In this article, we shall focus on general relativity and 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. The reasons are several-fold:
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 g . By varying the action with respect to , we obtain the field equations
where is the matter energy-momentum tensor. General covariance of the matter action implies the equations of motion ; varying with respect to yields the matter field equations. By virtue of the 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 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 .
Table 3: Metric theories and their PPN parameter values ( for all cases).
where is the Ricci scalar of the ``Einstein'' metric . (Apart from the scalar potential term , this corresponds to Eq. (20) with , , and .) This representation is a ``non-metric'' 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
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, post-Newtonian limit and PPN parameters are discussed in TEGP 5.3 , and the values of the PPN parameters are listed in Table 3 .
The parameters that enter the post-Newtonian limit are
where is the value of today far from the system being studied, as determined by appropriate cosmological boundary conditions. The following formula is also useful: . In Brans-Dicke theory ( constant), 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 weak-field 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 strong-field regions such as the interiors of neutron stars, and could take on values that would lead to significant differences from GR. Indeed, Damour and Nordtvedt have shown that in such general scalar-tensor theories, GR is a natural ``attractor'': Regardless of how different the theory may be from GR in the early universe (apart from special cases), cosmological evolution naturally drives the fields toward small values of the function , thence to large . Estimates of the expected relic deviations from GR today in such theories depend on the cosmological model, but range from to a few times for [47, 48].
Scalar fields coupled to gravity or matter are also ubiquitous in particle-physics-inspired models of unification, such as string theory. In some models, the coupling to matter may lead to violations of WEP, which are tested by Eötvös-type experiments. In many models the scalar field is massive; if the Compton wavelength is of macroscopic scale, its effects are those of a ``fifth force''. Only if the theory can be cast as a metric theory with a scalar field of infinite range or of range long compared to the scale of the system in question (solar system) can the PPN framework be strictly applied. If the mass of the scalar field is sufficiently large that its range is microscopic, then, on solar-system scales, the scalar field is suppressed, and the theory is essentially equivalent to general relativity. This is the case, for example, in the ``oscillating-G'' models of Accetta, Steinhardt and Will (see ), in which the potential function contains both quadratic (mass) and quartic (self-interaction) terms, causing the scalar field to oscillate (the initial amplitude of oscillation is provided by an inflationary epoch); high-frequency oscillations in the ``effective'' Newtonian constant then result. The energy density in the oscillating scalar field can be enough to provide a cosmological closure density without resorting to dark matter, yet the value of today is so large that the theory's local predictions are experimentally indistinguishable from GR. In other models, explored by Damour and Esposito-Farèse , non-linear scalar-field couplings can lead to ``spontaneous scalarization'' inside strong-field objects such as neutron stars, leading to large deviations from GR, even in the limit of very large .
|The Confrontation between General Relativity and
Clifford M. Will
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