×

Universal black holes. (English) Zbl 1460.83046

The (\(n+2\))-dimensional Schwarzschild spacetime has the Lorentzian metric \( g= - f(t) dt^2 + \frac{1}{f(t)} dr^2 + r^2 h \), where \(h\) is the metric of the \(n\)-dimensional sphere.
In the paper under review, the authors consider a family of Schwarzschild-like metrics, by multiplying the previous metric on the (\(t,r\))-plane with a (positive) conformal factor function \(e^{a(r)}\), and by relaxing the hypothesis on \(h\), which is a Riemannian metric on an \(n\)-dimensional “universal” space. The main purpose of the paper is to obtain sufficient conditions on the metric \(h\) and on the functions \(f\) and \(a\), which enable the new Lorentzian metric to be consistently employed in theories of gravity modelling a static vacuum black hole. Examples are constructed in concrete contexts as solutions in particular theories, such as Gauss-Bonnet, quadratic, F(R) and F(Lovelock) gravity, and certain conformal gravities.

MSC:

83C57 Black holes
83C15 Exact solutions to problems in general relativity and gravitational theory
83E15 Kaluza-Klein and other higher-dimensional theories
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
53Z05 Applications of differential geometry to physics
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Tangherlini, FR, Schwarzschild field in n dimensions and the dimensionality of space problem, Nuovo Cim., 27, 636 (1963) · Zbl 0114.21302
[2] Gibbons, GW; Wiltshire, DL, Space-Time as a Membrane in Higher Dimensions, Nucl. Phys., B 287, 717 (1987)
[3] Birmingham, D., Topological black holes in Anti-de Sitter space, Class. Quant. Grav., 16, 1197 (1999) · Zbl 0933.83025
[4] C. Cadeau and E. Woolgar, New five-dimensional black holes classified by horizon geometry and a Bianchi VI brane world, Class. Quant. Grav.18 (2001) 527 [gr-qc/0011029] [INSPIRE]. · Zbl 0983.83039
[5] Hervik, S., Einstein metrics: Homogeneous solvmanifolds, generalized Heisenberg groups and black holes, J. Geom. Phys., 52, 298 (2004) · Zbl 1094.53039
[6] R. Emparan and H.S. Reall, Generalized Weyl solutions, Phys. Rev.D 65 (2002) 084025 [hep-th/0110258] [INSPIRE].
[7] V. Pravda and A. Pravdová, WANDs of the black ring, Gen. Rel. Grav.37 (2005) 1277 [gr-qc/0501003] [INSPIRE]. · Zbl 1091.83007
[8] G. Gibbons and S.A. Hartnoll, A Gravitational instability in higher dimensions, Phys. Rev.D 66 (2002) 064024 [hep-th/0206202] [INSPIRE].
[9] H. Weyl, Gravitation and Elektrizit¨at, Sitzungsber. Preuss. Akad. Wiss. (1918) 465.
[10] A. Eddington, The Mathematical Theory of Relativity, Cambridge University Press, Cambridge, second ed. (1930). · JFM 56.1359.04
[11] DeWitt, BS, Dynamical Theory of Groups and Fields (1965), New York: Gordon and Breach, New York
[12] Scherk, J.; Schwarz, JH, Dual Models for Nonhadrons, Nucl. Phys., B 81, 118 (1974)
[13] Dotti, G.; Gleiser, RJ, Obstructions on the horizon geometry from string theory corrections to Einstein gravity, Phys. Lett., B 627, 174 (2005) · Zbl 1247.83198
[14] N. Farhangkhah and M.H. Dehghani, Lovelock black holes with nonmaximally symmetric horizons, Phys. Rev.D 90 (2014) 044014 [arXiv:1409.1410] [INSPIRE].
[15] Ray, S., Birkhoff’s theorem in Lovelock gravity for general base manifolds, Class. Quant. Grav., 32, 195022 (2015) · Zbl 1327.83234
[16] S. Ohashi and M. Nozawa, Lovelock black holes with a nonconstant curvature horizon, Phys. Rev.D 92 (2015) 064020 [arXiv:1507.04496] [INSPIRE].
[17] Buchdahl, HA, On a set of conform-invariant equations of the gravitational field, Proc. Edinburgh Math. Soc., 10, 16 (1953) · Zbl 0051.20103
[18] Bleecker, DD, Critical Riemannian manifolds, J. Diff. Geom., 14, 599 (1979) · Zbl 0462.53023
[19] Robinson, I.; Trautman, A., Some spherical gravitational waves in general relativity, Proc. Roy. Soc. Lond., A 265, 463 (1962) · Zbl 0099.42902
[20] J. Podolský and M. Ortaggio, Robinson-Trautman spacetimes in higher dimensions, Class. Quant. Grav.23 (2006) 5785 [gr-qc/0605136] [INSPIRE]. · Zbl 1111.83056
[21] Pravda, V.; Pravdová, A.; Ortaggio, M., Type D Einstein spacetimes in higher dimensions, Classical and Quantum Gravity, 24, 17, 4407-4428 (2007) · Zbl 1128.83012
[22] Hervik, S.; Ortaggio, M.; Wylleman, L., Minimal tensors and purely electric or magnetic spacetimes of arbitrary dimension, Class. Quant. Grav., 30, 165014 (2013) · Zbl 1470.83021
[23] Ortaggio, Marcello; Pravda, Vojtěch; Pravdová, Alena, Algebraic classification of higher dimensional spacetimes based on null alignment, Classical and Quantum Gravity, 30, 1, 013001 (2012) · Zbl 1261.83004
[24] Ortaggio, M.; Podolský, J.; Žofka, M., Static and radiating p-form black holes in the higher dimensional Robinson-Trautman class, JHEP, 02, 045 (2015) · Zbl 1388.83601
[25] 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]. · Zbl 1309.83032
[26] Jacobson, T., When is g_ttg_rr = −1?, Class. Quant. Grav., 24, 5717 (2007) · Zbl 1148.83322
[27] Besse, AL, Einstein Manifolds (1987), Berlin: Springer-Verlag, Berlin
[28] V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev.D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
[29] Wolf, JA, The geometry and structure of isotropy irreducible homogeneous spaces, Acta Math., 120, 59 (1968) · Zbl 0157.52102
[30] Kunduri, HK; Lucietti, J., Classification of near-horizon geometries of extremal black holes, Living Rev. Rel., 16, 8 (2013) · Zbl 1320.83005
[31] Gürses, M., Signatures of black holes in string theory, Phys. Rev., D 46, 2522 (1992)
[32] Zwiebach, B., Curvature Squared Terms and String Theories, Phys. Lett., 156B, 315 (1985)
[33] Dotti, G.; Oliva, J.; Troncoso, R., Vacuum solutions with nontrivial boundaries for the Einstein-Gauss-Bonnet theory, Int. J. Mod. Phys., A 24, 1690 (2009) · Zbl 1170.83317
[34] Bogdanos, C.; Charmousis, C.; Goutéraux, B.; Zegers, R., Einstein-Gauss-Bonnet metrics: black holes, black strings and a staticity theorem, Journal of High Energy Physics, 2009, 10, 037-037 (2009)
[35] Maeda, H., Gauss-Bonnet black holes with non-constant curvature horizons, Phys. Rev., D 81, 124007 (2010)
[36] G. Dotti, J. Oliva and R. Troncoso, Static solutions with nontrivial boundaries for the Einstein-Gauss-Bonnet theory in vacuum, Phys. Rev.D 82 (2010) 024002 [arXiv:1004.5287] [INSPIRE]. · Zbl 1170.83317
[37] Boulware, DG; Deser, S., String Generated Gravity Models, Phys. Rev. Lett., 55, 2656 (1985)
[38] Wheeler, JT, Symmetric Solutions to the Gauss-Bonnet Extended Einstein Equations, Nucl. Phys., B 268, 737 (1986)
[39] R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev.D 65 (2002) 084014 [hep-th/0109133] [INSPIRE].
[40] Pons, JM; Dadhich, N., On static black holes solutions in Einstein and Einstein-Gauss-Bonnet gravity with topology S^n × S^n, Eur. Phys. J., C 75, 280 (2015)
[41] Cai, R-G, A Note on thermodynamics of black holes in Lovelock gravity, Phys. Lett., B 582, 237 (2004) · Zbl 1246.83101
[42] R.-G. Cai and N. Ohta, Black Holes in Pure Lovelock Gravities, Phys. Rev.D 74 (2006) 064001 [hep-th/0604088] [INSPIRE].
[43] P. Bueno, P.A. Cano, A.Ó. Lasso and P.F. Ramírez, f(Lovelock) theories of gravity, JHEP04 (2016) 028 [arXiv:1602.07310] [INSPIRE]. · Zbl 1388.83569
[44] Wheeler, JT, Symmetric Solutions to the Maximally Gauss-Bonnet Extended Einstein Equations, Nucl. Phys., B 273, 732 (1986) · Zbl 0992.83522
[45] Whitt, B., Spherically Symmetric Solutions of General Second Order Gravity, Phys. Rev., D 38, 3000 (1988)
[46] Charmousis, C.; Dufaux, J-F, General Gauss-Bonnet brane cosmology, Class. Quant. Grav., 19, 4671 (2002) · Zbl 1027.83037
[47] Maeda, H.; Willison, S.; Ray, S., Lovelock black holes with maximally symmetric horizons, Class. Quant. Grav., 28, 165005 (2011) · Zbl 1225.83067
[48] Lovelock, D., The Einstein tensor and its generalizations, J. Math. Phys., 12, 498 (1971) · Zbl 0213.48801
[49] Oliva, J.; Ray, S., Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions, Phys. Rev., D 82, 124030 (2010)
[50] Dadhich, N.; Pons, JM, Static pure Lovelock black hole solutions with horizon topology S^(n) × S^(n), JHEP, 05, 067 (2015)
[51] Lanczos, C., A remarkable property of the Riemann-Christoffel tensor in four dimensions, Annals Math., 39, 842 (1938) · JFM 64.0767.03
[52] Gregory, C., Non-Linear Invariants and the Problem of Motion, Phys. Rev., 72, 72 (1947) · Zbl 0029.28704
[53] Buchdahl, HA, The Hamiltonian derivatives of a class of fundamental invariants, Quart. J. Math. Oxford, 19, 150 (1948) · Zbl 0031.07802
[54] S. Deser and B. Tekin, Energy in generic higher curvature gravity theories, Phys. Rev.D 67 (2003) 084009 [hep-th/0212292] [INSPIRE]. · Zbl 1045.83029
[55] Stelle, KS, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev., D 16, 953 (1977)
[56] Buchdahl, HA, On Eddington’s higher order equations of the gravitational field, Proc. Edinburgh Math. Soc., 8, 89 (1948) · Zbl 0036.42607
[57] Buchdahl, HA, A special class of solutions of the equations of the gravitational field arising from certain gauge-invariant action principles, Proc. Natl. Acad. Sci., 34, 66 (1948) · Zbl 0029.38301
[58] T. Málek and V. Pravda, Type III and N solutions to quadratic gravity, Phys. Rev.D 84 (2011) 024047 [arXiv:1106.0331] [INSPIRE].
[59] Mignemi, S.; Wiltshire, DL, Black holes in higher derivative gravity theories, Phys. Rev., D 46, 1475 (1992)
[60] S. Nojiri and S.D. Odintsov, Anti-de Sitter black hole thermodynamics in higher derivative gravity and new confining deconfining phases in dual CFT, Phys. Lett.B 521 (2001) 87 [Erratum ibid.B 542 (2002) 301] [hep-th/0109122] [INSPIRE]. · Zbl 1020.83023
[61] Buchdahl, HA, Non-linear Lagrangians and cosmological theory, Mon. Not. Roy. Astron. Soc., 150, 1 (1970)
[62] Pauli, W., Zur Theorie der Gravitation und der Elektrizität von Hermann Weyl, Phys. Z., 20, 457 (1919) · JFM 47.0791.02
[63] S. Deser and B. Tekin, Shortcuts to high symmetry solutions in gravitational theories, Class. Quant. Grav.20 (2003) 4877 [gr-qc/0306114] [INSPIRE]. · Zbl 1170.83438
[64] Hendi, SH, The Relation between F(R) gravity and Einstein-conformally invariant Maxwell source, Phys. Lett., B 690, 220 (2010)
[65] M. Hassaïne and C. Martínez, Higher-dimensional black holes with a conformally invariant Maxwell source, Phys. Rev.D 75 (2007) 027502 [hep-th/0701058] [INSPIRE].
[66] Bardoux, Y.; Caldarelli, MM; Charmousis, C., Shaping black holes with free fields, JHEP, 05, 054 (2012)
[67] S. Capozziello, A. Stabile and A. Troisi, Spherical symmetry in f(R)-gravity, Class. Quant. Grav.25 (2008) 085004 [arXiv:0709.0891] [INSPIRE]. · Zbl 1140.83393
[68] Kehagias, A.; Kounnas, C.; Lüst, D.; Riotto, A., Black hole solutions in R^2gravity, JHEP, 05, 143 (2015)
[69] Hendi, SH; Eslam Panah, B.; Mousavi, SM, Some exact solutions of F(R) gravity with charged (a)dS black hole interpretation, Gen. Rel. Grav., 44, 835 (2012) · Zbl 1238.83018
[70] Barrow, JD; Ottewill, AC, The Stability of General Relativistic Cosmological Theory, J. Phys., A 16, 2757 (1983)
[71] A. de la Cruz-Dombriz, A. Dobado and A.L. Maroto, Black Holes in f(R) theories, Phys. Rev.D 80 (2009) 124011 [Erratum ibid.D 83 (2011) 029903] [arXiv:0907.3872] [INSPIRE]. · Zbl 1183.83132
[72] Bueno, P.; Cano, PA, On black holes in higher-derivative gravities, Class. Quant. Grav., 34, 175008 (2017) · Zbl 1372.83032
[73] Nojiri, S.; Odintsov, SD, Instabilities and anti-evaporation of Reissner-Nordström black holes in modified F(R) gravity, Phys. Lett., B 735, 376 (2014) · Zbl 1380.83159
[74] M. Calzà, M. Rinaldi and L. Sebastiani, A special class of solutions in F(R)-gravity, Eur. Phys. J.C 78 (2018) 178 [arXiv:1802.00329] [INSPIRE].
[75] Lü, H.; Pang, Y.; Pope, CN, Black Holes in Six-dimensional Conformal Gravity, Phys. Rev., D 87, 104013 (2013)
[76] Riegert, RJ, Birkhoff’s Theorem in Conformal Gravity, Phys. Rev. Lett., 53, 315 (1984)
[77] Mannheim, PD; Kazanas, D., Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves, Astrophys. J., 342, 635 (1989)
[78] D. Klemm, Topological black holes in Weyl conformal gravity, Class. Quant. Grav.15 (1998) 3195 [gr-qc/9808051] [INSPIRE]. · Zbl 0942.83053
[79] Oliva, J.; Ray, S., A new cubic theory of gravity in five dimensions: Black hole, Birkhoff’s theorem and C-function, Class. Quant. Grav., 27, 225002 (2010) · Zbl 1205.83058
[80] Myers, RC; Robinson, B., Black Holes in Quasi-topological Gravity, JHEP, 08, 067 (2010) · Zbl 1291.83113
[81] Dehghani, MH; Bazrafshan, A.; Mann, RB; Mehdizadeh, MR; Ghanaatian, M.; Vahidinia, MH, Black Holes in Quartic Quasitopological Gravity, Phys. Rev., D 85, 104009 (2012)
[82] Cisterna, A.; Guajardo, L.; Hassaine, M.; Oliva, J., Quintic quasi-topological gravity, JHEP, 04, 066 (2017)
[83] P. Bueno, P.A. Cano and R.A. Hennigar, (Generalized) quasi-topological gravities at all orders, Class. Quant. Grav.37 (2020) 015002 [arXiv:1909.07983] [INSPIRE].
[84] Coley, AA; Gibbons, GW; Hervik, S.; Pope, CN, Metrics With Vanishing Quantum Corrections, Class. Quant. Grav., 25, 145017 (2008) · Zbl 1145.83014
[85] Thomas, TY, The Differential Invariants of Generalized Spaces (1934), London: Cambridge University Press, London
[86] Anderson, IM, Natural variational principles on Riemannian manifolds, Annals Math., 120, 329 (1984) · Zbl 0565.58019
[87] Gilkey, PB, Curvature and the eigenvalues of the Laplacian for elliptic complexes, Adv. Math., 10, 344 (1973) · Zbl 0259.58010
[88] M. Atiyah, R. Bott and V.K. Patodi, On the heat equation and the index theorem Invent. Math.19 (1973) 279. · Zbl 0364.58016
[89] Epstein, DBA, Natural tensors on Riemannian manifolds, J. Diff. Geom., 168, 631 (1975) · Zbl 0321.53039
[90] Hervik, Sigbjørn; Pravda, Vojtěch; Pravdová, Alena, Type III and N universal spacetimes, Classical and Quantum Gravity, 31, 21, 215005 (2014) · Zbl 1304.83016
[91] Prüfer, F.; Tricerri, F.; Vanhecke, L., Curvature invariants, differential operators and local homogeneity, Trans. Am. Math. Soc., 348, 4643 (1996) · Zbl 0867.53032
[92] M. Farhoudi, On higher order gravities, their analogy to GR and dimensional dependent version of Duff’s trace anomaly relation, Gen. Rel. Grav.38 (2006) 1261 [physics/0509210] [INSPIRE]. · Zbl 1104.83032
[93] Nakasone, M.; Oda, I., On Unitarity of Massive Gravity in Three Dimensions, Prog. Theor. Phys., 121, 1389 (2009) · Zbl 1176.83116
[94] İ. Güllü and B. Tekin, Massive Higher Derivative Gravity in D-dimensional Anti-de Sitter Spacetimes, Phys. Rev.D 80 (2009) 064033 [arXiv:0906.0102] [INSPIRE].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.