General relativity is a local theory. It only defines motion directly with respect to local reference matter.
It is known that prior to developing general relativity, <PERSON> thought very deeply about what he termed <PERSON>’s principle, but controversy has surrounded the question as to whether general relativity actually incorporates the principle, perhaps largely because it was never given clear expression. If the principle merely means that we can only talk of acceleration relative to other matter, then that is clearly the case in general relativity. However, the referenced matter is always local to the matter under consideration and, generally, discussion of <PERSON>’s principle seems to invoke a suggestion that rotation only makes sense in the context of the distribution of matter in the universe as a whole.
The origins of the discussion lie in <PERSON>’s rotating bucket argument. <PERSON> had observed experimentally that a concave meniscus forms in a spinning bucket hung from a rope, as the water starts to rotate with the bucket. He argued that in the absence of absolute space it would not make sense to say that the water in the bucket is rotating, and therefore that no concave meniscus would form in its surface.
<PERSON> appears to suggest that the answer lies in the motion of the water relative to distant stars. 469
This idea is certainly not directly expressed in the assumptions of general relativity, which is essentially a local theory. According to the restatement, N1*, of <PERSON>’s first law as a local law, we need consider only the motion of the particles of water relative to each other:
* N1*: An inertial body will locally remain at rest or in uniform motion with respect to other local inertial matter.
The local structure of spacetime is determined from the interactions of particles locally. An inertial frame is one in which inertial particles can maintain a state of rest with respect to each other while transmitting no net force — in effect this is the situation when there is no meniscus signifying rotation of the water.
On the other hand a converse argument can be presented. If spacetime is divided into local, and overlapping, regions, each described in inertial coordinates, then no rotation is possible in the global structure resulting by conjoining the regions. Thus we cannot say that the frame of the non-rotating bucket is determined from the “frame of the fixed stars”, but rather must say that the “frame of the fixed stars” is determined from local structures. In other words, <PERSON>’s principle is a consequence, not an underlying assumption in general relativity. 562