Energy-Momentum Tensors and Motion in Special Relativity
The notion of “motion” and “conserved quantities”, if applied to extended objects, is already quite non-trivial in Special Relativity. This contribution is meant to remind us on all the relevant mathematical structures and constructions that underlie thes
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Abstract The notion of “motion” and “conserved quantities”, if applied to extended objects, is 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 Poincaré symmetry—so called Poincaré 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 Møller Radius, of which we also list some typical values. Two Appendices present some mathematical background material on Hodge duality and group actions on manifolds.
1 Introduction This contribution deals with the “problem of motion” in Special Relativity. Thus we work entirely in Minkowski space M (to be defined below) and represent a material system by an energy-momentum tensor T the support of which is to be identified with the set of events (points) in Minkowski space where matter “exists”: supp(T) := { p ∈ M | T( p) = 0}.
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D. Giulini (B) Institute for Theoretical Physics, Riemann Center for Geometry and Physics, Leibniz University Hannover, Appelstrasse 2, 30167 Hannover, Germany e-mail: [email protected] D. Giulini Center for Applied Space Technology and Microgravity, University of Bremen, Am Fallturm, 28359 Bremen, Germany © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_3
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A central assumption will be that the material system is spatially well localised, which here shall mean that supp(T) has compact intersection with any Cauchy hypersurface in M. Note that Cauchy hypersurfaces end at spatial infinity I0 and that supp(T) need not have compact intersection with asymptotically hyperboloidal spacelike hypersurfaces which tend to lightlike rather than spacelike infinity. This is depicted in Fig. 1. Definition 1 We say that an energy-momentum tensor T describes a body iff the intersection of supp(T) with any Cauchy hypersurface in Minkowski space is compact. Hence we identify the event-set of a body with supp(T), which, in the sense made precise above, is of finite spatial extent, though it clearly will extend to timelike infinity. This is visualised as the a tubular neighbourhood stretching all the way from past-timelike to future-timelike infinity, as indicated by the shaded vertical tube in Fig. 1. It is also clear from Fig. 1 that we generally cannot require compact support of T on spacelike hypersurfaces which are not Cauchy, like L. In fact, if the body radiated in the finite past, given by the lighter-shade
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