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]).
. The Parametrized Post-Newtonian formalism
. Three-dimensional, Euclidean vector
notation is used throughout. All coordinate arbitrariness
(``gauge freedom'') has been removed by specialization of the
coordinates to the standard PPN gauge (TEGP 4.2 [147]).
Units are chosen so that G = c = 1, where G is the physically
measured Newtonian constant far from the solar system.
:
density of rest mass as measured in a local freely falling
frame momentarily comoving with the gravitating matter;
: coordinate velocity of the matter;
:
coordinate velocity of the PPN coordinate system relative
to the mean rest-frame of the universe;
:
internal energy per unit rest mass. It includes all forms
of non-rest-mass, non-gravitational energy, e.g. energy
of compression and thermal energy.
,
,
,
.
|
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 |
