Abstract
A characterization of the Kerr-NUT-(A)de Sitter metric among four dimensional Λ-vacuum spacetimes admitting a Killing vector ξ is obtained in terms of the proportionality of the self-dual Weyl tensor and a natural self-dual double two-form constructed from the Killing vector. This result recovers and extends a previous characterization of the Kerr and Kerr-NUT metrics (Mars, Class Quant Grav 16:2507–2523, 1999). The method of proof is based on (i) the presence of a second Killing vector field which is built in terms of geometric information arising from the Killing vector ξ exclusively, and (ii) the existence of an interesting underlying geometric structure involving a Riemannian submersion of a conformally related metric, both of which may be of independent interest. Other related metrics can also be similarly characterized, in particular the Λ < 0 “black branes” recently used in AdS/CFT correspondence to describe, via holography, the physics of Quark–Gluon plasma.
Article PDF
Similar content being viewed by others
References
Alexakis S., Ionescu A.D., Klainerman S.: Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces. Commun. Math. Phys. 299, 89–127 (2010)
Alexakis S., Ionescu A.D., Klainerman S.: Hawking’s local rigidity theorem without analyticity. Geom. Funct. Anal. 20, 845–869 (2010)
Alexakis, S., Ionescu, A.D., Klainerman, S.: Rigidity of stationary black holes with small angular momentum on the horizon. arXiv:1304.0487 [gr-qc] (2013)
Bäckdahl T., Valiente Kroon J.A.: On the construction of a geometric invariant measuring the deviation from Kerr data. Ann. Henri Poincaré 11, 1225–1271 (2010)
Cahen M., Defrise L.: Lorentzian 4 dimensional manifolds with ‘local isotropy’. Commun. Math. Phys. 11, 56–76 (1968)
Chen W., Lü H., Pope C.N.: General Kerr-NUT-AdS metrics in all dimensions. Class. Quant. Grav. 23, 5323–5340 (2006)
Chong Z.-W., Gibbons G.W., Lü H., Pope C.N.: Separability and Killing tensors in Kerr-Taub-Nut-De Sitter metrics in higher dimensions. Phys. Lett. B 609, 124–132 (2005)
Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. In: Evolution Equations, Clay Mathematics Proceedings, vol. 17. Amer. Math. Soc., Providence, pp. 97–205 (2013) (online at http://www.arxiv.org/abs/0811.0354)
Debever R., Kamran N., McLenaghan R.G.: Exhaustive integration and a single expression for the general solution of the type D vacuum and electrovac field equations with cosmological constant for a nonsingular aligned Maxwell field. J. Math. Phys. 25, 1955–1972 (1984)
Dias O.J.C., Horowitz G.T., Marolf D., Santos J.E.: On the nonlinear stability of asymptotically anti-de Sitter solutions. Class. Quant. Grav. 29, 230519 (2012)
Dyatlov S.: Exponential energy decay for Kerr-de Sitter black holes beyond event horizons. Math. Res. Lett. 18, 1023–1035 (2011)
Ferrando J.J., Sáez J.A.: On the invariant symmetries of the D-metrics. J. Math. Phys. 48, 102504 (2007)
García-Díaz A.: Electrovac type D solutions with cosmological constant. J. Math. Phys. 25, 1951–1954 (1984)
García-Parrado Gómez-Lobo A., Senovilla J.M.M.: A set of invariant quality factors measuring the deviation from the Kerr metric. Gen. Relat. Grav. 45, 1095–1127 (2013)
García-Parrado Gómez-Lobo A., Valiente Kroon J.A.: Kerr initial data. Class. Quant. Grav. 25, 205018 (2008)
Griffiths J.B., Podolský J.: A new look at the Plebański–Demiański family of solutions. Int. J. Mod. Phys. D 15, 335–370 (2006)
Griffiths J.B., Podolský J.: A note on the parameters of the Kerr-NUT-(anti-) de Sitter spacetime. Class. Quant. Grav. 24, 1687–1689 (2007)
Griffiths, J.B., Podolský, J.: Exact Space-Times in Eintein’s General Relativity. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2009)
Gromoll, D., Walschap, G.: Metric foliations and curvature. Progress in Mathematics, vol. 268. Birkhäuser Verlag AG, Basel (2009)
Holzegel G.: On the massive wave equation on slowly rotating Kerr-AdS spacetimes. Commun. Math. Phys. 294, 169–197 (2010)
Holzegel, G., Smulevici, J.: Decay properties of Klein–Gordon fields on Kerr-AdS spacetimes. arXiv:1110.6794 [gr-qc] (2011)
Houri T., Oota T., Yasui Y.: Closed conformal Killing–Yano tensor and Kerr-NUT-de Sitter space-time uniqueness. Phys. Lett. B 656, 214–216 (2007)
Houri T., Oota T., Yasui Y.: Closed conformal Killing–Yano tensor and the uniqueness of generalized Kerr-NUT-de Sitter spacetime. Clas. Quant. Grav. 26, 045015 (2009)
Ionescu A.D., Klainerman S.: On the uniqueness of smooth, stationary black holes in vacuum. Invent. Math. 175, 35–102 (2009)
Ionescu A.D., Klainerman S.: On the local extension of Killing vector-fields in Ricci flat manifolds. J. Am. Math. Soc. 26, 563–593 (2013)
Israel W.: Differential forms in general relativity. Commun. Dublin Inst. Adv. Stud. Ser. A 19, 1–100 (1970)
Klemm, D., Moretti, V., Vanzo, L.: Rotating topological black holes. Phys. Rev. D 57, 6127–6137 (1998) [Erratum, Phys. Rev. D 60, 109902 (1999)]
Krtous P., Frolov V.P., Kubiznak D.: Hidden symmetries of higher-dimensional black holes and uniqueness of the Kerr-NUT-(A)dS spacetime. Phys. Rev. D 78, 064022 (2008)
Kubiznak D., Frolov V.P.: The hidden symmetry of higher dimensional Kerr-NUT-AdS spacetimes. Class. Quant. Grav. 24, F1–F6 (2007)
Mars M.: A spacetime characterization of the Kerr metric. Class. Quant. Grav. 16, 2507–2523 (1999)
Mars M.: Uniqueness properties of the Kerr metric. Class. Quant. Grav. 17, 3353–3373 (2000)
Mars M.: Spacetime Ehlers group: transformation law for the Weyl tensor. Clas. Quant. Grav. 18, 719–738 (2001)
Mars M.: Wahlquist–Newman solution. Phys. Rev. D 63, 064022 (2001)
Mars M., Reiris M.: Global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces. Commun. Math. Phys. 322, 633–666 (2013)
McInnes B.: Fragile black holes and an angular momentum cutoff in peripheral heavy ion collision. Nucl. Phys. B 861, 236 (2012)
McInnes B.: Universality of the holographic angular momentum cutoff. Nucl. Phys. B 864, 722 (2012)
McInnes B., Teo E.: Generalised planar black holes and the holography of hydrodynamic shear. Nucl. Phys. Sect. B 878, 186–213 (2014)
O’Neill B.: fundamental equations of a submersion. Michigan Math. J. 13, 459–469 (1966)
Papapetrou A.: Champs gravitationnels stationnaires à symétrie axiale. Annales de l’institut Henri Poincaré (A) Physique théorique 4, 83–105 (1966)
Perjés, Z.: An improved characterization of the Kerr metric. KFKI-1984-115 preprint. In: Markov M.A. (ed.) Quantum Gravity 3. World Scientific Publishing Co., Singapore (1985)
Senovilla J.M.M.: Super-energy tensors. Class. Quant. Grav. 17, 2799–2841 (2000)
Simon W.: Characterizations of the Kerr metric. Gen. Relat. Grav. 16, 465–476 (1984)
Simon W.: Nuts have no hair. Class. Quant. Grav. 12, L125–L130 (1995)
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact solutions of Einstein’s field equations, 2nd edn. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (2003)
Wong W.W.-Y.: A space-time characterization of the Kerr–Newman metric. Ann. Henri Poincaré 10, 453–484 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by James A. Isenberg.
Rights and permissions
About this article
Cite this article
Mars, M., Senovilla, J.M.M. A Spacetime Characterization of the Kerr-NUT-(A)de Sitter and Related Metrics. Ann. Henri Poincaré 16, 1509–1550 (2015). https://doi.org/10.1007/s00023-014-0343-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-014-0343-3