Abstract
The Schwarzschild-de Sitter black hole solution, which has two event horizons, is considered to examine the relation between the energy component of quasi-localized energy-momentum complexes on and the heat flows passing through its boundary . Here is the patch between cosmological event horizon and black hole event horizon of the SdS black hole solution. Conclusively, the relation, like the Legendre transformation, between the energy component of quasi-localized Einstein and Møller energy-momentum complex and the heat flows passing through the boundary is obeyed, and these two energy components of quasi-localized energy-momentum complexes could be corresponding to thermodynamic potentials.
1. Introduction
Recently, Yang et al. [1–3] have inferred that the formula about the quasi-localized Einstein and Møller energy-momentum complexes on and the heat flows passing through the boundary of are related asHere is the patch between event horizon located at and inner Cauchy horizon located at for the spherically symmetric black hole with two separate horizons. Equation (1) is similar to the Legendre transformation between Helmholtz free energy and internal energy or between Gibbs free energy and enthalpy , and therefore these energy components of quasi-localized energy-momentum complexes and could correspond to thermodynamic potentials. But, in previous studies of Yang et al. [1–3], is a Cauchy horizon. There was a controversy that an acceptable definition exists for the temperature and entropy of black hole at Cauchy horizon. In this article, I consider the Schwarzschild-de Sitter (SdS) black hole solution, which has two separate event horizons, and a review of relation between the energy component of quasi-localized energy-momentum complexes on a patch between two event horizons and the heat flows passing through its boundary.
2. The Schwarzschild-de Sitter Black Hole Metric
The SdS black hole solution [4], describing a spherically symmetric solution of the vaccum Einstein field equations in the presence of a positive cosmological constant is given in the static formand its metric function was found aswhere . Here, I shall consider the metric function in a factorization formWhen , the metric function has two distinct positive real roots and , and the smaller one and the larger one can be regarded as the position of the black hole event horizon and the cosmological event horizon for observers moving on the world lines of constant between and . Compared with (4), the relations for these three roots are given byBecause of (6), , the metric function is taken to beand (7) and (8) are also reorganized asLet be a 2-sphere of radius . Thus we suggest that is the patch between cosmological event horizon and black hole event horizon , and the boundary of is . The patch is region I of Penrose diagram for SdS black hole solution (shown as Figure 1).
3. The Thermodynamics of SdS Black Hole
In the study of the thermodynamics of SdS black hole by Gibbons and Hawking [5], the Hawking temperatures [6] of and areand the Bekenstein-Hawking entropies [7, 8] of and areFor those two event horizons, the heat flows are evaluated asHence, the heat flow passing through the boundary would be expressed by
4. The Quasi-Localized Energy-Momentum Complexes
Subsequently, on , the quasi-localized energy-momentum complexes in the Trautman [9], Einstein [10] and Møller [11, 12] prescription should be considered. The energy component of the Einstein energy-momentum complex [9, 10] is given bywhereand is the outward unit normal vector over the infinitesimal surface element . The energy component within radius obtained by the Einstein energy-momentum complex isTherefore, the energy component of quasi-localized Einstein energy-momentum complex on isMoreover, according to the definition of the Møller energy-momentum complex [11, 12] and Gauss’s theorem, its energy component is given aswhereSo the energy component with radius obtained using the Møller energy-momentum complex isand the energy component of quasi-localized Møller energy-momentum complex on is
5. Conclusion
Consequently, the difference of energy between the Einstein and Møller prescription [13] is defined asAccording to (18) and (22), the difference of energy in the patch isand its value is three times the heat flow passing through the boundary In this way, the energy components of quasi-localized Einstein energy-momentum complex and Møller energy-momentum complex will combine with the heat flow passing through the boundary although the factor “” of the heat flow passing through in (26) is different from the factor “” in (1). As a matter of fact, must be positive if is dominated by attractive gravitation. For that reason, I prefer that is replaced by its absolute value . Finally, the difference of energy between the Einstein and Møller prescription on the patch is equal to the heat flow passing through its boundary , as the formula previously pointed out [1]Because all boundaries of are event horizons, the summation of the heat flows passing through those boundaries is well defined. In conclusion, for the SdS black hole solution, the establishment of Legendre transformation in (27) exhibits that and would play the role of thermodynamic potential. This result conforms with the viewpoint of Chang et al. [14] and our latest studies [1–3].
Conflicts of Interest
There are no conflicts of interest related to this paper.
Acknowledgments
The author would like to thank Professor Ching-Tang Tsao for useful suggestions. This work was partially supported by the Ministry of Science and Technology (Taiwan) under Contract no. MOST103-2633-M143-001.