In this review, we shall focus on GR, the general class of scalar-tensor modifications of it, of which the Jordan–Fierz–Brans–Dicke theory (Brans–Dicke, for short) is the classic example, and vector-tensor theories. The reasons are several-fold:
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 non-zero 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 solar-system 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 , whereR 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 absence of prior-geometric elements, the equations of motion are also a consequence of the field equations via the Bianchi identities .
The general procedure for deriving the post-Newtonian limit of metric theories is spelled out in TEGP 5.1 , and is described in detail for GR in TEGP 5.2 . 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 ). For some purposes, the action is conveniently written in a non-metric representation, sometimes denoted the “Einstein frame”, in which the gravitational action looks exactly like that of GR:, and the values of the PPN parameters are listed in Table 3.
The parameters that enter the post-Newtonian limit are
Damour and Esposito-Farèse  have adopted an alternative parametrization of scalar-tensor theories, in which one expands about a cosmological background field value :[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 non-linear 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 particle-physics-inspired 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 solar-system 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, four-vector field . In some models, the four-vector is unconstrained, while in others, called Einstein-Æther 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
Unconstrained theories were studied during the 1970s as “straw-man” 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 Einstein-Æther 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 .
Analyzing the post-Newtonian limit, they were able to infer values of the PPN parameters and as follows :
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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