The Relation between the Quasi-Localized Energy-Momentum Complexes and the Thermodynamic Potential for the Schwarzschild-de Sitter Black Hole

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 M and the heat flows passing through its boundary ∂M. HereM 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 energymomentum complex and the heat flows passing through the boundary is obeyed, and these two energy components of quasilocalized energy-momentum complexes could be corresponding to thermodynamic potentials.


Introduction
Recently, Yang et al. [1][2][3] have inferred that the formula about the quasi-localized Einstein and Møller energy-momentum complexes on M * and the heat flows passing through the boundary of M * are related as Here M * is the patch between event horizon H + located at  =  + and inner Cauchy horizon H − 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 energymomentum complexes  E | M * and  M | M * could correspond to thermodynamic potentials.But, in previous studies of Yang et al. [1][2][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.

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 form and its metric function was found as where  2 = 3/Λ.Here, I shall consider the metric function in a factorization form When 3 √ 3/ < 1, 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 by Because of ( 6),  0 = −(  +   ), the metric function is taken to be and ( 7) and ( 8) are also reorganized as Let S 2 () be a 2-sphere of radius .Thus we suggest that M = {S 2 () |   >  >   } is the patch between cosmological event horizon H  = S 2 (  ) and black hole event horizon H  = S 2 (  ), and the boundary of M is M = H  ∪ H  .The patch M is region I of Penrose diagram for SdS black hole solution (shown as Figure 1).

The Thermodynamics of SdS Black Hole
In the study of the thermodynamics of SdS black hole by Gibbons and Hawking [5], the Hawking temperatures T [6] of H  and H  are and the Bekenstein-Hawking entropies S [7,8] of H  and H  are For those two event horizons, the heat flows are evaluated as Hence, the heat flow passing through the boundary M would be expressed by

The Quasi-Localized Energy-Momentum Complexes
Subsequently, on M, 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 by where and n is the outward unit normal vector over the infinitesimal surface element   →  .The energy component within radius  obtained by the Einstein energy-momentum complex is Therefore, the energy component of quasi-localized Einstein energy-momentum complex on M is Moreover, according to the definition of the Møller energymomentum complex [11,12] and Gauss's theorem, its energy component is given as where So the energy component with radius  obtained using the Møller energy-momentum complex is and the energy component of quasi-localized Møller energymomentum complex on M is

Conclusion
Consequently, the difference of energy between the Einstein and Møller prescription [13] is defined as According to (18) and ( 22), the difference of energy in the patch M is and its value is three times the heat flow passing through the boundary M In this way, the energy components of quasi-localized Einstein energy-momentum complex

Figure 1 :
Figure 1: Penrose diagram for SdS black hole solution.
E | M and Møller energymomentum complex  M | M will combine with the heat flow passing through the boundary M  M | M must be positive if M is dominated by attractive gravitation.For that reason, I prefer that  M | M is replaced by its absolute value | M | M |.Finally, the difference of energy between the Einstein and Møller prescription on the patch M is equal to the heat flow passing through its boundary M, as the formula previously pointed out[1]Because all boundaries of M are event horizons, the summation of the heat flows passing through those boundaries ∑ M TS is well defined.In conclusion, for the SdS black hole solution, the establishment of Legendre transformation in (27) exhibits that  E | M and | M | [1][2][3] play the role of thermodynamic potential.This result conforms with the viewpoint of Chang et al.[14]and our latest studies[1][2][3].