Approximate spacetime symmetries and conservation laws

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.

26 pages, 1 figure. Added an example and simplified presentation. Accepted by Classical and Quantum Gravity

References in corpus (8)

• The motion of point particles in curved spacetime
• Introduction to Isolated Horizons in Numerical Relativity
• Self-Similarity in General Relativity \endtitle
• Circular orbits and spin in black-hole initial data
• Approximate Killing Vectors on S^2
• A new geometric invariant on initial data for Einstein equations
• Almost-stationary motions and gauge conditions in General Relativity
• Incorporation of Spacetime Symmetries in Einstein's Field Equations

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• Invariant conserved currents in generalized gravity
• Self-forces on static bodies in arbitrary dimensions
• Extended-body motion in black hole spacetimes: What is possible?
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• Prescriptions for measuring and transporting local angular momenta in general relativity
• Foundations of the self-force problem in arbitrary dimensions
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• Almost-Killing equation: Stability, hyperbolicity, and black hole Gauss law
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• Spin Hall effects and the localization of massless spinning particles
• Geodesic deviation to all orders via a tangent bundle formalism