General Covariance and Conservation Laws

Abstract

After strenuous efforts, having followed several dead-ends, Einstein had succeeded in establishing a consistent theory of general relativity. The mathematicians, whose illustrious ancestors had laid its mathematical foundation, happily took up the development and further interpretation of the new theory. Some disagreement arose on a basic question concerning its physical interpretation. Einstein, in this case, stuck to a more formal approach, putting a formal energy conservation law in the forefront. The reason is to be found in his earlier procedure, in which full covariance was rather doubtful [see Section II.4]. Hilbert, on the other hand, tried to exploit covariance physically. Finally it was Klein, the author of the Erlangen Programme and grand old master of geometry, who, in his last important work, completed the synthesis of the views of Einstein and Hilbert — the synthesis, in a sense, of the theory of relativity and the theory of invariants. In 1917, he published two memoirs dealing with ‘energy-momentum conservation’ in general relativity. Shortly before this, in an exchange of letters with Hilbert, Klein clarified the meaning of the four conditions which Hilbert had discovered in this context. What escaped Klein was the fact that there existed an even more elegant connection with the equations of general geometries, the Bianchi identities.