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Avenues for analytic exploration in axisymmetric spacetimes: Foundations and the triad formalism

Brink, Jeandrew and Zimmerman, Aaron and Hinderer, Tanja (2013) Avenues for analytic exploration in axisymmetric spacetimes: Foundations and the triad formalism. Physical Review D, 88 (4). Art. No. 044039. ISSN 2470-0010. doi:10.1103/PhysRevD.88.044039.

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Axially symmetric spacetimes are the only vacuum models for isolated systems with continuous symmetries that also include dynamics. For such systems, we review the reduction of the vacuum Einstein field equations to their most concise form by dimensionally reducing to the three-dimensional space of orbits of the Killing vector, followed by a conformal rescaling. The resulting field equations can be written as a problem in three-dimensional gravity with a complex scalar field as source. This scalar field, the Ernst potential, is constructed from the norm and twist of the spacelike Killing field. In the case where the axial Killing vector is twist-free, we discuss the properties of the axis and simplify the field equations using a triad formalism. We study two physically motivated triad choices that further reduce the complexity of the equations and exhibit their hierarchical structure. The first choice is adapted to a harmonic coordinate that asymptotes to a cylindrical radius and leads to a simplification of the three-dimensional Ricci tensor and the boundary conditions on the axis. We illustrate its properties by explicitly solving the field equations in the case of static axisymmetric spacetimes. The other choice of triad is based on geodesic null coordinates adapted to null infinity as in the Bondi formalism. We then explore the solution space of the twist-free axisymmetric vacuum field equations, identifying the known (unphysical) solutions together with the assumptions made in each case. This singles out the necessary conditions for obtaining physical solutions to the equations.

Item Type:Article
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URLURL TypeDescription Paper DOIArticle
Hinderer, Tanja0000-0002-3394-6105
Additional Information:© 2013 American Physical Society. Received 7 March 2013; published 21 August 2013. We thank Yanbei Chen and Anıl Zenginoğlu for valuable discussions. J. B. would like to thank Y. Chen and C. Ott for their hospitality while at Caltech. T. H. and A. Z. would like to thank NITheP of South Africa for their hospitality during much of this work. A. Z. is supported by NSF Grant No. PHY-1068881, CAREER Grant No. PHY-0956189, and the David and Barbara Groce Startup fund at Caltech. T. H. acknowledges support from NSF Grants No. PHY-0903631 and No. PHY-1208881, and the Maryland Center for Fundamental Physics.
Funding AgencyGrant Number
Caltech David and Barbara Groce Startup FundUNSPECIFIED
Maryland Center for Fundamental PhysicsUNSPECIFIED
Issue or Number:4
Classification Code:PACS: 04.20.-q, 04.20.Cv, 04.20.Jb
Record Number:CaltechAUTHORS:20130920-133125056
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:41450
Deposited By: Tony Diaz
Deposited On:20 Sep 2013 20:45
Last Modified:10 Nov 2021 04:30

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