The Confrontation between General Relativity and Experiment
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Abstract
The status of experimental tests of general relativity and of theoretical frameworks for analysing them are reviewed. Einstein’s equivalence principle (EEP) is well supported by experiments such as the Eötvös experiment, tests of special relativity, and the gravitational redshift experiment. Future tests of EEP and of the inverse square law will search for new interactions arising from unification or quantum gravity. Tests of general relativity at the postNewtonian level have reached high precision, including the light defl ection the Shapiro time delay, the perihelion advance of Mercury, and the Nordtvedt effect in lunar motion. Gravitational wave damping has been detected in an amount that agrees with general relativity to half a percent using the HulseTaylor binary pulsar, and new binary pulsar systems may yield further improvements. When direct observation of gravitational radiation from astrophysical sources begins, new tests of general relativity will be possible.
1 Introduction
At the time of the birth of general relativity (GR), experimental confirmation was almost a side issue. Einstein did calculate observable effects of general relativity, such as the perihelion advance of Mercury, which he knew to be an unsolved problem, and the deflection of light, which was subsequently verified, but compared to the inner consistency and elegance of the theory, he regarded such empirical questions as almost peripheral. But today, experimental gravitation is a major component of the field, characterized by continuing efforts to test the theory’s predictions, to search for gravitational imprints of highenergy particle interactions, and to detect gravitational waves from astronomical sources.
The modern history of experimental relativity can be divided roughly into four periods: Genesis, Hibernation, a Golden Era, and the Quest for Strong Gravity. The Genesis (1887–1919) comprises the period of the two great experiments which were the foundation of relativistic physics — the MichelsonMorley experiment and the Eötvös experiment — and the two immediate confirmations of GR — the deflection of light and the perihelion advance of Mercury. Following this was a period of Hibernation (1920–1960) during which theoretical work temporarily outstripped technology and experimental possibilities, and, as a consequence, the field stagnated and was relegated to the backwaters of physics and astronomy.
But beginning around 1960, astronomical discoveries (quasars, pulsars, cosmic background radiation) and new experiments pushed GR to the forefront. Experimental gravitation experienced a Golden Era (1960–1980) during which a systematic, worldwide effort took place to understand the observable predictions of GR, to compare and contrast them with the predictions of alternative theories of gravity, and to perform new experiments to test them. The period began with an experiment to confirm the gravitational frequency shift of light (1960) and ended with the reported decrease in the orbital period of the HulseTaylor binary pulsar at a rate consistent with the general relativity prediction of gravity wave energy loss (1979). The results all supported GR, and most alternative theories of gravity fell by the wayside (for a popular review, see [148]).
Since 1980, the field has entered what might be termed a Quest for Strong Gravity. Many of the remaining interesting weakfield predictions of the theory are extremely small and diffcult to check, in some cases requiring further technological development to bring them into detectable range. The sense of a systematic assault on the weakfield predictions of GR has been supplanted to some extent by an opportunistic approach in which novel and unexpected (and sometimes inexpensive) tests of gravity have arisen from new theoretical ideas or experimental techniques, often from unlikely sources. Examples include the use of lasercooled atom and ion traps to perform ultraprecise tests of special relativity; the proposal of a “fifth” force, which led to a host of new tests of the weak equivalence principle; and recent ideas of large extra dimensions, which have motived new tests of the inverse square law of gravity at submillimeter scales. Several major ongoing efforts also continue, principally the Stanford Gyroscope experiment, known as Gravity ProbeB.
Instead, much of the focus has shifted to experiments which can probe the effects of strong gravitational fields. The principal figure of merit that distinguishes strong from weak gravity is the quantity ∊∼GM/Rc^{2}, where G is the Newtonian gravitational constant, M is the characteristic mass scale of the phenomenon, R is the characteristic distance scale, and c is the speed of light. Near the event horizon of a nonrotating black hole, or for the expanding observable universe, ∊∼0.5; for neutron stars, ∊∼0.2. These are the regimes of strong gravity. For the solar system ∊<10^{5}; this is the regime of weak gravity. At one extreme are the strong gravitational fields associated with Planckscale physics. Will unification of the forces, or quantization of gravity at this scale leave observable effects accessible by experiment? Dramatically improved tests of the equivalence principle or of the inverse square law are being designed, to search for or bound the imprinted effects of Planckscale phenomena. At the other extreme are the strong fields associated with compact objects such as black holes or neutron stars. Astrophysical observations and gravitational wave detectors are being planned to explore and test GR in the strongfield, highlydynamical regime associated with the formation and dynamics of these objects.
In this Living Review, we shall survey the theoretical frameworks for studying experimental gravitation, summarize the current status of experiments, and attempt to chart the future of the subject. We shall not provide complete references to early work done in this field but instead will refer the reader to the appropriate review articles and monographs, specifically to Theory and Experiment in Gravitational Physics [147], hereafter referred to as TEGP. Additional recent reviews in this subject are [142, 145, 150, 139, 37, 117]. References to TEGP will be by chapter or section, e.g. “TEGP 8.9 [147]”.
2 Tests of the Foundations of Gravitation Theory
2.1 The Einstein equivalence principle
The principle of equivalence has historically played an important role in the development of gravitation theory. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraph of the Principia to it. In 1907, Einstein used the principle as a basic element of general relativity. We now regard the principle of equivalence as the foundation, not of Newtonian gravity or of GR, but of the broader idea that spacetime is curved.
One elementary equivalence principle is the kind Newton had in mind when he stated that the property of a body called “mass” is proportional to the “weight”, and is known as the weak equivalence principle (WEP). An alternative statement of WEP is that the trajectory of a freely falling body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition. In the simplest case of dropping two different bodies in a gravitational field, WEP states that the bodies fall with the same acceleration (this is often termed the Universality of Free Fall, or UFF).
 1.
WEP is valid.
 2.
The outcome of any local nongravitational experiment is independent of the velocity of the freelyfalling reference frame in which it is performed.
 3.
The outcome of any local nongravitational experiment is independent of where and when in the universe it is performed.
The second piece of EEP is called local Lorentz invariance (LLI), and the third piece is called local position invariance (LPI).
For example, a measurement of the electric force between two charged bodies is a local nongravitational experiment; a measurement of the gravitational force between two bodies (Cavendish experiment) is not.
 1.
Spacetime is endowed with a symmetric metric.
 2.
The trajectories of freely falling bodies are geodesics of that metric.
 3.
In local freely falling reference frames, the nongravitational laws of physics are those written in the language of special relativity.
The argument that leads to this conclusion simply notes that, if EEP is valid, then in local freely falling frames, the laws governing experiments must be independent of the velocity of the frame (local Lorentz invariance), with constant values for the various atomic constants (in order to be independent of location). The only laws we know of that fulfill this are those that are compatible with special relativity, such as Maxwell’s equations of electromagnetism. Furthermore, in local freely falling frames, test bodies appear to be unaccelerated, in other words they move on straight lines; but such “locally straight” lines simply correspond to “geodesics” in a curved spacetime (TEGP 2.3 [147]).
General relativity is a metric theory of gravity, but then so are many others, including the BransDicke theory. The nonsymmetric gravitation theory (NGT) of Moffat is not a metric theory. Neither, in this narrow sense, is superstring theory (see Sec. 2.3), which, while based fundamentally on a spacetime metric, introduces additional fields (dilatons, moduli) that can couple to material stressenergy in a way that can lead to violations, say, of WEP. So the notion of curved spacetime is a very general and fundamental one, and therefore it is important to test the various aspects of the Einstein Equivalence Principle thoroughly.
Recent advances in atomic spectroscopy and atomic timekeeping have made it possible to test LLI by checking the isotropy of the speed of light using oneway propagation (as opposed to roundtrip propagation, as in the MichelsonMorley experiment). In one experiment, for example, the relative phases of two hydrogen maser clocks at two stations of NASA’s Deep Space Tracking Network were compared over five rotations of the Earth by propagating a light signal oneway along an ultrastable fiberoptic link connecting them (see Sec. 2.2.3). Although the bounds from these experiments are not as tight as those from massanisotropy experiments, they probe directly the fundamental postulates of special relativity, and thereby of LLI (TEGP 14.1 [147], [144]).
The most precise standard redshift test to date was the VessotLevine rocket experiment that took place in June 1976 [131]. A hydrogenmaser clock was own on a rocket to an altitude of about 10,000 km and its frequency compared to a similar clock on the ground. The experiment took advantage of the masers’ frequency stability by monitoring the frequency shift as a function of altitude. A sophisticated data acquisition scheme accurately eliminated all effects of the firstorder Doppler shift due to the rocket’s motion, while tracking data were used to determine the payload’s location and the velocity (to evaluate the potential difference ΔU, and the special relativistic time dilation). Analysis of the data yielded a limit ∣α∣<2×10^{4}.
A “null” redshift experiment performed in 1978 tested whether the relative rates of two different clocks depended upon position. Two hydrogen maser clocks and an ensemble of three superconductingcavity stabilized oscillator (SCSO) clocks were compared over a 10day period. During the period of the experiment, the solar potential U/c^{2} changed sinusoidally with a 24hour period by 3×10^{13} because of the Earth’s rotation, and changed linearly at 3×10^{12} per day because the Earth is 90 degrees from perihelion in April. However, analysis of the data revealed no variations of either type within experimental errors, leading to a limit on the LPI violation parameter ∣α^{H}−α^{SCSO}∣<2×10^{2} [130]. This bound has been improved using more stable frequency standards [68, 109]. The varying gravitational redshift of Earthbound clocks relative to the highly stable Millisecond Pulsar PSR 1937+21, caused by the Earth’s motion in the solar gravitational field around the EarthMoon center of mass (amplitude 4000 km), has been measured to about 10 percent, and the redshift of stable oscillator clocks on the Voyager spacecraft caused by Saturn’s gravitational field yielded a one percent test. The solar gravitational redshift has been tested to about two percent using infrared oxygen triplet lines at the limb of the Sun, and to one percent using oscillator clocks on the Galileo spacecraft (TEGP 2.4 (c) [147] and 14.1 (a) [147]).
Modern advances in navigation using Earthorbiting atomic clocks and accurate timetransfer must routinely take gravitational redshift and timedilation effects into account. For example, the Global Positioning System (GPS) provides absolute accuracies of around 15 m (even better in its military mode) anywhere on Earth, which corresponds to 50 nanoseconds in time accuracy at all times. Yet the difference in rate between satellite and ground clocks as a result of special and general relativistic effects is a whopping 39 microseconds per day (46 μs from the gravitational redshift, and 7 μs from time dilation). If these effects were not accurately accounted for, GPS would fail to function at its stated accuracy. This represents a welcome practical application of GR! (For the role of GR in GPS, see [8]; for a popular essay, see [140].)
Bounds on cosmological variation of fundamental constants of nongravitational physics. For references to earlier work, see TEGP 2.4 (c) [147].
Constant k  Limit on k/k per  Method 

Hubble time 1.2×  
10^{10} yr  
Fine structure constant α=e^{2}/ħc  4×10^{4}  Hmaser vs. Hg ion clock [109] 
9×10^{5}  ^{87}Rb fountain vs. Cs clock [115]  
6×10^{7}  Oklo Natural Reactor [41]  
6×10^{5}  21cm vs. molecular absorption at Z=0.7 [57]  
Weak interaction constant β=G_{f}m _{P} ^{2} c/ħ^{3}  1  ^{187}Re, ^{40}K decay rates 
0.1  Oklo Natural Reactor [41]  
0.06  
ep mass ratio  1  Mass shift in quasar spectra at Z∼2 
Proton gfactor (g_{P})  10^{5}  21cm vs. molecular absorption at Z=0.7 [57] 
2.2 Theoretical Frameworks for Analyzing EEP
2.2.1 Schiff’s conjecture
Because the three parts of the Einstein equivalence principle discussed above are so very different in their empirical consequences, it is tempting to regard them as independent theoretical principles. On the other hand, any complete and selfconsistent gravitation theory must possess suffcient mathematical machinery to make predictions for the outcomes of experiments that test each principle, and because there are limits to the number of ways that gravitation can be meshed with the special relativistic laws of physics, one might not be surprised if there were theoretical connections between the three subprinciples. For instance, the same mathematical formalism that produces equations describing the free fall of a hydrogen atom must also produce equations that determine the energy levels of hydrogen in a gravitational field, and thereby the ticking rate of a hydrogen maser clock. Hence a violation of EEP in the fundamental machinery of a theory that manifests itself as a violation of WEP might also be expected to show up as a violation of local position invariance. Around 1960, Schiff conjectured that this kind of connection was a necessary feature of any selfconsistent theory of gravity. More precisely, Schiff’s conjecture states that any complete, selfconsistent theory of gravity that embodies WEP necessarily embodies EEP. In other words, the validity of WEP alone guarantees the validity of local Lorentz and position invariance, and thereby of EEP.
If Schiff’s conjecture is correct, then Eötvös experiments may be seen as the direct empirical foundation for EEP, hence for the interpretation of gravity as a curvedspacetime phenomenon. Of course, a rigorous proof of such a conjecture is impossible (indeed, some special counterexamples are known), yet a number of powerful “plausibility” arguments can be formulated.
2.2.1.1 Box 1. The TH∊μ formalism
 1.
Coordinate system and conventions: x^{0}=t=time coordinate associated with the static nature of the static spherically symmetric (SSS) gravitational field; x=(x, y, z)=isotropic quasiCartesian spatial coordinates; spatial vector and gradient operations as in Cartesian space.
 2.Matter and field variables:

m_{0a}=rest mass of particle a.

e_{a}=charge of particle a.

x _{ a} ^{ μ} (t)=world line of particle a.

v _{ a} ^{ μ} =dx _{ a} ^{ μ} /dt=coordinate velocity of particle a.

A_{μ}=electromagnetic vector potential; E=∇A_{0}−∂A/∂t, B=∇×A

 3.
Gravitational potential: U(x)
 4.
Arbitrary functions: T(U), H(U), ∊(U), μ(U); EEP is satisfied if ∊=¼=(H/T)^{1/2} for all U.
 5.Action:$$ I =  \sum\limits_a {{m_{0a}}\int {{{\left( {T  Hv_a^2} \right)}^{1/2}}dt + \sum\limits_a {{e_a}} \int {{A_\mu }\left( {x_a^\nu } \right)v_a^\mu dt + {{\left( {8\pi } \right)}^{  1}}\int {\left( { \epsilon {E^2}  {\mu ^{  1}}{B^2}} \right){d^4}x.} } } } $$
 6.NonMetric parameters:where c_{0}=(T_{0}/H_{0})^{1/2} and subscript “0” refers to a chosen point in space. If EEP is satisfied, Γ_{0}≡Λ_{0}≡ϒ_{0}≡0.$$ \begin{array}{*{20}{l}} {{\Gamma _0} =  c_0^2\left( {\partial /\partial U} \right)\ln {{\left[ { \epsilon {{\left( {T/H} \right)}^{1/2}}} \right]}_0},}\\ {{\Lambda _0} =  c_0^2\left( {\partial /\partial U} \right)\ln {{\left[ {\mu {{\left( {T/H} \right)}^{1/2}}} \right]}_0},}\\ {{\Upsilon _0} = 1  {{\left( {T{H^{  1}} \epsilon \mu } \right)}_0},} \end{array} $$
2.2.2 The TH∊μ formalism
Lightman and Lee then calculated explicitly the rate of fall of a “test” body made up of interacting charged particles, and found that the rate was independent of the internal electromagnetic structure of the body (WEP) if and only if Eq. (7) was satisfied. In other words, WEP→EEP and Schiff’s conjecture was verified, at least within the restrictions built into the formalism.
The redshift is the standard one (α=0), independently of the nature of the clock if and only if Γ_{0}≡Λ_{0}≡0. Thus the VessotLevine rocket redshift experiment sets a limit on the parameter combination 3Γ_{0}−Λ_{0} (Figure 3); the nullredshift experiment comparing hydrogenmaser and SCSO clocks sets a limit on \( \left {{\alpha _{\rm{H}}}  {\alpha _{{\rm{SCSO}}}}} \right = \frac{3}{2}\left {{\Gamma _0}  {\Lambda _0}} \right \) . Alvarez and Mann [4, 3, 5, 6, 7] extended the TH∊μ formalism to permit analysis of such effects as the Lamb shift, anomalous magnetic moments and nonbaryonic effects, and placed interesting bounds on EEP violations.
2.2.3 The c^{2} formalism
The electrodynamics given by Eq. (15) can also be quantized, so that we may treat the interaction of photons with atoms via perturbation theory. The energy of a photon is ћ times its frequency ω, while its momentum is ћω/c. Using this approach, one finds that the difference in round trip travel times of light along the two arms of the interferometer in the MichelsonMorley experiment is given by L_{0}(v^{2}/c)(c^{2}−1). The experimental null result then leads to the bound on (c^{2}−1) shown on Figure 2. Similarly the anisotropy in energy levels is clearly illustrated by the tensorial terms in Eqs. (16) and (18); by evaluating E͂ _{B} ^{ES ij} for each nucleus in the various HughesDrevertype experiments and comparing with the experimental limits on energy differences, one obtains the extremely tight bounds also shown on Figure 2.
The behavior of moving atomic clocks can also be analysed in detail, and bounds on (c^{2}−1) can be placed using results from tests of time dilation and of the propagation of light. In some cases, it is advantageous to combine the c^{2} framework with a “kinematical” viewpoint that treats a general class of boost transformations between moving frames. Such kinematical approaches have been discussed by Robertson, Mansouri and Sexl, and Will (see [144]).
2.3 EEP, particle physics, and the search for new interactions
In 1986, as a result of a detailed reanalysis of Eötvös’ original data, Fischbach et al. [62] suggested the existence of a fifth force of nature, with a strength of about a percent that of gravity, but with a range (as defined by the range λ of a Yukawa potential, e^{r/λ}/r) of a few hundred meters. This proposal dovetailed with earlier hints of a deviation from the inversesquare law of Newtonian gravitation derived from measurements of the gravity profile down deep mines in Australia, and with ideas from particle physics suggesting the possible presence of very lowmass particles with gravitationalstrength couplings. During the next four years numerous experiments looked for evidence of the fifth force by searching for compositiondependent differences in acceleration, with variants of the Eötvös experiment or with freefall Galileotype experiments. Although two early experiments reported positive evidence, the others all yielded null results. Over the range between one and 104 meters, the null experiments produced upper limits on the strength of a postulated fifth force between 10^{3} and 10^{6} of the strength of gravity. Interpreted as tests of WEP (corresponding to the limit of infiniterange forces), the results of two representative experiments from this period, the freefall Galileo experiment and the early EötWash experiment, are shown in Figure 1. At the same time, tests of the inversesquare law of gravity were carried out by comparing variations in gravity measurements up tall towers or down mines or boreholes with gravity variations predicted using the inverse square law together with Earth models and surface gravity data mathematically “continued” up the tower or down the hole. Despite early reports of anomalies, independent tower, borehole and seawater measurements now show no evidence of a deviation. Analyses of orbital data from planetary range measurements, lunar laser ranging, and laser tracking of the LAGEOS satellite verified the inversesquare law to parts in 10^{8} over scales of 10^{3} to 10^{5} km, and to parts in 10^{9} over planetary scales of several astronomical units [122]. The consensus at present is that there is no credible experimental evidence for a fifth force of nature. For reviews and bibliographies, see [61, 63, 64, 2, 143].
Nevertheless, theoretical evidence continues to mount that EEP is likely to be violated at some level, whether by quantum gravity effects, by effects arising from string theory, or by hitherto undetected interactions, albeit at levels well below those that motivated the fifthforce searches. Roughly speaking, in addition to the pure Einsteinian gravitational interaction, which respects EEP, theories such as string theory predict other interactions which do not. In string theory, for example, the existence of such EEPviolating fields is assured, but the theory is not yet mature enough to enable calculation of their strength (relative to gravity), or their range (whether they are long range, like gravity, or short range, like the nuclear and weak interactions, and thus too shortrange to be detectable).
Thus, EEP and related tests are now viewed as ways to discover or place constraints on new physical interactions, or as a branch of “nonaccelerator particle physics”, searching for the possible imprints of highenergy particle effects in the lowenergy realm of gravity. Whether current or proposed experiments can actually probe these phenomena meaningfully is an open question at the moment, largely because of a dearth of firm theoretical predictions. Despite this uncertainty, a number of experimental possibilities are being explored.
Concepts for an equivalence principle experiment in space have been developed. The project MICROSCOPE, designed to test WEP to 10^{15} has been approved by the French space agency CNES for a possible 2004 launch. Another, known as Satellite Test of the Equivalence Principle (STEP), is under consideration as a possible joint effort of NASA and the European Space Agency (ESA), with the goal of a 10^{18} test. The gravitational redshift could be improved to the 10^{10} level using atomic clocks on board a spacecraft which would travel to within four solar radii of the Sun. Laboratory tests of the gravitational inverse square law at submillimeter scales are being developed as ways to search for new shortrange interactions or for the existence of large extra dimensions; the challenge of these experiments is to distinguish gravitationlike interactions from electromagnetic and quantum mechanical (Casimir) effects [88].
3 Tests of PostNewtonian Gravity
3.1 Metric theories of gravity and the strong equivalence principle
3.1.1 Universal coupling and the metric postulates
The overwhelming empirical evidence supporting the Einstein equivalence principle, discussed in the previous section, supports the conclusion that the only theories of gravity that have a hope of being viable are metric theories, or possibly theories that are metric apart from possible weak or shortrange nonmetric couplings (as in string theory). Therefore for the remainder of this article, we shall turn our attention exclusively to metric theories of gravity, which assume that (i) there exists a symmetric metric, (ii) test bodies follow geodesics of the metric, and (iii) in local Lorentz frames, the nongravitational laws of physics are those of special relativity.
The property that all nongravitational fields should couple in the same manner to a single gravitational field is sometimes called “universal coupling”. Because of it, one can discuss the metric as a property of spacetime itself rather than as a field over spacetime. This is because its properties may be measured and studied using a variety of different experimental devices, composed of different nongravitational fields and particles, and, because of universal coupling, the results will be independent of the device. Thus, for instance, the proper time between two events is a characteristic of spacetime and of the location of the events, not of the clocks used to measure it.
Consequently, if EEP is valid, the nongravitational laws of physics may be formulated by taking their special relativistic forms in terms of the Minkowski metric η and simply “going over” to new forms in terms of the curved spacetime metric g, using the mathematics of differential geometry. The details of this “going over” can be found in standard textbooks ([94, 136], TEGP 3.2. [147]).
3.1.2 The strong equivalence principle
In any metric theory of gravity, matter and nongravitational fields respond only to the spacetime metric g. In principle, however, there could exist other gravitational fields besides the metric, such as scalar fields, vector fields, and so on. If, by our strict definition of metric theory, matter does not couple to these fields, what can their role in gravitation theory be? Their role must be that of mediating the manner in which matter and nongravitational fields generate gravitational fields and produce the metric; once determined, however, the metric alone acts back on the matter in the manner prescribed by EEP.
What distinguishes one metric theory from another, therefore, is the number and kind of gravitational fields it contains in addition to the metric, and the equations that determine the structure and evolution of these fields. From this viewpoint, one can divide all metric theories of gravity into two fundamental classes: “purely dynamical” and “priorgeometric”.
By “purely dynamical metric theory” we mean any metric theory whose gravitational fields have their structure and evolution determined by coupled partial differential field equations. In other words, the behavior of each field is influenced to some extent by a coupling to at least one of the other fields in the theory. By “prior geometric” theory, we mean any metric theory that contains “absolute elements”, fields or equations whose structure and evolution are given a priori, and are independent of the structure and evolution of the other fields of the theory. These “absolute elements” typically include flat background metrics η, cosmic time coordinates t, and algebraic relationships among otherwise dynamical fields, such as g_{μν}=h_{μν}+k_{μ}k_{ν}, where h_{μν} and k_{μ} may be dynamical fields.
General relativity is a purely dynamical theory since it contains only one gravitational field, the metric itself, and its structure and evolution are governed by partial differential equations (Einstein’s equations). BransDicke theory and its generalizations are purely dynamical theories; the field equation for the metric involves the scalar field (as well as the matter as source), and that for the scalar field involves the metric. Rosen’s bimetric theory is a priorgeometric theory: It has a nondynamical, Riemannflat background metric η, and the field equations for the physical metric g involve η.
By discussing metric theories of gravity from this broad point of view, it is possible to draw some general conclusions about the nature of gravity in different metric theories, conclusions that are reminiscent of the Einstein equivalence principle, but that are subsumed under the name “strong equivalence principle”.
 (i)
A theory which contains only the metric g yields local gravitational physics which is independent of the location and velocity of the local system. This follows from the fact that the only field coupling the local system to the environment is g, and it is always possible to find a coordinate system in which g takes the Minkowski form at the boundary between the local system and the external environment. Thus the asymptotic values of g_{μν} are constants independent of location, and are asymptotically Lorentz invariant, thus independent of velocity. General relativity is an example of such a theory.
 (ii)
A theory which contains the metric g and dynamical scalar fields ϕ_{A} yields local gravitational physics which may depend on the location of the frame but which is independent of the velocity of the frame. This follows from the asymptotic Lorentz invariance of the Minkowski metric and of the scalar fields, but now the asymptotic values of the scalar fields may depend on the location of the frame. An example is BransDicke theory, where the asymptotic scalar field determines the effective value of the gravitational constant, which can thus vary as ϕ varies. On the other hand, a form of velocity dependence in local physics can enter indirectly if the asymptotic values of the scalar field vary with time cosmologically. Then the rate of variation of the gravitational constant could depend on the velocity of the frame.
 (iii)
A theory which contains the metric g and additional dynamical vector or tensor fields or priorgeometric fields yields local gravitational physics which may have both location and velocitydependent effects.
 1.
WEP is valid for selfgravitating bodies as well as for test bodies.
 2.
The outcome of any local test experiment is independent of the velocity of the (freely falling) apparatus.
 3.
The outcome of any local test experiment is independent of where and when in the universe it is performed.
The above discussion of the coupling of auxiliary fields to local gravitating systems indicates that if SEP is strictly valid, there must be one and only one gravitational field in the universe, the metric g. These arguments are only suggestive however, and no rigorous proof of this statement is available at present. Empirically it has been found that every metric theory other than GR introduces auxiliary gravitational fields, either dynamical or prior geometric, and thus predicts violations of SEP at some level (here we ignore quantumtheory inspired modifications to GR involving “R^{2}” terms). General relativity seems to be the only metric theory that embodies SEP completely. This lends some credence to the conjecture SEP→General Relativity. In Sec. 3.6, we shall discuss experimental evidence for the validity of SEP.
3.2 The parametrized postNewtonian formalism
Despite the possible existence of longrange gravitational fields in addition to the metric in various metric theories of gravity, the postulates of those theories demand that matter and nongravitational fields be completely oblivious to them. The only gravitational field that enters the equations of motion is the metric g. The role of the other fields that a theory may contain can only be that of helping to generate the spacetime curvature associated with the metric. Matter may create these fields, and they plus the matter may generate the metric, but they cannot act back directly on the matter. Matter responds only to the metric.
Thus the metric and the equations of motion for matter become the primary entities for calculating observable effects, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric.
The PPN Parameters and their significance (note that α_{3} has been shown twice to indicate that it is a measure of two effects).
Pameter  What it means relative to GR  Value in GR  Value in semiconservative theories  Value in fullyconservative theories 

γ  How much spacecurvature produced by unit rest mass?  1  γ  γ 
β  How much “nonlinearity” in the superposition law for gravity?  1  β  β 
ξ  Preferredlocation effects?  0  ξ  ξ 
α _{1}  Preferredframe effects?  0  α _{1}  0 
α _{2}  0  α _{2}  0  
α _{3}  0  0  0  
α _{3}  Violation of conservation of total momentum?  0  0  0 
ζ _{1}  0  0  0  
ζ _{2}  0  0  0  
ζ _{3}  0  0  0  
ζ _{4}  0  0  0 
The parameters γ and β are the usual EddingtonRobertsonSchiff parameters used to describe the “classical” tests of GR, and are in some sense the most important; they are the only nonzero parameters in GR and scalartensor gravity. The parameter ξ is nonzero in any theory of gravity that predicts preferredlocation effects such as a galaxyinduced anisotropy in the local gravitational constant G_{L} (also called “Whitehead” effects); α_{1}, α_{2}, α_{3} measure whether or not the theory predicts postNewtonian preferredframe effects; α3, ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4} measure whether or not the theory predicts violations of global conservation laws for total momentum. Next to γ and β, the parameters α_{1} and α_{2} occur most frequently with nontrivial null values. In Table 2 we show the values these parameters take (i) in GR, (ii) in any theory of gravity that possesses conservation laws for total momentum, called “semiconservative” (any theory that is based on an invariant action principle is semiconservative), and (iii) in any theory that in addition possesses six global conservation laws for angular momentum, called “fully conservative” (such theories automatically predict no postNewtonian preferredframe effects). Semiconservative theories have five free PPN parameters (γ, β, ξ, α_{1}, α_{2}) while fully conservative theories have three (γ, β, ξ).
The PPN formalism was pioneered by Kenneth Nordtvedt [98], who studied the postNewtonian metric of a system of gravitating point masses, extending earlier work by Eddington, Robertson and Schiff (TEGP 4.2 [147]). A general and unified version of the PPN formalism was developed by Will and Nordtvedt. The canonical version, with conventions altered to be more in accord with standard textbooks such as [94], is discussed in detail in TEGP 4 [147]. Other versions of the PPN formalism have been developed to deal with point masses with charge, fluid with anisotropic stresses, bodies with strong internal gravity, and postpostNewtonian effects (TEGP 4.2, 14.2 [147]).
3.2.1 Box 2. The Parametrized PostNewtonian formalism
 1.
Coordinate system: The framework uses a nearly globally Lorentz coordinate system in which the coordinates are (t, x^{1}, x^{2}, x^{3}). Threedimensional, Euclidean vector notation is used throughout. All coordinate arbitrariness (“gauge freedom”) has been removed by specialization of the coordinates to the standard PPN gauge (TEGP 4.2 [147]). Units are chosen so that G=c=1, where G is the physically measured Newtonian constant far from the solar system.
 2.Matter variables:

ρ: density of rest mass as measured in a local freely falling frame momentarily comoving with the gravitating matter;

v^{i}=(dxi/dt): coordinate velocity of the matter;

w^{i}: coordinate velocity of the PPN coordinate system relative to the mean restframe of the universe;

p: pressure as measured in a local freely falling frame momentarily comoving with the matter;

П internal energy per unit rest mass. It includes all forms of nonrestmass, nongravitational energy, e.g. energy of compression and thermal energy.

 3.
PPN parameters: γ, β, ξ, α_{1}, α_{2}, α_{3}, ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4}.
 4.$$ \begin{array}{*{20}{l}} {{g_{00}} =  1 + 2U  2\beta {U^2}  2\xi {\Phi_W} + \left( {2\gamma + 2 + {\alpha _3} + {\zeta _1}  2\xi } \right){\Phi _1} + 2\left( {3\gamma  2\beta + 1 + {\zeta _2} + \xi } \right){\Phi _2} + 2\left( {1 + {\zeta _3}} \right){\Phi _3} + 2\left( {3\gamma + 3{\zeta _4}  2\xi } \right){\Phi _4}  \left( {{\zeta _1}  2\xi } \right){\mathcal A}  \left( {{\alpha _1}  {\alpha _2}  {\alpha _3}} \right){w^2}U  {\alpha _2}{w^i}{w^j}{U_{ij}} + \left( {2{\alpha _3}  {\alpha _1}} \right){w^i}{V_i} + O\left( {{ \epsilon ^3}} \right),}\\ {{g_{0i}} =  \frac{1}{2}\left( {4\gamma + 3 + {\alpha _1}  {\alpha _2} + {\zeta _1}  2\xi } \right){V_i}  \frac{1}{2}\left( {1 + {\alpha _2}  {\zeta _1} + 2\xi } \right){W_i}  \frac{1}{2}\left( {{\alpha _1}  2{\alpha _2}} \right){w^i}U  {\alpha _2}{w^j}{U_{ij}} + O\left( {{ \epsilon ^{5/2}}} \right),}\\ {{g_{ij}} = \left( {1 + 2\gamma U + O\left( {{ \epsilon ^2}} \right)} \right){\delta _{ij.}}} \end{array} $$
 5.Metric potentials:$$ \begin{array}{*{20}{c}} {U = \int {\frac{{\rho '}}{{\left {{\rm{x}}  {\rm{x'}}} \right}}{d^3}x',} }\\ {{U_{ij}} = \int {\frac{{\rho '{{\left( {x  x'} \right)}_i}{{\left( {x  x'} \right)}_j}}}{{{{\left {{\rm{x}}  {\rm{x'}}} \right}^3}}}{d^3}x',} }\\ {{\Phi _W} = \int {\frac{{\rho '\rho ''\left( {{\rm{x}}  {\rm{x'}}} \right)}}{{{{\left {{\rm{x}}  {\rm{x'}}} \right}^3}}} \cdot \left( {\frac{{{\rm{x'}}  {\rm{x''}}}}{{\left {{\rm{x}}  {\rm{x''}}} \right}}  \frac{{{\rm{x}}  {\rm{x''}}}}{{\left {{\rm{x'}}  {\rm{x''}}} \right}}} \right){d^3}x'{d^3}x'',} }\\ {{\mathcal A} = \int {\frac{{\rho '{{\left[ {{\rm{v'}} \cdot \left( {{\rm{x}}  {\rm{x'}}} \right)} \right]}^2}}}{{{{\left {{\rm{x}}  {\rm{x'}}} \right}^3}}}{d^3}x',} }\\ {{\Phi _1} = \int {\frac{{\rho '{{v'}^2}}}{{\left {{\rm{x}}  {\rm{x'}}} \right}}{d^3}x',} }\\ {{\Phi _2} = \int {\frac{{\rho 'U'}}{{\left {{\rm{x}}  {\rm{x'}}} \right}}{d^3}x',} }\\ {{\Phi _3} = \int {\frac{{\rho '\Pi '}}{{\left {{\rm{x}}  {\rm{x'}}} \right}}{d^3}x',} }\\ {{\Phi _4} = \int {\frac{{p'}}{{\left {{\rm{x}}  {\rm{x'}}} \right}}{d^3}x',} }\\ {{V_i} = \int {\frac{{\rho '{{v'}_i}}}{{\left {{\rm{x}}  {\rm{x'}}} \right}}{d^3}x',} }\\ {{W_i} = \int {\frac{{\rho '\left[ {{\rm{v'}} \cdot \left( {{\rm{x}}  {\rm{x'}}} \right)} \right]{{\left( {x  x'} \right)}_i}}}{{{{\left {{\rm{x}}  {\rm{x'}}} \right}^3}}}{d^3}x'.} } \end{array} $$
 6.Stressenergy tensor (perfect fluid):$$ \begin{array}{*{20}{l}} {{T^{00}} = \rho \left( {1 + \Pi + {v^2} + 2U} \right),}\\ {{T^{0i}} = \rho {v^i}\left( {1 + \Pi + {v^2} + 2U + p/\rho } \right),}\\ {{T^{ij}} = \rho {v^i}{v^j}\left( {1 + \Pi + {v^2} + 2U + p/\rho } \right) + p{\delta ^{ij}}\left( {1  2\gamma U} \right).} \end{array} $$
 7.

Stressed matter: \( {T^{\mu \nu }}_{;\nu } = 0, \)

Test bodies: \( {d^2}{x^\mu }/d{\lambda ^2} + {\Gamma ^\mu }_{\nu \lambda }\left( {d{x^\nu }/d\lambda } \right)\left( {d{x^\lambda }/d\lambda } \right) = 0, \)

Maxwell’s equations: \( \begin{array}{*{20}{c}} {{F^{\mu \nu }}_{;\nu } = 4\pi {J^\mu },}&{{F_{\mu \nu }} = {A_{\nu ;\mu }}  {A_{\mu ;\nu }}.} \end{array} \)

3.3 Competing theories of gravity
One of the important applications of the PPN formalism is the comparison and classification of alternative metric theories of gravity. The population of viable theories has fluctuated over the years as new effects and tests have been discovered, largely through the use of the PPN framework, which eliminated many theories thought previously to be viable. The theory population has also fluctuated as new, potentially viable theories have been invented.

A full compendium of alternative theories is given in TEGP 5 [147].

Many alternative metric theories developed during the 1970s and 1980s could be viewed as “strawman” theories, invented to prove that such theories exist or to illustrate particular properties. Few of these could be regarded as wellmotivated theories from the point of view, say, of field theory or particle physics. Examples are the vectortensor theories studied by Will, Nordtvedt and Hellings.

A number of theories fall into the class of “priorgeometric” theories, with absolute elements such as a flat background metric in addition to the physical metric. Most of these theories predict “preferredframe” effects, that have been tightly constrained by observations (see Sec. 3.6.2). An example is Rosen’s bimetric theory.

A large number of alternative theories of gravity predict gravitational wave emission substantially different from that of general relativity, in strong disagreement with observations of the binary pulsar (see Sec. 7).

Scalartensor modifications of GR have recently become very popular in unification schemes such as string theory, and in cosmological model building. Because the scalar fields are generally massive, the potentials in the postNewtonian limit will be modified by Yukawalike terms.
3.3.1 General relativity
The metric g is the sole dynamical field, and the theory contains no arbitrary functions or parameters, apart from the value of the Newtonian coupling constant G, which is measurable in laboratory experiments. Throughout this article, we ignore the cosmological constant λ. Although λ has significance for quantum field theory, quantum gravity, and cosmology, on the scale of the solarsystem or of stellar systems its effects are negligible, for values of λ corresponding to a cosmological closure density.
Metric theories and their PPN parameter values (α_{3}=ζ_{i}=0 for all cases).
Theory  Arbitrary Functions or Constants  Cosmic Matching Parameters  PPN Parameters  

γ  β  ξ  α _{1}  α _{2}  
General Relativity  none  none  1  1  0  0  0 
ScalarTensor  
BransDicke  ω  φ _{0}  \(\frac{{1 + \omega }}{{2 + \omega }}\)  1  0  0  0 
General  A(ϕ), V(ϕ)  ϕ _{0}  \(\frac{{1 + \omega }}{{2 + \omega }}\)  1+Λ  0  0  0 
Rosen’s Bimetric  none  c_{0}, c_{1}  1  1  0  0  \(\frac{{{c_0}}}{{{c_1}}}  1\) 
3.3.2 Scalartensor theories
Scalar fields coupled to gravity or matter are also ubiquitous in particlephysicsinspired models of unification, such as string theory. In some models, the coupling to matter may lead to violations of WEP, which are tested by Eötvöstype experiments. In many models the scalar field is massive; if the Compton wavelength is of macroscopic scale, its effects are those of a “fifth force”. Only if the theory can be cast as a metric theory with a scalar field of infinite range or of range long compared to the scale of the system in question (solar system) can the PPN framework be strictly applied. If the mass of the scalar field is sufficiently large that its range is microscopic, then, on solarsystem scales, the scalar field is suppressed, and the theory is essentially equivalent to general relativity. This is the case, for example, in the “oscillatingG” models of Accetta, Steinhardt and Will (see [120]), in which the potential function V(ϕ) contains both quadratic (mass) and quartic (selfinteraction) terms, causing the scalar field to oscillate (the initial amplitude of oscillation is provided by an inflationary epoch); highfrequency oscillations in the “effective” Newtonian constant G_{eff}≡G/φ=GA(ϕ)^{2} then result. The energy density in the oscillating scalar field can be enough to provide a cosmological closure density without resorting to dark matter, yet the value of ω today is so large that the theory’s local predictions are experimentally indistinguishable from GR. In other models, explored by Damour and EspositoFarèse [43], nonlinear scalarfield couplings can lead to “spontaneous scalarization” inside strongfield objects such as neutron stars, leading to large deviations from GR, even in the limit of very large ω.
3.4 Tests of the parameter γ
With the PPN formalism in hand, we are now ready to confront gravitation theories with the results of solarsystem experiments. In this section we focus on tests of the parameter γ, consisting of the deflection of light and the time delay of light.
3.4.1 The deflection of light
It is interesting to note that the classic derivations of the deflection of light that use only the principle of equivalence or the corpuscular theory of light yield only the “1/2” part of the coefficient in front of the expression in Eq. (30). But the result of these calculations is the deflection of light relative to local straight lines, as established for example by rigid rods; however, because of space curvature around the Sun, determined by the PPN parameter γ, local straight lines are bent relative to asymptotic straight lines far from the Sun by just enough to yield the remaining factor “γ/2”. The first factor “1/2” holds in any metric theory, the second “γ/2” varies from theory to theory. Thus, calculations that purport to derive the full deflection using the equivalence principle alone are incorrect.
However, the development of VLBI, verylongbaseline radio interferometry, produced greatly improved determinations of the deflection of light. These techniques now have the capability of measuring angular separations and changes in angles as small as 100 microarcseconds. Early measurements took advantage of a series of heavenly coincidences: Each year, groups of strong quasistellar radio sources pass very close to the Sun (as seen from the Earth), including the group 3C273, 3C279, and 3C48, and the group 0111+02, 0119+11 and 0116+08. As the Earth moves in its orbit, changing the lines of sight of the quasars relative to the Sun, the angular separation δθ between pairs of quasars varies (Eq. (32)). The time variation in the quantities d, d_{r}, χ and Ф_{r} in Eq. (32) is determined using an accurate ephemeris for the Earth and initial directions for the quasars, and the resulting prediction for δθ as a function of time is used as a basis for a leastsquares fit of the measured δθ, with one of the fitted parameters being the coefficient 1/2(1+γ). A number of measurements of this kind over the period 1969–1975 yielded an accurate determination of the coefficient 1/2(1+γ) which has the value unity in GR. A 1995 VLBI measurement using 3C273 and 3C279 yielded (1+γ)/2=0.9996±0.0017 [85].
A recent series of transcontinental and intercontinental VLBI quasar and radio galaxy observations made primarily to monitor the Earth’s rotation (“VLBI” in Figure 5) was sensitive to the deflection of light over almost the entire celestial sphere (at 90° from the Sun, the deflection is still 4 milliarcseconds). A recent analysis of over 2 million VLBI observations yielded (1+γ)/2=0.99992±0.00014 [59]. Analysis of observations made by the Hipparcos optical astrometry satellite yielded a test at the level of 0.3 percent [66]. A VLBI measurement of the deflection of light by Jupiter was reported; the predicted deflection of about 300 microarcseconds was seen with about 50 percent accuracy [129]. The results of lightdeflection measurements are summarized in Figure 5.
3.4.2 The time delay of light
In the two decades following Irwin Shapiro’s 1964 discovery of this effect as a theoretical consequence of general relativity, several highprecision measurements were made using radar ranging to targets passing through superior conjunction. Since one does not have access to a “Newtonian” signal against which to compare the roundtrip travel time of the observed signal, it is necessary to do a differential measurement of the variations in roundtrip travel times as the target passes through superior conjunction, and to look for the logarithmic behavior of Eq. (34). In order to do this accurately however, one must take into account the variations in roundtrip travel time due to the orbital motion of the target relative to the Earth. This is done by using radarranging (and possibly other) data on the target taken when it is far from superior conjunction (i.e. when the timedelay term is negligible) to determine an accurate ephemeris for the target, using the ephemeris to predict the PPN coordinate trajectory x_{e}(t) near superior conjunction, then combining that trajectory with the trajectory of the Earth x_{⊕}(t) to determine the Newtonian roundtrip time and the logarithmic term in Eq. (34). The resulting predicted roundtrip travel times in terms of the unknown coefficient 1/2(1+γ) are then fit to the measured travel times using the method of leastsquares, and an estimate obtained for 1/2(1+γ).
The targets employed included planets, such as Mercury or Venus, used as passive reflectors of the radar signals (“passive radar”), and artificial satellites, such as Mariners 6 and 7, Voyager 2, and the Viking Mars landers and orbiters, used as active retransmitters of the radar signals (“active radar”).
The results for the coefficient 1/2(1+γ) of all radar timedelay measurements performed to date (including a measurement of the oneway time delay of signals from the millisecond pulsar PSR 1937+21) are shown in Figure 5 (see TEGP 7.2 [147] for discussion and references). The Viking experiment resulted in a 0.1 percent measurement [111].
From the results of VLBI lightdeflection experiments, we can conclude that the coeficient 1/2(1+γ) must be within at most 0.014 percent of unity. Scalartensor theories must have ω>3500 to be compatible with this constraint.
3.5 The perihelion shift of Mercury
The explanation of the anomalous perihelion shift of Mercury’s orbit was another of the triumphs of GR. This had been an unsolved problem in celestial mechanics for over half a century, since the announcement by Le Verrier in 1859 that, after the perturbing effects of the planets on Mercury’s orbit had been accounted for, and after the effect of the precession of the equinoxes on the astronomical coordinate system had been subtracted, there remained in the data an unexplained advance in the perihelion of Mercury. The modern value for this discrepancy is 43 arcseconds per century. A number of ad hoc proposals were made in an attempt to account for this excess, including, among others, the existence of a new planet Vulcan near the Sun, a ring of planetoids, a solar quadrupole moment and a deviation from the inversesquare law of gravitation, but none was successful. General relativity accounted for the anomalous shift in a natural way without disturbing the agreement with other planetary observations.
3.6 Tests of the strong equivalence principle
The next class of solarsystem experiments that test relativistic gravitational effects may be called tests of the strong equivalence principle (SEP). In Sec. 3.1.2 we pointed out that many metric theories of gravity (perhaps all except GR) can be expected to violate one or more aspects of SEP. Among the testable violations of SEP are a violation of the weak equivalence principle for gravitating bodies that leads to perturbations in the EarthMoon orbit; preferredlocation and preferredframe effects in the locally measured gravitational constant that could produce observable geophysical effects; and possible variations in the gravitational constant over cosmological timescales.
3.6.1 The Nordtvedt effect and the lunar Eötvös experiment
Since August 1969, when the first successful acquisition was made of a laser signal reflected from the Apollo 11 retroreflector on the Moon, the lunar laserranging experiment (LURE) has made regular measurements of the roundtrip travel times of laser pulses between a network of observatories and the lunar retroreflectors, with accuracies that are approaching 50 ps (1 cm). These measurements are fit using the method of leastsquares to a theoretical model for the lunar motion that takes into account perturbations due to the Sun and the other planets, tidal interactions, and postNewtonian gravitational effects. The predicted roundtrip travel times between retroreflector and telescope also take into account the librations of the Moon, the orientation of the Earth, the location of the observatory, and atmospheric effects on the signal propagation. The “Nordtvedt” parameter η along with several other important parameters of the model are then estimated in the leastsquares method.
Several independent analyses of the data found no evidence, within experimental uncertainty, for the Nordtvedt effect (for recent results see [56, 154, 96]). Their results can be summarized by the bound η<0.001. These results represent a limit on a possible violation of WEP for massive bodies of 5 parts in 10^{13} (compare Figure 1). For BransDicke theory, these results force a lower limit on the coupling constant ω of 1000. Note that, at this level of precision, one cannot regard the results of lunar laser ranging as a “clean” test of SEP until one eliminates the possibility of a compensating violation of WEP for the two bodies, because the chemical compositions of the Earth and Moon differ in the relative fractions of iron and silicates. To this end, the EötWash group carried out an improved test of WEP for laboratory bodies whose chemical compositions mimic that of the Earth and Moon. The resulting bound of four parts in 10^{13} [10] reduces the ambiguity in the Lunar laser ranging bound, and establishes the firm limit on the universality of acceleration of gravitational binding energy at the level of η<1.3×10^{3}.
In GR, the Nordtvedt effect vanishes; at the level of several centimeters and below, a number of nonnull general relativistic effects should be present [102].
3.6.2 Preferredframe and preferredlocation effects
Some theories of gravity violate SEP by predicting that the outcomes of local gravitational experiments may depend on the velocity of the laboratory relative to the mean rest frame of the universe (preferredframe effects) or on the location of the laboratory relative to a nearby gravitating body (preferredlocation effects). In the postNewtonian limit, preferredframe effects are governed by the values of the PPN parameters α_{1}, α_{2}, and α_{3}, and some preferredlocation effects are governed by ξ (see Table 2).
Current limits on the PPN parameters. Here η is a combination of other parameters given by η=4β−γ−3−10ξ/3−α_{1}−2_{α2}/3−2ζ_{1}/3−ζ_{2}/3.
Parameter  Effect  Limit  Remarks 

γ−1  time delay  2×10^{3}  Viking ranging 
light deflection  3×10^{4}  VLBI  
β−1  perihelion shift  3×10^{3}  J_{2}=10^{7} from helioseismology 
Nordtvedt effect  6×10^{4}  η=4β−γ−3 assumed  
ξ  Earth tides  10^{3}  gravimeter data 
α _{1}  orbital polarization  10^{4}  Lunar laser ranging 
2×10^{4}  PSR J2317+1439  
α _{2}  spin precession  4×10^{7}  solar alignment with ecliptic 
α _{3}  pulsar acceleration  2×10^{20}  pulsar Ṗ statistics 
η  Nordtvedt effect  10^{3}  lunar laser ranging 
ζ _{1}    2×10^{2}  combined PPN bounds 
ζ _{2}  binary acceleration  4×10^{5}  P̈_{p} for PSR 1913+16 
ζ _{3}  Newton’s 3rd law  10^{8}  Lunar acceleration 
ζ _{4}      not independent 
3.6.3 Constancy of the Newtonian gravitational constant
Most theories of gravity that violate SEP predict that the locally measured Newtonian gravitational constant may vary with time as the universe evolves. For the scalartensor theories listed in Table 3, the predictions for Ġ/G can be written in terms of time derivatives of the asymptotic scalar field. Where G does change with cosmic evolution, its rate of variation should be of the order of the expansion rate of the universe, i.e. Ġ/G∼H_{0}, where H_{0} is the Hubble expansion parameter and is given by H_{0}=100 h km s^{1} M_{pc}^{1}=h×10^{10} yr^{1}, where current observations of the expansion of the universe give h≈0.7.
Constancy of the gravitational constant. For the pulsar data, the bounds are dependent upon the theory of gravity in the strongfield regime and on neutron star equation of state.
Method  Ġ/G(10^{12} yr^{1} 

Lunar Laser Ranging  0±8 
3±5  
Viking Radar  2±4 
2±10  
Binary Pulsar  11±11 
Pulsar PSR 0655+64  <55 
The best limits on Ġ/G still come from ranging measurements to the Viking landers and Lunar laser ranging measurements [56, 154, 96]. It has been suggested that radar observations of a Mercury orbiter over a twoyear mission (30 cm accuracy in range) could yield Δ(Ġ/G)∼10^{14} yr^{1}.
Although bounds on Ġ/G from solarsystem measurements can be correctly obtained in a phenomenological manner through the simple expedient of replacing G by G_{0}+Ġ_{0}(t−t_{0}) in Newton’s equations of motion, the same does not hold true for pulsar and binary pulsar timing measurements. The reason is that, in theories of gravity that violate SEP, such as scalartensor theories, the “mass” and moment of inertia of a gravitationally bound body may vary with variation in G. Because neutron stars are highly relativistic, the fractional variation in these quantities can be comparable to ΔG/G, the precise variation depending both on the equation of state of neutron star matter and on the theory of gravity in the strongfield regime. The variation in the moment of inertia affects the spin rate of the pulsar, while the variation in the mass can affect the orbital period in a manner that can subtract from the direct effect of a variation in G, given by Ṗ_{b}/P_{b}=2Ġ/G [101]. Thus, the bounds quoted in Table 5 for the binary pulsar PSR 1913+16 [51] and the pulsar PSR 0655+64 [69] are theorydependent and must be treated as merely suggestive.
3.7 Other tests of postNewtonian gravity
3.7.1 Tests of postNewtonian conservation laws
A nonzero value for any of these parameters would result in a violation of conservation of momentum, or of Newton’s third law in gravitating systems. An alternative statement of Newton’s third law for gravitating systems is that the “active gravitational mass”, that is the mass that determines the gravitational potential exhibited by a body, should equal the “passive gravitational mass”, the mass that determines the force on a body in a gravitational field. Such an equality guarantees the equality of action and reaction and of conservation of momentum, at least in the Newtonian limit.
A classic test of Newton’s third law for gravitating systems was carried out in 1968 by Kreuzer, in which the gravitational attraction of fluorine and bromine were compared to a precision of 5 parts in 10^{5}.
3.7.2 Geodetic precession
For a gyroscope orbiting the Earth, the precession is about 8 arcseconds per year. The Stanford Gyroscope Experiment has as one of its goals the measurement of this effect to 5×10^{5} (see below); if achieved, this would substantially improve the accuracy of the parameter γ.
3.7.3 Search for gravitomagnetism
Another proposal to look for an effect of gravitomagnetism is to measure the relative precession of the line of nodes of a pair of laserranged geodynamics satellites (LAGEOS), ideally with supplementary inclination angles; the inclinations must be supplementary in order to cancel the dominant nodal precession caused by the Earth’s Newtonian gravitational multipole moments. Unfortunately, the two existing LAGEOS satellites are not in appropriately inclined orbits, and no plans exist at present to launch a third satellite in a supplementary orbit. Nevertheless, by combing nodal precession data from LAGEOS I and II with perigee advance data from the slightly eccentric orbit of LAGEOS II, Ciufolini et al. reported a partial cancellation of multipole effects, and a resulting 20 percent confirmation of GR [34].
3.7.4 Improved PPN parameter values
A number of advanced space missions have been proposed in which spacecraft orbiters or landers and improved tracking capabilities could lead to significant improvements in values of the PPN parameters, of J_{2} of the Sun, and of Ġ/G. Doppler tracking of the Cassini spacecraft (launched to orbit and study Saturn in 1997) during its 2003 superior conjunction could measure γ to a few parts in 10^{5}, by measuring the time variation of the Shapiro delay [77]. A Mercury orbiter, in a twoyear experiment, with 3 cm range capability, could yield improvements in the perihelion shift to a part in 10^{4}, in γ to 4×10^{5}, in Ġ/G to 10^{14} yr^{1}, and in J_{2} to a few parts in 10^{8}. Proposals are being developed, primarily in Europe, for advanced space missions which will have tests of PPN parameters as key components, including GAIA, a highprecision astrometric telescope (successor to Hipparcos), which could measure lightdeflection and γ to the 10^{6} level [9]. Nordtvedt [103] has argued that “grand fits” of large solar system range data sets, including ranging to Mercury, Mars and the Moon, could yield substantially improved measurements of PPN parameters.
4 Strong Gravity and Gravitational Waves: A New Testing Ground
4.1 Strongfield systems in general relativity
4.1.1 Defining weak and strong gravity

The system may contain strongly relativistic objects, such as neutron stars or black holes, near and inside which ∊∼1, and the postNewtonian approximation breaks down. Nevertheless, under some circumstances, the orbital motion may be such that the interbody potential and orbital velocities still satisfy ∊≪1 so that a kind of postNewtonian approximation for the orbital motion might work; however, the strongfield internal gravity of the bodies could (especially in alternative theories of gravity) leave imprints on the orbital motion.

The evolution of the system may be affected by the emission of gravitational radiation. The 1PN approximation does not contain the effects of gravitational radiation backreaction. In the expression for the metric given in Box 2, radiation backreaction effects do not occur until O(∊^{7/2}) in g_{00}, O(∊^{3}) in g_{0i}, and O(∊^{5/2}) in g_{ij}. Consequently, in order to describe such systems, one must carry out a solution of the equations substantially beyond 1PN order, sufficient to incorporate the leading radiation damping terms at 2.5PN order.

The system may be highly relativistic in its orbital motion, so that U∼v^{2}∼1 even for the interbody field and orbital velocity. Systems like this include the late stage of the inspiral of binary systems of neutron stars or black holes, driven by gravitational radiation damping, prior to a merger and collapse to a final stationary state. Binary inspiral is one of the leading candidate sources for detection by a worldwide network of laser interferometric gravitational wave observatories nearing completion. A proper description of such systems requires not only equations for the motion of the binary carried to extraordinarily high PN orders (at least 3.5PN), but also requires equations for the farzone gravitational waveform measured at the detector, that are equally accurate to high PN orders beyond the leading “quadrupole” approximation.
Of course, some systems cannot be properly described by any postNewtonian approximation because their behavior is fundamentally controlled by strong gravity. These include the imploding cores of supernovae, the final merger of two compact objects, the quasinormalmode vibrations of neutron stars and black holes, the structure of rapidly rotating neutron stars, and so on. Phenomena such as these must be analysed using different techniques. Chief among these is the full solution of Einstein’s equations via numerical methods. This field of “numerical relativity” is a rapidly growing and maturing branch of gravitational physics, whose description is beyond the scope of this article. Another is black hole perturbation theory (see [93] for a review).
4.1.2 Compact bodies and the strong equivalence principle
When dealing with the motion and gravitational wave generation by orbiting bodies, one finds a remarkable simplification within general relativity. As long as the bodies are sufficiently wellseparated that one can ignore tidal interactions and other effects that depend upon the finite extent of the bodies (such as their quadrupole and higher multipole moments), then all aspects of their orbital behavior and gravitational wave generation can be characterized by just two parameters: mass and angular momentum. Whether their internal structure is highly relativistic, as in black holes or neutron stars, or nonrelativistic as in the Earth and Sun, only the mass and angular momentum are needed. Furthermore, both quantities are measurable in principle by examining the external gravitational field of the bodies, and make no reference whatsoever to their interiors.
Damour [39] calls this the “effacement” of the bodies’ internal structure. It is a consequence of the strong equivalence principle (SEP), described in Section 3.1.2.
General relativity satisfies SEP because it contains one and only one gravitational field, the spacetime metric g_{μν}. Consider the motion of a body in a binary system, whose size is small compared to the binary separation. Surround the body by a region that is large compared to the size of the body, yet small compared to the separation. Because of the general covariance of the theory, one can choose a freelyfalling coordinate system which comoves with the body, whose spacetime metric takes the Minkowski form at its outer boundary (ignoring tidal effects generated by the companion). There is thus no evidence of the presence of the companion body, and the structure of the chosen body can be obtained using the field equations of GR in this coordinate system. Far from the chosen body, the metric is characterized by the mass and angular momentum (assuming that one ignores quadrupole and higher multipole moments of the body) as measured far from the body using orbiting test particles and gyroscopes. These asymptotically measured quantities are oblivious to the body7#x2019;s internal structure. A black hole of mass m and a planet of mass m would produce identical spacetimes in this outer region.
The geometry of this region surrounding the one body must be matched to the geometry provided by the companion body. Einstein’s equations provide consistency conditions for this matching that yield constraints on the motion of the bodies. These are the equations of motion. As a result the motion of two planets of mass and angular momentum m_{1}, m_{2}, J_{1} and J_{2} is identical to that of two black holes of the same mass and angular momentum (again, ignoring tidal effects).
This effacement does not occur in an alternative gravitional theory like scalartensor gravity. There, in addition to the spacetime metric, a scalar field φ is generated by the masses of the bodies, and controls the local value of the gravitational coupling constant (i.e. G is a function of φ). Now, in the local frame surrounding one of the bodies in our binary system, while the metric can still be made Minkowskian far away, the scalar field will take on a value φ_{0} determined by the companion body. This can affect the value of G inside the chosen body, alter its internal structure (specifically its gravitational binding energy) and hence alter its mass. Effectively, each mass becomes several functions m_{A}(φ) of the value of the scalar field at its location, and several distinct masses come into play, inertial mass, gravitational mass, “radiation” mass, etc. The precise nature of the functions will depend on the body, specifically on its gravitational binding energy, and as a result, the motion and gravitational radiation may depend on the internal structure of each body. For compact bodies such as neutron stars, and black holes these internal structure effects could be large; for example, the gravitational binding energy of a neutron star can be 40 percent of its total mass. At 1PN order, the leading manifestation of this effect is the Nordtvedt effect.
This is how the study of orbiting systems containing compact objects provides strongfield tests of general relativity. Even though the strongfield nature of the bodies is effaced in GR, it is not in other theories, thus any result in agreement with the predictions of GR constitutes a kind of “null” test of strongfield gravity.
4.2 Motion and gravitational radiation in general relativity
The motion of bodies and the generation of gravitational radiation are longstanding problems that date back to the first years following the publication of GR, when Einstein calculated the gravitational radiation emitted by a laboratoryscale object using the linearized version of GR, and de Sitter calculated Nbody equations of motion for bodies in the 1PN approximation to GR. It has at times been controversial, with disputes over such issues as whether Einstein’s equations alone imply equations of motion for bodies (Einstein, Infeld and Hoffman demonstrated explicitly that they do, using a matching procedure similar to the one described above), whether gravitational waves are real or are artifacts of general covariance (Einstein waffled; Bondi and colleagues proved their reality rigorously in the 1950s), and even over algebraic errors (Einstein erred by a factor of 2 in his first radiation calculation; Eddington found the mistake). Shortly after the discovery of the binary pulsar PSR 1913+16 in 1974, questions were raised about the foundations of the “quadrupole formula” for gravitational radiation damping (and in some quarters, even about its quantitative validity). These questions were answered in part by theoretical work designed to shore up the foundations of the quadrupole approximation, and in part (perhaps mostly) by the agreement between the predictions of the quadrupole formula and the observed rate of damping of the pulsar’s orbit (see Section 5.1). Damour [39] gives a thorough review of this subject.
The problem of motion and radiation has received renewed interest since 1990, with proposals for construction of largescale laser interferometric gravitationalwave observatories, such as the LIGO project in the US, VIRGO and GEO600 in Europe, and TAMA300 in Japan, and the realization that a leading candidate source of detectable waves would be the inspiral, driven by gravitational radiation damping, of a binary system of compact objects (neutron stars or black holes) [1, 127]. The analysis of signals from such systems will require theoretical predictions from GR that are extremely accurate, well beyond the leadingorder prediction of Newtonian or even postNewtonian gravity for the orbits, and well beyond the leadingorder formulae for gravitational waves.
This presented a major theoretical challenge: to calculate the motion and radiation of systems of compact objects to very high PN order, a formidable algebraic task, while addressing a number of issues of principle that have historically plagued this subject, sufficiently well to ensure that the results were physically meaningful. This challenge is in the process of being met, so that we may soon see a remarkable convergence between observational data and accurate predictions of gravitational theory that could provide new, strongfield tests of GR.
Here we give a brief overview of the problem of motion and gravitational radiation.
4.3 Einstein’s equations in “relaxed” form
At the same time, just as in electromagnetism, the formal integral (47) must be handled differently, depending on whether the field point is in the far zone or the near zone. For field points in the far zone or radiation zone, x>λ̴>x′ (λ̴ is the gravitational wavelength/2π), the field can be expanded in inverse powers of R=x in a multipole expansion, evaluated at the “retarded time” t−R. The leading term in 1/R is the gravitational waveform. For field points in the near zone or induction zone, x∼x′<λ̴, the field is expanded in powers of x−x′ about the local time t, yielding instantaneous potentials that go into the equations of motion.
However, because the source τ^{αβ} contains h^{αβ} itself, it is not confined to a compact region, but extends over all spacetime. As a result, there is a danger that the integrals involved in the various expansions will diverge or be illdefined. This consequence of the nonlinearity of Einstein’s equations has bedeviled the subject of gravitational radiation for decades. Numerous approaches have been developed to try to handle this difficulty. The “postMinkowskian” method of Blanchet, Damour and Iyer [19, 20, 21, 45, 22, 15] solves Einstein’s equations by two different techniques, one in the near zone and one in the far zone, and uses the method of singular asymptotic matching to join the solutions in an overlap region. The method provides a natural “regularization” technique to control potentially divergent integrals. The “Direct Integration of the Relaxed Einstein Equations” (DIRE) approach of Will, Wiseman and Pati [152, 105] retains Eq. (47) as the global solution, but splits the integration into one over the near zone and another over the far zone, and uses different integration variables to carry out the explicit integrals over the two zones. In the DIRE method, all integrals are finite and convergent.
These methods assume from the outset that gravity is sufficiently weak that ‖h^{αβ}‖<1 and harmonic coordinates exists everywhere, including inside the bodies. Thus, in order to apply the results to cases where the bodies may be neutron stars or black holes, one relies upon the strong equivalence principle to argue that, if tidal forces are ignored, and equations are expressed in terms of masses and spins, one can simply extrapolate the results unchanged to the situation where the bodies are ultrarelativistic. While no general proof of this exists, it has been shown to be valid in specific circumstances, such as at 2PN order in the equations of motion, and for black holes moving in a Newtonian background field [39].
Methods such as these have resolved most of the issues that led to criticism of the foundations of gravitational radiation theory during the 1970s.
4.4 Equations of motion and gravitational waveform
These formalisms have also been generalized to include the leading effects of spinorbit and spinspin coupling between the bodies [83, 82].
Another approach to gravitational radiation is applicable to the special limit in which one mass is much smaller than the other. This is the method of blackhole perturbation theory. One begins with an exact background spacetime of a black hole, either the nonrotating Schwarzschild or the rotating Kerr solution, and perturbs it according to g_{μν}=g _{ μν} ^{(0)} +h_{μν}. The particle moves on a geodesic of the background spacetime, and a suitably defined source stressenergy tensor for the particle acts as a source for the gravitational perturbation and wave field h_{μν}. This method provides numerical results that are exact in v, as well as analytical results expressed as series in powers of v, both for nonrotating and for rotating black holes. For nonrotating holes, the analytical expansions have been carried to 5.5PN order, or ∊^{5.5} beyond the quadrupole approximation. All results of black hole perturbation agree precisely with the m_{1}→0 limit of the PN results, up to the highest PN order where they can be compared (for a detailed review see [93]).
4.5 Gravitational wave detection
5 Stellar System Tests of Gravitational Theory
5.1 The binary pulsar and general relativity
The 1974 discovery of the binary pulsar PSR 1913+16 by Joseph Taylor and Russell Hulse during a routine search for new pulsars provided the first possibility of probing new aspects of gravitational theory: the effects of strong relativistic internal gravitational fields on orbital dynamics, and the effects of gravitational radiation reaction. For reviews of the discovery and current status, see the published Nobel Prize lectures by Hulse and Taylor [76, 123]. For a thorough review of pulsars, including binary and millisecond pulsars, see the Living Review by Dunc Lorimer [89].
Parameters of the binary pulsar PSR 1913+16. The numbers in parentheses denote errors in the last digit. Data taken from an online catalogue of pulsars maintained by Stephen Thorsett of the University of California, Santa Cruz, see [ 128 ].
Parameter  Symbol  Value 

(units)  
(i) “Physical” parameters  
Right Ascension  α  19^{h}15^{m}28.^{s}00018(15) 
Declination  δ  16°06′27.″4043(3) 
Pulsar Period  P_{P} (ms)  59.029997929613(7) 
Derivative of Period  Ṗ _{P}  8.62713(8)×10^{18} 
(ii) “Keplerian” parameters  
Projected semimajor axis  a_{p} sin i (s)  2.3417592(19) 
Eccentricity  e  0.6171308(4) 
Orbital Period  P_{b} (day)  0.322997462736(7) 
Longitude of periastron  ω_{0} (°)  226.57528(6) 
Julian date of periastron  T_{0} (MJD)  46443.99588319(3) 
(iii) “PostKeplerian” parameters  
Mean rate of periastron advance  〈ω̇〉 (° yr^{1})  4.226621(11) 
Redshift/time dilation  γ′ (ms)  4.295(2) 
Orbital period derivative  Ṗ_{b} (10^{12})  2.422(6) 
Three factors make this system an arena where relativistic celestial mechanics must be used: the relatively large size of relativistic effects [v_{orbit}≈(m/r)^{1/2}≈10^{3}], a factor of 10 larger than the corresponding values for solarsystem orbits; the short orbital period, allowing secular effects to build up rapidly; and the cleanliness of the system, allowing accurate determinations of small effects. Because the orbital separation is large compared to the neutron stars’ compact size, tidal effects can be ignored. Just as Newtonian gravity is used as a tool for measuring astrophysical parameters of ordinary binary systems, so GR is used as a tool for measuring astrophysical parameters in the binary pulsar.
The observational parameters that are obtained from a leastsquares solution of the arrivaltime data fall into three groups: (i) nonorbital parameters, such as the pulsar period and its rate of change (defined at a given epoch), and the position of the pulsar on the sky; (ii) five “Keplerian” parameters, most closely related to those appropriate for standard Newtonian binary systems, such as the eccentricity e and the orbital period P_{b}; and (iii) five “postKeplerian” parameters. The five postKeplerian parameters are: 〈ω̇〉, the average rate of periastron advance; γ′, the amplitude of delays in arrival of pulses caused by the varying effects of the gravitational redshift and time dilation as the pulsar moves in its elliptical orbit at varying distances from the companion and with varying speeds; Ṗ_{b}, the rate of change of orbital period, caused predominantly by gravitational radiation damping; and r and s=sin i, respectively the “range” and “shape” of the Shapiro time delay of the pulsar signal as it propagates through the curved spacetime region near the companion, where i is the angle of inclination of the orbit relative to the plane of the sky.
Because f_{b} and e are separately measured parameters, the measurement of the three postKeplerian parameters provide three constraints on the two unknown masses. The periastron shift measures the total mass of the system, Ṗ_{b} measures the chirp mass, and γ′ measures a complicated function of the masses. GR passes the test if it provides a consistent solution to these constraints, within the measurement errors.
The consistency among the constraints provides a test of the assumption that the two bodies behave as “point” masses, without complicated tidal effects, obeying the general relativistic equations of motion including gravitational radiation. It is also a test of strong gravity, in that the highly relativistic internal structure of the neutron stars does not influence their orbital motion, as predicted by the strong equivalence principle of GR.
Recent observations [84, 138] indicate variations in the pulse profile, which suggests that the pulsar is undergoing precession as it moves through the curved spacetime generated by its companion, an effect known as geodetic precession. The amount is consistent with GR, assuming that the pulsar’s spin is suitably misaligned with the orbital angular momentum. Unfortunately, the evidence suggests that the pulsar beam may precess out of our line of sight by 2020.
5.2 A population of binary pulsars?
 PSR 1534+12. This is a binary pulsar system in our galaxy. Its pulses are significantly stronger and narrower than those of PSR 1913+16, so timing measurements are more precise, reaching 3 μs accuracy. Its parameters are listed in Table 7 [118, 119]. The orbital plane appears to be almost edgeon relative to the line of sight (i≃80°); as a result the Shapiro delay is substantial, and separate values of the parameters r and s have been obtained with interesting accuracy. Assuming general relativity, one infers that the two masses are m_{1}=1.335±0.002M_{⊙} and m_{2}=1.344±0.002M_{⊙}. The rate of orbit decay Ṗ_{b} agrees with GR to about 15 percent, the precision limited by the poorly known distance to the pulsar, which introduces a significant uncertainty into the subtraction of galactic acceleration. Independently of Ṗ_{b}, measurement of the four other postKeplerian parameters gives two tests of strongfield gravity in the nonradiative regime [124].
Parameter
B1534+12
B2127+11C
B1855+09
B0655+64
(i) “Keplerian” parameters
a_{p} sin i (s)
3.729464(3)
2.520(3)
9.2307802(4)
4.125612(5)
e
0.2736775(5)
0.68141(2)
0.00002168(5)
0.0000075(11)
P_{b} (day)
0.42073729933(3)
0.335282052(6)
12.3271711905(6)
1.028669703(1)
(ii) “PostKeplerian” parameters
〈ω̇〉 (° yr^{1})
1.755794(19)
4.457(12)
γ′ (ms)
2.071(6)
4.9(1.1)
Ṗ_{b} (10^{12})
0.131(9)
<0.5
r(μs)
6.3(1.3)
1.27(10)
s=sin i
0.983(8)
0.9992(5)

PSR 2127+11C. This system appears to be a clone of the HulseTaylor binary pulsar, with very similar values for orbital period and eccentricity (see Table 7). The inferred total mass of the system is 2.706±0.011M_{⊙}. Because the system is in the globular cluster M15 (NGC 7078), it suffers Doppler shifts resulting from local accelerations, either by the mean cluster gravitational field or by nearby stars, that are more difficult to estimate than was the case with the galactic system PSR 1913+16. This may make a separate, precision measurement of the relativistic contribution to Ṗ_{b} impossible.

PSR 1855+09. This binary pulsar system is not particularly relativistic, with a long period (12 days) and highly circular orbit. However, because we observe the orbit nearly edge on, the Shapiro delay is large and measurable, as reflected in the postKeplerian parameters r and s.

PSR 0655+64. This system consists of a pulsar and a white dwarf companion in a nearly circular orbit. Only an upper limit on Ṗ_{b} has been placed.
5.3 Binary pulsars and alternative theories
Soon after the discovery of the binary pulsar it was widely hailed as a new testing ground for relativistic gravitational effects. As we have seen in the case of GR, in most respects, the system has lived up to, indeed exceeded, the early expectations.
On the other hand, the early observations of PSR 1913+16 already indicated that, in GR, the masses of the two bodies were nearly equal, so that, in theories of gravity that are in some sense “close” to GR, dipole gravitational radiation would not be a strong effect, because of the apparent symmetry of the system. The Rosen theory, and others like it, are not “close” to general relativity, except in their predictions for the weakfield, slowmotion regime of the solar system. When relativistic neutron stars are present, theories like these can predict strong effects on the motion of the bodies resulting from their internal highly relativistic gravitational structure (violations of SEP). As a consequence, the masses inferred from observations of the periastron shift and γ′ may be significantly different from those inferred using general relativity, and may be different from each other, leading to strong dipole gravitational radiation damping. By contrast, the BransDicke theory is “close” to GR, roughly speaking within 1/ω_{BD} of the predictions of the latter, for large values of the coupling constant ω_{BD} (here we use the subscript BD to distinguish the coupling constant from the periastron advance ω̇). Thus, despite the presence of dipole gravitational radiation, the binary pulsar provides at present only a weak test of BransDicke theory, not yet competitive with solarsystem tests.
5.4 Binary pulsars and scalartensor gravity
In order to estimate the sensitivities s_{a} and κ _{a} ^{*} , one must adopt an equation of state for the neutron stars. It is sufficient to restrict attention to relatively stiff neutron star equations of state in order to guarantee neutron stars of sufficient mass, approximately 1.4M_{⊙}. The lower limit on ω_{BD} required to give consistency among the constraints on 〈ω̇〉, γ and Ṗ_{b} as in Figure 6 is several hundred [153]. The combination of 〈ω̇〉 and γ give a constraint on the masses that is relatively weakly dependent on ξ, thus the constraint on ξ is dominated by Ṗ_{b} and is directly proportional to the measurement error in Ṗ_{b}; in order to achieve a constraint comparable to the solar system value of 3×10^{4}, the error in Ṗ _{b} ^{OBS} would have to be reduced by more than a factor of ten.
Alternatively, a binary pulsar system with dissimilar objects, such as a white dwarf or black hole companion, would provide potentially more promising tests of dipole radiation. Unfortunately, none has been discovered to date; the dissimilar system B0655+64, with a white dwarf companion is in a highly circular orbit, making measurement of the periastron shift meaningless, and is not as relativistic as 1913+16. From the upper limit on Ṗ_{b} (Table 7), one can infer at best the weak bound ω_{BD}>100.
6 Gravitational Wave Tests of Gravitational Theory
6.1 Gravitational wave observatories
Some time in the next decade, a new opportunity for testing relativistic gravity will be realized, with the commissioning and operation of kilometerscale, laser interferometric gravitational wave observatories in the U.S. (LIGO project), Europe (VIRGO and GEO600 projects) and Japan (TAMA300 project). Gravitationalwave searches at these observatories are scheduled to commence around 2002. The LIGO broadband antennas will have the capability of detecting and measuring the gravitational waveforms from astronomical sources in a frequency band between about 10 Hz (the seismic noise cutoff) and 500 Hz (the photon counting noise cutoff), with a maximum sensitivity to strain at around 100 Hz of h∼Δl/l∼10^{22} (rms). The most promising source for detection and study of the gravitational wave signal is the “inspiralling compact binary” — a binary system of neutron stars or black holes (or one of each) in the final minutes of a death dance leading to a violent merger. Such is the fate, for example, of the HulseTaylor binary pulsar PSR 1913+16 in about 300 million years. Given the expected sensitivity of the “advanced LIGO” (around 2007), which could see such sources out to hundreds of megaparsecs, it has been estimated that from 3 to 100 annual inspiral events could be detectable. Other sources, such as supernova core collapse events, instabilities in rapidly rotating nascent neutron stars, signals from nonaxisymmetric pulsars, and a stochastic background of waves, may be detectable (for reviews, see [1, 127]; for updates on the status of various projects, see [65, 32]).
A similar network of cryogenic resonantmass gravitational antennas have been in operation for many years, albeit at lower levels of sensitivity (h∼10^{19}). While modest improvements in sensitivity may be expected in the future, these resonant detectors are not expected to be competitive with the large interferometers, unless new designs involving bars of spherical, or nearly spherical shape come to fruition. These systems are primarily sensitive to waves in relatively narrow bands about frequencies in the hundreds to thousands of Hz range [104, 73, 14, 110].
In addition, plans are being developed for an orbiting laser interferometer space antenna (LISA for short). Such a system, consisting of three spacecraft separated by millions of kilometers, would be sensitive primarily in the very low frequency band between 10^{4} and 10^{1} Hz, with peak strain sensitivity of order h∼10^{23} [54].
In addition to opening a new astronomical window, the detailed observation of gravitational waves by such observatories may provide the means to test general relativistic predictions for the polarization and speed of the waves, and for gravitational radiation damping.
6.2 Polarization of gravitational waves
A suitable array of gravitational antennas could delineate or limit the number of modes present in a given wave. The strategy depends on whether or not the source direction is known. In general there are eight unknowns (six polarizations and two direction cosines), but only six measurables (R_{0i0j}). If the direction can be established by either association of the waves with optical or other observations, or by timeofflight measurements between separated detectors, then six suitably oriented detectors suffice to determine all six components. If the direction cannot be established, then the system is underdetermined, and no unique solution can be found. However, if one assumes that only transverse waves are present, then there are only three unknowns if the source direction is known, or five unknowns otherwise. Then the corresponding number (three or five) of detectors can determine the polarization. If distinct evidence were found of any mode other than the two transverse quadrupolar modes of GR, the result would be disastrous for GR. On the other hand, the absence of a breathing mode would not necessarily rule out scalartensor gravity, because the strength of that mode depends on the nature of the source.
Some of the details of implementing such polarization observations have been worked out for arrays of resonant cylindrical, diskshaped, spherical and truncated icosahedral detectors (TEGP 10.2 [147], for recent reviews see [87, 133]); initial work has been done to assess whether the groundbased or spacebased laser interferometers (or combinations of the two types) could perform interesting polarization measurements [134, 33, 90, 67]. Unfortunately for this purpose, the two LIGO observatories (in Washington and Louisiana states, respectively) have been constructed to have their respective arms as parallel as possible, apart from the curvature of the Earth; while this maximizes the joint sensitivity of the two detectors to gravitational waves, it minimizes their ability to detect two modes of polarization.
6.3 Gravitational radiation backreaction
In the binary pulsar, a test of GR was made possible by measuring at least three relativistic effects that depended upon only two unknown masses. The evolution of the orbital phase under the damping effect of gravitational radiation played a crucial role. Another situation in which measurement of orbital phase can lead to tests of GR is that of the inspiralling compact binary system. The key differences are that here gravitational radiation itself is the detected signal, rather than radio pulses, and the phase evolution alone carries all the information. In the binary pulsar, the first derivative of the binary frequency ḟ_{b}, was measured; here the full nonlinear variation of f_{b} as a function of time is measured.
Broadband laser interferometers are especially sensitive to the phase evolution of the gravitational waves, which carry the information about the orbital phase evolution. The analysis of gravitational wave data from such sources will involve some form of matched filtering of the noisy detector output against an ensemble of theoretical “template” waveforms which depend on the intrinsic parameters of the inspiralling binary, such as the component masses, spins, and so on, and on its inspiral evolution. How accurate must a template be in order to “match” the waveform from a given source (where by a match we mean maximizing the crosscorrelation or the signaltonoise ratio)? In the total accumulated phase of the wave detected in the sensitive bandwidth, the template must match the signal to a fraction of a cycle. For two inspiralling neutron stars, around 16,000 cycles should be detected during the final few minutes of inspiral; this implies a phasing accuracy of 10^{5} or better. Since v∼1/10 during the late inspiral, this means that correction terms in the phasing at the level of v^{5} or higher are needed. More formal analyses confirm this intuition [35, 60, 36, 108].
Because it is a slowmotion system (v∼10^{3}), the binary pulsar is sensitive only to the lowestorder effects of gravitational radiation as predicted by the quadrupole formula. Nevertheless, the first correction terms of order v and v^{2} to the quadrupole formula were calculated as early as 1976 [135] see TEGP 10.3 [147]).
But for laser interferometric observations of gravitational waves, the bottom line is that, in order to measure the astrophysical parameters of the source and to test the properties of the gravitational waves, it is necessary to derive the gravitational waveform and the resulting radiation backreaction on the orbit phasing at least to 2PN order beyond the quadrupole approximation, and probably to 3PN order.
Similar expressions can be derived for the loss of angular momentum and linear momentum. (For explicit formulas for noncircular orbits, see [70].) These losses react back on the orbit to circularize it and cause it to inspiral. The result is that the orbital phase (and consequently the gravitational wave phase) evolves nonlinearly with time. It is the sensitivity of the broadband LIGO and VIRGOtype detectors to phase that makes the higherorder contributions to df/dt so observationally relevant. A readytouse set of formulae for the 2PN gravitational waveform template, including the nonlinear evolution of the gravitational wave frequency (not including spin effects) have been published [28] and incorporated into the Gravitational Radiation Analysis and Simulation Package (GRASP), a software toolkit used in LIGO.
If the coefficients of each of the powers of f in Eq. (72) can be measured, then one again obtains more than two constraints on the two unknowns m_{1} and m_{2}, leading to the possibility to test GR. For example, Blanchet and Sathyaprakash [30, 29] have shown that, by observing a source with a sufficiently strong signal, an interesting test of the 4π coefficient of the “tail” term could be performed.
Another possibility involves gravitational waves from a small mass orbiting and inspiralling into a (possibly supermassive) spinning black hole. A general noncircular, nonequatorial orbit will precess around the hole, both in periastron and in orbital plane, leading to a complex gravitational waveform that carries information about the nonspherical, strongfield spacetime around the hole. According to GR, this spacetime must be the Kerr spacetime of a rotating black hole, uniquely specified by its mass and angular momentum, and consequently, observation of the waves could test this fundamental hypothesis of GR [114, 107].
6.4 Speed of gravitational waves
According to GR, in the limit in which the wavelength of gravitational waves is small compared to the radius of curvature of the background spacetime, the waves propagate along null geodesics of the background spacetime,i.e. they have the same speed c as light (in this section, we do not set c=1). In other theories, the speed could differ from c because of coupling of gravitation to “background” gravitational fields. For example, in the Rosen bimetric theory with a flat background metric η, gravitational waves follow null geodesics of η, while light follows null geodesics of g (TEGP 10.1 [147]).
The foregoing discussion assumes that the source emits both gravitational and electromagnetic radiation in detectable amounts, and that the relative time of emission can be established to sufficient accuracy, or can be shown to be sufficiently small.
However, there is a situation in which a bound on the graviton mass can be set using gravitational radiation alone [151]. That is the case of the inspiralling compact binary. Because the frequency of the gravitational radiation sweeps from low frequency at the initial moment of observation to higher frequency at the final moment, the speed of the gravitons emitted will vary, from lower speeds initially to higher speeds (closer to c) at the end. This will cause a distortion of the observed phasing of the waves and result in a shorter than expected overall time Δt_{a} of passage of a given number of cycles. Furthermore, through the technique of matched filtering, the parameters of the compact binary can be measured accurately, (assuming that GR is a good approximation to the orbital evolution, even in the presence of a massive graviton), and thereby the emission time Δt_{e} can be determined accurately. Roughly speaking, the “phase interval” fΔt in Eq. (76) can be measured to an accuracy 1/ρ, where ρ is the signaltonoise ratio.
Thus one can estimate the bounds on λ_{g} achievable for various compact inspiral systems, and for various detectors. For stellarmass inspiral (neutron stars or black holes) observed by the LIGO/VIRGO class of groundbased interferometers, D≈200 Mpc, f≈100 Hz, and fΔtρ^{1}≈1/10. The result is λ_{g}>10^{13} km. For supermassive binary black holes (10^{4} to 10^{7}M_{⊙}) observed by the proposed laser interferometer space antenna (LISA), D≈3 Gpc, f≈10^{3} Hz, and fΔt∼ρ^{1}1/1000. The result is λ_{g}>10^{17} km.
6.5 Other stronggravity tests
One of the central difficulties of testing general relativity in the strongfield regime is the possibility of contamination by uncertain or complex physics. In the solar system, weakfield gravitational effects could in most cases be measured cleanly and separately from nongravitational effects. The remarkable cleanliness of the binary pulsar permitted precise measurements of gravitational phenomena in a strongfield context.
Unfortunately, nature is rarely so kind. Still, under suitable conditions, qualitative and even quantitative strongfield tests of general relativity can be carried out.
One example is in cosmology. From a few seconds after the big bang until the present, the underlying physics of the universe is well understood, although significant uncertainties remain (amount of dark matter, value of the cosmological constant, the number of light neutrino families, etc.). Some alternative theories of gravity that are qualitatively different from GR fail to produce cosmologies that meet even the minimum requirements of agreeing qualitatively with bigbang nucleosynthesis (BBN) or the properties of the cosmic microwave background (TEGP 13.2 [147]). Others, such as BransDicke theory, are sufficiently close to GR (for large enough ω_{BD}) that they conform to all cosmological observations, given the underlying uncertainties. The generalized scalartensor theories, however, could have small ω_{BD} at early times, while evolving through the attractor mechanism to large ω_{BD} today. One way to test such theories is through bigbang nucleosynthesis, since the abundances of the light elements produced when the temperature of the universe was about 1 MeV are sensitive to the rate of expansion at that epoch, which in turn depends on the strength of interaction between geometry and the scalar field. Because the universe is radiationdominated at that epoch, uncertainties in the amount of cold dark matter or of the cosmological constant are unimportant. The nuclear reaction rates are reasonably well understood from laboratory experiments and theory, and the number of light neutrino families (3) conforms to evidence from particle accelerators. Thus, within modest uncertainties, one can assess the quantitative difference between the BBN predictions of GR and scalartensor gravity under strongfield conditions and compare with observations. The most sophisticated recent analysis [49] places bounds on the parameters α_{0} and β_{0} of the generalized framework of Damour and EspositoFarèse (see Sec. 5.4 and Fig. 8) that are weaker than solarsystem bounds for β_{0}<0.3, but substantially stronger for β>0.3.
Another example is the exploration of the spacetime near black holes via accreting matter. Observations of lowluminosity binary Xray sources suggest that a form of accretion known as advectiondominated accretion flow (ADAF) may be important. In this kind of flow, the accreting gas is too thin to radiate its energy efficiently, but instead transports (advects) it inward toward the central object. If the central object is a neutron star, the matter hits the surface and radiates the energy away; if it is a black hole, the matter and its advected energy disappear. Systems in which the accreting object is believed to be a black hole from estimates of its mass are indeed observed to be underluminous, compared to systems where the object is believe to be a neutron star. This has been regarded as the first astrophysical evidence for the existence of black hole event horizons (for a review, see [92]). While supporting one of the critical strongfield predictions of GR, the observations and models are not likely any time soon to be able to distinguish one gravitational theory from another (except for theories that do not predict black holes at all).
Another example involving accretion purports to explore the strongfield region just outside massive black holes in active galactic nuclei. Here, iron in the inner region of a thin accretion disk is irradiated by Xrayemitting material above or below the disk, and fluoresces in the K_{α} line. The spectral shape of the line depends on relativistic Doppler and curvedspacetime effects as the iron orbits the black hole near the innermost stable circular orbit, and could be used to determine whether the hole is a nonrotating Schwarzschild black hole, or a rotating Kerr black hole. Because of uncertainties in the detailed models, the results are inconclusive to date, but the combination of higherresolution observations and better modelling could lead to striking tests of strongfield predictions of GR.
7 Conclusions
We find that general relativity has held up under extensive experimental scrutiny. The question then arises, why bother to continue to test it? One reason is that gravity is a fundamental interaction of nature, and as such requires the most solid empirical underpinning we can provide. Another is that all attempts to quantize gravity and to unify it with the other forces suggest that the standard general relativity of Einstein is not likely to be the last word. Furthermore, the predictions of general relativity are fixed; the theory contains no adjustable constants so nothing can be changed. Thus every test of the theory is either a potentially deadly test or a possible probe for new physics. Although it is remarkable that this theory, born 80 years ago out of almost pure thought, has managed to survive every test, the possibility of finding a discrepancy will continue to drive experiments for years to come.
Notes
Acknowledgements
This work was supported in part by the National Science Foundation, Grant Number PHY 9600049.
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