Energy-Momentum Tensors and Motion in Special Relativity

TL;DR

This paper reviews the mathematical structures underlying the concepts of motion and conserved quantities for extended objects in Special Relativity, including Poincaré charges, angular momentum decomposition, and center of mass, with detailed geometric and algebraic insights.

Contribution

It provides a comprehensive review of the mathematical framework for motion and conserved quantities in Special Relativity, emphasizing geometric and algebraic structures.

Findings

Clarifies the role of Poincaré charges in conserved quantities

Details the decomposition of angular momentum into Spin and orbital parts

Describes the geometric notion of center of mass and Moeller Radius

Abstract

The notions of "motion" and "conserved quantities", if applied to extended objects, are already quite non-trivial in Special Relativity. This contribution is meant to remind us on all the relevant mathematical structures and constructions that underlie these concepts, which we will review in some detail. Next to the prerequisites from Special Relativity, like Minkowski space and its automorphism group, this will include the notion of a body in Minkowski space, the momentum map, a characterisation of the habitat of globally conserved quantities associated with Poincar\'e symmetry -- so called Poincar\'e charges --, the frame-dependent decomposition of global angular momentum into Spin and an orbital part, and, last not least, the likewise frame-dependent notion of centre of mass together with a geometric description of the Moeller Radius, of which we also list some typical values. Two Appendices present some mathematical background material on Hodge duality and group actions on manifolds. This is a contribution to the book: "Equations of Motion in Relativistic Gravity", edited by Dirk Puetzfeld and Claus Laemmerzahl, to be published by Springer Verlag.