Approximate spacetime symmetries and conservation laws

Abstract

Abstract. A notion of geometric symmetry is introduced that generalizes the classical concepts of Killing fields and other affine collineations. There is a sense in which flows under these new vector fields minimize deformations of the connection near a specified observer. Any exact affine collineations that may exist are special cases. The remaining vector fields can all be interpreted as analogs of Poincaré and other well-known symmetries near timelike worldlines. Approximate conservation laws generated by these objects are discussed for both geodesics and extended matter distributions. One example is a generalized Komar integral that may be taken to define the linear and angular momenta of a spacetime volume as seen by a particular observer. This is evaluated explicitly for a gravitational plane wave spacetime.

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…its tensorial nature) and its uniqueness property (see Komar [322] quoting Sachs) is especially attractive, the vector field K a is still to be determined. A new suggestion for the approximate spacetime symmetries that can, in principle, be used in Komar’s expression, both near a point and a world line, is given in [235]. This is a generalization of the affine collineations (including the homotheties and the Killing symmetries).…”

“…A slightly different framework for calculations in small regions was used in [327, 170, 235]. Instead of the Newman-Penrose (or the GHP) formalism and the spin coefficient equations, holonomic (Riemann or Fermi type normal) coordinates on an open neighborhood U of a point p ∈ M or a timelike curve γ are used, in which the metric, as well as the Christoffel symbols on U , are expressed by the coordinates on U and the components of the Riemann tensor at p or on γ .…”

“…In [235] a new kind of approximate symmetries, namely approximate affine collineations, are introduced both near a point and a world line, and used to introduce Komar-type ‘conserved’ currents. (For a readable text on the non-Killing type symmetries see, e.g., [233]. )…”

Properties of an affine transport equation and its holonomy

“…We treat the generalized holonomy of the parallelogramoid in Sec. 4, and in Sec. 5 we discuss the implications of these calculations for the program described in [5].…”

“…For example, along an accelerating worldline, it is often useful to carry vectors using Fermi-Walker transport (see, e.g., [1]). Similarly, for a spinning point particle, its 4-momentum and angular-momentum tensor are jointly transported through the coupled Mathisson-Papapetrou equations [2,3] (the dual of these equations, the Killing transport equations, also transport a vector and antisymmetric tensor in a related way-see, e.g., [4]). In [5], one of the authors and a collaborator introduced an affine transport equation for vectors that proved useful for measuring physical effects related to the gravitationalwave memory and for transporting a type of special-relativistic linear and angular momentum in general relativity.…”