Laue's Theorem Revisited: Energy-Momentum Tensors, Symmetries, and the Habitat of Globally Conserved Quantities

TL;DR

This paper revisits Laue's theorem on energy-momentum tensors, providing a geometric generalization from special relativity to arbitrary space-times, clarifying its assumptions and broadening its applicability.

Contribution

It offers a new, geometric formulation of Laue's theorem applicable to general space-times, extending and clarifying its original scope in special relativity.

Findings

Generalizes Laue's theorem to arbitrary space-times

Provides a geometric proof using differential geometry

Clarifies the theorem's assumptions and physical interpretation

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 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 space-time 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 space-times with significantly less symmetries. This result, which seems to be new, not only generalises 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.