Thus the metric and the equations of motion for matter become the primary entities for calculating observable effects, and all that distinguishes one metric theory from another is the particular way in which matter and possibly other gravitational fields generate the metric.
The comparison of metric theories of gravity with each other and with experiment becomes particularly simple when one takes the slow-motion, weak-field limit. This approximation, known as the post-Newtonian limit, is sufficiently accurate to encompass most solar-system tests that can be performed in the foreseeable future. It turns out that, in this limit, the spacetime metric g predicted by nearly every metric theory of gravity has the same structure. It can be written as an expansion about the Minkowski metric () in terms of dimensionless gravitational potentials of varying degrees of smallness. These potentials are constructed from the matter variables (Box 2) in imitation of the Newtonian gravitational potential
The ``order of smallness'' is determined according to the rules , , and so on (we use units in which G = c =1; see Box 2).
Table 2: The PPN Parameters and their significance (note that has been shown twice to indicate that it is a measure of two effects).
A consistent post-Newtonian limit requires determination of correct through , through and through (for details see TEGP 4.1 ). The only way that one metric theory differs from another is in the numerical values of the coefficients that appear in front of the metric potentials. The parametrized post-Newtonian (PPN) formalism inserts parameters in place of these coefficients, parameters whose values depend on the theory under study. In the current version of the PPN formalism, summarized in Box 2, ten parameters are used, chosen in such a manner that they measure or indicate general properties of metric theories of gravity (Table 2). Under reasonable assumptions about the kinds of potentials that can be present at post-Newtonian order (basically only Poisson-like potentials), one finds that ten PPN parameters exhaust the possibilities.
The parameters and are the usual Eddington-Robertson-Schiff parameters used to describe the ``classical'' tests of GR, and are in some sense the most important; they are the only non-zero parameters in GR and scalar-tensor gravity. The parameter is non-zero in any theory of gravity that predicts preferred-location effects such as a galaxy-induced anisotropy in the local gravitational constant (also called ``Whitehead'' effects); , , measure whether or not the theory predicts post-Newtonian preferred-frame effects; , , , , measure whether or not the theory predicts violations of global conservation laws for total momentum. Next to and , the parameters and occur most frequently with non-trivial null values. In Table 2 we show the values these parameters take (i) in GR, (ii) in any theory of gravity that possesses conservation laws for total momentum, called ``semi-conservative'' (any theory that is based on an invariant action principle is semi-conservative), and (iii) in any theory that in addition possesses six global conservation laws for angular momentum, called ``fully conservative'' (such theories automatically predict no post-Newtonian preferred-frame effects). Semi-conservative theories have five free PPN parameters (, , , , ) while fully conservative theories have three (, , ).
The PPN formalism was pioneered by Kenneth Nordtvedt , who studied the post-Newtonian metric of a system of gravitating point masses, extending earlier work by Eddington, Robertson and Schiff (TEGP 4.2 ). A general and unified version of the PPN formalism was developed by Will and Nordtvedt. The canonical version, with conventions altered to be more in accord with standard textbooks such as , is discussed in detail in TEGP 4 . Other versions of the PPN formalism have been developed to deal with point masses with charge, fluid with anisotropic stresses, bodies with strong internal gravity, and post-post-Newtonian effects (TEGP 4.2, 14.2 ).
|The Confrontation between General Relativity and
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
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