The Role of the Volume in Black Hole Thermodynamics

Abstract

Gibbons et al. [42] found the energy E of Kerr–anti-de Sitter black holes by integrating the first law of black hole thermodynamics, δE=P_iΩiδJ_i+TδS, with black hole angular momenta J_i, angular velocity Ω_i, temperature T and entropy S. They showed that E corresponds to the Ashtekar–Magnon–Das (AMD) energy, calculated in frame adapted to the Killing vector ξ^a which is asymptotically timelike and hypersurface-orthogonal. In Cvetič et al. [27], the first law was extended by interpreting E as an enthalpy and the cosmological constant Λ as being proportional to a pressure P according to Λ=−(D−2)P/16π. The modified first law is δE=P_iΩ_iδJ_i+TδS+V_thδP with “thermodynamic volume” V_th. Due to scaling symmetry, the Smarr relation (D−3)E=(D−2)(P_iΩ_iJ_i+TS)−2PV_th is automatically satisfied. In a frame adapted to the Killing vector β^a=∇_bh^ba/(D-1) where h is the Principal Conformal Killing–Yano tensor, the corresponding AMD energy F and angular velocities ω_i satisfy the Smarr relation (D−3)F=(D − 2)(P_iΩ_iJ_i+TS)−2V_geo with “geometric volume” V_geo. I extend the work of Parikh [89] to define the vector volume V_C of a D-dimensional stationary black hole to be equal to the rate of growth of the D-volume of the black hole along the flow of the stationarity Killing vector. I show that V_geo=V_C. These papers and my work suggest the following questions: why is it necessary to use a frame adapted to ξ^a rather than β^a to recover the first law? Why does V_C appear more naturally in the β^a frame? Adapting Barnich and Compère [14], I define a (D−2)-form I_χ associated with each Killing vector χ^a. The integral of I_χ over an arbitrary (D−2)-surface enclosing the black hole gives a conserved quantity H_χ = ∫I_χ, with E = H_ξ and F = H_β. I show that the first law will be satisfied with quantities constructed from I_χ if the background anti-de Sitter metric and the vector χ^a both have unvarying components. This holds for ξ^a but not β^a, explaining why the first law works for E but not F. I show that V_C appears in the β-associated Smarr relation due to simplifications related to h.