A No-Go Theorem about Rotation
in Relativity Theory1
DAVID B. MALAMENT
Within the framework of general relativity, in some cases at least, it is a rather delicate and interesting question just what it means to say that a body is or is not “rotating.” Moreover, the reasons for this—at least the ones I have in mind—do not have much to do with traditional controversy over “absolute vs. relative” conceptions of motion. Rather they concern particular geometric complexities that arise when one allows for the possibility of spacetime curvature. The relevant distinction for my purposes is not that between attributions of “relative” and “absolute” rotation, but between attributions of rotation than can and cannot be analyzed in terms of a motion (in the limit) at a point. It is the latter—ones that make essential reference to extended regions of spacetime—that can be problematic.
The problem has two parts. First, one can easily think of different criteria for when a body is rotating. The criteria agree if the background spacetime structure is sufficiently simple, for example, in Minkowski spacetime (the regime of “special relativity”). But they do not do so in general. Second, none of the criteria fully answers to our classical intuitions. Each one exhibits some feature or other that violates those intuitions in a significant and interesting way.
My principle goal in what follows is to make the second claim precise in the form of a modest no-go theorem. To keep tilings simple, I'll limit attention to a special case. I'll consider (one-dimensional) rings centered about an axis of rotational symmetry, and consider what it could mean to say that the rings are not rotating around the axis. (It is convenient to work with the negative formulation.) The discussion will have several parts.
First, for purposes of motivation, I'll describe two standard criteria of nonrotation that seem particularly simple and natural. (1 could assemble a longer list of proposed criteria, but I am more interested in formulating a general negative claim that applies to all.)2 One involves considerations of