| GENERAL |
|
|
|
|
Foliation and the First Law of Black Hole Thermodynamics |
| Azad A. Siddiqui1**, Syed Muhammad Jawwad Riaz2, M. Akbar2
|
1Department of Basic Sciences and Humanities, EME College, National University of Sciences and Technology (NUST), Peshawar Road, Rawalpindi, Pakistan
2Centre for Advanced Mathematics and Physics, National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan
|
|
| Cite this article: |
|
Azad A. Siddiqui, Syed Muhammad Jawwad Riaz, M. Akbar 2011 Chin. Phys. Lett. 28 050401 |
|
|
|
|
Abstract There has been lots of interest in exploring the thermodynamic properties at the horizon of a black hole spacetime. It has been shown earlier that for different spacetimes, the Einstein field equations at the horizon can be expressed as the first law of black hole thermodynamics. Using the idea of foliation, we develop a simpler procedure to obtain such results. We consider r= constant slices, for the Schwarzschild and Reissner–Nordstrom black hole spacetimes. The Einstein field equations for the induced 3−dimensional metrics of the hypersurfaces are expressed in thermodynamic quantities under the virtual displacements of the hypersurfaces. As expected, it is found that the field equations of the induced metric corresponding to the horizon can be written as a first law of black hole thermodynamics. It is to be mentioned here that our procedure is much easier, to obtain such results, as here one has to essentially deal with (n-1)−dimensional induced metric for an n-dimensional spacetime.
|
| Keywords:
04.70.Dy
05.70.-a
98.80.Jk
|
|
|
Received: 10 January 2011
Published: 26 April 2011
|
|
| PACS: |
04.70.Dy
|
(Quantum aspects of black holes, evaporation, thermodynamics)
|
| |
05.70.-a
|
(Thermodynamics)
|
| |
98.80.Jk
|
(Mathematical and relativistic aspects of cosmology)
|
|
|
|
|
|
[1] Misner C W, Thorne K S and Wheeler K A 1973 Gravitation (Cranbury NJ: W H Freeman and Sons)
[2] Stephani H, Kramer D, MacCallum M A H, Hoenselaers C and Herlt E 2002 Exact Solutions of Einstein's Field Equations (Cambridge: Cambridge University)
[3] York J 1972 Phys. Rev. Lett. 28 1082
[4] Estabrook F et al 1973 Phys. Rev. D 7 2814
[5] Smarr L and York J W Jr 1978 Phys. Rev. D 17 2529
[6] Eardley D M and Smarr L 1978 Phys. Rev. D 19 2239
[7] Marsden J E and Tipler F J 1980 Phys. Rep. 66 109
[8] Iriondo M, Malec E and Murchadha N O 1996 Phys. Rev. D 54 4792
[9] Beig R and Nurchadha N O 1998 Phys. Rev. D 57 4728
[10] Guven J and Murchadha N O 1999 Phys. Rev. D 60 104015
[11] Goddard A 1977 Commun. Math. Phys. 54 279
[12] Brill D R and Flaherty F 1976 Commun. Math. Phys. 50 157
[13] Brill D R, Cavallo J A and Isenberg J A 1980 J. Math. Phys. 21 2789
[14] Pervez A, Qadir A and Siddiqui A A 1995 Phys. Rev. D 51 4598
[15] Kraus P and Wilczek F 1995 Nucl. Phys. B 433 403
[16] Rendall A D 1996 Helv. Phys. Acta 69 490
[17] Beig R and Heinzle J M 2005 Commun. Math. Phys. 260 673 arXiv:gr-qc/0501020
[18] Qadir A and Siddiqui A A 1999 J. Math. Phys. 40 5883
[19] Hussain V, Qadir A and Siddiqui A A 2002 Phys. Rev. D 65 027501 arXiv:gr-qc/0110068
[20] Qadir A and Siddiqui A A 2002 Nuovo Cimento B 117 909
[21] Qadir A and Siddiqui A A 2006 Int. J. Mod. Phys. D 15 1419
[22] Beig R and Siddiqui A A 2007 Class. Quantum Grav. 24 5435
[23] Jacobson T 1995 Phys. Rev. Lett. 75 1260
[24] Padmanabhan T 2002 Class Quant. Grav. 19 5387 arXiv:gr-qc/0204019
[25] Carlip S 1995 Class Quant. Grav. 12 2853
[26] Cai R G, Lu Z J and Zhang Y Z 1997 Phys. Rev. D 55 853
[27] Ashtekar A 2002 Adv. Theor. Math. Phys. 6 507
[28] Padmanabhan T 2002 Mod. Phys. Lett. A 17 923 arXiv:gr-qc/0202078
[29] Paranjape A, Sarkar S and Padmanabhan T 2006 Phys. Rev. D 74 104015 arXiv:hep-th/0607240
[30] Akbar M and Vai R G 2006 Phys. Lett. B 635 7 arXiv:hep-th/0602156
[31] Akbar M and Cai R G 2007 Phys. Rev. D 75 084003 arXiv:hep-th/0609128
[32] Eling C, Guedens R and Jacobson T 2006 Phys. Rev. Lett. 96 121301
[33] Akbar M 2007 Chin. Phys. Lett. 24 1158 arXiv:hep-th/0702029
[34] Akbar M and Siddiqui A A 2007 Phys. Lett. B 656 217
[35] Cai R G and Cao L M 2007 Phys. Rev. D 75 064008
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
