3.4 Tests of the parameter 3 Tests of Post-Newtonian Gravity3.2 The parametrized post-Newtonian formalism

3.3 Competing theories of gravity 

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

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:

3.3.1 General relativity 

The metric g is the sole dynamical field, and the theory contains no arbitrary functions or parameters, apart from the value of the Newtonian coupling constant G, which is measurable in laboratory experiments. Throughout this article, we ignore the cosmological constant tex2html_wrap_inline4183 . Although tex2html_wrap_inline4183 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 values of tex2html_wrap_inline4183 corresponding to a cosmological closure density.

The field equations of GR are derivable from an invariant action principle tex2html_wrap_inline4437, where


where R is the Ricci scalar, and tex2html_wrap_inline4441 is the matter action, which depends on matter fields tex2html_wrap_inline4443 universally coupled to the metric g . By varying the action with respect to tex2html_wrap_inline4263, we obtain the field equations


where tex2html_wrap_inline4449 is the matter energy-momentum tensor. General covariance of the matter action implies the equations of motion tex2html_wrap_inline4419 ; varying tex2html_wrap_inline4441 with respect to tex2html_wrap_inline4443 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 tex2html_wrap_inline4457 .

The general procedure for deriving the post-Newtonian limit is spelled out in TEGP 5.1 [147Jump To The Next Citation Point In The Article], and is described in detail for GR in TEGP 5.2 [147Jump To The Next Citation Point In The Article]. The PPN parameter values are listed in Table  3 .

Table 3: Metric theories and their PPN parameter values (tex2html_wrap_inline3767 for all cases).

3.3.2 Scalar-tensor theories 

These theories contain the metric g , a scalar field tex2html_wrap_inline4205, a potential function tex2html_wrap_inline4497, and a coupling function tex2html_wrap_inline4499 (generalizations to more than one scalar field have also been carried out [42Jump To The Next Citation Point In The Article]). 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:


where tex2html_wrap_inline4501 is the Ricci scalar of the ``Einstein'' metric tex2html_wrap_inline4201 . (Apart from the scalar potential term tex2html_wrap_inline4497, this corresponds to Eq. (20Popup Equation) with tex2html_wrap_inline4507, tex2html_wrap_inline4509, and tex2html_wrap_inline4511 .) This representation is a ``non-metric'' one because the matter fields tex2html_wrap_inline4443 couple to a combination of tex2html_wrap_inline4205 and tex2html_wrap_inline4201 . 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 [147Jump To The Next Citation Point In The Article], and the values of the PPN parameters are listed in Table  3 .

The parameters that enter the post-Newtonian limit are


where tex2html_wrap_inline4519 is the value of tex2html_wrap_inline4521 today far from the system being studied, as determined by appropriate cosmological boundary conditions. The following formula is also useful: tex2html_wrap_inline4523 . In Brans-Dicke theory (tex2html_wrap_inline4525 constant), the larger the value of tex2html_wrap_inline3833, the smaller the effects of the scalar field, and in the limit tex2html_wrap_inline4529 (tex2html_wrap_inline4531), the theory becomes indistinguishable from GR in all its predictions. In more general theories, the function tex2html_wrap_inline4533 could have the property that, at the present epoch, and in weak-field situations, the value of the scalar field tex2html_wrap_inline4519 is such that tex2html_wrap_inline3833 is very large and tex2html_wrap_inline4539 is very small (theory almost identical to GR today), but that for past or future values of tex2html_wrap_inline4521, or in strong-field regions such as the interiors of neutron stars, tex2html_wrap_inline3833 and tex2html_wrap_inline4539 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 tex2html_wrap_inline3757, thence to large tex2html_wrap_inline3833 . Estimates of the expected relic deviations from GR today in such theories depend on the cosmological model, but range from tex2html_wrap_inline3921 to a few times tex2html_wrap_inline4553 for tex2html_wrap_inline4555  [47Jump To The Next Citation Point In The Article, 48Jump To The Next Citation Point In The Article].

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 [120]), in which the potential function tex2html_wrap_inline4497 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 tex2html_wrap_inline4559 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 tex2html_wrap_inline3833 today is so large that the theory's local predictions are experimentally indistinguishable from GR. In other models, explored by Damour and Esposito-Farèse [43], 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 tex2html_wrap_inline3833 .

3.4 Tests of the parameter 3 Tests of Post-Newtonian Gravity3.2 The parametrized post-Newtonian formalism

image The Confrontation between General Relativity and Experiment
Clifford M. Will
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