Experimental Tests of General Relativity:
Recent Progress and Future Directions
Einstein’s general theory of relativity is the standard theory of gravity, especially where the needs of astronomy, astrophysics, cosmology and fundamental physics are concerned. As such, this theory is used for many practically important purposes involving spacecraft navigation, geodesy, time transfer, etc. Here I review the foundations of general relativity, discuss recent progress in the tests of relativistic gravity, and present motivations for the new generation of high-accuracy tests of new physics beyond general relativity. Space-based experiments in fundamental physics are capable today to uniquely address important questions related to the fundamental laws of nature. I discuss the advances in our understanding of fundamental physics that are anticipated in the near future and evaluate the discovery potential of a number of the recently proposed space-based gravitational experiments.
November 25, 2015 will mark the Centennial of general theory of relativity, which was developed by Albert Einstein between 1905 and 1915 Frederiks (1999); Vizgin (2001). Ever since its original publication Einstein (1907, 1915, 1916), the theory continues to be an active area of both theoretical and experimental research Turyshev et al. (2007); Turyshev (2008a).
The theory first demonstrated its empirical success in 1915 by explaining the anomalous perihelion precession of Mercury’s orbit. This anomaly was known long before Einstein; it amounts to 43 arcsec per century (”/cy) and cannot be explained within Newton’s gravity, thereby presenting a challenge for physicists and astronomers. In 1855, Urbain LeVerrier, who in 1846 predicted the existence of Neptune, a planet on an extreme orbit, wrote that the anomalous residue of the Mercurial precession would be accounted for if yet another planet, the as-yet undiscovered planet Vulcan, revolves inside the Mercurial orbit. Because of the proximity of the sun Vulcan would not be easily observed, but LeVerrier thought he, nevertheless, had detected it. However, no confirmation came in the decades that followed. It took another 60 years to solve this puzzle. In 1915, before publishing the historical paper containing the field equations of general relativity (e.g. Einstein (1915)), Einstein computed the expected perihelion precession of Mercury’s orbit. When he obtained the famous 43 ”/cy needed to account for the anomaly, he realized that a new era in gravitational physics had just begun!
Shortly thereafter, Sir Arthur Eddington’s 1919 observations of star lines-of-sight during a solar eclipse Dyson et al. (1920) confirmed the doubling of the deflection angles predicted by general relativity as compared to Newtonian and equivalence principle (EP) arguments.111Eddington had been aware of several alternative predictions for his experiment. In 1801 Johann Georg von Soldner Lenard (1921) had pointed out that Newtonian gravity predicts that the trajectory of starlight bends in the vicinity of a massive object, but the predicted effect is only half the value predicted by general relativity as calculated by Einstein Einstein (1911). Other investigators claimed that gravity will did affect light propagation. Eddington’s experiment settled these claims by pronouncing general relativity a winner. Observations were made simultaneously in the city of Sobral in Brazil and on the island of Principe off the west coast of Africa; these observations focused on determining the change in position of stars as they passed near the Sun on the celestial sphere. The results were presented on November 6, 1919 at a special joint meeting of the Royal Astronomical Society and the Royal Society of London Coles (2001). The data from Sobral, with measurements of seven stars in good visibility, yielded deflections of arcsec. The data from Principe were less convincing. Only five stars were reliably measured, and the conditions there led to a much larger error. Nevertheless, the obtained value was arcsec. Both were within of Einstein’s value of 1.74 and were more than two standard deviations away from both zero and the Newtonian value of 0.87. These observations became the first dedicated experiment to test general theory of relativity.222The early accuracy, however, was poor. Dyson et al. Dyson et al. (1920) quoted an optimistically low uncertainty in their measurement, which was argued by some to have been plagued with systematic error and possibly confirmation bias, although modern re-analysis of the dataset suggests that Eddington’s analysis was accurate Kennefick (2007). However, considerable uncertainty remained in these measurements for almost 50 years, until first interplanetary spacecraft and microwave tracking techniques became available. It was not until the late 1960s that it was definitively shown that the angle of deflection was the full value predicted by general relativity. In Europe, which was still recovering from the World War I, this result was considered spectacular news and occupied the front pages of most major newspapers making general relativity an instant success.
From these beginnings, the general theory of relativity has been verified at ever higher accuracy; presently, it successfully accounts for all data gathered to date. The true renaissance in the tests of general relativity began in 1970s with major advances in several disciplines notably in microwave spacecraft tracking, high precision astrometric observations, and lunar laser ranging (LLR) (see Fig.1).
Thus, analysis of 14 months’s worth of data obtained from radio ranging to the Viking spacecraft verified, to an estimated accuracy of 0.1%, the prediction of the general theory of relativity that the round-trip times of light signals traveling between the earth and Mars are increased by the direct effect of solar gravity Shapiro et al. (1976, 1977); Reasenberg et al. (1979). The corresponding value for the Eddington’s metric parameter333To describe the accuracy achieved in the solar system experiments, it is useful to refer to the parameterized post-Newtonian (PPN) formalism (see discussion in Sec. II.2 and Will (1993)). Two parameters are of interest here, the PPN parameters and , whose values in general relativity are . The introduction of and is useful with regard to measurement accuracies Turyshev (1996a). In the PPN formalism (originally developed by Eddington), the angle of light deflection is proportional to ( + 1)/2, so that astrometric measurements might be used for a precise determination of the parameter . The parameter contributes to the relativistic perihelion precession of a body’s orbit (see Will (1993)).. In 1978 the value of the PPN parameter was obtained at the level of . Spacecraft and planetary radar observations have reached an accuracy of 0.15% Williams et al. (2001); Anderson et al. (2002a); Pitjeva (2005).
Meanwhile, very long baseline interferometry (VLBI) has achieved accuracies of better than 0.1 mas (milliarcseconds of arc), and regular geodetic VLBI measurements have frequently been used to determine the space curvature parameter . Detailed analyses of VLBI data have yielded a consistent stream of improvements Robertson and Carter (1984); Robertson et al. (1991), Lebach et al. (1995), Eubanks et al. (1997) and Shapiro et al. (2004), resulting in the accuracy of better than 0.045% in the tests of gravity via astrometric VLBI observations.
LLR, a continuing legacy of the Apollo program, provided improved constraint on the combination of parameters Williams et al. (1996, 2001, 2004, 2004, 2008) In 2004, the analysis of LLR data Williams et al. (2004) constrained this combination as , leading to an accuracy of 0.011% in verification of general relativity via precision measurements of the lunar orbit.
Finally, microwave tracking of the Cassini spacecraft on its approach to Saturn improved the measurement accuracy of the parameter to , thereby reaching the current best accuracy of 0.002% provided by tests of gravity in the solar system Iess et al. (1999); Bertotti et al. (2003).
To date, general relativity is also in agreement with data from the binary and double pulsars. Recently, investigators have shown a considerable interest in the physical processes occurring in the strong gravitational field regime with relativistic pulsars providing a promising possibility to test gravity in this qualitatively different dynamical environment. Although, strictly speaking, the binary pulsars move in the weak gravitational field of a companion, they do provide precision tests of the strong-field regime Lange et al. (2001). This becomes clear when considering strong self-field effects which are predicted by the majority of alternative theories. Such effects would, clearly affect the pulsars’ orbital motion, allowing us to search for these effects and hence providing us with a unique precision strong-field test of gravity. The general theoretical framework for pulsar tests of strong-field gravity was introduced in Damour and Taylor (1992); the observational data for the initial tests were obtained with PSR1534 Taylor et al. (1992). An analysis of strong-field gravitational tests and their theoretical justification is presented in Damour and Esposito-Farese (1996a, b); Damour and Esposito-Farese (1998). By measuring relativistic corrections to the Keplerian description of the orbital motion, the recent analysis of the data collected from the double pulsar system, PSR J0737-3039A/B, found agreement with the general relativity within an uncertainty of at a confidence level Kramer et al. (2006), to date the most precise pulsar test of gravity yet obtained.
As a result, both in the weak field limit (as in our solar system) and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested. It is remarkable that more than 90 years after general relativity was conceived, Einstein’s theory has survived every test Will (2006). Such longevity and success make general relativity the de-facto “standard” theory of gravitation for all practical purposes involving spacecraft navigation and astrometry, astronomy, astrophysics, cosmology and fundamental physics Turyshev et al. (2007).
However, despite its remarkable success, there are many important reasons to question the validity of general relativity and to determine the level of accuracy at which it is violated. On the theoretical front, problems arise from several directions, most concerning the strong-gravitational field regime. This challenges include the appearance of spacetime singularities and the inability of classical description to describe the physics of very strong gravitational fields. A way out of this difficulty may be through gravity quantization. However, despite the success of modern gauge-field theories in describing the electromagnetic, weak, and strong interactions, we do not yet understand how gravity should be described at the quantum level.
The continued inability to merge gravity with quantum mechanics along with recent cosmological observations, indicates that the pure tensor gravity of general relativity needs modification. In theories that attempt to include gravity, new long-range forces arise as an addition to the Newtonian inverse-square law. Regardless of whether the cosmological constant should be included, there are also important reasons to consider additional fields, especially scalar fields. Although the latter naturally appear in these modern theories, their inclusion predicts a non-Einsteinian behavior of gravitating systems. These deviations from general relativity lead to violation of the EP, and to modification of large-scale gravitational phenomena, and they cast doubt upon the constancy of the fundamental constants. These predictions motivate new searches for very small deviations of relativistic gravity from the behavior prescribed by general relativity; they also provide a new theoretical paradigm and guidance for future space-based gravity experiments Turyshev et al. (2007); Turyshev (2008a).
Note that on the largest spatial scales, such as the galactic and cosmological scales, general relativity has not yet been subject to precision tests. Some researchers have interpreted observations supporting the presence of dark matter and dark energy as a failure of general relativity at large distances, at small accelerations, or at small curvatures (see discussion in Refs. Turyshev (2008a); Milgrom (2001); Bekenstein (2004); Moffat (2006); Dvali et al. (2000); Rubakov and Tinyakov (2008); Sotiriou and Faraoni (2008)). Fig. 2 shows our present knowledge of gravity at various distance scales; it also indicates the theories that have been proposed to explain various observed phenomena and the techniques that have been used to conduct experimental studies of gravity at various regimes. The very strong gravitational fields that must be present close to black holes, especially those supermassive black holes that are thought to power quasars and less active active galactic nuclei, belong to a field of intensely active research. Observations of these quasars and active galactic nuclei are difficult to obtain, and the interpretation of the observations is heavily dependent upon astrophysical models other than general relativity and competing fundamental theories of gravitation; however, such interpretations are qualitatively consistent with the black-hole concept as modeled in general relativity.
Today physics stands at the threshold of major discoveries Witten (2001); Turyshev (2008a); Wilczek (2008). Growing observational evidence points to the need for new physics. Efforts to discover new fundamental symmetries, investigations of the limits of established symmetries, tests of the general theory of relativity, searches for gravitational waves, and attempts to understand the nature of dark matter were among the topics that had been the focus of the scientific research at the end of the last century. These efforts have further intensified with the discovery of dark energy made in the late 1990s, which triggered many new activities aimed at answering important questions related to the most fundamental laws of Nature Wilczek (2006, 2008); Turyshev et al. (2007).
Historically, the nature of matter on Earth and the laws governing it were discovered in laboratories on Earth. Scientific progress in these experiments depends on clever experimental strategy and the use of advanced technologies needed to overcome the limits imposed by the environment typically present in Earth-based laboratories. Since recently, in many cases the conditions in these laboratories cannot be improved to the required levels of purity, thus, a space deployment becomes a natural and well justified next step.
Space deployment offers access to the conditions with a particular dynamical “purity” that are not available on Earth, but are pivotal for many pioneering investigations exploring the limits of modern physics. In particular, for many fundamental physics experiments, especially those aiming at exploring gravitation, cosmology, atomic physics, while achieving uttermost measurement precision, and increasingly so for high energy and particle astrophysics, a space-based location is the ultimate destination.
There are two approaches to physics research in space: one can detect and study signals from remote astrophysical objects (the “observatory” mode) or one can perform carefully designed experiments in space (the “laboratory” mode). Fig. 2 emphasizes the “areas of responsibility” of these two disciplines - observational and space-based laboratory research in fundamental physics. The two methods are complementary and the latter, which is the focus of this paper, has the advantage of utilizing the well-understood and controlled environments of a space-based laboratory.
Considering gravitation and fundamental physics, our solar system is the laboratory that offers many opportunities to improve the tests of relativistic gravity. A carefully designed gravitational experiment has the advantage to conduct tests in a controlled and well-understood environment and can achieve accuracies superior to its ground-based counterpart. Existing technologies allow one to take advantage of the unique environments found only in space, including variable gravity potentials, large distances, high velocity and low acceleration regimes, availability of pure geodetic trajectories, microgravity and thermally-stable environments (see discussion in Turyshev et al. (2007)). Among the favorable factors are also the greatly reduced contribution of non-gravitational sources of noise (presently down to ), access to significant variations of gravitational potential (thus, compared with terrestrial conditions, the Sun offers a factor of 3,000 increase in the strength of gravitational effects) and corresponding gravitational acceleration (for instance, a highly eccentric solar orbit with apoapsis of 5 AU and periapsis of 10 solar radii offers more than 2 orders of magnitude in variation of the solar gravity potential and 4 orders of magnitude in variation in the corresponding gravitational acceleration, clearly not available otherwise), and also access to large distances, velocities, separations, availability of remote benchmarks and inertial references — the conditions that simply can not be achieved on the ground.
With recent advances in several applied physics disciplines new instruments and technologies have become available. These include highly accurate atomic clocks, optical frequency combs, atom interferometers, drag-free technologies, low-thrust micro-propulsion techniques, optical transponders, long-baseline optical interferometers, etc. Turyshev (ed.) (2007); Phillips (2007). Today, a new generation of high performance quantum sensors (ultra-stable atomic clocks, accelerometers, gyroscopes, gravimeters, gravity gradiometers, etc.) is surpassing previous state-of-the-art instruments, demonstrating the high potential of these techniques based on the engineering and manipulation of atomic systems. Atomic clocks and inertial quantum sensors represent a key technology for accurate frequency measurements and ultra-precise monitoring of accelerations and rotations (see discussion in Turyshev et al. (2007)).
New quantum devices based on ultra-cold atoms will enable fundamental physics experiments testing quantum physics, physics beyond the Standard Model of fundamental particles and interactions, special relativity, gravitation and general relativity. Some of these instruments are already space-qualified, thereby enabling a number of high-precision investigations in laboratory fundamental physics in space. Therefore, in conjunction with the new high-precision measurement technologies, the unique conditions of space deployment are critical for the progress in the gravitational research. As a result, modern-day space-based experiments are capable of reaching very high accuracies in testing the foundations of modern physics and are well positioned to provide major advances in this area Turyshev (2008a).
In this paper I discuss recent solar-system gravitational experiments that have contributed to the progress in relativistic gravity research by providing important guidance in the search for the next theory of gravity. I also present theoretical motivation for new generation of high-precision gravitational experiments and discuss a number of recently proposed space-based tests of relativistic gravity.
The paper is organized as follows. Section II discusses the foundations of the general theory of relativity and reviews the results of recent experiments designed to test the foundations of this theory. We present the parameterized post-Newtonian formalism – a phenomenological framework that is used to facilitate experimental tests of relativistic gravity. Section III presents motivations for extending the theoretical model of gravity provided by general relativity; it presents models arising from string theory, discusses the scalar-tensor theories of gravity, and highlights the phenomenological implications of these proposals. I briefly review recent proposals to modify gravity on large scales and review their experimental implications. Section IV discusses future space-based experiments that aim to expand our knowledge of gravity. I focus on the space-based tests of general theory of relativity and discuss the experiments aiming to test the EP, local Lorentz and position invariances, the search for variability of the fundamental constants, tests of the gravitational inverse-square law and tests of alternative and modified-gravity theories. I present a list of the proposed space missions, focusing only on the most representative and viable concepts. I conclude in Section V.
Ii Testing The Foundations of General Relativity
General relativity is a tensor field theory of gravitation with universal coupling to the particles and fields of the Standard Model. It describes gravity as a universal deformation of the flat spacetime Minkowski metric, :
Alternatively, it can also be defined as the unique, consistent, local theory of a massless spin-2 field , whose source is the total, conserved energy-momentum tensor (see Yao et al. (2006) and references therein).
Classically Einstein (1915, 1916), the general theory of relativity is defined by two postulates. One of the postulates states that the action describing the propagation and self-interaction of the gravitational field is given by
where is Newton’s universal gravitational constant, is the matrix inverse of , and , is the Ricci scalar given as with the quantity being the Ricci tensor and are the Christoffel symbols.
The second postulate states that couples universally, and minimally, to all the fields of the Standard Model by replacing the Minkowski metric everywhere. Schematically (suppressing matrix indices and labels for the various gauge fields and fermions and for the Higgs doublet), this postulate can be given by
where , the covariant derivative contains, besides the usual gauge field terms, a (spin-dependent) gravitational contribution Weinberg (1972) and is the vacuum energy density. Applying the variational principle with respect to to the total action
one obtains the well-known Einstein’s field equations of the general theory of relativity,
where with being the (symmetric) energy-momentum tensor of the matter as described by the Standard Model with the Lagrangian density . With the value for the vacuum energy density , as measured by recent cosmological observations Weinberg (1989); Spergel et al. (2007), the cosmological constant is too small to be observed by solar system experiments, but is clearly important for grater scales.
The theory is invariant under arbitrary coordinate transformations: . To solve the field equations Eq. (5), one needs to fix this coordinate gauge freedom. E.g., the “harmonic gauge” (which is the analogue of the Lorentz gauge, , in electromagnetism) corresponds to imposing the condition .
Einstein’s equations Eq. (5) link the geometry of a four-dimensional, Riemannian manifold representing space-time with the energy-momentum contained in that space-time. Phenomena that, in classical mechanics, are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity.
ii.1 Scalar-Tensor Extensions to General Relativity
Metric theories have a special place among alternative theories of gravity. The reason for this is that, independently of the different principles at their foundations, the gravitational field in these theories affects the matter directly through the metric tensor , which is determined from the particular theory’s field equations. As a result, in contrast to Newtonian gravity, this tensor expresses the properties of a particular gravitational theory and carries information about the gravitational field of the bodies.
In many alternative theories of gravity, the gravitational coupling strength exhibits a dependence on a field of some sort; in scalar-tensor theories, this is a scalar field . A general action for these theories can be written as
where , , and are generic functions, are coupling functions, and is the Lagrangian density of the matter fields of the Standard Model Eq. (3).
The Brans–Dicke theory Brans and Dicke (1961) is the best known alternative theory of gravity. It corresponds to the choice
Note that in the Brans–Dicke theory the kinetic energy term of the field is non-canonical and that the latter has a dimension of energy squared. In this theory, the constant marks observational deviations from general relativity, which is recovered in the limit . In the context of the Brans-Dicke theory, one can operationally introduce Mach’s Principle which states that the inertia of bodies is due to their interaction with the matter distribution in the Universe. Indeed, in this theory the gravitational coupling is proportional to , which depends on the energy-momentum tensor of matter through the field equations. The stringent observational bound resulting from the 2003 experiment with the Cassini spacecraft require that Bertotti et al. (2003); Will (2006). There exist additional alternative theories that provide guidance for gravitational experiments (see Will (2006) for review).
ii.2 Metric Theories of Gravity and PPN Formalism
Generalizing on a phenomenological parameterization of the gravitational metric tensor field, which Eddington originally developed for a special case, a method called the parameterized post-Newtonian (PPN) formalism has been developed Nordtvedt (1968, 1968, 1969); Thorne and Will (1971); Will (1971a, b, c, d); Will and Nordtvedt (1972); Nordtvedt and Will (1972); Will (1973, 1993). This method represents the gravity tensor’s potentials for slowly moving bodies and weak inter-body gravity, and is valid for a broad class of metric theories, including general relativity as a unique case. The several parameters in the PPN metric expansion vary from theory to theory, and they are individually associated with various symmetries and invariance properties of the underlying theory (see Will (1993) for details).
If (for the sake of simplicity) one assumes that Lorentz invariance, local position invariance and total momentum conservation hold, the metric tensor for a system of point-like gravitational sources in four dimensions may be written as
where the indices and refer to the bodies and where includes body , whose motion is being investigated. is the gravitational constant for body given as , where is the universal Newtonian gravitational constant and is the isolated rest mass of a body . In addition, the vector is the barycentric radius-vector of this body, the vector is the vector directed from body to body , , and the vector is the unit vector along this direction.
Although general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases: For weak gravitational fields and slow speed (relative to the speed of light), the theory’s predictions converge on those of Newton’s law of gravity with some post-Newtonian corrections. The term in is the Newtonian limit; the terms multiplied by the parameters and , are post-Newtonian terms. The term multiplied by the post-post-Newtonian parameter also enters the calculation of the relativistic light propagation for some modern-day experiments Turyshev et al. (2007); Turyshev (2008a) (such as LATOR and BEACON, see Sec. IV.6).
In this special case, when only two PPN parameters (, ) are considered, these parameters have a clear physical meaning. The parameter represents the measure of the curvature of the space-time created by a unit rest mass; parameter represents a measure of the non-linearity of the law of superposition of the gravitational fields in the theory of gravity. General relativity, when analyzed in standard PPN gauge, gives , and the other eight parameters vanish; the theory is thus embedded in a two-dimensional space of theories.
The Brans–Dicke theory Brans and Dicke (1961) in addition to the metric tensor, contains a scalar field and an arbitrary coupling constant , which yields the two PPN parameter values, , , where is an unknown dimensionless parameter of this theory. Other general scalar-tensor theories yield different values of Damour and Nordtvedt (1993a, b); Turyshev (1996a).
Note that in the complete PPN framework, a particular metric theory of gravity in the PPN formalism with a specific coordinate gauge is fully characterized by means of 10 PPN parameters Turyshev (1996a); Will (1993). Thus, in addition to the parameters and , there are eight other parameters and (not included in Eqs. (II.2), see Will (1993) for details). The formalism uniquely prescribes the values of these parameters for the particular theory under study. Gravity experiments can be analyzed in terms of the PPN metric, and an ensemble of experiments determine the unique value for these parameters, (and hence the metric field itself).
To analyze the motion of an -body system one derives the Lagrangian function Einstein et al. (1938); Turyshev (1996a); Will (1993). Within the accuracy sufficient for most of the gravitational experiments in the solar system, this function for the motion of an -body system can be presented in the following form Landau and Lifshitz (1988); Moyer (2003); Turyshev (2008a):
The Lagrangian in Eq. (9), leads to the point-mass Newtonian and relativistic perturbative accelerations in the solar system’s barycentric frame444When describing the motion of spacecraft in the solar system the models also include forces from asteroids and planetary satellites Standish and Williams (in press, 2008).:
To determine the orbits of the planets and spacecraft one must also describe propagation of electro-magnetic signals between any of the two points in space. The corresponding light-time equation can be derived from the metric tensor Eq. (II.2) as below
where refers to the signal transmission time, and refers to the reception time. are the barycentric positions of the transmitter and receiver, and is their spatial separation. The terms proportional to are important only for the Sun and are negligible for all other bodies in the solar system.
This PPN expansion serves as a useful framework to test relativistic gravitation in the context of the gravitational experiments. The main properties of the PPN metric tensor given by Eqs. (II.2) are well established and are widely used in modern astronomical practice Moyer (1981a, b); Turyshev (1996a); Brumberg (1972, 1991); Standish et al. (1992); Will (1993). For practical purposes one uses the metric to derive the Lagrangian function of an -body gravitating system Turyshev (1996a); Will (1993) which is then used to derive the equations of motion for gravitating bodies and light, namely Eqs. (10) and (11). The general relativistic equations of motion Eq. (10) are then used to produce numerical codes for the purposes of construction solar system’s ephemerides, spacecraft orbit determination Moyer (2003); Standish et al. (1992); Turyshev (1996a), and analysis of the gravitational experiments in the solar system Turyshev (2008a); Will (1993); Turyshev et al. (2004a).
ii.3 PPN-Renormalized Extension of General Relativity
Given the phenomenological success of general relativity, it is convenient to use this theory to describe experiments. In this sense, any possible deviation from general relativity would appear as a small perturbation to this general relativistic background. Such perturbations are proportional to re-normalized PPN parameters (i.e., , etc.), which are zero in general relativity, but which may have non-zero values for some gravitational theories. In terms of the metric tensor, this PPN-perturbative procedure may be conceptually presented as
where metric is derived from Eq. (II.2) by taking the general relativistic values of the PPN parameters and where is the PPN metric perturbation. The PPN-renormalized metric perturbation for a system of point-like gravitational sources in four dimensions may be given as
Given the smallness of the current values for the PPN parameters and , the PPN metric perturbation represents a very small deformation of the general relativistic background . The expressions in Eqs. (II.3) embody the “spirit” of many gravitational tests assuming that general relativity provides the correct description of the experimental situation and enables the searches for small non-Einsteinian deviations.
Similarly, one can derive the PPN-renormalized version of the Lagrangian in Eq. (9):
where is Eq. (9) taken with general relativistic values of the PPN parameters and is
The equations of motion from Eq. (10) may also be presented in the PPN-renormalized form with explicit dependence on the PPN-perturbative acceleration terms:
where are the equations of motion from Eq. (10) with the values of the PPN parameters and set to their general relativistic values of unity. Then the PPN-perturbative acceleration term is given as
Eq. (17) provides a useful framework for gravitational research. Thus, besides the terms with PPN-renormalized parameters and , it also contains , the parameter that signifies a possible inequality between the gravitational and inertial masses and that is needed to facilitate investigation of a possible violation of the EP (see Sec. IV.1.2). In addition, Eq. (17) also includes parameter , which is needed to investigate possible temporal variation in the gravitational constant (see Sec. IV.4.2). Note that in general relativity .
Finally, we present the similar expressions for the light-time equation Eq. (11). This equation that can be written as with the PPN perturbation given as below
Eqs. (16), (17) and (18) are used to focus the science objectives and to describe gravitational experiments (especially those to be conducted in the solar system) that well be discussed below. So far, general theory of relativity survived every test Turyshev et al. (2007), yielding the ever improving values for the PPN parameters , namely using the data from the Cassini spacecraft taken during solar conjunction experiment Bertotti et al. (2003) and , which resulted from the recent analysis of the LLR data Williams et al. (2004) (see Fig. 1).
Iii Search for New Physics Beyond General Relativity
The fundamental physical laws of Nature, as we know them today, are described by the Standard Model of particles and fields and the general theory of relativity. The Standard Model specifies the families of fermions (i.e., leptons and quarks) and their interactions by vector fields that transmit the strong, electromagnetic, and weak forces. General relativity is a tensor-field theory of gravity with universal coupling to the particles and fields of the Standard Model.
However, despite the beauty and simplicity of general relativity and the success of the Standard Model, our present understanding of the fundamental laws of physics has several shortcomings. Although recent progress in string theory Witten (2001, 2003) is very encouraging, the search for a realistic theory of quantum gravity remains a challenge. This continued inability to merge gravity with quantum mechanics indicates that the pure tensor gravity of general relativity needs modification or augmentation. The recent remarkable progress in observational cosmology has subjected the general theory of relativity to increased scrutiny by suggesting a non-Einsteinian scenario of the Universe’s evolution. Researchers now believed that new physics is needed to resolve these issues.
Theoretical models of the kinds of new physics that can solve the problems described above typically involve new interactions, some of which could manifest themselves as violations of the EP, variation of fundamental constants, modification of the inverse-square law of gravity at short distances, Lorenz symmetry breaking, or large-scale gravitational phenomena. Each of these manifestations offers an opportunity for space-based experimentation and, hopefully, a major discovery.
In this Section I present motivations for the new generation of gravitational experiments that are expected to advance the relativistic gravity research up to five orders of magnitude below the level that is currently tested by experiments Turyshev et al. (2004b, 2006a); Turyshev et al. (2007). Specifically, I discuss theoretical models that predict non-Einsteinian behavior which can be investigated in the experiments conducted in the solar system. Such an interesting behavior led to a number of space-based experiments proposed recently to investigate the corresponding effects (see Sec. IV for details).
iii.1 String/M-Theory and Tensor-Scalar Extensions of General Relativity
An understanding of gravity at the quantum level will allow us to ascertain whether the gravitational “constant” is a running coupling constant like those of other fundamental interactions of Nature. String/M-theory Green et al. (1987) hints at a negative answer to this question, given the non-renormalization theorems of supersymmetry, a symmetry at the core of the underlying principle of string/M-theory and brane models, Polchinski (1995); Horava and Witten (1996a, b); Lukas et al. (1999); Randall and Sundrum (1999). One-loop higher–derivative quantum gravity models may permit a running gravitational coupling, as these models are asymptotically free – a striking property Julve and Tonin (1978); Fradkin and Tseytlin (1982); Avramidi and Barvinsky (1985). In the absence of a screening mechanism for gravity, asymptotic freedom may imply that quantum gravitational corrections take effect on macroscopic and even cosmological scales, which has some bearing on the dark matter problem Goldman et al. (1992) and, in particular, on the subject of the large scale structure of the Universe. Either way, it seems plausible to assume that quantum gravity effects manifest themselves only on cosmological scales.
Both consistency between a quantum description of matter and a geometric description of space-time, and the appearance of singularities involving minute curvature length scales indicate that a full theory of quantum gravity is needed for an adequate description of the interior of black holes and time evolution close to the Big Bang: a theory in which gravity and the associated geometry of space-time are described in the language of quantum theory. Despite major efforts in this direction, no complete and consistent theory of quantum gravity is currently available; there are, however, a number of promising candidates.
String theory is viewed as the most promising means of making general relativity compatible with quantum mechanics Green et al. (1987). The closed-string theory has a spectrum that contains as zero-mass eigenstates the graviton , the dilaton , and the antisymmetric second-order tensor . There are various ways to extract the physics of our four-dimensional world, and a major difficulty lies in finding a natural mechanism that fixes the value of the dilaton field, as it does not acquire a potential at any order in string-perturbation theory. However, although the usual quantum-field theories used in elementary particle physics to describe interactions, do lead to an acceptable effective (quantum) field theory of gravity at low energies, they result in models devoid of all predictive power at very high energies.
Damour & Polyakov Damour and Polyakov (1994a, b) have studied a possible a mechanism to circumvent the above-mentioned difficulty by suggesting string loop-contributions, which are counted by dilaton interactions instead of by a potential. They proposed the least coupling principle (LCP), realized via a cosmological attractor mechanism (CAM) (see e.g., Refs. Damour and Polyakov (1994a, b); Damour et al. (2002a, b)), which can reconcile the existence of a massless scalar field in the low energy world with existing tests of general relativity (and with cosmological inflation). However, it is not yet known whether this mechanism can be realized in string theory. The authors assumed the existence of a massless scalar field (i.e., of a flat direction in the potential), with gravitational strength coupling to matter. A priori, this appears phenomenologically forbidden; however, the CAM tends to drive towards a value where its coupling to matter becomes naturally . After dropping the antisymmetric second-order tensor and introducing fermions and Yang–Mills fields , with field strength , in a spacetime described by the metric , the relevant effective low-energy four-dimensional action in the string frame can be written in the generic form as
where555In the general case, one expects that each mater field would have a different coupling function, e.g., , etc. , is the inverse of the string tension, is a gauge group constant, is the inflation field and the constants , , …, can, in principle, be determined via computation.
To recover Einsteinian gravity, one performs a conformal transformation with , that leads to an effective action where the coupling constants and masses are functions of the rescaled dilaton, :
It follows that and the coupling constants and masses are now dilaton-dependent, through and .
The CAM leads to some generic predictions even without the knowledge of the specific structure of the various coupling functions, namely . The basic assumption is that the string-loop corrections are such that there exists a minimum in (some of) the functions at some (finite or infinite) value . During inflation, the dynamics is governed by a set of coupled differential equations for the scale factors and . In particular, the equation of motion for contains a term . At this state of evolution (i.e., during inflation, when has a large vacuum expectation value) this coupling drives towards the special point where reaches a minimum. Once has been attracted near , essentially (classically) decouples from so that inflation proceeds as if were not there. A similar attractor mechanism exists during the other phases of cosmological evolution, and tends to decouple from the dominant cosmological matter. For this mechanism to efficiently decouple from all types of matter, there must be a special point to approximately minimize all the important coupling functions. A way of having such a special point in field space is to assume that is a limiting point where all coupling functions have finite limits. This leads to the so-called runaway dilaton scenario, in which the mere assumption that as implies that is an attractor where all couplings vanish.
This mechanism also predicts (approximately composition-independent) values for the post-Einstein parameters and , which parametrize deviations from general relativity. For simplicity, I discuss only the theories for which . Hence, for a theory for which the can be locally neglected, given that its mass is small enought that it acts cosmologically, it has been shown that in the PPN limit, that if one writes
where is the coupling function to matter and the factor that allows one to write the theory in the Einstein frame in this model . Then, the two post-Einstein parameters are of the form
where is the dilaton coupling to hadronic matter. In this model, all tests of general relativity are violated. However, all these violations are correlated. For instance, the following link between EP violations and solar-system deviations is established
Given that present tests of the EP place a limit on the ratio of the order of (see Sec. IV.1), one finds . Note that the upper limit given on by the Cassini experiment was , so in this case the necessary sensitivity has not yet been reached to test the CAM.
It is also possible that the dynamics of the quintessence field evolves to a point of minimal coupling to matter. In Ref. Damour and Polyakov (1994b) the authors showed that could be attracted towards a value during the matter dominated era that decoupled the dilaton from matter. For universal coupling, (see Eq. (6)), this would motivate improvements in the accuracy of the EP and other tests of general relativity. The authors of Ref. Veneziano (2002) suggested that with a large number of non-self-interacting matter species, the coupling constants are determined by the quantum corrections of the matter species, and would evolve as a run-away dilaton with asymptotic value . Due to the LCP, the dependence of the masses on the dilaton implies that particles fall differently in a gravitational field, and hence are in violation of the weak form of the EP (WEP). Although, the effect (on the order of ) is rather small in the solar system conditions, application of already available technology can potentially test predictions that represent a distinct experimental signature of string/M-theory.
These recent theoretical findings suggest that the present agreement between general relativity and experiment may be naturally compatible with the existence of a scalar contribution to gravity. In particular, Damour & Nordtvedt Damour and Nordtvedt (1993a, b) (see also Damour and Polyakov (1994a, b) for non-metric versions of this mechanism and Damour et al. (2002a, b) for a recent summary of the runaway-dilaton scenario) found that a scalar-tensor theory of gravity may contain a built-in CAM toward general relativity. Scenarios considered by these authors assume that the scalar coupling parameter was of order one in the early universe (i.e., before inflation), and show that this parameter then evolves to be close to (but not exactly equal to) zero at the present time. Fig. 3 illustrates this mechanism in greater details.
The Eddington parameter , whose value in general relativity is unity, is perhaps the most fundamental PPN parameter, in that is a measure of the fractional strength of the scalar gravity interaction in scalar-tensor theories of gravity Damour and Esposito-Farese (1996a, b). Within perturbation theory for such theories, all other PPN parameters to all relativistic orders collapse to their general relativistic values in proportion to . Under some assumptions (see, e.g. Damour and Nordtvedt (1993b)) one can estimate the likely order of magnitude of the left-over coupling strength at the present time which; this value, depending on the total mass density of the universe, can be given as , where is the ratio of the current density to the closure density and where is the Hubble constant in units of 100 km/sec/Mpc. Compared to the cosmological constant, these scalar-field models are consistent with the supernovae observations for a lower matter density, , and a higher age, . If this is indeed the case, the level would be the lower bound for the present value of PPN parameter Damour and Nordtvedt (1993a, b).
Recently, Damour et al. Damour et al. (2002a, b) have estimated within the framework compatible with string theory and modern cosmology, confirming the results of Refs. Damour and Nordtvedt (1993a, b). This recent analysis discusses a scenario wherein a composition-independent coupling of a dilaton to hadronic matter produces detectable deviations from general relativity in high-accuracy light-deflection experiments in the solar system. This work assumes only some general property of the coupling functions (for large values of the field, i.e. for an “attractor at infinity”) and then assumes that is on the order of one at the beginning of the controllably classical part of inflation. Damour et al. Damour et al. (2002b) showed that one can relate the present value of to the cosmological density fluctuations. For the simplest inflationary potentials (favored by Wilkinson Microwave Anisotropy Probe (WMAP) mission, i.e. Bennett et al. (2003)), these authors found that the present value of could be just below . In particular, within this framework , where is the dilaton coupling to hadronic matter. Its value depends on the model taken for the inflation potential , with again being the inflation field; the level of the expected deviations from general relativity is for Damour et al. (2002b). These predictions are based on the work in scalar-tensor extensions of gravity that are consistent with, and indeed often part of, present cosmological models.
For runaway-dilaton scenario, comparison with the minimally coupled scalar-field action,
reveals that the negative scalar kinetic term leads to an action equivalent to a “ghost” in quantum-field theory, that is referred to as “phantom energy” in the cosmological context Caldwell (2002). Such a scalar-field model could in theory generate acceleration by the field evolving up the potential toward the maximum. Phantom fields are plagued by catastrophic ultraviolet instabilities, as particle excitations have a negative mass Cline et al. (2004); Rubakov and Tinyakov (2008); Sergienko and Rubakov (2008); the fact that their energy is unbounded from below allows vacuum decay through the production of high-energy real particles as well as negative-energy ghosts that are in contradiction with the constraints on ultrahigh-energy cosmic rays Sreekumar et al. (1998).
Such runaway behavior can potentially be avoided by the introduction of higher-order kinetic terms in the action. One implementation of this idea is known as “ghost condensation” Arkani-Hamed et al. (2004). In this scenario, the scalar field has a negative kinetic energy near , but the quantum instabilities are stabilized by the addition of higher-order corrections to the scalar field Lagrangian of the form . The “ghost” energy is then bounded from below, and stable evolution of the dilaton occurs with Piazza and Tsujikawa (2004). The gradient is non-vanishing in the vacuum, violating Lorentz invariance; this may have important consequences in cosmology and in laboratory experiments.
The analyses discussed above predict very small (ranging from to for ) observable post-Newtonian deviations from general relativity in the solar system, thereby motivating new generation of advanced gravity experiments. In many cases, such tests would require reaching the accuracy needed to measure effects of the next post-Newtonian order () Turyshev et al. (2007); Turyshev et al. (2004c), promising important outcomes for the twenty-first century fundamental physics.
iii.2 Observational Motivations for Higher Accuracy Tests of Gravity
Recent astrophysical measurements of the angular structure of the cosmic microwave background (CMB) de Bernardis et al. (2000), the masses of large-scale structures Peacock et al. (2001), and the luminosity distances of type Ia supernovae Perlmutter et al. (1999); Riess et al. (1998) have placed stringent constraints on the cosmological constant and have also led to a revolutionary conclusion: The expansion of the universe is accelerating. The implication of these observations for cosmological models is that a classically evolving scalar field currently dominates the energy density of the universe. Such models have been shown to share the advantages of , namely (a) compatibility with the spatial flatness predicted inflation, (b) a universe older than the standard Einstein-de Sitter model, and, (c) combined with cold dark matter (CDM), predictions for large-scale structure formation in good agreement with data from galaxy surveys. As well as imprinting their distinctive signature on the CMB anisotropy, scalar-field models remain viable and should be testable in the near future. This completely unexpected discovery demonstrates the importance of testing important ideas about the nature of gravity. We are presently in the “discovery” phase of this new physics, and although there are many theoretical conjectures as to the origin of a non-zero , it is essential that we exploit every available opportunity to elucidate the physics at the root of the observed phenomena.
Description of quantum matter in a classical gravitational background poses interesting challenges, notably the possibility that the zero-point fluctuations of the matter fields generate a non-vanishing vacuum energy density , that corresponds to the term , in Eq. (3) Weinberg (1989). This is equivalent to adding a “cosmological constant” term on the left-hand side of Einstein’s equations Eq. (5), with . Recent cosmological observations suggest a positive value of corresponding to Spergel et al. (2007). Such a small value has a negligible effect on the solar system dynamics and relevant gravitational tests. Quantizing the gravitational field itself poses a challenge because of the perturbative non-renormalizability of Einstein’s Lagrangian Weinberg (1989); Wilczek (2006). Superstring theory offers a promising avenue toward solving this challenge.
There is now a great deal of evidence indicating that over 70% of the critical density of the universe is in the form of a “negative-pressure” dark energy component; we have no understanding of its origin or nature. The fact that the expansion of the universe is currently undergoing a period of acceleration has been well tested: The expansion has been directly measured from the light curves of several hundred type Ia supernovae Perlmutter et al. (1999); Riess et al. (1998); Tonry et al. (2003), and has been independently inferred from observations of CMB by the WMAP satellite Bennett et al. (2003) and other CMB experiments Halverson et al. (2002); Netterfield et al. (2002). Cosmic speed-up can be accommodated within general relativity by invoking a mysterious cosmic fluid with large negative pressure, dubbed dark energy. The simplest possibility for dark energy is a cosmological constant; unfortunately, the smallest estimates for its value are 55 orders of magnitude too large (for reviews see Carroll (2001); Peebles and Ratra (2003)). Most of the theoretical studies operate in the shadow of the cosmological constant problem, the most embarrassing hierarchy problem in physics. This fact has motivated a host of other possibilities, most of which assume , with the dynamical dark energy being associated with a new scalar field (see Carroll et al. (2004, 2005) and references therein). However, none of these suggestions is compelling and most have serious drawbacks. Given the magnitude of this problem, a number of authors have considered the possibility that cosmic acceleration is not due to a particular substance, but rather that it arises from new gravitational physics (see discussion in Carroll (2001); Peebles and Ratra (2003); Carroll et al. (2004)). In particular, certain extensions to general relativity in a low energy regime Carroll et al. (2004); Capozziello and Troisi (2005); Carroll et al. (2005) were shown to predict an experimentally consistent universe evolution without the need for dark energy Bertolami et al. (2007). These dynamical models are expected to explain the observed acceleration of the universe without dark energy, but may produce measurable gravitational effects on the scales of the solar system.
iii.3 Modified Gravity as an Alternative to Dark Energy
Certain modifications of the Einstein–Hilbert action Eq. (2) by introducing terms that diverge as the scalar curvature goes to zero could mimic dark energy Carroll et al. (2004, 2005). Recently, models involving inverse powers of the curvature have been proposed as an alternative to dark energy. In these models there are more propagating degrees of freedom in the gravitational sector than the two contained in the massless graviton in general relativity. The simplest models of this kind add inverse powers of the scalar curvature to the action (), thereby introducing a new scalar excitation in the spectrum. For the values of the parameters required to explain the acceleration of the Universe this scalar field is almost massless in vacuum; this could lead to a possible conflict with the solar system experiments.
However, models that involve inverse powers of other invariants, in particular those that diverge for in the Schwarzschild solution, generically recover an acceptable weak-field limit at short distances from sources by means of a screening or shielding of the extra degrees of freedom at short distances Navarro and Van Acoleyen (2006). Such theories can lead to late-time acceleration, but they typically result in one of two problems: either they are in conflict with tests of general relativity in the solar system, due to the existence of additional dynamical degrees of freedom Chiba (2003), or they contain ghost-like degrees of freedom that seem difficult to reconcile with fundamental theories.
The idea that the cosmic acceleration of the Universe may be caused by modification of gravity at very large distances, and not by a dark energy source, has recently received a great deal of attention (see Rubakov and Tinyakov (2008); Sotiriou and Faraoni (2008)). Such a modification could be triggered by extra space dimensions, to which gravity spreads over cosmic distances. In addition to being testable by cosmological surveys, modified gravity predicts testable deviations in planetary motions, providing new motivations for a new generation of advanced gravitational experiments in space Turyshev et al. (2007); Turyshev (2008a). An example of recent theoretical progress is the Dvali-Gabadadze-Porrati (DGP) brane-world model, which explores the possibility that we live on a brane embedded in a large extra dimension, and where the strength of gravity in the bulk is substantially less than that on the brane Dvali et al. (2000). Although such a theory can lead to perfectly conventional gravity on large scales, it is also possible to choose the dynamics in such a way that new effects show up exclusively in the far infrared, thereby providing a mechanism to explain the acceleration of the universe Perlmutter et al. (1999); Riess et al. (1998). Interestingly, DGP gravity and other modifications of general relativity hold out the possibility of having interesting and testable predictions that distinguish them from models of dynamical dark energy. One outcome of this work is that the physics of the accelerating universe may be deeply tied to the properties of gravity on relatively short scales, from millimeters to astronomical units Dvali et al. (2000, 2003).
Although many effects predicted by gravity modification models are suppressed within the solar system, there are measurable effects induced by some long-distance modifications of gravity Dvali et al. (2000). For instance, in the case of the precession of the planetary perihelion in the solar system, the anomalous perihelion advance, , induced by a small correction, , to Newton’s potential, , is given in radians per revolution Dvali et al. (2003) by The most reliable data regarding the planetary perihelion advances come from the inner planets of the solar system, where a majority of the corrections are negligible. However, LLR offers an interesting possibility to test for these new effects Williams et al. (2004). Evaluating the expected magnitude of the effect to the Earth-Moon system, one predicts an anomalous shift of Dvali et al. (2003), compared with the achieved accuracy of . Therefore, the theories of gravity modification raise an intriguing possibility of discovering new physics that could be addressed with the new generation of astrometric measurements Turyshev (2008a).
iii.4 Scalar Field Models as Candidates for Dark Energy
One of the simplest candidates for dynamical dark energy is a scalar field, , with an extremely low-mass and an effective potential . If the field is rolling slowly, its persistent potential energy is responsible for creating the late epoch of inflation we observe today. For the models that include only inverse powers of the curvature, other than the Einstein-Hilbert term, it is possible that in regions where the curvature is large the scalar has a large mass that could make the dynamics similar to those of general relativity Cembranos (2006). At the same time, the scalar curvature, although larger than its mean cosmological value, is very small in the solar system, thereby satisfying constraints set by the gravitational tests performed to date Erickcek et al. (2006); Nojiri and Odintsov (2006); Turyshev (1995, 1996b); Silaev and Turyshev (1997). Nevertheless, it is not clear whether these models may be regarded as a viable alternative to dark energy.
Effective scalar fields are prevalent in supersymmetric field theories and string/M-theory. For example, string theory predicts that the vacuum expectation value of a scalar field, the dilaton, determines the relationship between the gauge and gravitational couplings. A general, low energy effective action for the massless modes of the dilaton can be cast as a scalar-tensor theory (as in Eq. (6)) with a vanishing potential, where , , and are the dilatonic couplings to gravity, the scalar kinetic term, and the gauge and matter fields, respectively, which encode the effects of loop effects and potentially non-perturbative corrections.
A string-scale cosmological constant or exponential dilaton potential in the string frame translates into an exponential potential in the Einstein frame. Such quintessence potentials Wetterich (1988); Ratra and Peebles (1988); Wetterich (2003); Peebles and Ratra (2003); Wetterich (2004) can have scaling Ferreira and Joyce (1997), and tracking Zlatev et al. (1999) properties that allow the scalar field energy density to evolve alongside the other matter constituents. A problematic feature of scaling potentials Ferreira and Joyce (1997) is that they do not lead to accelerative expansion, as the energy density simply scales with that of matter. Alternatively, certain potentials can predict a dark energy density that alternately dominates the Universe and decays away; in such models, the acceleration of the Universe is transient Albrecht and Skordis (2000); Dodelson et al. (2000); Bento et al. (2002). Collectively, quintessence potentials predict that the density of the dark energy dynamically evolves over time, in contrast to the cosmological constant. Similar to a cosmological constant, however, the scalar field is expected to have no significant density perturbations within the causal horizon, so they contribute little to the evolution of the clustering of matter in large-scale structure Ferreira and Joyce (1998).
In addition to couplings to ordinary matter, the quintessence field may have nontrivial couplings to dark matter Farrar and Peebles (2004); Bertolami et al. (2007). Non-perturbative string-loop effects do not lead to universal couplings, although it is possible that the dilaton decouples more slowly from dark matter than it does from gravity and fermions. This coupling can provide a mechanism to generate acceleration with a scaling potential while also being consistent with EP tests. It can also explain why acceleration began to occur only relatively recently, by being triggered by the non-minimal coupling to the CDM, rather than by a feature in the effective potential Bean and Magueijo (2001); Gasperini et al. (2002). Such couplings can not only generate acceleration, but can also modify structure formation through the coupling to CDM density fluctuations Bean (2001) and adiabatic instabilities Bean et al. (2008a, b), in contrast to minimally coupled quintessence models. Dynamical observables that are sensitive to the evolution in matter perturbations as well as to the expansion of the Universe, such as (a) the matter power spectrum as measured by large-scale surveys and (b) weak lensing convergence spectra, could distinguish non-minimal couplings from theories with minimal effect on clustering.
In the next section I will discuss the new effects predicted by the theories and models discussed above. I will also present a list of experiments that were proposed to test these important predictions in dedicated space experiments.
Iv Search for a new theory of gravity with space-based experiments
It is well known that work on the general theory of relativity began with the EP, in which gravitational acceleration was a priori held indistinguishable from acceleration caused by mechanical forces; as a consequence, gravitational mass was therefore identical to inertial mass. Since Newton’s time, the question about the equality of inertial and passive gravitational masses has risen in almost every theory of gravitation. Einstein elevated this identity, which was implicit in Newton’s gravity, to a guiding principle in his attempts to explain both electromagnetic and gravitational acceleration according to the same set of physical laws Einstein (1907, 1915, 1916); Frederiks (1999); Kobzarev (1990). Thus, almost 100 years ago Einstein postulated that not only mechanical laws of motion, but also all non-gravitational laws should behave in freely falling frames as if gravity were absent. It is this principle that predicts identical accelerations of compositionally different objects in the same gravitational field, and it also allows gravity to be viewed as a geometrical property of spacetime–leading to the general relativistic interpretation of gravitation.
Remarkably, the EP has been (and still is!) a focus of gravitational research for more than 400 years Williams et al. (2008). Since the time of Galileo we have known that objects of different mass and composition accelerate at identical rates in the same gravitational field. From 1602 to 1604, through his study of inclined planes and pendulums, Galileo formulated a law of falling bodies that led to an early empirical version of the EP. However, these famous results were not published for another 35 years. It took an additional 50 years before a theory of gravity describing these and other early gravitational experiments was published by Newton in his Principia in 1687. On the basis of his second law, Newton concluded that the gravitational force is proportional to the mass of the body on which it acted; from his third law, he postulated the gravitational force is proportional to the mass of its source.
Newton was aware that the inertial mass in his second law, , might not be the same as the gravitational mass relating force to gravitational field . Indeed, after rearranging these two equations, we find and thus, in principle, materials with different values of the ratio could accelerate at different rates in the same gravitational field. Newton tested this possibility with simple pendulums of the same length but with different masses and compositions, but he found no difference in their periods. On this basis Newton concluded that was constant for all matter; and that, by a suitable choice of units, the ratio could always be set to one, i.e. . Bessel subsequently tested this ratio more accurately, and then in a definitive 1889 experiment Eötvös was able to experimentally verify this equality of the inertial and gravitational masses to an accuracy of one part in Eötvös (1890); Eötvös et al. (1922); Bod et al. (1991).
Today, more than 320 years after Newton proposed a comprehensive approach to studying the relation between the two masses of a body, this relation remains to be the subject of numerous theoretical and experimental investigations. The question regarding the equality of inertial and passive gravitational masses has arisen in almost every theory of gravitation. In 1915 the EP became a part of the foundation of Einstein’s general theory of relativity; subsequently, many experimental efforts focused on testing the EP in the search for limits of general relativity. Thus, the early tests of the EP were further improved by Dicke and his colleagues Roll et al. (1964) to one part in . Most recently, a University of Washington group Baeßler et al. (1999); Adelberger (2001) improved upon Dicke’s verification of the EP by several orders of magnitude, reporting , thereby confirming Einstein’s intuition.
In a 1907 paper, using the early version of the EP Einstein (1907) Einstein made important preliminary predictions regarding the influence of gravity on light propagation; these predictions presented the next important step in the development of his theory. He realized that a ray of light coming from a distant star would appear to be attracted by solar mass while passing close to the Sun. As a result, the ray’s trajectory is bent twice as much in the direction towards the Sun compared to the same trajectory analyzed with Newton’s theory (see discussion in Sec. I). In addition, light radiated by a star would interact with the star’s gravitational potential, resulting in the radiation shifting slightly toward the infrared end of the spectrum. So the original EP, as described by Einstein, concluded that free-fall and inertial motion were physically equivalent which constitutes the weak the form of the EP (WEP).
In about 1912, Einstein (with the help of mathematician Marcel Grossmann) began a new phase of his gravitational research by phrasing his work in terms of the tensor calculus of Tullio Levi-Civita and Gregorio Ricci-Curbastro. The tensor calculus greatly facilitated calculations in four-dimensional space-time, a notion that Einstein had obtained from Hermann Minkowski’s 1907 mathematical elaboration of Einstein’s own special theory of relativity. Einstein called his new theory the general theory of relativity. After a number of false starts, he published the definitive field equations of his theory in late 1915 Einstein (1915, 1916). Since that time, physicists have endeavored to understand and verify various predictions of the general theory of relativity with ever increasing accuracy (for review of various gravitational experiments available during the period of 1970-2000, see Braginskiǐ and Rudenko (1970); Konopleva (1977); Rudenko (1978); Kobzarev (1990); Will (1993))
Note that, although the EP guided the development of general relativity, it is not a founding principle of relativity but rather a simple consequence of the geometrical nature of the theory. In general relativity, test objects in free-fall follow geodesics of spacetime, and what we perceive as the force of gravity is instead a result of our being unable to follow those geodesics of spacetime, because the mechanical resistance of matter prevents us from doing so.
Below we discuss on the space-based gravitational experiments aiming to test the various aspects of the EP, tests of Lorentz and position invatiancies, search for variability of the fundamental constants, tests of the gravitational inverse square law and tests of alternative and modified-gravity theories.
iv.1 Tests of the Equivalence Principle
Since Einstein developed general relativity, there was a need to develop a framework to test the theory against other possible theories of gravity compatible with special relativity. This was developed by Robert Dicke Dicke (1959, 1961) as part of his program to test general relativity. Two new principles were suggested, the so-called Einstein EP (EEP) and the strong EP (SEP), each of which assumes the WEP as a starting point. They only differ in whether or not they apply to gravitational experiments.
The EEP states that the result of a local non-gravitational experiment in an inertial frame of reference is independent of the velocity or location in the universe of the experiment. This is a kind of Copernican extension of Einstein’s original formulation, which requires that suitable frames of reference all over the universe behave identically. It is an extension of the postulates of special relativity in that it requires that dimensionless physical values such as the fine-structure constant and electron-to-proton mass ratio be constant. From the theoretical standpoint, the EEP Damour and Nordtvedt (1993a, b); Williams et al. (2004, 2008) is at the foundation of the general theory of relativity; therefore, testing the principle is very important. As far as the experiment is concerned, the EEP includes three testable hypotheses:
Universality of free fall (UFF), which states that freely falling bodies have the same acceleration in the same gravitational field independent of their compositions (see Sec. IV.1),
Local Lorentz invariance (LLI), which suggests that clocks’ rates are independent of the clock’s velocities (see Sec. IV.2), and
Local position invariance (LPI), which postulates that clocks’ rates are also independent of their spacetime positions (see Sec. IV.3).
Using these three hypotheses Einstein deduced that gravity is a geometric property of spacetime Dicke (1959); Will (2006). One can test the validity of both the EP and the field equations that determine the geometric structure created by a mass distribution. There are two different “flavors” of the EP, the weak and the strong forms (WEP and SEP, respectively), which are being tested in various experiments performed with laboratory test masses and with bodies of astronomical sizes Williams et al. (2008).
iv.1.1 The Weak Equivalence Principle
The weak form of the EP (the WEP, also known as the UFF) holds that the gravitational properties of strong and electro-weak interactions obey the EP. In this case the relevant test-body differences are their fractional nuclear-binding differences, their neutron-to-proton ratios, their atomic charges, etc. Furthermore, the equality of gravitational and inertial masses implies that different neutral massive test bodies have the same free-fall acceleration in an external gravitational field, and therefore in freely falling inertial frames the external gravitational field appears only in the form of a tidal interaction Singe (1960). Apart from these tidal corrections, freely falling bodies behave as though external gravity were absent Anderson et al. (1996).
General relativity and other metric theories of gravity assume that the WEP is exact. However, many extensions of the Standard Model that contain new macroscopic-range quantum fields predict quantum exchange forces that generically violate the WEP because they couple to generalized “charges” rather than to mass/energy as does gravity Damour and Nordtvedt (1993a, b); Damour and Polyakov (1994a, b); Damour et al. (2002a, b).
In a laboratory, precise tests of the EP can be made by comparing the free fall accelerations, and , of different test bodies. When the bodies are at the same distance from the source of the gravity, the expression for the EP takes the elegant form
where and are the gravitational and inertial masses of each body, respectively. The sensitivity of the EP test is determined by the precision of the differential acceleration measurement divided by the degree to which the test bodies differ (e.g. composition).
Various experiments have been performed to measure the ratios of gravitational to inertial masses of bodies. Recent experiments on bodies of laboratory dimensions have verified the WEP to a fractional precision of by Roll et al. (1964), to by Braginsky and Panov (1972); Su et al. (1994) and more recently to a precision of Adelberger (2001). The accuracy of these experiments is high enough to confirm that the strong, weak, and electromagnetic interactions each contribute equally to the passive gravitational and inertial masses of the laboratory bodies.
Currently, the most accurate results in testing the WEP have been reported by ground-based laboratories Williams et al. (2008); Baeßler et al. (1999). The most recent result Adelberger (2001); Schlamminger et al. (2008) for the fractional differential acceleration between beryllium and titanium test bodies was given by the Eöt-Wash group666The Eöt-Wash group at the University of Washington in Seattle has developed new techniques in high-precision studies of weak-field gravity and searches for possible new interactions weaker than gravity. See http://www.npl.washington.edu/eotwash/ for details. as . A review of the most recent laboratory tests of gravity can be found in Ref. Gundlach et al. (2007). Significant improvements in the tests of the EP are expected from dedicated space-based experiments Turyshev et al. (2007).
The composition-independence of acceleration rates of various masses toward the Earth can be tested in space-based laboratories to a precision of many additional orders of magnitude, down to levels at which some models of the unified theory of quantum gravity, matter, and energy suggest a possible violation of the EP Damour and Nordtvedt (1993a, b); Damour and Polyakov (1994a, b); Damour et al. (2002a, b). In some scalar-tensor theories, the strength of EP violations and the magnitude of the fifth force mediated by the scalar can be drastically larger in space than on the ground Mota and Barrow (2004), providing further justification for space deployment. Importantly, many of these theories predict observable violations of the EP at various levels of accuracy ranging from to . Therefore, even a confirmation of no EP-violation will be exceptionally valuable, as it will place useful constraints on the range of possibilities in the development of a unified physical theory.
Compared with Earth-based laboratories, experiments in space can benefit from a range of conditions, including free fall and significantly reduced contributions due to seismic, thermal, and other non-gravitational noise (see Appendix A in Turyshev et al. (2007)). As a result, many experiments have been proposed to test the EP in space. Below I present only a partial list of these missions. Furthermore, to illustrate the use of different technologies, I discuss only the most representative concepts while not going into technical details of these experiments.
The Micro-Satellite à traînée Compensée pour l’Observation du Principe d’Equivalence (MicroSCOPE) mission777See http://microscope.onera.fr/ for details on MicroSCOPE mission. is a room-temperature EP experiment in space that utilizes electrostatic differential accelerometers Touboul and Rodrigues (2001). The mission is currently under development by Centre National d’Etudes Spatiales (CNES)888Centre National d’Etudes Spatiales (CNES) – the French Space Agency, see website at: http://www.cnes.fr/ and the European Space Agency (ESA), and is scheduled for launch in 2010. The design goal is to achieve a differential acceleration accuracy of . MicroSCOPE’s electrostatic differential accelerometers are based on flight heritage designs from the CHAMP, GRACE, and GOCE missions.999Several gravity missions were recently developed by the German National Research Center for Geosciences (GFZ). Among them are CHAMP (Gravity and Magnetic Field Mission); GRACE (Gravity Recovery And Climate Experiment Mission, together with NASA); and GOCE (Global Ocean Circulation Experiment), together with ESA and other European countries. See http://www.gfz-potsdam.de/pb1/op/index_GRAM.html
The Principle of Equivalence Measurement (POEM) experiment Reasenberg and Phillips (2007) is a ground-based test of the WEP and now is under development. It will be able to detect a violation of the EP with a fractional acceleration accuracy of 5 parts in 10 in a short experiment (i.e., a few days long) and with a three- to tenfold better accuracy in a longer experiment. The experiment makes use of optical distance measurement (by tracking frequency gauge (TFG) laser gauge Phillips and Reasenberg (2005)) and will be advantageously sensitive to short-range forces with a characteristic length scale of km. SR-POEM, a POEM-based proposed room-temperature test of the WEP during a sub-orbital flight on a sounding rocket, was also recently proposed Turyshev et al. (2007). It is anticipated to be able to search for a violation of the EP with a single-flight accuracy of 1 part in 10. Extension to higher accuracy in an orbital mission is under study. Additionally, the Space Test of Universality of Free Fall (STUFF) Turyshev et al. (2007) is a recent study of a space-based experiment that relies on optical metrology and proposes to reach an accuracy of 1 part in 10 in testing the EP in space.
The Quantum Interferometer Test of the Equivalence Principle (QuITE) Kasevich and Maleki (2003) is a proposed cold-atom-based test of the EP in space. QuITE intends to measure the absolute single axis differential acceleration, with an accuracy of 1 part in , by utilizing two co-located matter wave interferometers with different atomic species.101010Compared to the ground-based conditions, space offers a factor of nearly improvement in the integration times in observation of the free-falling atoms (i.e., progressing from ms to sec). The longer integration times translate into the accuracy improvements Turyshev et al. (2007). QuITE will improve the current EP limits set in similar experiments conducted in ground-based laboratory conditions111111Its ground-based analog, called “Atomic Equivalence Principle Test (AEPT)”, is currently being built at Stanford University. AEPT is designed to reach sensitivity of one part in . Peters et al. (2001); Fray et al. (2004) by nearly seven to nine orders of magnitude. Similarly, the Interférométrie à Source Cohérente pour Applications dans l’Espace (I.C.E.) project121212Interférométrie à Source Cohérente pour Applications dans l’Espace (I.C.E.), see http://www.ice-space.fr supported by CNES in France aims to develop a high-precision accelerometer based on coherent atomic sources in space Nyman et al. (2006) with an accurate test of the EP as one of its main objectives.
The Galileo Galilei (GG) mission Nobili et al. (2007) is an Italian space experiment131313Galileo Galilei (GG) website: http://eotvos.dm.unipi.it/nobili proposed to test the EP at room temperature with an accuracy of 1 part in . The key instrument of GG is a differential accelerometer made of weakly coupled coaxial, concentric test cylinders that spin rapidly around the symmetry axis and are sensitive in the plane perpendicular to it. GG is included in the National Aerospace Plan of the Italian Space Agency (ASI) for implementation in the near future.
The Satellite Test of Equivalence Principle (STEP) mission Mester et al. (2001); Worden et al. (2001) is a proposed test of the EP to be conducted from a free-falling platform in space provided by a drag-free spacecraft orbiting the Earth. STEP will test the composition independence of gravitational acceleration for cryogenically controlled test masses by searching for a violation of the EP with a fractional acceleration accuracy of 1 part in 10. As such, this ambitious experiment will be able to test very precisely for the presence of any new non-metric, long range physical interactions.
This impressive evidence and the future prospects of testing the WEP for laboratory bodies is incomplete for astronomical body scales. The experiments searching for WEP violations are conducted in laboratory environments that utilize test masses with negligible amounts of gravitational self-energy; therefore, a large-scale experiment is needed to test the postulated equality of gravitational self-energy contributions to the inertial and passive gravitational masses of the bodies Nordtvedt (1968). Once the self-gravity of the test bodies is non-negligible (which is currently true only for bodies of astronomical sizes), the corresponding experiment test the ultimate version of the EP – the SEP.
iv.1.2 The Strong Equivalence Principle
In its strong form the EP (the SEP) is extended to cover the gravitational properties resulting from gravitational energy itself Williams et al. (2008). It is an assumption about the way that gravity begets gravity, i.e. about the non-linear property of gravitation. Although general relativity assumes that the SEP is exact, alternate metric theories of gravity—such as those involving scalar fields and other extensions of gravity theory—typically violate the SEP. For the SEP case, the relevant test-body differences are the fractional contributions to their masses by gravitational self-energy. Because of the extreme weakness of gravity, SEP test bodies must have astronomical sizes.
The SEP states that the results of any local experiment, gravitational or not, in an inertial frame of reference are independent of where and when in the universe it is conducted. This is the only form of the EP that applies to self-gravitating objects (such as stars), which have substantial internal gravitational interactions. It requires that the gravitational constant be the same everywhere in the universe and is incompatible with a fifth force. It is much more restrictive than the EEP. General relativity is the only known theory of gravity compatible with this form of the EP.
Nordtvedt Nordtvedt (1968, 1968, 1970) suggested several solar system experiments for testing the SEP. One of these was the lunar test. Another, a search for the SEP effect in the motion of the Trojan asteroids, was carried out by Orellana and Vucetich (1993). Interplanetary spacecraft tests have been considered by Anderson et al. (1996) and discussed Anderson and Williams (2001). An experiment employing existing binary pulsar data has been proposed Damour and Schäfer (1991). It was pointed out that binary pulsars may provide an excellent possibility for testing the SEP in the new regime of strong self-gravity Damour and Esposito-Farese (1996b), however the corresponding tests have yet to reach competitive accuracy Kramer et al. (2006).
The PPN formalism Nordtvedt (1968); Will (1971c, d); Will and Nordtvedt (1972); Nordtvedt and Will (1972) describes the motion of celestial bodies in a theoretical framework common to a wide class of metric theories of gravity. To facilitate investigation of a possible violation of the SEP, Eq. (10) allows for a possible inequality of the gravitational and inertial masses, given by the parameter , which in the PPN formalism is expressed Nordtvedt (1968, 1968) as
where is the mass of a body, is the body’s (negative) gravitational self-energy, is its total mass-energy, and is a dimensionless constant for SEP violation Nordtvedt (1968, 1968). Any SEP violation is quantified by the parameter : in fully-conservative, Lorentz-invariant theories of gravity Will (1993, 2006) the SEP parameter is related to the PPN parameters by . In general relativity , so that (see Williams et al. (2008); Will (1993, 2006)).
The quantity is the body’s gravitational self-energy , which for a body is given by
For a sphere with a radius and uniform density, , where is the escape velocity. Accurate evaluation for solar system bodies requires numerical integration of the expression of Eq. (27). Evaluating the standard solar model Ulrich (1982) results in . Because gravitational self-energy is proportional to