general relativity - Gravity theories with the equivalence principle but different from GR

Einstein's general relativity assumes the equivalence of acceleration and gravitation. Is there a general class of gravity theories that have this property but disagree with general relativity? Will such theories automatically satisfy any of the tests of general relativity such as the precession of mercury or the bending of light?

Answer

Dear Carl, my overall moral answer is the opposite one to the first two answers, so let me write a separate answer. My overall message is that with the right minimal extra assumptions, a correct theory respecting the equivalence principle has to agree with GR at cosmological distances.

There are various modifications or competitors of GR, see e.g. this list:

http://en.wikipedia.org/wiki/Classical_theories_of_gravitation#Articles_on_specific_classical_field_theories_of_gravitation

First of all, most of the theories tend to be given by an action. This is needed to preserve the Noether conservation laws for symmetries, and so on. No interesting (viable enough and new enough) non-action theory is known, as far as I know.

The equivalence principle requires that local physics must know how it's mapped to the flat Minkowski space - so there must exist the metric tensor at each point. However, there may also exist other fields or degrees of freedom.

One class you find in the list above is Brans-Dicke theory. It's an example of a broader class of theories that contain new scalar or tensor fields, besides the metric tensor. Torsion is a popular, frequently discussed addition by the people working on "competition to GR".

If such fields are massless, they may modify the long-distance physics. But they effectively destroy the equivalence principle, too. If the extra fields are massless (like moduli, massless scalar fields), their values should be viewed as a part of the gravitational field but these values influence local physics. That's really forbidden.

So when understood sufficiently strictly, only the metric tensor should form the spectrum of massless (or light) fields.

Then, you may have many theories which are GR plus additional fields. The additional fields are then treated "matter fields". They can be anything. An interesting subclass are Kaluza-Klein theories. In them, many new massive fields may be understood as coming from a higher-dimensional general relativity.

With the equivalence principle fully respected, we really deal with theories defined by an action where the metric tensor is the only massive "gravitational" field. The action for the other fields - matter - is a separate question and there are of course many choices. The action for the metric tensor must be diffeomorphism invariant.

One may prove that such an action, to have the invariance, must be a function of the Riemann tensor and its covariant derivatives, with properly contracted indices. For example, terms like $$\nabla_\alpha R_{\beta\gamma\delta\epsilon} \nabla^\alpha R^{\beta\gamma\delta\epsilon}$$ are tolerable. However, all such terms contain either extra covariant derivatives or extra powers of the Riemann tensor, relatively to the Ricci scalar $R$. For dimensional reasons, all such extra terms have to be multiplied by a positive power of a distance $L$ and it seems likely that all such distances $L$ that could occur are microscopic or ultramicroscopic. So all such terms become negligible at cosmological (or astrophysical) distances. In this long-distance limit, the gravitational theory has to be given by the Einstein-Hilbert action e.g. Einstein's equations and the predictions for Mercury and bending of light are inevitable.

It wasn't an accident that Einstein ended up with the right theory. He had to - and many of his "heuristic" arguments (especially "minimality") may be justified by the more valid RG-like $L$-argument above.