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Can all of the predictions made in Special Relativity (SR) also be made in General Relativity (GR)?

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2 Answers 2

up vote 8 down vote accepted

Yes. There is a set of metric tensors that describe flat spacetime--that is, the spacetime of special relativity. General relativity allows us to consider many kinds of metrics, but limiting ourselves only to those that are flat reproduces all the basic predictions of special relativity.

A big thing that separates SR from GR is that GR demands that matter and energy couple to the underlying curvature of spacetime. In the absence of such coupling, spacetime would simply be flat.

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On the hypothetical level, it might not be flat. In the absence of coupling, geometry won't respond to the behavior or matter, but one could still have a complicated geometry affecting the behavior of matter. Even with restrictive global symmetries analogous to Minkowski spacetime, it might be de Sitter or something (e.g., turning off matter coupling but keeping a cosmological constant). Without any such conditions, it's arbitrary. –  Stan Liou Dec 6 '12 at 6:56
As Stan said, your last sentence is not correct. There are non flat solution of GR with no matter at all: black holes for example. –  Curious George Dec 6 '12 at 20:14
I agree that what form spacetime takes in absence of matter is arbitrary. Perhaps it would be better to say that GR contains all the predictions for flat metrics, as well as for others that have vanishing stress-energy everywhere. –  Muphrid Dec 6 '12 at 21:27

Yes, but not only general relativity must reduce to special relativity, but any physical law used in general relativity must reduce to the physical law in special relativity. For instance, curved spacetime electrodynamics must reduce to the ordinary Maxwellian electrodynamics when the spacetime is flat.

Precisely Wald emphasizes this fact in his well-known textbook on general relativity (p.68):

The laws of physics in general relativity are governed by two basic principles: (1) the principle of general covariance [...] (2) the requirement that equations must reduce to the equations satisfied in special relativity in the case where $g_{ab}$ is flat.

In the above quote $g_{ab}$ is the spacetime metric.

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