The property that all non-gravitational fields should couple in the same manner to a single gravitational field is sometimes called ``universal coupling''. Because of it, one can discuss the metric as a property of spacetime itself rather than as a field over spacetime. This is because its properties may be measured and studied using a variety of different experimental devices, composed of different non-gravitational fields and particles, and, because of universal coupling, the results will be independent of the device. Thus, for instance, the proper time between two events is a characteristic of spacetime and of the location of the events, not of the clocks used to measure it.

Consequently, if EEP is valid, the non-gravitational laws of
physics may be formulated by taking their special relativistic
forms in terms of the Minkowski metric
**
**
and simply ``going over'' to new forms in terms of the curved
spacetime metric
**
g
**, using the mathematics of differential geometry. The details of
this ``going over'' can be found in standard textbooks ([94,
136], TEGP 3.2. [147]).

What distinguishes one metric theory from another, therefore, is the number and kind of gravitational fields it contains in addition to the metric, and the equations that determine the structure and evolution of these fields. From this viewpoint, one can divide all metric theories of gravity into two fundamental classes: ``purely dynamical'' and ``prior-geometric''.

By ``purely dynamical metric theory'' we mean any metric
theory whose gravitational fields have their structure and
evolution determined by coupled partial differential field
equations. In other words, the behavior of each field is
influenced to some extent by a coupling to at least one of the
other fields in the theory. By ``prior geometric'' theory, we
mean any metric theory that contains ``absolute elements'',
fields or equations whose structure and evolution are given
*a priori*, and are independent of the structure and evolution of the other
fields of the theory. These ``absolute elements'' typically
include flat background metrics
**
**, cosmic time coordinates
*t*, and algebraic relationships among otherwise dynamical fields,
such as
, where
and
may be dynamical fields.

General relativity is a purely dynamical theory since it
contains only one gravitational field, the metric itself, and its
structure and evolution are governed by partial differential
equations (Einstein's equations). Brans-Dicke theory and its
generalizations are purely dynamical theories; the field equation
for the metric involves the scalar field (as well as the matter
as source), and that for the scalar field involves the metric.
Rosen's bimetric theory is a prior-geometric theory: It has a
non-dynamical, Riemann-flat background metric
**
**, and the field equations for the physical metric
**
g
**
involve

By discussing metric theories of gravity from this broad point of view, it is possible to draw some general conclusions about the nature of gravity in different metric theories, conclusions that are reminiscent of the Einstein equivalence principle, but that are subsumed under the name ``strong equivalence principle''.

Consider a local, freely falling frame in any metric theory of
gravity. Let this frame be small enough that inhomogeneities in
the external gravitational fields can be neglected throughout its
volume. On the other hand, let the frame be large enough to
encompass a system of gravitating matter and its associated
gravitational fields. The system could be a star, a black hole,
the solar system or a Cavendish experiment. Call this frame a
``quasi-local Lorentz frame''. To determine the behavior of the
system we must calculate the metric. The computation proceeds in
two stages. First we determine the external behavior of the
metric and gravitational fields, thereby establishing boundary
values for the fields generated by the local system, at a
boundary of the quasi-local frame ``far'' from the local system.
Second, we solve for the fields generated by the local system.
But because the metric is coupled directly or indirectly to the
other fields of the theory, its structure and evolution will be
influenced by those fields, and in particular by the boundary
values taken on by those fields far from the local system. This
will be true even if we work in a coordinate system in which the
asymptotic form of
in the boundary region between the local system and the external
world is that of the Minkowski metric. Thus the gravitational
environment in which the local gravitating system resides can
influence the metric generated by the local system via the
boundary values of the auxiliary fields. Consequently, the
results of local gravitational experiments may depend on the
location and velocity of the frame relative to the external
environment. Of course, local
*non*
-gravitational experiments are unaffected since the gravitational
fields they generate are assumed to be negligible, and since
those experiments couple only to the metric, whose form can
always be made locally Minkowskian at a given spacetime event.
Local gravitational experiments might include Cavendish
experiments, measurement of the acceleration of massive
self-gravitating bodies, studies of the structure of stars and
planets, or analyses of the periods of ``gravitational clocks''.
We can now make several statements about different kinds of
metric theories.

(i) A theory which contains only the metric
**
g
**
yields local gravitational physics which is independent of the
location and velocity of the local system. This follows from the
fact that the only field coupling the local system to the
environment is

(ii) A theory which contains the metric
**
g
**
and dynamical scalar fields
yields local gravitational physics which may depend on the
location of the frame but which is independent of the velocity of
the frame. This follows from the asymptotic Lorentz invariance of
the Minkowski metric and of the scalar fields, but now the
asymptotic values of the scalar fields may depend on the location
of the frame. An example is Brans-Dicke theory, where the
asymptotic scalar field determines the effective value of the
gravitational constant, which can thus vary as
varies. On the other hand, a form of velocity dependence in
local physics can enter indirectly if the asymptotic values of
the scalar field vary with time cosmologically. Then the

(iii) A theory which contains the metric
**
g
**
and additional dynamical vector or tensor fields or
prior-geometric fields yields local gravitational physics which
may have both location and velocity-dependent effects.

These ideas can be summarized in the strong equivalence principle (SEP), which states that:

- WEP is valid for self-gravitating bodies as well as for test bodies.
- The outcome of any local test experiment is independent of the velocity of the (freely falling) apparatus.
- The outcome of any local test experiment is independent of where and when in the universe it is performed.

The above discussion of the coupling of auxiliary fields to
local gravitating systems indicates that if SEP is strictly
valid, there must be one and only one gravitational field in the
universe, the metric
**
g
**
. These arguments are only suggestive however, and no rigorous
proof of this statement is available at present. Empirically it
has been found that every metric theory other than GR introduces
auxiliary gravitational fields, either dynamical or prior
geometric, and thus predicts violations of SEP at some level
(here we ignore quantum-theory inspired modifications to GR
involving ``
'' terms). General relativity seems to be the only metric theory
that embodies SEP completely. This lends some credence to the
conjecture SEP
General Relativity. In Sec.
3.6, we shall discuss experimental evidence for the validity of
SEP.

The Confrontation between General Relativity and
Experiment
Clifford M. Will
http://www.livingreviews.org/lrr-2001-4
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