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Title: Spacetime encodings. III. Second order Killing tensors

Abstract

This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV) spacetimes that were found by Carter to have separable Hamilton-Jacobi equations, and thus admit a second-order Killing tensor. The derivation of the spacetimes presented in this paper borrows from ideas about dynamical systems, and illustrates concepts that can be generalized to higher-order Killing tensors. The relationship between the components of the Killing equations and metric functions are given explicitly. The origin of the four separable coordinate systems found by Carter is explained and classified in terms of the analytic structure associated with the Killing equations. A geometric picture of what the orbital invariants may represent is built. Requiring that a SAV spacetime admits a second-order Killing tensor is very restrictive, selecting very few candidates from the group of all possible SAV spacetimes. This restriction arises due to the fact that the consistency conditions associated with the Killing equations require that the field variables obey a second-order differential equation, as opposed to a fourth-order differential equation that imposes the weaker condition that the spacetime be SAV. This paper introduces ideas that could lead to the explicit computation of more general orbital invariants in the form of higher-order Killing tensors.

Authors:
 [1]
  1. Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91103 (United States)
Publication Date:
OSTI Identifier:
21409022
Resource Type:
Journal Article
Journal Name:
Physical Review. D, Particles Fields
Additional Journal Information:
Journal Volume: 81; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevD.81.022001; (c) 2010 The American Physical Society; Journal ID: ISSN 0556-2821
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; AXIAL SYMMETRY; HAMILTON-JACOBI EQUATIONS; METRICS; SPACE-TIME; TENSORS; DIFFERENTIAL EQUATIONS; EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; SYMMETRY

Citation Formats

Brink, Jeandrew. Spacetime encodings. III. Second order Killing tensors. United States: N. p., 2010. Web. doi:10.1103/PHYSREVD.81.022001.
Brink, Jeandrew. Spacetime encodings. III. Second order Killing tensors. United States. https://doi.org/10.1103/PHYSREVD.81.022001
Brink, Jeandrew. 2010. "Spacetime encodings. III. Second order Killing tensors". United States. https://doi.org/10.1103/PHYSREVD.81.022001.
@article{osti_21409022,
title = {Spacetime encodings. III. Second order Killing tensors},
author = {Brink, Jeandrew},
abstractNote = {This paper explores the Petrov type D, stationary axisymmetric vacuum (SAV) spacetimes that were found by Carter to have separable Hamilton-Jacobi equations, and thus admit a second-order Killing tensor. The derivation of the spacetimes presented in this paper borrows from ideas about dynamical systems, and illustrates concepts that can be generalized to higher-order Killing tensors. The relationship between the components of the Killing equations and metric functions are given explicitly. The origin of the four separable coordinate systems found by Carter is explained and classified in terms of the analytic structure associated with the Killing equations. A geometric picture of what the orbital invariants may represent is built. Requiring that a SAV spacetime admits a second-order Killing tensor is very restrictive, selecting very few candidates from the group of all possible SAV spacetimes. This restriction arises due to the fact that the consistency conditions associated with the Killing equations require that the field variables obey a second-order differential equation, as opposed to a fourth-order differential equation that imposes the weaker condition that the spacetime be SAV. This paper introduces ideas that could lead to the explicit computation of more general orbital invariants in the form of higher-order Killing tensors.},
doi = {10.1103/PHYSREVD.81.022001},
url = {https://www.osti.gov/biblio/21409022}, journal = {Physical Review. D, Particles Fields},
issn = {0556-2821},
number = 2,
volume = 81,
place = {United States},
year = {Fri Jan 15 00:00:00 EST 2010},
month = {Fri Jan 15 00:00:00 EST 2010}
}