Abstract
Based on a mathematical lemma related to the Vandermonde determinant and two theorems derived from the first law of black hole thermodynamics, we investigate the angular momentum independence of the entropy sum as well as the entropy product of general rotating Kaluza-Klein black holes in higher dimensions. We show that for both non-charged rotating Kaluza-Klein black holes and non-charged rotating Kaluza-Klein-AdS black holes, the angular momentum of the black holes will not be present in entropy sum relation in dimensions d≥4, while the independence of angular momentum of the entropy product holds provided that the black holes possess at least one zero rotation parameter a j = 0 in higher dimensions d≥5, which means that the cosmological constant does not affect the angular momentum-free property of entropy sum and entropy product under the circumstances that charge δ=0. For the reason that the entropy relations of charged rotating Kaluza-Klein black holes as well as the non-charged rotating Kaluza-Klein black holes in asymptotically flat spacetime act the same way, it is found that the charge has no effect in the angular momentum-independence of entropy sum and product in asymptotically flat spactime.
Similar content being viewed by others
References
Cvetič, M., Gibbons G.W., Pope, C.N.: Universal area product formulas for rotating and charged black holes in four and higher dimensions. Phys. Rev. Lett. 106, 121301 (2011)
Toldo, C., Vandoren, S.: Static nonextremal AdS4 black hole solutions. JHEP 1209, 048 (2012)
Cvetič, M., Lü, H., Pope, C.N.: Entropy-product rules for charged rotating black holes. Phys. Rev. D 88, 044046 (2013)
Abdolrahimi, S., Shoom, A.A.: Distorted five-dimensional electrically charged black holes. Phys. Rev. D 89, 024040 (2014)
Lü, H., Pang, Y., Pope, C. N.: AdS dyonic black hole and its thermodynamics. J. High Energy Phys. 2013, 1–19 (2013)
Chow, D.D.K., Compre, G.: Seed for general rotating non-extremal black holes of N=8 supergravity. Classical and Quantum Gravity 31, 022001 (2014)
Castro, A., Rodriguez, M.J.: Universal properties and the first law of black hole inner mechanics. Phys. Rev. D 86, 024008 (2012)
Visser, M.: Quantization of area for event and cauchy horizons of the Kerr-Newman black hole. J. High Energy Phys. 2012, 1–13 (2012)
Chen, B., Liu, S., Zhang, J.-j.: Thermodynamics of black hole horizons and Kerr/CFT correspondence. J. High Energy Phys. 2012, 1–27 (2012)
Detournay, S.: Inner mechanics of three-dimensional black holes. Phys. Rev. Lett. 109, 031101 (2012)
Visser, M.: Area products for stationary black hole horizons. Phys. Rev. D 88, 044014 (2013)
Faraoni, V., Moreno, A.F.Z.: Are quantization rules for horizon areas universal?. Phys. Rev. D 88, 044011 (2013)
Castro, A., Dehmami, N., Giribet, G., Kastor, D.: On the universality of inner black hole mechanics and higher curvature gravity. J. High Energy Phys. 2013, 1–26 (2013)
Wang, J., Xu, W., Meng, X. H.: The universal property of horizon entropy sum of black holes in four dimensional asymptotical (anti-)de Sitter spacetime background. JHEP 1401, 031 (2014)
Zhang, Y., Gao, S.: Mass dependence of the entropy product and sum. Phys. Rev. D 91, 064032 (2015)
Liu, H., he Meng, X.: Angular momentum independence of the entropy sum and entropy product for AdS rotating black holes in all dimensions. Phys. Lett. B (2016). doi:10.1016/j.physletb.2016.05.084, arXiv:1605.00066v2 [gr–qc]
Wu, S.-Q.: General rotating charged Kaluza-Klein-AdS black holes in higher dimensions. Phys. Rev. D 83, 121502 (2011)
Chen, B., Zhang, J.-J.: Holographic descriptions of black rings. J. High Energy Phys. 2012, 1–22 (2012)
Larsen, F.: String model of black hole microstates. Phys. Rev. D 56, 1005–1008 (1997)
Cvetič, M., Larsen, F.: General rotating black holes in string theory: Greybody factors and event horizons. Phys. Rev. D 56, 4994–5007 (1997)
Cvetic, M., Larsen, F.: Greybody factors for rotating black holes in four dimensions. Nucl. Phys. B 506, 107–120 (1997)
Ansorg, M., Hennig, J.: The inner cauchy horizon of axisymmetric and stationary black holes with surrounding matter. Classical and Quantum Gravity 25, 222001 (2008)
Ansorg, M., Hennig, J.: Inner cauchy horizon of axisymmetric and stationary black holes with surrounding matter in Einstein-Maxwell theory. Phys. Rev. Lett. 102, 221102 (2009)
Wang, J., Xu, W., Meng, X.-H.: Entropy relations of black holes with multihorizons in higher dimensions. Phys. Rev. D 89, 044034 (2014)
Liu, H., Meng, X.-H., Xu, W., Zhu, B.: Universal entropy relations: entropy formulae and entropy bound. arXiv:1605.00764 [gr–qc]
Kastor, D., Ray, S., Traschen, J.: Enthalpy and the mechanics of AdS black holes. Class. Quant. Grav. 26, 195011 (2009)
Dolan, B.P.: Pressure and volume in the first law of black hole thermodynamics. Class. Quant. Grav. 28, 235017 (2011)
Cvetic, M., Gibbons, G.W., Kubiznak, D., Pope, C.N.: Black hole enthalpy and an entropy inequality for the thermodynamic volume. Phys. Rev. D 84, 024037 (2011)
Dolan, B.P., Kastor, D., Kubiznak, D., Mann, R.B., Traschen, J.: Thermodynamic volumes and isoperimetric inequalities for de Sitter black holes. Phys. Rev. D 87, 104017 (2013)
Altamirano, N., Kubiznak, D., Mann, R. B., Sherkatghanad, Z.: Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume. Galaxies 2, 89 (2014)
Xu, W., Wang, J., he Meng, X.: Entropy relations and the application of black holes with the cosmological constant and Gauss-Bonnet term. Physic Letters B 742, 225–230 (2015)
Gibbons, G.W., Lü, H., Page, D.N., Pope, C.N.: Rotating black holes in higher dimensions with a cosmological constant. Phys. Rev. Lett. 93, 171102 (2004)
Gibbons, G., Lu, H., Page, D. N., Pope, C.: The general Kerr-de Sitter metrics in all dimensions. J. Geom. Phys. 53, 49–73 (2005)
Kunz, J., Maison, D., Navarro-Lrida, F., Viebahn, J.: Rotating Einstein-Maxwellaxwell-Dilaton black holes in d dimensions. Phys. Lett. B 639, 95–100 (2006)
Meng, X.-H., Xu, W., Wang, J.: A note on entropy relations of black hole horizons. Int. J. Mod. Phys. A 29, 1450088 (2014). http://www.worldscientific.com/doi/pdf/10.1142/S0217751X14500882
Xu, W., Wang, J., Meng, X.-H.: The entropy sum of AdS black holes in four and higher dimensions. Int. J. Mod. Phys. A 29, 1450172 (2014). http://www.worldscientific.com/doi/pdf/10.1142/S0217751X14501723
Xu, W., Wang, J., Meng, X.-h.: Thermodynamic relations for the entropy and temperature of multi-horizon black holes. Galaxies 3, 53 (2015)
Acknowledgments
For the present work we thank Xu Wei, Wang Deng and Yao Yanhong for helpful discussions. We would also like to thank the anonymous referee for helpful comments to make this work improved greatly. This project is partially supported by NSFC.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, H., Meng, Xh. Angular Momentum-Free of the Entropy Relations for Rotating Kaluza-Klein Black Holes. Int J Theor Phys 56, 437–449 (2017). https://doi.org/10.1007/s10773-016-3185-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-016-3185-6