# General covariance from the equivalence principle

Einstein's equivalence principle (EEP) tells us that there is no way in principle to locally distinguish between inertial acceleration and the effects of a gravitational field by carrying out any non-gravitational physics experiments. Furthermore, in a free-falling frame of reference the laws of physics should reduce to those of special relativity (SR).

My question is, can it be inferred from the EEP that the laws of physics must be the identical in all frames of reference, i.e. they must be generally covariant?

Having thought about it, I think it’s the case that although the principle of equivalence and general covariance do not imply one another, it is the case that the principle of equivalence, in particular, the EEP, that gives physical content to the principle of general covariance. The reason being that, by itself, the principle of general covariance is vacuous since any physical theory can be expressed in a general covariant form. What gives it physical content is that the EEP implies that the (non-gravitational) laws of physics reduce to those of special relativity in the presence of an arbitrary gravitational field, so long as one is in a local inertial frame of reference. Given this, we can use the notion of general covariance to re-express the equations in this frame in tensorial form. These equations will then reduce to their special relativistic form whenever one is in a local inertial reference frame. Now that they are in tensorial form, if they hold in one frame of reference, they must hold in all frames of reference (both inertial and non-inertial). This is exactly the requirement of general covariance.

• You might find this paper insightful: pitt.edu/~jdnorton/papers/decades.pdf – bapowell Jan 6 '18 at 0:35
• In Weinberg's Gravitation, Chapter 4, section 1, there is a beautiful and lucid discussion about t. his. I strongly advice you to read it. – Run like hell Sep 1 '18 at 1:15
• Other theories of physics can be expressed in a coordinate-independent way. In fact, if they couldn't be, I would find that extremely strange -- it would mean that the theory cared about what names you gave to points in spacetime. The equivalence principle, on the other hand, is highly specific to GR. If you demand the e.p. plus consistency with SR, you get GR. – Ben Crowell May 7 at 0:21

## 2 Answers

The Einstein equivalence principle (EEP) states that in small enough regions of spacetime the laws of physics reduce to those of special relativity. The general covariance principle assumes that the laws of physics have the same form in any reference frame on the simple consideration that the coordinates systems are just artifacts of the human mind and finds its expression in the tensorial formalism. The two principles are not related, but together allow to generalize the physical laws stated in an inertial reference frame to an arbitrary one.

My understanding of the situation is that we should separate mathematical and physical aspects in our thinking.

In a mathematical theory we can formulate sentences such as "Can we infer this theorem from that set of axioms". Ideally, theorems and axioms are formalized to an extent, that we can attempt to infer by a systematic method of applying logical or mathematical operations.

In physics, statements should be open to experimental testing. I will not able to prove a statement or infer it in a mathematical sense.

In your case, EPP is a physical statement: It is open to experimental verification. Someone might come up with an experiment which distinguishes acceleration from gravity. Thus far, no one came up with such an experiment (in a local sense; you might note that there are tidal forces in gravity, which allow us to distinguish gravity from acceleration using non-local experiments).

Covariance is a mathematical property. It is not open to experimental verification but follows from the definition of a vector field on a manifold. If we introduce the concept of bases and of coordinate transforms, this leads us to representations of the vector field in the various bases. If we do this correctly, the resulting transformation rules automatically are covariant.

The two spheres touch each other when physicists formulate "forces can be described by vector fields".

In do not see, however, a mathematical property could be inferred from a physical statement.