Formal solution of the fourth order Killing equations for stationary axisymmetric vacuum spacetimes
- Creators
- Brink, Jeandrew
Abstract
An analytic understanding of the geodesic structure around non-Kerr spacetimes will result in a powerful tool that could make the mapping of spacetime around massive quiescent compact objects possible. To this end, I present an analytic closed form expression for the components of a the fourth order Killing tensor for stationary axisymmetric vacuum (SAV) spacetimes. It is as yet unclear what subset of SAV spacetimes admit this solution. The solution is written in terms of an integral expression involving the metric functions and two specific Green's functions. A second integral expression has to vanish in order for the solution to be exact. In the event that the second integral does not vanish it is likely that the best fourth order approximation to the invariant has been found. This solution can be viewed as a generalized Carter constant providing an explicit expression for the fourth invariant, in addition to the energy, azimuthal angular momentum and rest mass, associated with geodesic motion in SAV spacetimes, be it exact or approximate. I further comment on the application of this result for the founding of a general algorithm for mapping the spacetime around compact objects using gravitational wave observatories.
Additional Information
© 2011 American Physical Society. Received 21 November 2009; published 7 November 2011. My sincere thanks to Frank Estabrook for many useful discussions. I am also indebted to Tanja Hinderer and Michele Vallisneri for their insightful comments on the manuscript. I gratefully acknowledge support from NSF Grant Nos. PHY-0653653 and PHY-0601459, NASA Grant No. NNX07AH06G, the Brinson Foundation, and the David and Barbara Groce startup fund at Caltech.Attached Files
Published - Brink2011p16363Phys_Rev_D.pdf
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Additional details
- Eprint ID
- 28305
- Resolver ID
- CaltechAUTHORS:20111206-075752577
- PHY-0653653
- NSF
- PHY-0601459
- NSF
- NNX07AH06G
- NASA
- Brinson Foundation
- Caltech David and Barbara Groce startup fund
- Created
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2011-12-06Created from EPrint's datestamp field
- Updated
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2021-11-09Created from EPrint's last_modified field