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Spacetime encodings. II. Pictures of integrability

Jeandrew Brink

  • Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91103, USA

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Phys. Rev. D 78, 102002 – Published 5 November, 2008

DOI: https://doi.org/10.1103/PhysRevD.78.102002

Abstract

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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See Also

Spacetime encodings. I. A spacetime reconstruction problem

Jeandrew Brink
Phys. Rev. D 78, 102001 (2008)

Spacetime encodings. III. Second order Killing tensors

Jeandrew Brink
Phys. Rev. D 81, 022001 (2010)

Spacetime encodings. IV. The relationship between Weyl curvature and Killing tensors in stationary axisymmetric vacuum spacetimes

Jeandrew Brink
Phys. Rev. D 81, 022002 (2010)

Article Text

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Outline

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Phys. Rev. D 78, 102002– Published 5 November, 2008
  • Received 8 July 2008

© 2008 American Physical Society

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