Substantive General Covariance: Another Decade of Dispute

Whether Einstein’s theory of general relativity (GR) satisfies a substantive principle deserving the name “general covariance ” is a notoriously controversial matter. John Norton’s masterful review of the matter, published in 1993, was aptly subtitled “eight decades of dispute ” (Norton, 1993). And yet, despite the continuing controversy, there has been broad agreement about a number of core issues. Two closely related theses are part of the orthodox position: (i) that general covariance does not distinguish general relativity from pre-relativistic theories when the latter are appropriately formulated and (ii) that general covariance, by itself, does not have any physical content. The first of these theses is almost as old as GR itself. Einstein had sought a gravitational theory that was compatible with special relativity (SR). Soon after 1905 he came to believe that what was required was a generalization of SR’s restricted relativity principle. According to SR, all inertial frames are on a par from the point of view of the fundamental laws. What Einstein sought was a theory according to which all frames are on a par. General covariance was supposed to implement this. A theory is general covariant if the equations that express its laws are left form-invariant by smooth but otherwise arbitrary coordinate transformations.1 Since these coordinate transformations include transformations between coordinate systems adapted to frames in arbitrary relative motion, it would seem that there can be no privileged frames of reference in a general covariant theory. This impression, however, is misleading. As Kretschmann famously pointed out, “by means of a purely mathematical reformulation of the equations representing the theory, an