General Relativity – How General Theory of Relativity Demonstrates Uniformity of Natural Laws Across All Frames of Reference

Question

The General Theory Of Relativity is often presented as both a theory that extends relativity to all frames of reference (or, at least, to inertial frames and uniformly accelerated ones) and, at the same time, a theory of gravity.

The gravity aspect of the theory is commonly well explained. I know that the details are mathematically intricate, but this is often summarized as " matter/energy tells space how to curve, space tells matter how to move". Under this aspect, the result of the theory seems to be Einstein's field equations.

However, it's more difficult to see, in ordinary popular presentations of the theory, in which way it realizes its primary goal. What I mean is: how does the GR theory manage to express the laws of nature in a way that keeps their form invariant in all reference frames?

One thing one could expect (maybe very naively) was some sort of extension of the Lorentz transformations, that is, a transformation that would be valid, not only from one inertial reference frame to another but, from an arbitrary inertial frame to any arbitrary ( uniformly) accelerated frame.

But, no popular presentation mentions anything like this. So, I guess that such a universal transformation/ mapping on reference frames is not what is to be expected from the GR theory.

Hence the question: what remains in the theory from its initial goal (namely, as I said, the extension of relativity to all frames of reference)?

Certainly, the Equivalence Principle ( under one form or another) is part of the answer to my question. (If gravity can be theorized as constant acceleration ( at least locally), I understand that a theory of gravity has something to say as to what the laws of nature look like in an accelerated frame. )

Another element is the fact that the theory makes use of tensors, that have to be invariant under any change of coordinates. And it seems (I took this in Norton's online lecture) that Einstein reduced the requisite of invariance under change of reference frame to a requisite of invariance under change of coordinates.

These are vague ideas I gathered from various sources, but it seems to me that I still cannot figure out in which way this theory of gravity achieves its primary goal in as much as it is a general theory of relativity.

Answer 1

Without gravity

• Inertial frames in Cartesian coordinates are related to each other by Lorentz transformations. A "typical" undergraduate course in special relativity will explain these in detail. We are interested in tensors which transform simply under Lorentz transformations. The partial derivative of a Lorentz tensor is a tensor.
• Non-inertial frames are essentially the same, mathematically, as frames in non-Cartesian coordinates. Namely, you take an inertial frame, and perform a general coordinate transformation. To properly handle these general coordinate transformations, it is useful to introduce several geometric objects. For example, the partial derivative of a tensor is no longer a tensor, essentially because under a coordinate transformation there an an extra term where the partial derivative acts on the Jacobian matrix of the transformation. Thus we introduce a covariant derivative, which generalizes the ordinary partial derivative in such a way that it transforms as a tensor under general coordinate transformations. While there is no curvature, developing the mathematical apparatus needed here brings you a long way toward understanding curved spacetimes. In particular, formulating, say, Maxwell's equations using covariant derivatives instead of partial derivatives, means their form will hold in any coordinate system, which includes non-inertial reference frames.

With gravity

• Spacetime is curved. All of the machinery developed to handle non-inertial frames in flat space-time (such as covariant derivatives) are needed, as well as additional objects (like the Riemann curvature tensor) that describe the spatial curvature. One way to think of curvature, is that it is an obstruction to finding coordinates where the metric takes the form of the Minkowski metric everywhere in spacetime.

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I would say that what distinguishes general relativity and special relativity is the presence of gravity, or space-time curvature. So I would consider non-inertial frames in the absence of gravity as part of special relativity. However, a lot of the mathematical tools you need to describe non-inertial frames, such as covariant derivatives, are also needed to describe curved space-times.