Approximate spacetime symmetries and conservation laws

TL;DR

The paper introduces a generalized notion of spacetime symmetries that extend classical concepts, leading to approximate conservation laws and a generalized Komar integral for defining momenta relative to an observer.

Contribution

It proposes a new geometric symmetry concept that generalizes Killing fields, enabling the derivation of approximate conservation laws and a generalized Komar integral in curved spacetimes.

Findings

Generalized symmetries minimize connection deformations near observers.

Approximate conservation laws are derived for geodesics and matter distributions.

Explicit evaluation of the generalized Komar integral for a gravitational wave spacetime.

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 Poincare 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.