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:
].
. Although
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
corresponding to a cosmological closure density.
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).
, a potential function 
, and a
coupling function 
(generalizations to more than one scalar
field have also been carried out [42]).
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 
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 .
|
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 |
