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Formal solution of the fourth order Killing equations for stationary axisymmetric vacuum spacetimes

Brink, Jeandrew (2011) Formal solution of the fourth order Killing equations for stationary axisymmetric vacuum spacetimes. Physical Review D, 84 (10). Art. No. 104015. ISSN 2470-0010. https://resolver.caltech.edu/CaltechAUTHORS:20111206-075752577

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1103/PhysRevD.84.104015DOIUNSPECIFIED
http://link.aps.org/doi/10.1103/PhysRevD.84.104015PublisherUNSPECIFIED
http://prd.aps.org/abstract/PRD/v84/i10/e104015PublisherUNSPECIFIED
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.
Funders:
Funding AgencyGrant Number
NSFPHY-0653653
NSFPHY-0601459
NASANNX07AH06G
Brinson FoundationUNSPECIFIED
Caltech David and Barbara Groce startup fundUNSPECIFIED
Issue or Number:10
Classification Code:PACS: 04.20.Jb, 04.30.Tv, 04.70.Bw
Record Number:CaltechAUTHORS:20111206-075752577
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20111206-075752577
Official Citation:Formal solution of the fourth order Killing equations for stationary axisymmetric vacuum spacetimes Jeandrew Brink Published 7 November 2011 (5 pages) 104015
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28305
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:06 Dec 2011 16:13
Last Modified:03 Oct 2019 03:31

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