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International Journal of Theoretical Physics

, Volume 56, Issue 2, pp 437–449 | Cite as

Angular Momentum-Free of the Entropy Relations for Rotating Kaluza-Klein Black Holes

  • Hang Liu
  • Xin-he Meng
Article
  • 114 Downloads

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.

Keywords

Black hole thermodynamics Angular momentum independence Entropy relations General Relativity 

Notes

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.

References

  1. 1.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Toldo, C., Vandoren, S.: Static nonextremal AdS4 black hole solutions. JHEP 1209, 048 (2012)ADSCrossRefGoogle Scholar
  3. 3.
    Cvetič, M., Lü, H., Pope, C.N.: Entropy-product rules for charged rotating black holes. Phys. Rev. D 88, 044046 (2013)ADSCrossRefGoogle Scholar
  4. 4.
    Abdolrahimi, S., Shoom, A.A.: Distorted five-dimensional electrically charged black holes. Phys. Rev. D 89, 024040 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    Lü, H., Pang, Y., Pope, C. N.: AdS dyonic black hole and its thermodynamics. J. High Energy Phys. 2013, 1–19 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Castro, A., Rodriguez, M.J.: Universal properties and the first law of black hole inner mechanics. Phys. Rev. D 86, 024008 (2012)ADSCrossRefGoogle Scholar
  8. 8.
    Visser, M.: Quantization of area for event and cauchy horizons of the Kerr-Newman black hole. J. High Energy Phys. 2012, 1–13 (2012)CrossRefGoogle Scholar
  9. 9.
    Chen, B., Liu, S., Zhang, J.-j.: Thermodynamics of black hole horizons and Kerr/CFT correspondence. J. High Energy Phys. 2012, 1–27 (2012)ADSMathSciNetGoogle Scholar
  10. 10.
    Detournay, S.: Inner mechanics of three-dimensional black holes. Phys. Rev. Lett. 109, 031101 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Visser, M.: Area products for stationary black hole horizons. Phys. Rev. D 88, 044014 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Faraoni, V., Moreno, A.F.Z.: Are quantization rules for horizon areas universal?. Phys. Rev. D 88, 044011 (2013)ADSCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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)ADSCrossRefGoogle Scholar
  15. 15.
    Zhang, Y., Gao, S.: Mass dependence of the entropy product and sum. Phys. Rev. D 91, 064032 (2015)ADSCrossRefGoogle Scholar
  16. 16.
    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]
  17. 17.
    Wu, S.-Q.: General rotating charged Kaluza-Klein-AdS black holes in higher dimensions. Phys. Rev. D 83, 121502 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    Chen, B., Zhang, J.-J.: Holographic descriptions of black rings. J. High Energy Phys. 2012, 1–22 (2012)ADSGoogle Scholar
  19. 19.
    Larsen, F.: String model of black hole microstates. Phys. Rev. D 56, 1005–1008 (1997)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Cvetič, M., Larsen, F.: General rotating black holes in string theory: Greybody factors and event horizons. Phys. Rev. D 56, 4994–5007 (1997)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Cvetic, M., Larsen, F.: Greybody factors for rotating black holes in four dimensions. Nucl. Phys. B 506, 107–120 (1997)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Ansorg, M., Hennig, J.: The inner cauchy horizon of axisymmetric and stationary black holes with surrounding matter. Classical and Quantum Gravity 25, 222001 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wang, J., Xu, W., Meng, X.-H.: Entropy relations of black holes with multihorizons in higher dimensions. Phys. Rev. D 89, 044034 (2014)ADSCrossRefGoogle Scholar
  25. 25.
    Liu, H., Meng, X.-H., Xu, W., Zhu, B.: Universal entropy relations: entropy formulae and entropy bound. arXiv:1605.00764 [gr–qc]
  26. 26.
    Kastor, D., Ray, S., Traschen, J.: Enthalpy and the mechanics of AdS black holes. Class. Quant. Grav. 26, 195011 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Dolan, B.P.: Pressure and volume in the first law of black hole thermodynamics. Class. Quant. Grav. 28, 235017 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    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)ADSCrossRefGoogle Scholar
  29. 29.
    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)ADSCrossRefGoogle Scholar
  30. 30.
    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)ADSCrossRefMATHGoogle Scholar
  31. 31.
    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)ADSCrossRefMATHGoogle Scholar
  32. 32.
    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)ADSCrossRefGoogle Scholar
  33. 33.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    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)ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    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 ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    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 ADSCrossRefMATHGoogle Scholar
  37. 37.
    Xu, W., Wang, J., Meng, X.-h.: Thermodynamic relations for the entropy and temperature of multi-horizon black holes. Galaxies 3, 53 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of PhysicsNankai UniversityTianjinChina
  2. 2.State Key Laboratory of Theoretical PhysicsInstitute of Theoretical Physics, Chinese Academy of ScienceBeijingChina

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