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
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
摘要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.
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.
(Mathematical and relativistic aspects of cosmology)
引用本文:
Azad A. Siddiqui**;Syed Muhammad Jawwad Riaz;M. Akbar
. Foliation and the First Law of Black Hole Thermodynamics[J]. 中国物理快报, 2011, 28(5): 50401-050401.
Azad A. Siddiqui**, Syed Muhammad Jawwad Riaz, M. Akbar
. Foliation and the First Law of Black Hole Thermodynamics. Chin. Phys. Lett., 2011, 28(5): 50401-050401.
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