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Spacetime encodings. II. Pictures of integrability

Brink, Jeandrew (2008) Spacetime encodings. II. Pictures of integrability. Physical Review D, 78 (10). Art. No. 102002. ISSN 2470-0010.

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I visually explore the features of geodesic orbits in arbitrary stationary axisymmetric vacuum (SAV) spacetimes that are constructed from a complex Ernst potential. Some of the geometric features of integrable and chaotic orbits are highlighted. The geodesic problem for these SAV spacetimes is rewritten as a 2 degree of freedom problem and the connection between current ideas in dynamical systems and the study of two manifolds sought. The relationship between the Hamilton-Jacobi equations, canonical transformations, constants of motion, and Killing tensors are commented on. Wherever possible I illustrate the concepts by means of examples from general relativity. This investigation is designed to build the readers' intuition about how integrability arises, and to summarize some of the known facts about 2 degree of freedom systems. Evidence is given, in the form of an orbit-crossing structure, that geodesics in SAV spacetimes might admit a fourth constant of motion that is quartic in momentum (by contrast with Kerr spacetime, where Carter's fourth constant is quadratic).

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Additional Information:© 2008 The American Physical Society. (Received 8 July 2008; published 5 November 2008) I would like to thank Frank Estabrook from whom I learnt a great deal. I would also like to thank Ilya Mandel, Yasushi Mino, Kip Thorne, and Michele Vallisneri for many useful discussions and good advice.
Issue or Number:10
Record Number:CaltechAUTHORS:BRIprd08b
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:12323
Deposited By: Archive Administrator
Deposited On:11 Nov 2008 19:12
Last Modified:03 Oct 2019 00:27

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