Abstract
We demonstrate separability of the Maxwell’s equations in the Myers-Perry-(A)dS geometry and derive explicit solutions for various polarizations. Application of our construction to the four-dimensional Kerr black hole leads to a new ansatz for the Maxwell field which has significant advantages over the previously known parameterization.
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S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199 [Erratum ibid. 46 (1976) 206] [INSPIRE].
S.S. Gubser and I.R. Klebanov, Emission of charged particles from four-dimensional and five-dimensional black holes, Nucl. Phys. B 482 (1996) 173 [hep-th/9608108] [INSPIRE].
S.S. Gubser and I.R. Klebanov, Four-dimensional grey body factors and the effective string, Phys. Rev. Lett. 77 (1996) 4491 [hep-th/9609076] [INSPIRE].
I.R. Klebanov and S.D. Mathur, Black hole grey body factors and absorption of scalars by effective strings, Nucl. Phys. B 500 (1997) 115 [hep-th/9701187] [INSPIRE].
I.R. Klebanov, World volume approach to absorption by nondilatonic branes, Nucl. Phys. B 496 (1997) 231 [hep-th/9702076] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.A. Tseytlin, String theory and classical absorption by three-branes, Nucl. Phys. B 499 (1997) 217 [hep-th/9703040] [INSPIRE].
M. Krasnitz and I.R. Klebanov, Testing effective string models of black holes with fixed scalars, Phys. Rev. D 56 (1997) 2173 [hep-th/9703216] [INSPIRE].
S.S. Gubser and I.R. Klebanov, Absorption by branes and Schwinger terms in the world volume theory, Phys. Lett. B 413 (1997) 41 [hep-th/9708005] [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett. 11 (1963) 237 [INSPIRE].
B. Carter, Global structure of the Kerr family of gravitational fields, Phys. Rev. 174 (1968) 1559 [INSPIRE].
B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].
M. Walker and R. Penrose, On quadratic first integrals of the geodesic equations for type [22] spacetimes, Commun. Math. Phys. 18 (1970) 265 [INSPIRE].
K. Yano, Some remarks on tensor fields and curvature, Ann. Math. 55 (1952) 328.
S. Tachibana, On conformal Killing tensor in a Riemannian space, Tohoku Math. J. 21 (1969) 56.
T. Kashiwada, On conformal Killing tensor, Nat. Sci. Rep. Ochanomizu Univ. 19 (1968) 67 [INSPIRE].
G.W. Gibbons, R.H. Rietdijk and J.W. van Holten, SUSY in the sky, Nucl. Phys. B 404 (1993) 42 [hep-th/9303112] [INSPIRE].
J.W. van Holten, Supersymmetry and the geometry of Taub-NUT, Phys. Lett. B 342 (1995) 47 [hep-th/9409139] [INSPIRE].
D. Kubiznak and P. Krtous, On conformal Killing-Yano tensors for Plebanski-Demianski family of solutions, Phys. Rev. D 76 (2007) 084036 [arXiv:0707.0409] [INSPIRE].
M. Durkee, Geodesics and symmetries of doubly-spinning black rings, Class. Quant. Grav. 26 (2009) 085016 [arXiv:0812.0235] [INSPIRE].
S.A. Teukolsky, Rotating black holes: separable wave equations for gravitational and electromagnetic perturbations, Phys. Rev. Lett. 29 (1972) 1114 [INSPIRE].
S.A. Teukolsky, Perturbations of a rotating black hole 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185 (1973) 635 [INSPIRE].
J.C. Breckenridge, D.A. Lowe, R.C. Myers, A.W. Peet, A. Strominger and C. Vafa, Macroscopic and microscopic entropy of near extremal spinning black holes, Phys. Lett. B 381 (1996) 423 [hep-th/9603078] [INSPIRE].
C.G. Callan and J.M. Maldacena, D-brane approach to black hole quantum mechanics, Nucl. Phys. B 472 (1996) 591 [hep-th/9602043] [INSPIRE].
G.T. Horowitz and A. Strominger, Counting states of near extremal black holes, Phys. Rev. Lett. 77 (1996) 2368 [hep-th/9602051] [INSPIRE].
J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [INSPIRE].
O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].
S.D. Mathur, The fuzzball proposal for black holes: an elementary review, Fortsch. Phys. 53 (2005) 793 [hep-th/0502050] [INSPIRE].
I. Bena and N.P. Warner, Black holes, black rings and their microstates, Lect. Notes Phys. 755 (2008) 1 [hep-th/0701216] [INSPIRE].
K. Skenderis and M. Taylor, The fuzzball proposal for black holes, Phys. Rept. 467 (2008) 117 [arXiv:0804.0552] [INSPIRE].
S.D. Mathur, Fuzzballs and the information paradox: a summary and conjectures, arXiv:0810.4525 [INSPIRE].
F.R. Tangherlini, Schwarzschild field in N dimensions and the dimensionality of space problem, Nuovo Cim. 27 (1963) 636 [INSPIRE].
H. Kodama and A. Ishibashi, A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, Prog. Theor. Phys. 110 (2003) 701 [hep-th/0305147] [INSPIRE].
A. Ishibashi and H. Kodama, Stability of higher dimensional Schwarzschild black holes, Prog. Theor. Phys. 110 (2003) 901 [hep-th/0305185] [INSPIRE].
H. Kodama and A. Ishibashi, Master equations for perturbations of generalized static black holes with charge in higher dimensions, Prog. Theor. Phys. 111 (2004) 29 [hep-th/0308128] [INSPIRE].
A.S. Cornell, W. Naylor and M. Sasaki, Graviton emission from a higher-dimensional black hole, JHEP 02 (2006) 012 [hep-th/0510009] [INSPIRE].
V. Cardoso, M. Cavaglia and L. Gualtieri, Black hole particle emission in higher-dimensional spacetimes, Phys. Rev. Lett. 96 (2006) 071301 [Erratum ibid. 96 (2006) 219902] [hep-th/0512002] [INSPIRE].
O.J.C. Dias, G.T. Horowitz, D. Marolf and J.E. Santos, On the nonlinear stability of asymptotically anti-de Sitter solutions, Class. Quant. Grav. 29 (2012) 235019 [arXiv:1208.5772] [INSPIRE].
Z.W. Chong, G.W. Gibbons, H. Lü and C.N. Pope, Separability and Killing tensors in Kerr-Taub-NUT-de Sitter metrics in higher dimensions, Phys. Lett. B 609 (2005) 124 [hep-th/0405061] [INSPIRE].
M. Vasudevan, K.A. Stevens and D.N. Page, Separability of the Hamilton-Jacobi and Klein-Gordon equations in Kerr-de Sitter metrics, Class. Quant. Grav. 22 (2005) 339 [gr-qc/0405125] [INSPIRE].
W. Chen, H. Lü and C.N. Pope, Separability in cohomogeneity-2 Kerr-NUT-AdS metrics, JHEP 04 (2006) 008 [hep-th/0602084] [INSPIRE].
H.K. Kunduri and J. Lucietti, Integrability and the Kerr-(A)dS black hole in five dimensions, Phys. Rev. D 71 (2005) 104021 [hep-th/0502124] [INSPIRE].
V.P. Frolov and D. Kubiznak, Hidden symmetries of higher dimensional rotating black holes, Phys. Rev. Lett. 98 (2007) 011101 [gr-qc/0605058] [INSPIRE].
D.N. Page, D. Kubiznak, M. Vasudevan and P. Krtous, Complete integrability of geodesic motion in general Kerr-NUT-AdS spacetimes, Phys. Rev. Lett. 98 (2007) 061102 [hep-th/0611083] [INSPIRE].
P. Krtous, D. Kubiznak, D.N. Page and V.P. Frolov, Killing-Yano tensors, rank-2 Killing tensors and conserved quantities in higher dimensions, JHEP 02 (2007) 004 [hep-th/0612029] [INSPIRE].
V.P. Frolov and D. Kubiznak, Higher-dimensional black holes: hidden symmetries and separation of variables, Class. Quant. Grav. 25 (2008) 154005 [arXiv:0802.0322] [INSPIRE].
P. Krtous, V.P. Frolov and D. Kubiznak, Hidden symmetries of higher dimensional black holes and uniqueness of the Kerr-NUT-(A)dS spacetime, Phys. Rev. D 78 (2008) 064022 [arXiv:0804.4705] [INSPIRE].
M. Cariglia, P. Krtous and D. Kubiznak, Commuting symmetry operators of the Dirac equation, Killing-Yano and Schouten-Nijenhuis brackets, Phys. Rev. D 84 (2011) 024004 [arXiv:1102.4501] [INSPIRE].
M. Cariglia, P. Krtous and D. Kubiznak, Dirac equation in Kerr-NUT-(A)dS spacetimes: intrinsic characterization of separability in all dimensions, Phys. Rev. D 84 (2011) 024008 [arXiv:1104.4123] [INSPIRE].
D. Kubiznak and M. Cariglia, On integrability of spinning particle motion in higher-dimensional black hole spacetimes, Phys. Rev. Lett. 108 (2012) 051104 [arXiv:1110.0495] [INSPIRE].
D. Kubiznak, H.K. Kunduri and Y. Yasui, Generalized Killing-Yano equations in D = 5 gauged supergravity, Phys. Lett. B 678 (2009) 240 [arXiv:0905.0722] [INSPIRE].
T. Houri, D. Kubiznak, C.M. Warnick and Y. Yasui, Generalized hidden symmetries and the Kerr-Sen black hole, JHEP 07 (2010) 055 [arXiv:1004.1032] [INSPIRE].
D. Kubiznak, C.M. Warnick and P. Krtous, Hidden symmetry in the presence of fluxes, Nucl. Phys. B 844 (2011) 185 [arXiv:1009.2767] [INSPIRE].
P. Krtous, Electromagnetic field in higher-dimensional black-hole spacetimes, Phys. Rev. D 76 (2007) 084035 [arXiv:0707.0002] [INSPIRE].
M. Durkee and H.S. Reall, Perturbations of higher-dimensional spacetimes, Class. Quant. Grav. 28 (2011) 035011 [arXiv:1009.0015] [INSPIRE].
I. Kolar and P. Krtous, Weak electromagnetic field admitting integrability in Kerr-NUT-(A)dS spacetimes, Phys. Rev. D 91 (2015) 124045 [arXiv:1504.00524] [INSPIRE].
V.P. Frolov, P. Krtous and D. Kubiznak, Weakly charged generalized Kerr-NUT-(A)dS spacetimes, Phys. Lett. B 771 (2017) 254 [arXiv:1705.00943] [INSPIRE].
M. Cvetič and F. Larsen, General rotating black holes in string theory: grey body factors and event horizons, Phys. Rev. D 56 (1997) 4994 [hep-th/9705192] [INSPIRE].
M. Cvetič and F. Larsen, Grey body factors for rotating black holes in four-dimensions, Nucl. Phys. B 506 (1997) 107 [hep-th/9706071] [INSPIRE].
J.M. Maldacena and A. Strominger, Universal low-energy dynamics for rotating black holes, Phys. Rev. D 56 (1997) 4975 [hep-th/9702015] [INSPIRE].
M. Cvetič and F. Larsen, Near horizon geometry of rotating black holes in five-dimensions, Nucl. Phys. B 531 (1998) 239 [hep-th/9805097] [INSPIRE].
O. Lunin and S.D. Mathur, The slowly rotating near extremal D1-D5 system as a ‘hot tube’, Nucl. Phys. B 615 (2001) 285 [hep-th/0107113] [INSPIRE].
B.D. Chowdhury and S.D. Mathur, Radiation from the non-extremal fuzzball, Class. Quant. Grav. 25 (2008) 135005 [arXiv:0711.4817] [INSPIRE].
M. Cvetič and F. Larsen, Greybody factors and charges in Kerr/CFT, JHEP 09 (2009) 088 [arXiv:0908.1136] [INSPIRE].
F. De Jonghe, K. Peeters and K. Sfetsos, Killing-Yano supersymmetry in string theory, Class. Quant. Grav. 14 (1997) 35 [hep-th/9607203] [INSPIRE].
C. Keeler and F. Larsen, Separability of black holes in string theory, JHEP 10 (2012) 152 [arXiv:1207.5928] [INSPIRE].
Y. Chervonyi and O. Lunin, (Non)-integrability of geodesics in D-brane backgrounds, JHEP 02 (2014) 061 [arXiv:1311.1521] [INSPIRE].
Y. Chervonyi and O. Lunin, Killing(-Yano) tensors in string theory, JHEP 09 (2015) 182 [arXiv:1505.06154] [INSPIRE].
R.C. Myers and M.J. Perry, Black holes in higher dimensional space-times, Annals Phys. 172 (1986) 304 [INSPIRE].
G.W. Gibbons, H. Lü, D.N. Page and C.N. Pope, The general Kerr-de Sitter metrics in all dimensions, J. Geom. Phys. 53 (2005) 49 [hep-th/0404008] [INSPIRE].
D.D.K. Chow, M. Cvetič, H. Lü and C.N. Pope, Extremal black hole/CFT correspondence in (gauged) supergravities, Phys. Rev. D 79 (2009) 084018 [arXiv:0812.2918] [INSPIRE].
D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav. 26 (2009) 195011 [arXiv:0904.2765] [INSPIRE].
M. Cvetič, G.W. Gibbons and C.N. Pope, Universal area product formulae for rotating and charged black holes in four and higher dimensions, Phys. Rev. Lett. 106 (2011) 121301 [arXiv:1011.0008] [INSPIRE].
M. Cvetič, G.W. Gibbons, D. Kubiznak and C.N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume, Phys. Rev. D 84 (2011) 024037 [arXiv:1012.2888] [INSPIRE].
V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].
P. Krtous, D. Kubiznak, D.N. Page and M. Vasudevan, Constants of geodesic motion in higher-dimensional black-hole spacetimes, Phys. Rev. D 76 (2007) 084034 [arXiv:0707.0001] [INSPIRE].
T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and geodesic integrability, J. Phys. A 41 (2008) 025204 [arXiv:0707.4039] [INSPIRE].
T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness, Phys. Lett. B 656 (2007) 214 [arXiv:0708.1368] [INSPIRE].
T. Houri, T. Oota and Y. Yasui, Closed conformal Killing-Yano tensor and uniqueness of generalized Kerr-NUT-de Sitter spacetime, Class. Quant. Grav. 26 (2009) 045015 [arXiv:0805.3877] [INSPIRE].
S. Chandrasekhar, The mathematical theory of black holes, Oxford University Press, Oxford U.K., (1983) [INSPIRE].
A.A. Starobinsky and S.M. Churilov, Amplification of electromagnetic and gravitational waves scattered by a rotating “black hole”, Sov. Phys. JETP 38 (1974) 1 [Zh. Eksp. Teor. Fiz. 65 (1973) 3] [INSPIRE].
W.H. Press and S.A. Teukolsky, Perturbations of a rotating black hole II. Dynamical stability of the Kerr metric, Astrophys. J. 185 (1973) 649 [INSPIRE].
S.A. Teukolsky and W.H. Press, Perturbations of a rotating black hole III. Interaction of the hole with gravitational and electromagnetic radiation, Astrophys. J. 193 (1974) 443 [INSPIRE].
S. Chandrasekhar, On a transformation of Teukolsky’s equation and the electromagnetic perturbations of the Kerr black hole, Proc. Roy. Soc. Lond. A 348 (1976) 39.
S. Chandrasekhar, The solution of Dirac’s equation in Kerr geometry, Proc. Roy. Soc. Lond. A 349 (1976) 571.
S. Chandrasekhar, The gravitational perturbations of the Kerr black hole I. The perturbations in the quantities which vanish in the stationary state, Proc. Roy. Soc. Lond. A 358 (1978) 421.
E. Newman and R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566 [INSPIRE].
R.C. Myers, Myers-Perry black holes, arXiv:1111.1903 [INSPIRE].
P. Moon and D.E. Spencer, Theorems on separability in Riemannian n-space, Proc. Amer. Math. Soc. 3 (1952) 635.
E.G. Kalnins and W. Miller, Killing tensors and nonorthogonal variable separation for Hamilton-Jacobi equations, SIAM J. Math. Anal. 12 (1981) 617.
E.G. Kalnins and W. Miller, Conformal Killing tensors and variable separation for Hamilton-Jacobi equations, SIAM J. Math. Anal. 14 (1983) 126.
E.G. Kalnins and W. Miller, The theory of orthogonal R-separation for Helmholtz equations, Adv. Math. 51 (1984) 91.
E.G. Kalnins, J.M. Kress and W. Miller, Jacobi, ellipsoidal coordinates and superintegrable systems, J. Nonlin. Math. Phys. 12 (2005) 209.
S.W. Hawking, C.J. Hunter and M. Taylor, Rotation and the AdS/CFT correspondence, Phys. Rev. D 59 (1999) 064005 [hep-th/9811056] [INSPIRE].
M. Günaydin, G. Sierra and P.K. Townsend, Gauging the D = 5 Maxwell-Einstein supergravity theories: more on Jordan algebras, Nucl. Phys. B 253 (1985) 573 [INSPIRE].
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Lunin, O. Maxwell’s equations in the Myers-Perry geometry. J. High Energ. Phys. 2017, 138 (2017). https://doi.org/10.1007/JHEP12(2017)138
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DOI: https://doi.org/10.1007/JHEP12(2017)138