Where is the PdV term in the first law of black-hole thermodynamics? Brian P Dolan National University of Ireland, Maynooth, Ireland and Dublin Institute for Advanced Studies, Ireland 9th Vienna Central European Seminar on Particle Physics and Quantum Field Theory, 30th November 2012
Outline Review of black hole thermodynamics Temperature and Entropy Laws of Thermodynamics Hawking radiation Pressure and Enthalpy Where is the PdV term? What is V? Critical behaviour Conclusions and Outlook
Temperature and Entropy Schwarzschild line element: ds 2 = (r)dt 2 + 1 (r) dr2 +r 2 (dθ 2 +sin 2 θdφ 2 ), ( (r) = 1 2G NM r ) r h = 2G N M Area: A = 16πG 2 N M 2 Entropy: S A (l 2 l 2 Pl = G N, c = 1) Bekenstein (1972) Pl Temperature, T = κ 2π : κ =surface gravity Hawking (1974) Schwarzschild, κ = 1 4G N M T = 8πG N M Solar mass black hole: T = 6 10 8 K, S 10 78
Temperature and Entropy Schwarzschild line element: ds 2 = (r)dt 2 + 1 (r) dr2 +r 2 (dθ 2 +sin 2 θdφ 2 ), ( (r) = 1 2G NM r ) r h = 2G N M Area: A = 16πG 2 N M 2 Entropy: S A (l 2 l 2 Pl = G N, c = 1) Bekenstein (1972) Pl Temperature, T = κ 2π : κ =surface gravity Hawking (1974) Schwarzschild, κ = 1 4G N M T = 8πG N M Solar mass black hole: T = 6 10 8 K, S 10 78
Temperature and Entropy Schwarzschild line element: ds 2 = (r)dt 2 + 1 (r) dr2 +r 2 (dθ 2 +sin 2 θdφ 2 ), ( (r) = 1 2G NM r ) r h = 2G N M Area: A = 16πG 2 N M 2 Entropy: S A (l 2 l 2 Pl = G N, c = 1) Bekenstein (1972) Pl Temperature, T = κ 2π : κ =surface gravity Hawking (1974) Schwarzschild, κ = 1 4G N M T = 8πG N M Solar mass black hole: T = 6 10 8 K, S 10 78
Temperature and Entropy Schwarzschild line element: ds 2 = (r)dt 2 + 1 (r) dr2 +r 2 (dθ 2 +sin 2 θdφ 2 ), ( (r) = 1 2G NM r ) r h = 2G N M Area: A = 16πG 2 N M 2 Entropy: S A (l 2 l 2 Pl = G N, c = 1) Bekenstein (1972) Pl Temperature, T = κ 2π : κ =surface gravity Hawking (1974) Schwarzschild, κ = 1 4G N M T = 8πG N M Solar mass black hole: T = 6 10 8 K, S 10 78
Temperature and Entropy Schwarzschild line element: ds 2 = (r)dt 2 + 1 (r) dr2 +r 2 (dθ 2 +sin 2 θdφ 2 ), ( (r) = 1 2G NM r ) r h = 2G N M Area: A = 16πG 2 N M 2 Entropy: S A (l 2 l 2 Pl = G N, c = 1) Bekenstein (1972) Pl Temperature, T = κ 2π : κ =surface gravity Hawking (1974) Schwarzschild, κ = 1 4G N M T = 8πG N M Solar mass black hole: T = 6 10 8 K, S 10 78
Temperature and Entropy Schwarzschild line element: ds 2 = (r)dt 2 + 1 (r) dr2 +r 2 (dθ 2 +sin 2 θdφ 2 ), ( (r) = 1 2G NM r ) r h = 2G N M Area: A = 16πG 2 N M 2 Entropy: S A (l 2 l 2 Pl = G N, c = 1) Bekenstein (1972) Pl Temperature, T = κ 2π : κ =surface gravity Hawking (1974) Schwarzschild, κ = 1 4G N M T = 8πG N M Solar mass black hole: T = 6 10 8 K, S 10 78
First and Second laws Internal energy U(S), T = U S : identify M = U(S) First Law of Black Hole Thermodynamics With S = a A G N dm = du = T ds r h = 2G N M, A = 16πGN 2M2, ds = 32πaG N TdS = 4adM a = 1 4 Bekenstein-Hawking Entropy S = 1 4 A l 2 Pl MdM, T = 8πG N M,
First and Second laws Internal energy U(S), T = U S : identify M = U(S) First Law of Black Hole Thermodynamics With S = a A G N dm = du = T ds r h = 2G N M, A = 16πGN 2M2, ds = 32πaG N TdS = 4adM a = 1 4 Bekenstein-Hawking Entropy S = 1 4 A l 2 Pl MdM, T = 8πG N M,
First and Second laws Internal energy U(S), T = U S : identify M = U(S) First Law of Black Hole Thermodynamics With S = a A G N dm = du = T ds r h = 2G N M, A = 16πGN 2M2, ds = 32πaG N TdS = 4adM a = 1 4 Bekenstein-Hawking Entropy S = 1 4 A l 2 Pl MdM, T = 8πG N M,
First and Second laws Internal energy U(S), T = U S : identify M = U(S) First Law of Black Hole Thermodynamics With S = a A G N dm = du = T ds r h = 2G N M, A = 16πGN 2M2, ds = 32πaG N TdS = 4adM a = 1 4 Bekenstein-Hawking Entropy S = 1 4 A l 2 Pl MdM, T = 8πG N M,
First and Second laws Internal energy U(S), T = U S : identify M = U(S) First Law of Black Hole Thermodynamics With S = a A G N dm = du = T ds r h = 2G N M, A = 16πGN 2M2, ds = 32πaG N TdS = 4adM a = 1 4 Bekenstein-Hawking Entropy S = 1 4 A l 2 Pl MdM, T = 8πG N M,
First and Second laws Internal energy U(S), T = U S : identify M = U(S) First Law of Black Hole Thermodynamics With S = a A G N dm = du = T ds r h = 2G N M, A = 16πGN 2M2, ds = 32πaG N TdS = 4adM a = 1 4 Bekenstein-Hawking Entropy S = 1 4 A l 2 Pl MdM, T = 8πG N M,
Hawking radiation Free energy, F(T), is Legendre transform of U(S) F = U TS = M κa 8πG N Schwarzschild: F = M 2 = 16πGT Heat capacity: C = U T = T 2 F T 2 = 8πG NM 2 = 2S < 0 Negative Heat capacity! Radiates with power P AT 4 3 G 2 n M2 Lifetime τ M P G2 N M3, M 1012 kg τ 10 10 years If the black hole is shrinking, is the heat capacity C V or C P?
Hawking radiation Free energy, F(T), is Legendre transform of U(S) F = U TS = M κa 8πG N Schwarzschild: F = M 2 = 16πGT Heat capacity: C = U T = T 2 F T 2 = 8πG NM 2 = 2S < 0 Negative Heat capacity! Radiates with power P AT 4 3 G 2 n M2 Lifetime τ M P G2 N M3, M 1012 kg τ 10 10 years If the black hole is shrinking, is the heat capacity C V or C P?
Hawking radiation Free energy, F(T), is Legendre transform of U(S) F = U TS = M κa 8πG N Schwarzschild: F = M 2 = 16πGT Heat capacity: C = U T = T 2 F T 2 = 8πG NM 2 = 2S < 0 Negative Heat capacity! Radiates with power P AT 4 3 G 2 n M2 Lifetime τ M P G2 N M3, M 1012 kg τ 10 10 years If the black hole is shrinking, is the heat capacity C V or C P?
Hawking radiation Free energy, F(T), is Legendre transform of U(S) F = U TS = M κa 8πG N Schwarzschild: F = M 2 = 16πGT Heat capacity: C = U T = T 2 F T 2 = 8πG NM 2 = 2S < 0 Negative Heat capacity! Radiates with power P AT 4 3 G 2 n M2 Lifetime τ M P G2 N M3, M 1012 kg τ 10 10 years If the black hole is shrinking, is the heat capacity C V or C P?
Hawking radiation Free energy, F(T), is Legendre transform of U(S) F = U TS = M κa 8πG N Schwarzschild: F = M 2 = 16πGT Heat capacity: C = U T = T 2 F T 2 = 8πG NM 2 = 2S < 0 Negative Heat capacity! Radiates with power P AT 4 3 G 2 n M2 Lifetime τ M P G2 N M3, M 1012 kg τ 10 10 years If the black hole is shrinking, is the heat capacity C V or C P?
Pressure and enthalpy First Law of thermodynamics du = T ds PdV Where is the P dv term in the first law? Include cosmological constant Λ, contributes pressure P and energy density ǫ = P = Λ 8πG N Henneaux+Teitelboim (1984); (1989); Teitelboim (1985) Thermal energy Enthalpy U = U(S,V) U = M +ǫv = M PV M = U +PV M = U +PV = H(S,P) Kastor, Ray+Traschen [09042765]
Pressure and enthalpy First Law of thermodynamics du = T ds PdV Where is the P dv term in the first law? Include cosmological constant Λ, contributes pressure P and energy density ǫ = P = Λ 8πG N Henneaux+Teitelboim (1984); (1989); Teitelboim (1985) Thermal energy Enthalpy U = U(S,V) U = M +ǫv = M PV M = U +PV M = U +PV = H(S,P) Kastor, Ray+Traschen [09042765]
Pressure and enthalpy First Law of thermodynamics du = T ds PdV Where is the P dv term in the first law? Include cosmological constant Λ, contributes pressure P and energy density ǫ = P = Λ 8πG N Henneaux+Teitelboim (1984); (1989); Teitelboim (1985) Thermal energy Enthalpy U = U(S,V) U = M +ǫv = M PV M = U +PV M = U +PV = H(S,P) Kastor, Ray+Traschen [09042765]
Pressure and enthalpy First Law of thermodynamics du = T ds PdV Where is the P dv term in the first law? Include cosmological constant Λ, contributes pressure P and energy density ǫ = P = Λ 8πG N Henneaux+Teitelboim (1984); (1989); Teitelboim (1985) Thermal energy Enthalpy U = U(S,V) U = M +ǫv = M PV M = U +PV M = U +PV = H(S,P) Kastor, Ray+Traschen [09042765]
Pressure and enthalpy First Law of thermodynamics du = T ds PdV Where is the P dv term in the first law? Include cosmological constant Λ, contributes pressure P and energy density ǫ = P = Λ 8πG N Henneaux+Teitelboim (1984); (1989); Teitelboim (1985) Thermal energy Enthalpy U = U(S,V) U = M +ǫv = M PV M = U +PV M = U +PV = H(S,P) Kastor, Ray+Traschen [09042765]
What is V? Include Λ, (r) = 1 2G NM r Λ 3 r2 (Λ < 0), (r h ) = 0 M(r h,λ) = H(S,P), Thermodynamic definition of V (BPD [10085023]): V = ( ) H P S S = πr2 h, P = Λ l 2 8πG Pl N V = 4 (l Pl S) 3 2 = 4πr3 h 3 π 3 Problem S = πr2 h and V = 4πr3 l 2 h 3 are not independent! Pl Resolved by introducing rotation, J
What is V? Include Λ, (r) = 1 2G NM r Λ 3 r2 (Λ < 0), (r h ) = 0 M(r h,λ) = H(S,P), Thermodynamic definition of V (BPD [10085023]): V = ( ) H P S S = πr2 h, P = Λ l 2 8πG Pl N V = 4 (l Pl S) 3 2 = 4πr3 h 3 π 3 Problem S = πr2 h and V = 4πr3 l 2 h 3 are not independent! Pl Resolved by introducing rotation, J
What is V? Include Λ, (r) = 1 2G NM r Λ 3 r2 (Λ < 0), (r h ) = 0 M(r h,λ) = H(S,P), Thermodynamic definition of V (BPD [10085023]): V = ( ) H P S S = πr2 h, P = Λ l 2 8πG Pl N V = 4 (l Pl S) 3 2 = 4πr3 h 3 π 3 Problem S = πr2 h and V = 4πr3 l 2 h 3 are not independent! Pl Resolved by introducing rotation, J
What is V? Include Λ, (r) = 1 2G NM r Λ 3 r2 (Λ < 0), (r h ) = 0 M(r h,λ) = H(S,P), Thermodynamic definition of V (BPD [10085023]): V = ( ) H P S S = πr2 h, P = Λ l 2 8πG Pl N V = 4 (l Pl S) 3 2 = 4πr3 h 3 π 3 Problem S = πr2 h and V = 4πr3 l 2 h 3 are not independent! Pl Resolved by introducing rotation, J
What is V? Include Λ, (r) = 1 2G NM r Λ 3 r2 (Λ < 0), (r h ) = 0 M(r h,λ) = H(S,P), Thermodynamic definition of V (BPD [10085023]): V = ( ) H P S S = πr2 h, P = Λ l 2 8πG Pl N V = 4 (l Pl S) 3 2 = 4πr3 h 3 π 3 Problem S = πr2 h and V = 4πr3 l 2 h 3 are not independent! Pl Resolved by introducing rotation, J
AdS-Kerr Thermodynamics Rotating black-hole in asymptotically AdS (Kerr-AdS) Angular momentum J = am (G N = = 1,Λ = 3 L 2 ) Area of event horizon: A = 4π(r2 h +a2 ) = 4S 1 a2 L 2 Mass and enthalpy (1+ M = 1 ){ 8PS ( 3 S 2 1+ 8PS ) 3 +4π 2 J 2} := H(S,P,J) 2 πs Sorkin (1982); Caldarelli, Gognola+Klemm [hep-th/9908022] Thermodynamic volume: V = H P = 2 ( {S 2 1+ 8PS } )+2π 2 J 2 S,J 3πH 3 Cvetic, Gibbons+Kubiz nák [10122888]; BPD [11066260] ( ) 3V 1/3 4π Equality only for J = 0 ( ) S 1/2 π
AdS-Kerr Thermodynamics Rotating black-hole in asymptotically AdS (Kerr-AdS) Angular momentum J = am (G N = = 1,Λ = 3 L 2 ) Area of event horizon: A = 4π(r2 h +a2 ) = 4S 1 a2 L 2 Mass and enthalpy (1+ M = 1 ){ 8PS ( 3 S 2 1+ 8PS ) 3 +4π 2 J 2} := H(S,P,J) 2 πs Sorkin (1982); Caldarelli, Gognola+Klemm [hep-th/9908022] Thermodynamic volume: V = H P = 2 ( {S 2 1+ 8PS } )+2π 2 J 2 S,J 3πH 3 Cvetic, Gibbons+Kubiz nák [10122888]; BPD [11066260] ( ) 3V 1/3 4π Equality only for J = 0 ( ) S 1/2 π
AdS-Kerr Thermodynamics Rotating black-hole in asymptotically AdS (Kerr-AdS) Angular momentum J = am (G N = = 1,Λ = 3 L 2 ) Area of event horizon: A = 4π(r2 h +a2 ) = 4S 1 a2 L 2 Mass and enthalpy (1+ M = 1 ){ 8PS ( 3 S 2 1+ 8PS ) 3 +4π 2 J 2} := H(S,P,J) 2 πs Sorkin (1982); Caldarelli, Gognola+Klemm [hep-th/9908022] Thermodynamic volume: V = H P = 2 ( {S 2 1+ 8PS } )+2π 2 J 2 S,J 3πH 3 Cvetic, Gibbons+Kubiz nák [10122888]; BPD [11066260] ( ) 3V 1/3 4π Equality only for J = 0 ( ) S 1/2 π
AdS-Kerr Thermodynamics Rotating black-hole in asymptotically AdS (Kerr-AdS) Angular momentum J = am (G N = = 1,Λ = 3 L 2 ) Area of event horizon: A = 4π(r2 h +a2 ) = 4S 1 a2 L 2 Mass and enthalpy (1+ M = 1 ){ 8PS ( 3 S 2 1+ 8PS ) 3 +4π 2 J 2} := H(S,P,J) 2 πs Sorkin (1982); Caldarelli, Gognola+Klemm [hep-th/9908022] Thermodynamic volume: V = H P = 2 ( {S 2 1+ 8PS } )+2π 2 J 2 S,J 3πH 3 Cvetic, Gibbons+Kubiz nák [10122888]; BPD [11066260] ( ) 3V 1/3 4π Equality only for J = 0 ( ) S 1/2 π
AdS-Kerr Thermodynamics Rotating black-hole in asymptotically AdS (Kerr-AdS) Angular momentum J = am (G N = = 1,Λ = 3 L 2 ) Area of event horizon: A = 4π(r2 h +a2 ) = 4S 1 a2 L 2 Mass and enthalpy (1+ M = 1 ){ 8PS ( 3 S 2 1+ 8PS ) 3 +4π 2 J 2} := H(S,P,J) 2 πs Sorkin (1982); Caldarelli, Gognola+Klemm [hep-th/9908022] Thermodynamic volume: V = H P = 2 ( {S 2 1+ 8PS } )+2π 2 J 2 S,J 3πH 3 Cvetic, Gibbons+Kubiz nák [10122888]; BPD [11066260] ( ) 3V 1/3 4π Equality only for J = 0 ( ) S 1/2 π
AdS-Kerr Thermodynamics Rotating black-hole in asymptotically AdS (Kerr-AdS) Angular momentum J = am (G N = = 1,Λ = 3 L 2 ) Area of event horizon: A = 4π(r2 h +a2 ) = 4S 1 a2 L 2 Mass and enthalpy (1+ M = 1 ){ 8PS ( 3 S 2 1+ 8PS ) 3 +4π 2 J 2} := H(S,P,J) 2 πs Sorkin (1982); Caldarelli, Gognola+Klemm [hep-th/9908022] Thermodynamic volume: V = H P = 2 ( {S 2 1+ 8PS } )+2π 2 J 2 S,J 3πH 3 Cvetic, Gibbons+Kubiz nák [10122888]; BPD [11066260] ( ) 3V 1/3 4π Equality only for J = 0 ( ) S 1/2 π
Critical behaviour 001 0008 JP 0006 0004 0002 0 0 002 004 006 T > 0 below green curve, locally stable above red curve C P on red curve: two values of SP for given JP ( small and large black holes), merge to one at critical point (JP) crit, (SP) crit (Caldarelli, Gognola+Klemm [hep-th/9908022]) (SP) crit 008204, (JP) crit 0002857, (VP 3 2) crit 001768 SP 008 01 012 014
Critical behaviour 001 0008 JP 0006 0004 0002 0 0 002 004 006 T > 0 below green curve, locally stable above red curve C P on red curve: two values of SP for given JP ( small and large black holes), merge to one at critical point (JP) crit, (SP) crit (Caldarelli, Gognola+Klemm [hep-th/9908022]) (SP) crit 008204, (JP) crit 0002857, (VP 3 2) crit 001768 SP 008 01 012 014
Equation of state and critical exponents Define t := T T c T c, v := V V c V c, p := P P c P c Expand the equation of state about the critical point: p = 242t 081tv 021v 3 +o(t 2,tv 2,v 4 ) cf van der Waals gas: p = 4t 6tv 3 2 v3 +o(t 2,tv 2,v 4 ) C V = T/ T S V,J t α ; At fixed p < 0, v > v < t β ; Isothermal compressibility, κ T = 1 ( V ) V P T,J t γ ; At t = 0, p v δ ; Mean Field Exponents α = 0, β = 1, γ = 1, δ = 3 2
Equation of state and critical exponents Define t := T T c T c, v := V V c V c, p := P P c P c Expand the equation of state about the critical point: p = 242t 081tv 021v 3 +o(t 2,tv 2,v 4 ) cf van der Waals gas: p = 4t 6tv 3 2 v3 +o(t 2,tv 2,v 4 ) C V = T/ T S V,J t α ; At fixed p < 0, v > v < t β ; Isothermal compressibility, κ T = 1 ( V ) V P T,J t γ ; At t = 0, p v δ ; Mean Field Exponents α = 0, β = 1, γ = 1, δ = 3 2
Equation of state and critical exponents Define t := T T c T c, v := V V c V c, p := P P c P c Expand the equation of state about the critical point: p = 242t 081tv 021v 3 +o(t 2,tv 2,v 4 ) cf van der Waals gas: p = 4t 6tv 3 2 v3 +o(t 2,tv 2,v 4 ) C V = T/ T S V,J t α ; At fixed p < 0, v > v < t β ; Isothermal compressibility, κ T = 1 ( V ) V P T,J t γ ; At t = 0, p v δ ; Mean Field Exponents α = 0, β = 1, γ = 1, δ = 3 2
Equation of state and critical exponents Define t := T T c T c, v := V V c V c, p := P P c P c Expand the equation of state about the critical point: p = 242t 081tv 021v 3 +o(t 2,tv 2,v 4 ) cf van der Waals gas: p = 4t 6tv 3 2 v3 +o(t 2,tv 2,v 4 ) C V = T/ T S V,J t α ; At fixed p < 0, v > v < t β ; Isothermal compressibility, κ T = 1 ( V ) V P T,J t γ ; At t = 0, p v δ ; Mean Field Exponents α = 0, β = 1, γ = 1, δ = 3 2
Equation of state and critical exponents Define t := T T c T c, v := V V c V c, p := P P c P c Expand the equation of state about the critical point: p = 242t 081tv 021v 3 +o(t 2,tv 2,v 4 ) cf van der Waals gas: p = 4t 6tv 3 2 v3 +o(t 2,tv 2,v 4 ) C V = T/ T S V,J t α ; At fixed p < 0, v > v < t β ; Isothermal compressibility, κ T = 1 ( V ) V P T,J t γ ; At t = 0, p v δ ; Mean Field Exponents α = 0, β = 1, γ = 1, δ = 3 2
Equation of state and critical exponents Define t := T T c T c, v := V V c V c, p := P P c P c Expand the equation of state about the critical point: p = 242t 081tv 021v 3 +o(t 2,tv 2,v 4 ) cf van der Waals gas: p = 4t 6tv 3 2 v3 +o(t 2,tv 2,v 4 ) C V = T/ T S V,J t α ; At fixed p < 0, v > v < t β ; Isothermal compressibility, κ T = 1 ( V ) V P T,J t γ ; At t = 0, p v δ ; Mean Field Exponents α = 0, β = 1, γ = 1, δ = 3 2
Gibbs Free Energy Gibbs Free Energy, G(T,P,J) = H(S,P,J) TS: (J = 1) P G T
Kerr-Reissner-Nordström-AdS Similar story for J = 0,Q 0 (Reissner-Nordström AdS) Gibbs free energy is a swallowtail, same as van der Waals Champlin, Emparan, Johnson+Myers [hep-th/9902170;9904197] Exponents are mean field, Kubiz nák+mann [arxiv:12050559] J 0, Q 0 Caldarelli, Gognola+Klemm [hep-th/9908022]
Kerr-Reissner-Nordström-AdS Similar story for J = 0,Q 0 (Reissner-Nordström AdS) Gibbs free energy is a swallowtail, same as van der Waals Champlin, Emparan, Johnson+Myers [hep-th/9902170;9904197] Exponents are mean field, Kubiz nák+mann [arxiv:12050559] J 0, Q 0 Caldarelli, Gognola+Klemm [hep-th/9908022]
Kerr-Reissner-Nordström-AdS Similar story for J = 0,Q 0 (Reissner-Nordström AdS) Gibbs free energy is a swallowtail, same as van der Waals Champlin, Emparan, Johnson+Myers [hep-th/9902170;9904197] Exponents are mean field, Kubiz nák+mann [arxiv:12050559] J 0, Q 0 Caldarelli, Gognola+Klemm [hep-th/9908022]
Q Kerr-Reissner-Nordström-AdS Similar story for J = 0,Q 0 (Reissner-Nordström AdS) Gibbs free energy is a swallowtail, same as van der Waals Champlin, Emparan, Johnson+Myers [hep-th/9902170;9904197] Exponents are mean field, Kubiz nák+mann [arxiv:12050559] J 0, Q 0 Caldarelli, Gognola+Klemm [hep-th/9908022] J Line of second order critical points (mean field) Unique black hole Large and small black holes
Conclusions Λ 0 P dv term in black hole 1st law First Law du = TdS +ΩdJ +ΦdQ PdV Black hole mass is identified with enthalpy, H(S,P): dm = dh = T ds +ΩdJ +ΦdQ +V dp Thermodynamic volume: V = ( ) H P S When J 0, V and S are not independent, ( ) 3V 2 ( ) S 3, 4π π reverse iso-perimetric inequality Gibbs free energy: G(T,P) = T lnz, (Z = e I E) Line of second order critical points in J Q plane with mean field exponents and van der Waals type equation of state
Outlook Heat capacity: C P = T T ; C V = S T T P Condensed matter applications? S V Higher dimensional rotating black holes; other backgrounds? de Sitter, Λ > 0, P < 0? Beyond mean field?
Smarr relation Ordinary thermodynamics: U(S,V,n) (n = number of moles) is a function of extensive variables U is also extensive λ d U(S,V,n) = U(λ d S,λ d V,λ d n) U = S U U +V S V +n U n Euler equation U = ST VP +nµ (µ = chemical potential) G = U +VP ST = nµ Gibbs-Duhem relation Black hole: S λ 2 S, P λ 2 P, J λ 2 J, M λm λh(s,p,j) = H(λ 2 S,λ 2 P,λ 2 J) M = H(S,P,J) H = 2S H S 2P H P +2J H J H = 2ST 2VP +2J Ω Smarr relation Smarr (1973); Kastor, Ray+Traschen [09042765]
Penrose Processes Mechanical energy: dw = du = TdS ΩdJ ΦdQ +PdV Most efficient is S = const and decrease J and Q Maximum work starting from an extremal black-hole Start from J = J max, Q = 0 and keep P constant maximum efficiency: η max = 518%, for PS = 1837 BPD [11066260] (compare η max = 1 1 2 = 29%, eg Wald (1984)) Charged-Kerr-AdS: η max = 75% (compare η max = 50%)
Compressibility BPD [arxiv:11090198] V P Adiabatic compressibility: κ = 1 V κ J=0 = 0 Maximum for J max (T = 0): κ max = eg P = 0, S,J 0 2S(1+8PS) (3+8PS) 2 (1+4PS) ( ) 2 κ max = 2S 9 = 4πM2 9 = 26 10 38 M M ms 2 kg 1 cf neutron star, M M, R 10km, degenerate Fermi gas κ 10 34 ms 2 kg 1 Very stiff equation of state! ρ = M V, speed of sound v 2 s = ρ P, S,J Speed of Sound vs 2 (2πJ) 4 = 1+ ) 2 (2S 2 +(2πJ) 2 1 2 v2 s 1, with v s = 1 for J = 0
Internal Energy U = H +PV U(S,V,J) = ( π ) 3 ( )( ) 3V S 2 (3V ) 2 S 4π 2π 2 +J2 J 2 4π ( ) S 3 π, BPD [11066260] As J 0, ( 3V 2 ( 4π) S 3 π) T = U J S V,J finite lim 2 J 0 is finite ( 3V 4π) 2 ( π) S 3 But lim 2 U J 0 C S 2 V = U T V = T S T V,J 0 as J 0 P = U V S,J Virial expansion at fixed T (large V), P = T 2 ( 4π 3V ) 1/3 1 ( ) 4π 2/3 + 4π J 2 ( ) J 4 8π 3V 3 V 2 +o V 11/3
P V diagram 0005 0004 0003 0002 0001 0 2 4 6 8 10 P as a function of ( 3V 4π) 1/3, curves of constant T for J = 1 Critical point at T c = 00418J 1/2, P c = 000286/J and V C = 1158J 3/2 (Caldarelli, Gognola+Klemm [hep-th/9908022])
Equation of state Equation of state T(V,P) = 4π { (3V 4π ) 1 3 +8πGP ( 3V 4π } )1 3 P V
BTZ black hole d 2 s = (r)dt 2 + 1 (r)dr 2 +r 2 dφ 2, (r) = 8G N M + r2 L 2 (Λ = 1 L 2 ) r h = 8G N ML, S = πr h 2l Pl (l Pl = G N ) Enthalpy H = M: H(S,P) = 4l Pl π S2 P Equation of state: Thermodynamic volume: PV 1 2 = π 4l Pl T V = πr 2 h Gibbs free energy: G = H TS = M
Heat capacity: C P = T = S > 0 P T S Phase transition Compare pure AdS 3 with (r) = 1+ r2 L 2 (Equivalent to setting M = 1 8G N in BTZ) Enthalpy, H = 1 8G N = constant, T = 0 Gibbs free energy: G = H = 1 8G N If M < 1 8G N in BTZ, pure AdS 3 has lower free energy Equivalent to temperature: T = 2G N P π
Quantum corrections to BTZ equation of state (Brown+Henneaux (1984); Maloney+Witten [07120155]) Rotating BTZ black hole: (angular momentum J) ds 2 = (r)dt 2 + 1 ( (r) dr2 +r 2 dφ 4G ) 2 NJ r 2 dt (r) = ) ( 8G N M + r2 L 2 + 16G2 N J2 r 2 Inner and outer event horizons: [ r± 2 = 4G N ML 2 1± 1 Hawking temperature: ( J ML ]1 ) 2 2 T = (r h ) 4π = (r2 h r2 ) 2πL 2 r h
Partition function Wick rotate to Euclidean time: t it E, J ij E Also r h r E,+ and r ir E, where [ ]1 ( ) 2 2 re,± 2 = 4G NML 2 JE 1+ ±1 ML Define τ = r E, +ir E,+ L Inverse Hawking temperature: 1 2πT = (Im(τ) > 0) r E,+ L 2 (re,+ 2 +r2 E, ) = L Im Define q = e 2πiτ in terms of which Z BTZ = (q q) L 16 G N to all orders in perturbation theory n=2 ( 1 ) τ 1 q n 2
Entropy corrections Set J = 0: T = r h 2π, τ = i r L 2 h L and q = e 4π2 LT The J = 0 partition function: Z BTZ = e π 2 TL 2 2 2 G N n=2 ) (1 e 4π2 n 2 TL Defining x = TL = r h 2πL, the Gibbs free energy is G(T,P) = T lnz BTZ = π2 x 2 Entropy, S < 1 4 (area) 2G N +2T n=2 ln (1 e ) 4π2 nx
Equation of state (quantum volume): V(T,P) = G [ P = πrh 2 1 8πG N T L (BPD [arxiv:10085023]) n=2 ] n e 4π2nx 1 P classical quantum
Higher dimensional black holes Line element: d 2 s = (r)dt 2 + 1 (r)dr 2 +r d d 2 Ω (d) Ω (d) = 2π d 2 : volume of d-dimensional unit sphere Γ( d 2) Generalisation: event horizon any d-dimensional Einstein space R ij = R d g ij with constant Ricci curvature R and unit radius volume Ω (d) (eg flat torus, or CP d 2 for even d) (r) = R d(d 1) 16πG N Ω (d) d Planck length l d Pl = G N : S = Ω (d) 4 r d h l d Pl M r d 1, P = Λ 8πG N 2Λ d(d +1) r2
Identify H(S,P) = M H(S,P) = S R 4π d 1 ( 4l d Pl S Ω (d) gives thermodynamic volume V = (BPD [arxiv:10085023]) Equation of state ( ) H P S ) 1 d + 16πG N P d +1 = Ω (d)r d+1 h d +1 ( 4l d Pl S Ω (d) { T = ( ) (d +1)V 1 ( ) 1 } d+1 (d +1)V d+1 R +16πGN P 4πd Ω (d) Ω (d) ) 1 d,
Heat capacity 16πG N P C P = Sd 16πG N P ( ) 2 4l d Pl S d Ω (d) ( ) 2 4l d Pl S d Ω (d) For R > 0, phase transition at T = 2 R(d) G N P d π For flat event horizon, R = 0, C P = Sd +R R,