Laue’s theorem revisited: Energy–momentum tensors, symmetries, and the habitat of globally conserved quantities

Abstract

The energy–momentum tensor for a particular matter component summarises its local energy–momentum distribution in terms of densities and current densities. We re-investigate under what conditions these local distributions can be integrated to meaningful global quantities. This leads us directly to a classic theorem by Max von Laue concerning integrals of components of the energy–momentum tensor, whose statement and proof we recall. In the first half of this paper, we do this within the realm of Special Relativity (SR) and in the traditional mathematical language using components with respect to affine charts, thereby focusing on the intended physical content and interpretation. In the second half, we show how to do all this in a proper differential-geometric fashion and on arbitrary spacetime manifolds, this time focusing on the group-theoretic and geometric hypotheses underlying these results. Based on this we give a proper geometric statement and proof of Laue’s theorem, which is shown to generalise from Minkowski space (which has the maximal number of isometries) to spacetimes with significantly less symmetries. This result, which seems to be new, not only generalizes but also clarifies the geometric content and hypotheses of Laue’s theorem. A series of three appendices lists our conventions and notation and summarises some of the conceptual and mathematical background needed in the main text.