### 3.1 Metric theories of gravity and the strong equivalence principle

#### 3.1.1 Universal coupling and the metric postulates

The empirical evidence supporting the Einstein equivalence principle, discussed in the previous Section 2, supports the conclusion that the only theories of gravity that have a hope of being viable are metric theories, or possibly theories that are metric apart from very weak or short-range non-metric couplings (as in string theory). Therefore for the remainder of this review, we shall turn our attention exclusively to metric theories of gravity, which assume that

1. there exists a symmetric metric,
2. test bodies follow geodesics of the metric, and
3. in local Lorentz frames, the non-gravitational laws of physics are those of special relativity.

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 , using the mathematics of differential geometry. The details of this “going over” can be found in standard textbooks (see [189270], TEGP 3.2. [281]).

#### 3.1.2 The strong equivalence principle

In any metric theory of gravity, matter and non-gravitational fields respond only to the spacetime metric . In principle, however, there could exist other gravitational fields besides the metric, such as scalar fields, vector fields, and so on. If, by our strict definition of metric theory, matter does not couple to these fields, what can their role in gravitation theory be? Their role must be that of mediating the manner in which matter and non-gravitational fields generate gravitational fields and produce the metric; once determined, however, the metric alone acts back on the matter in the manner prescribed by EEP.

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 or cosmic time coordinates .

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 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.

• A theory which contains only the metric 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 , and it is always possible to find a coordinate system in which takes the Minkowski form at the boundary between the local system and the external environment (neglecting inhomogeneities in the external gravitational field). Thus the asymptotic values of are constants independent of location, and are asymptotically Lorentz invariant, thus independent of velocity. General relativity is an example of such a theory.
• A theory which contains the metric 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 rate of variation of the gravitational constant could depend on the velocity of the frame.
• A theory which contains the metric 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:

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

The distinction between SEP and EEP is the inclusion of bodies with self-gravitational interactions (planets, stars) and of experiments involving gravitational forces (Cavendish experiments, gravimeter measurements). Note that SEP contains EEP as the special case in which local gravitational forces are ignored.

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 . These arguments are only suggestive however, and no rigorous proof of this statement is available at present. Empirically it has been found that almost 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 “R2” terms). The one exception is Nordström’s 1913 conformally-flat scalar theory [195], which can be written purely in terms of the metric; the theory satisfies SEP, but unfortunately violates experiment by predicting no deflection of light. General relativity seems to be the only viable metric theory that embodies SEP completely. In Section 3.6, we shall discuss experimental evidence for the validity of SEP.