In this review, we shall focus on GR, the general class of scalartensor modifications of it, of which the Jordan–Fierz–Brans–Dicke theory (Brans–Dicke, for short) is the classic example, and vectortensor theories. The reasons are severalfold:
The metric 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 . We do this despite recent evidence, from supernova data, of an accelerating universe, which would indicate either a nonzero cosmological constant or a dynamical “dark energy” contributing about 70 percent of the critical density. Although 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 the values of inferred from supernova observations.
The field equations of GR are derivable from an invariant action principle , where
where R is the Ricci scalar, and is the matter action, which depends on matter fields universally coupled to the metric . By varying the action with respect to , we obtain the field equations where is the matter energymomentum tensor. General covariance of the matter action implies the equations of motion ; varying with respect to yields the matter field equations of the Standard Model. By virtue of the absence of priorgeometric elements, the equations of motion are also a consequence of the field equations via the Bianchi identities .The general procedure for deriving the postNewtonian limit of metric theories is spelled out in TEGP 5.1 [281], and is described in detail for GR in TEGP 5.2 [281]. The PPN parameter values are listed in Table 3.

These theories contain the metric , a scalar field , a potential function , and a coupling function (generalizations to more than one scalar field have also been carried out [73]). For some purposes, the action is conveniently written in a nonmetric representation, sometimes denoted the “Einstein frame”, in which the gravitational action looks exactly like that of GR:
where is the Ricci scalar of the “Einstein” metric . (Apart from the scalar potential term , this corresponds to Equation (28) with , , and .) This representation is a “nonmetric” one because the matter fields couple to a combination of and . Despite appearances, however, it is a metric theory, because it can be put into a metric representation by identifying the “physical metric” The action can then be rewritten in the metric form where The Einstein frame is useful for discussing general characteristics of such theories, and for some cosmological applications, while the metric representation is most useful for calculating observable effects. The field equations, postNewtonian limit and PPN parameters are discussed in TEGP 5.3 [281], and the values of the PPN parameters are listed in Table 3.The parameters that enter the postNewtonian limit are
where is the value of today far from the system being studied, as determined by appropriate cosmological boundary conditions. In Brans–Dicke theory (), the larger the value of , the smaller the effects of the scalar field, and in the limit (), the theory becomes indistinguishable from GR in all its predictions. In more general theories, the function could have the property that, at the present epoch, and in weakfield situations, the value of the scalar field is such that is very large and is very small (theory almost identical to GR today), but that for past or future values of , or in strongfield regions such as the interiors of neutron stars, and could take on values that would lead to significant differences from GR. It is useful to point out that all versions of scalartensor gravity predict that (see Table 3).Damour and EspositoFarèse [73] have adopted an alternative parametrization of scalartensor theories, in which one expands about a cosmological background field value :
A precisely linear coupling function produces Brans–Dicke theory, with , or . The function acts as a potential for the scalar field within matter, and, if , then during cosmological evolution, the scalar field naturally evolves toward the minimum of the potential, i.e. toward , , or toward a theory close to, though not precisely GR [80, 81]. Estimates of the expected relic deviations from GR today in such theories depend on the cosmological model, but range from 10^{–5} to a few times 10^{–7} for .Negative values of correspond to a “locally unstable” scalar potential (the overall theory is still stable in the sense of having no tachyons or ghosts). In this case, objects such as neutron stars can experience a “spontaneous scalarization”, whereby the interior values of can take on values very different from the exterior values, through nonlinear interactions between strong gravity and the scalar field, dramatically affecting the stars’ internal structure and leading to strong violations of SEP. On the other hand, in the case , one must confront that fact that, with an unstable potential, cosmological evolution would presumably drive the system away from the peak where , toward parameter values that could be excluded by solar system experiments.
Scalar fields coupled to gravity or matter are also ubiquitous in particlephysicsinspired models of unification, such as string theory [254, 176, 85, 82, 83]. In some models, the coupling to matter may lead to violations of EEP, which could be tested or bounded by the experiments described in Section 2.1. In many models the scalar field could be 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.
These theories contain the metric and a dynamical, typically timelike, fourvector field . In some models, the fourvector is unconstrained, while in others, called EinsteinÆther theories it is constrained to be timelike with unit norm. The most general action for such theories that is quadratic in derivatives of the vector is given by
where The coefficients are arbitrary. In the unconstrained theories, and is arbitrary. In the constrained theories, is a Lagrange multiplier, and by virtue of the constraint , the factor in front of the Ricci scalar can be absorbed into a rescaling of ; equivalently, in the constrained theories, we can set . Note that the possible term can be shown under integration by parts to be equivalent to a linear combination of the terms involving and .Unconstrained theories were studied during the 1970s as “strawman” alternatives to GR. In addition to having up to four arbitrary parameters, they also left the magnitude of the vector field arbitrary, since it satisfies a linear homogenous vacuum field equation of the form ( in all such cases studied). Indeed, this latter fact was one of most serious defects of these theories. Each theory studied corresponds to a special case of the action (39), all with :
The EinsteinÆther theories were motivated in part by a desire to explore possibilities for violations of Lorentz invariance in gravity, in parallel with similar studies in matter interactions, such as the SME. The general class of theories was analyzed by Jacobson and collaborators [137, 183, 138, 99, 113], motivated in part by [156].
Analyzing the postNewtonian limit, they were able to infer values of the PPN parameters and as follows [113]:
where , , , subject to the constraints , , .By requiring that gravitational wave modes have real (as opposed to imaginary) frequencies, one can impose the bounds and . Considerations of positivity of energy impose the constraints , and .
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