{"id":{"repo_id":"queens","oai_identifier":"oai:queensu.scholaris.ca:1974/34358"},"canonical_url":"https://search.dev.ndltd.org/etd/queens/oai:queensu.scholaris.ca:1974/34358","repository":{"repo_id":"queens","name":"Queens University","base_url":"https://qspace.library.queensu.ca/server/oai/request"},"display":{"title":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Ballik, William John Victor"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Physics, Engineering Physics and Astronomy","school":null,"contributors":[],"advisors":["Lake, Kayll"],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-02-25","date_published":"2025-02-25","updated_at":"2026-07-27T20:35:21Z","subjects":["general relativity","black holes","black hole thermodynamics","black hole volumes","black hole mechanics","anti-de sitter","kerr-anti-de sitter"],"languages":["eng"],"rights":["Attribution-NonCommercial-NoDerivatives 4.0 International"],"rights_urls":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1974/34358","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Physics, Engineering Physics and Astronomy"]},{"key":"dc:contributor.supervisor","label":"Supervisor","values":["Lake, Kayll"]},{"key":"dc:creator","label":"Author","values":["Ballik, William John Victor"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2025-02-25T13:42:22Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2025-02-25T13:42:22Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-02-25"]},{"key":"dc:type","label":"Dc Type","values":["thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["general relativity","black holes","black hole thermodynamics","black hole volumes","black hole mechanics","anti-de sitter","kerr-anti-de sitter"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Attribution-NonCommercial-NoDerivatives 4.0 International"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://creativecommons.org/licenses/by-nc-nd/4.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/1974/34358"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["PhD"]},{"key":"dc:title","label":"Title","values":["The Role of the Volume in Black Hole Thermodynamics"]}]}],"canonical_facts":{"dc:contributor.department":["Physics, Engineering Physics and Astronomy"],"dc:contributor.supervisor":["Lake, Kayll"],"dc:creator":["Ballik, William John Victor"],"dc:date.accessioned":["2025-02-25T13:42:22Z"],"dc:date.available":["2025-02-25T13:42:22Z"],"dc:date.issued":["2025-02-25"],"dc:description.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."],"dc:description.degree":["PhD"],"dc:identifier.uri":["https://hdl.handle.net/1974/34358"],"dc:language.iso":["eng"],"dc:rights":["Attribution-NonCommercial-NoDerivatives 4.0 International"],"dc:rights.uri":["http://creativecommons.org/licenses/by-nc-nd/4.0/"],"dc:subject":["general relativity","black holes","black hole thermodynamics","black hole volumes","black hole mechanics","anti-de sitter","kerr-anti-de sitter"],"dc:title":["The Role of the Volume in Black Hole Thermodynamics"],"dc:type":["thesis"]},"updated_at":"2026-07-27T20:35:21Z"}