Abstract
We study Robinson-Trautman spacetimes in the presence of an aligned p-form Maxwell field and an arbitrary cosmological constant in n ≥ 4 dimensions. As it turns out, the character of these exact solutions depends significantly on the (relative) value of n and p. In odd dimensions the solutions reduce to static black holes dressed with an electric and a magnetic field, with an Einstein space horizon (further constrained by the Einstein-Maxwell equations) — both the Weyl and Maxwell types are D. Even dimensions, however, open up more possibilities. In particular, when 2p = n there exist non-static solutions describing black holes gaining (or losing) mass by receiving (or emitting) electromagnetic radiation. In this case the Weyl type is II (D) and the Maxwell type can be II (D) or N. Conditions under which the Maxwell field is self-dual (for odd p) are also discussed, and a few explicit examples presented. Finally, the case p = 1 is special in all dimensions and leads to static metrics with a non-Einstein transverse space.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Emparan and H.S. Reall, Black holes in higher dimensions, Living Rev. Rel. 11 (2008) 6 [arXiv:0801.3471] [INSPIRE].
S. Hollands and A. Ishibashi, Black hole uniqueness theorems in higher dimensional spacetimes, Class. Quant. Grav. 29 (2012) 163001 [arXiv:1206.1164] [INSPIRE].
G.T. Horowitz ed., Black holes in higher dimensions, Cambridge University Press, Cambridge U.K. (2012) [INSPIRE].
F.R. Tangherlini, Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cim. 27 (1963) 636 [INSPIRE].
G.W. Gibbons and D.L. Wiltshire, Space-time as a membrane in higher dimensions, Nucl. Phys. B 287 (1987) 717 [hep-th/0109093] [INSPIRE].
R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [INSPIRE].
R. Emparan, S. Ohashi and T. Shiromizu, No-dipole-hair theorem for higher-dimensional static black holes, Phys. Rev. D 82 (2010) 084032 [arXiv:1007.3847] [INSPIRE].
R. Güven, Hertz potentials in higher dimensions, Class. Quant. Grav. 6 (1989) 1961 [INSPIRE].
T. Shiromizu, S. Ohashi and K. Tanabe, Perturbative no-hair property of form fields for higher dimensional static black holes, Phys. Rev. D 83 (2011) 084016 [arXiv:1101.1121] [INSPIRE].
M. Durkee, V. Pravda, A. Pravdová and H.S. Reall, Generalization of the Geroch-Held-Penrose formalism to higher dimensions, Class. Quant. Grav. 27 (2010) 215010 [arXiv:1002.4826] [INSPIRE].
M. Ortaggio, Asymptotic behavior of Maxwell fields in higher dimensions, Phys. Rev. D 90 (2014) 124020 [arXiv:1406.3186] [INSPIRE].
I. Robinson and A. Trautman, Some spherical gravitational waves in general relativity, Proc. Roy. Soc. Lond. A 265 (1962) 463 [INSPIRE].
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, second ed., Cambridge University Press, Cambridge U.K. (2003) [INSPIRE].
J.B. Griffiths and J. Podolský, Exact space-times in Einstein’s general relativity, Cambridge University Press, Cambridge U.K. (2009) [INSPIRE].
R. Emparan, G.T. Horowitz and R.C. Myers, Exact description of black holes on branes, JHEP 01 (2000) 007 [hep-th/9911043] [INSPIRE].
G.B. de Freitas and H.S. Reall, Algebraically special solutions in AdS/CFT, JHEP 06 (2014) 148 [arXiv:1403.3537] [INSPIRE].
N. Van den Bergh, Einstein-Maxwell null fields of Petrov type D, Class. Quant. Grav. 6 (1989) 1871.
M. Cahen and J. Sengier, Espaces de classe D admettant un champ électromagnetique (in French), Bull. Acad. Roy. Belg. Cl. Sci. 53 (1967) 801.
J. Leroy, Champs électromagnétiques à rayons intégrables, divergents et sans distorsion (in French), Bull. Acad. Roy. Belg. Cl. Sci. 62 (1976) 259.
R. Debever, N. Van den Bergh and J. Leroy, Diverging Einstein-Maxwell null fields of Petrov type D, Class. Quant. Grav. 6 (1989) 1373.
J.M.M. Senovilla, Black hole formation by incoming electromagnetic radiation, Class. Quant. Grav. 32 (2015) 017001 [arXiv:1408.2778] [INSPIRE].
J.P.S. Lemos, Gravitational collapse to toroidal, cylindrical, and planar black holes, Phys. Rev. D 57 (1998) 4600 [gr-qc/9709013] [INSPIRE].
J.P.S. Lemos, Collapsing shells of radiation in anti-de Sitter spacetimes and the hoop and cosmic censorship conjectures, Phys. Rev. D 59 (1999) 044020 [gr-qc/9812078] [INSPIRE].
P. Dadras, J.T. Firouzjaee and R. Mansouri, A concrete anti-de Sitter black hole with dynamical horizon having toroidal cross-sections and its characteristics, Europhys. Lett. 100 (2012) 39001 [arXiv:1207.3673] [INSPIRE].
J. Podolský and M. Ortaggio, Robinson-Trautman spacetimes in higher dimensions, Class. Quant. Grav. 23 (2006) 5785 [gr-qc/0605136] [INSPIRE].
M. Ortaggio, Higher dimensional spacetimes with a geodesic, shearfree, twistfree and expanding null congruence, in Proceedings of the XVII SIGRAV Conference, Torino Italy September 4-7 2006 [gr-qc/0701036] [INSPIRE].
M. Ortaggio, J. Podolský and M. Žofka, Robinson-Trautman spacetimes with an electromagnetic field in higher dimensions, Class. Quant. Grav. 25 (2008) 025006 [arXiv:0708.4299] [INSPIRE].
V. Pravda, A. Pravdová and M. Ortaggio, Type D Einstein spacetimes in higher dimensions, Class. Quant. Grav. 24 (2007) 4407 [arXiv:0704.0435] [INSPIRE].
R. Švarc and J. Podolský, Absence of gyratons in the Robinson-Trautman class, Phys. Rev. D 89 (2014) 124029 [arXiv:1406.0729] [INSPIRE].
J. Podolský and R. Švarc, Algebraic structure of Robinson-Trautman and Kundt geometries in arbitrary dimension, Class. Quant. Grav. 32 (2015) 015001 [arXiv:1406.3232] [INSPIRE].
Y. Bardoux, M.M. Caldarelli and C. Charmousis, Shaping black holes with free fields, JHEP 05 (2012) 054 [arXiv:1202.4458] [INSPIRE].
M. Henneaux and C. Teitelboim, Dynamics of chiral (self-dual) p-forms, Phys. Lett. B 206 (1988) 650 [INSPIRE].
M.J. Duff and P. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys. Lett. B 94 (1980) 179 [INSPIRE].
S. Deser, A. Gomberoff, M. Henneaux and C. Teitelboim, Duality, self-duality, sources and charge quantization in abelian N-form theories, Phys. Lett. B 400 (1997) 80 [hep-th/9702184] [INSPIRE].
M.S. Bremer, H. Lü, C.N. Pope and K.S. Stelle, Dirac quantization conditions and Kaluza-Klein reduction, Nucl. Phys. B 529 (1998) 259 [hep-th/9710244] [INSPIRE].
R. Milson, A. Coley, V. Pravda and A. Pravdová, Alignment and algebraically special tensors in Lorentzian geometry, Int. J. Geom. Meth. Mod. Phys. 2 (2005) 41 [gr-qc/0401010] [INSPIRE].
G. Bergqvist and J.M.M. Senovilla, Null cone preserving maps, causal tensors and algebraic Rainich theory, Class. Quant. Grav. 18 (2001) 5299 [gr-qc/0104090] [INSPIRE].
A. Coley, R. Milson, V. Pravda and A. Pravdová, Vanishing scalar invariant spacetimes in higher dimensions, Class. Quant. Grav. 21 (2004) 5519 [gr-qc/0410070] [INSPIRE].
R. Milson, Alignment and the classification of Lorentz-signature tensors, gr-qc/0411036 [INSPIRE].
S. Hervik, M. Ortaggio and L. Wylleman, Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension, Class. Quant. Grav. 30 (2013) 165014 [arXiv:1203.3563] [INSPIRE].
A.Z. Petrov, Einstein spaces, translation of the 1961 Russian ed., Pergamon Press, Oxford U.K. (1969).
S. Bochner, Curvature and Betti numbers, Ann. Math. 49 (1948) 379.
K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies 32, Princeton University Press, Princeton U.S.A. (1953).
S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 2, Interscience, New York U.S.A. (1969).
H.W. Brinkmann, Einstein spaces which are mapped conformally on each other, Math. Ann. 94 (1925) 119 [INSPIRE].
M. Ortaggio, V. Pravda and A. Pravdová, Algebraic classification of higher dimensional spacetimes based on null alignment, Class. Quant. Grav. 30 (2013) 013001 [arXiv:1211.7289] [INSPIRE].
M. Ortaggio, V. Pravda and A. Pravdová, Higher dimensional Kerr-Schild spacetimes, Class. Quant. Grav. 26 (2009) 025008 [arXiv:0808.2165] [INSPIRE].
M. Henneaux and C. Teitelboim, p-form electrodynamics, Found. Phys. 16 (1986) 593 [INSPIRE].
M. Ortaggio, V. Pravda and A. Pravdová, Ricci identities in higher dimensions, Class. Quant. Grav. 24 (2007) 1657 [gr-qc/0701150] [INSPIRE].
M. Ortaggio, V. Pravda and A. Pravdová, On the Goldberg-Sachs theorem in higher dimensions in the non-twisting case, Class. Quant. Grav. 30 (2013) 075016 [arXiv:1211.2660] [INSPIRE].
S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946) 776.
M. Bañados, C. Teitelboim and J. Zanelli, Black hole in three-dimensional spacetime, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].
M.J. Bowick, S.B. Giddings, J.A. Harvey, G.T. Horowitz and A. Strominger, Axionic black holes and an Aharonov-Bohm effect for strings, Phys. Rev. Lett. 61 (1988) 2823 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1411.1943
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Ortaggio, M., Podolský, J. & Žofka, M. Static and radiating p-form black holes in the higher dimensional Robinson-Trautman class. J. High Energ. Phys. 2015, 45 (2015). https://doi.org/10.1007/JHEP02(2015)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)045