3.3 Competing theories of gravity
One of the important applications of the PPN formalism is the comparison and classification of
alternative metric theories of gravity. The population of viable theories has fluctuated over the years as new
effects and tests have been discovered, largely through the use of the PPN framework, which eliminated
many theories thought previously to be viable. The theory population has also fluctuated as new,
potentially viable theories have been invented.
In this review, we shall focus on GR, 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, and vector-tensor
theories. The reasons are several-fold:
- A full compendium of alternative theories circa 1981 is given in TEGP 5 [281
].
- Many alternative metric theories developed during the 1970s and 1980s could be viewed as
“straw-man” theories, invented to prove that such theories exist or to illustrate particular
properties. Few of these could be regarded as well-motivated theories from the point of view,
say, of field theory or particle physics.
- A number of theories fall into the class of “prior-geometric” theories, with absolute elements
such as a flat background metric in addition to the physical metric. Most of these theories
predict “preferred-frame” effects, that have been tightly constrained by observations (see
Section 3.6.2). An example is Rosen’s bimetric theory.
- A large number of alternative theories of gravity predict gravitational wave emission
substantially different from that of general relativity, in strong disagreement with observations
of the binary pulsar (see Section 7).
- Scalar-tensor modifications of GR have become very popular in unification schemes such as
string theory, and in cosmological model building. Because the scalar fields could be massive,
the potentials in the post-Newtonian limit could be modified by Yukawa-like terms.
- Vector-tensor theories have attracted recent attention, in the spirit of the SME (see
Section 2.2.4), as models for violations of Lorentz invariance in the gravitational sector.
3.3.1 General relativity
The metric
is the sole dynamical field, and the theory contains no arbitrary functions or parameters,
apart from the value of the Newtonian coupling constant
, which is measurable in laboratory
experiments. Throughout this article, we ignore the cosmological constant
. We do this despite recent
evidence, from supernova data, of an accelerating universe, which would indicate either a non-zero
cosmological constant or a dynamical “dark energy” contributing about 70 percent of the critical density.
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 the values of
inferred from
supernova observations.
The field equations of GR are derivable from an invariant action principle
, where
where
is the Ricci scalar, and
is the matter action, which depends on matter fields
universally coupled to the metric
. 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 of the Standard Model. 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 of metric theories is spelled out in
TEGP 5.1 [281
], and is described in detail for GR in TEGP 5.2 [281
]. The PPN parameter values are
listed in Table 3.
3.3.2 Scalar-tensor theories
These theories contain the metric
, a scalar field
, a potential function
, and a coupling
function
(generalizations to more than one scalar field have also been carried out [73
]). 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 Equation (28) 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 [281
], 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. In Brans-Dicke theory (
), 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. It is useful to point out that all versions of scalar-tensor gravity predict that
(see
Table 3).
Damour and Esposito-Farèse [73] have adopted an alternative parametrization of scalar-tensor
theories, in which one expands
about a cosmological background field value
:
A precisely linear coupling function produces Brans-Dicke theory, with
, or
. The function
acts as a potential for the scalar field
within
matter, and, if
, then during cosmological evolution, the scalar field naturally evolves toward the
minimum of the potential, i.e. toward
,
, or toward a theory close to, though
not precisely GR [80, 81]. 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
.
Negative values of
correspond to a “locally unstable” scalar potential (the overall theory is still
stable in the sense of having no tachyons or ghosts). In this case, objects such as neutron stars can
experience a “spontaneous scalarization”, whereby the interior values of
can take on values very
different from the exterior values, through non-linear interactions between strong gravity and the
scalar field, dramatically affecting the stars’ internal structure and leading to strong violations
of SEP. On the other hand, in the case
, one must confront that fact that, with an
unstable
potential, cosmological evolution would presumably drive the system away from
the peak where
, toward parameter values that could be excluded by solar system
experiments.
Scalar fields coupled to gravity or matter are also ubiquitous in particle-physics-inspired models of
unification, such as string theory [254, 176, 85, 82, 83]. In some models, the coupling to matter may lead
to violations of EEP, which could be tested or bounded by the experiments described in Section 2.1. In
many models the scalar field could be 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.
3.3.3 Vector-tensor theories
These theories contain the metric
and a dynamical, typically timelike, four-vector field
. In some
models, the four-vector is unconstrained, while in others, called Einstein-Æther theories it is constrained to
be timelike with unit norm. The most general action for such theories that is quadratic in derivatives of the
vector is given by
where
The coefficients
are arbitrary. In the unconstrained theories,
and
is arbitrary. In the
constrained theories,
is a Lagrange multiplier, and by virtue of the constraint
, the factor
in front of the Ricci scalar can be absorbed into a rescaling of
; equivalently, in the
constrained theories, we can set
. Note that the possible term
can be shown under
integration by parts to be equivalent to a linear combination of the terms involving
and
.
Unconstrained theories were studied during the 1970s as “straw-man” alternatives to GR. In
addition to having up to four arbitrary parameters, they also left the magnitude of the vector field
arbitrary, since it satisfies a linear homogenous vacuum field equation of the form
(
in all such cases studied). Indeed, this latter fact was one of most serious defects of
these theories. Each theory studied corresponds to a special case of the action (39), all with
:
-
General vector-tensor theory;
,
,
,
(see TEGP 5.4 [281
])
-
The gravitational Lagrangian for this class of theories had
the form
, where
,
corresponding to the values
,
,
,
. In these
theories
,
,
, and
are complicated functions of the parameters and of
, while the rest vanish.
-
Will-Nordtvedt theory (see [290])
-
This is the special case
,
. In this theory, the PPN parameters are
given by
,
, and zero for the rest.
-
Hellings-Nordtvedt theory;
(see [128])
-
This is the special case
,
,
. Here
,
,
and
are complicated functions of the parameters and of
, while the rest vanish.
The Einstein-Æther theories were motivated in part by a desire to explore possibilities for violations of
Lorentz invariance in gravity, in parallel with similar studies in matter interactions, such as the SME. The
general class of theories was analyzed by Jacobson and collaborators [137, 183, 138, 99, 113
], motivated in
part by [156].
Analyzing the post-Newtonian limit, they were able to infer values of the PPN parameters
and
as follows [113]:
where
,
,
, subject to the constraints
,
,
.
By requiring that gravitational wave modes have real (as opposed to imaginary) frequencies, one can
impose the bounds
and
. Considerations of positivity of
energy impose the constraints
,
and
.