleradiation will be present in black-hole neutron-star systems, however.

For example, for a 1.4 M� neutron star and a 10 M� (3 M�) black hole at200 Mpc observed by an advanced LIGO detector, the bound on ω could be 600(1800). The bound increases linearly with signal-to-noise ratio, although the eventrate decreases as the inverse cube of S/N (Will, 1994). Significantly strongerbounds could be obtained using observations of low-frequency gravitational waves

[ 113 ]

552 C. WILL

by a space-based LISA-type detector. For example, observations of a 1.4M� NSinspiralling to a 103M� BH with a signal-to-noise ratio of 10 could yield a boundof ω > 2.4 × 105, substantially greater than the current experimental bound ofω > 3000 (Scharre and Will, 2002). The results are summarized in Table IV.

For a follow-on mission to LISA whose peak sensitivity to gravitational waveswould be at frequencies between those of LISA and the ground-based detectors, thebounds could be even stronger: for a NS inspiralling into a 200 M� BH, the boundcould exceed 12 million. Recent discoveries of such intermediate-mass black holesin globular clusters make such tests potentially feasible (Miller, 2002).

References

Baessler, S. et al.: 1999, Phys. Rev. Lett. 83, 3585.Barish, B.C. and Weiss, R.: 1999, Phys. Today 52, 44 (October).Cutler, C. et al.: 1993, Phys. Rev. Lett. 70, 2984.Damour, T. and Dyson, F.: 1996, Nucl. Phys. B 480, 37–54.Haugan, M.P. and Will, C.M.: 1987, Phys. Today 40, 69 (May).Hoyle, C.D. et al.: 2001, Phys. Rev. Lett. 86, 1418.Long, J.C., Chan H.W. and Price, J.C.: 1999, Nucl. Phys. B 539, 23–34.Miller, M.C.: 2002, Astrophys. J., in press (astro-ph/0206404).Salomon, C. et al.: 1999, in: M. Inguscio and E. Arimondo (eds.), Proceedings of the International

Conference on Atomic Physics 2000, World Scientific, Singapore, in press.Scharre, P.D. and Will, C.M.: 2002, Phys. Rev. D 65, 042002.Talmadge, C. et al.: 1988, Phys. Rev. Lett. 61, 1159.Thorne, K.S.: 1987, in: S.W. Hawking and W. Israel (eds.), 300 Years of Gravitation, Cambridge

University Press, Cambridge, p. 330.Visser, M.: 1998, Gen. Rel. Grav. 30, 1717.Will, C.M.: 1993, Theory and Experiment in Gravitational Physics, Cambridge University Press,

Cambridge.Will, C.M.: 1994, Phys. Rev. D 50, 6058.Will, C.M.: 1998, Phys. Rev. D 57, 2061.Will, C.M.: 1999, Phys. Today 52, 38 (October).Will, C.M.: 2001, Living Reviews in Relativity 4, 2001-4,

(http://www.livingreviews.org/Articles/Volume4/2001-4will).

[ 114 ]