The Role of the Volume in Black Hole Thermodynamics
Loading...
Authors
Ballik, William John Victor
Date
2025-02-25
Type
thesis
Language
eng
Keyword
general relativity , black holes , black hole thermodynamics , black hole volumes , black hole mechanics , anti-de sitter , kerr-anti-de sitter
Alternative Title
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.
Description
Citation
Publisher
License
Queen's University's Thesis/Dissertation Non-Exclusive License for Deposit to QSpace and Library and Archives Canada
ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
Attribution-NonCommercial-NoDerivatives 4.0 International
ProQuest PhD and Master's Theses International Dissemination Agreement
Intellectual Property Guidelines at Queen's University
Copying and Preserving Your Thesis
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.
Attribution-NonCommercial-NoDerivatives 4.0 International