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 [147], and is described in detail for GR in TEGP 5.2 [147
]. 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
where
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 [147], 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 [120]), 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 [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
.
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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 |