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 [147
]). 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 [98], who studied the post-Newtonian metric of a system of
gravitating point masses, extending earlier work by Eddington,
Robertson and Schiff (TEGP 4.2 [147]). 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 [94
], is discussed in detail in TEGP 4 [147
]. 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 [147
]).
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The Confrontation between General Relativity and
Experiment
Clifford M. Will http://www.livingreviews.org/lrr-2001-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |