Breakdown of the semi-classical approximation in the path integral and black hole thermodynamics Based upon work with Bob McNees
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1 Breakdown of the semi-classical approximation in the path integral and black hole thermodynamics Based upon work with Bob McNees Daniel Grumiller Center for Theoretical Physics Massachusetts Institute of Technology University of Washington, June 2007 hep-th/
2 Outline Introduction Euclidean Path Integral Dilaton Gravity in 2D Free Energy Applications D. Grumiller Black Hole Thermodynamics 2/32
3 Outline Introduction Euclidean Path Integral Dilaton Gravity in 2D Free Energy Applications D. Grumiller Black Hole Thermodynamics Introduction 3/32
4 Black Hole Thermodynamics - Why? Black Hole Thermodynamics D. Grumiller Black Hole Thermodynamics Introduction 4/32
5 Black Hole Thermodynamics - Why? B-H: S = A 4G N, 1 st : de = T ds + work, 2 nd : ds 0 Classical General Relativity Four Laws (Bardeen, Carter, Hawking, 1973) Gedankenexperiments with entropy (Bekenstein, 1973) Black Hole Thermodynamics D. Grumiller Black Hole Thermodynamics Introduction 4/32
6 Black Hole Thermodynamics - Why? B-H: S = A 4G N, 1 st : de = T ds + work, 2 nd : ds 0 Classical General Relativity Four Laws (Bardeen, Carter, Hawking, 1973) Gedankenexperiments with entropy (Bekenstein, 1973) Black Hole Thermodynamics Quantum Gravity Semiclassical approximation? Microstate counting (Strominger, Vafa, 1996; Ashtekar, Corichi, Baez, Krasnov, 1997) D. Grumiller Black Hole Thermodynamics Introduction 4/32
7 Black Hole Thermodynamics - Why? B-H: S = A 4G N, 1 st : de = T ds + work, 2 nd : ds 0 Classical General Relativity Four Laws (Bardeen, Carter, Hawking, 1973) Gedankenexperiments with entropy (Bekenstein, 1973) Black Hole Analogues Sonic Black Holes (Unruh, 1981) Hawking effect in condensed matter? Black Hole Thermodynamics Quantum Gravity Semiclassical approximation? Microstate counting (Strominger, Vafa, 1996; Ashtekar, Corichi, Baez, Krasnov, 1997) D. Grumiller Black Hole Thermodynamics Introduction 4/32
8 Black Hole Thermodynamics - Why? B-H: S = A 4G N, 1 st : de = T ds + work, 2 nd : ds 0 Classical General Relativity Four Laws (Bardeen, Carter, Hawking, 1973) Gedankenexperiments with entropy (Bekenstein, 1973) Black Hole Analogues Sonic Black Holes (Unruh, 1981) Hawking effect in condensed matter? Black Hole Thermodynamics Quantum Gravity Semiclassical approximation? Microstate counting (Strominger, Vafa, 1996; Ashtekar, Corichi, Baez, Krasnov, 1997) Dual Formulations AdS/CFT (Maldacena 1997, Gubser, Klebanov, Polyakov 1998, Witten 1998) Hawking-Page transition D. Grumiller Black Hole Thermodynamics Introduction 4/32
9 Black Hole Thermodynamics - How? Many different approaches available... Approach: Advantage: Drawback: D. Grumiller Black Hole Thermodynamics Introduction 5/32
10 Black Hole Thermodynamics - How? Many different approaches available... Approach: Physical arguments Advantage: Very simple Drawback: ad-hoc! D. Grumiller Black Hole Thermodynamics Introduction 5/32
11 Black Hole Thermodynamics - How? Many different approaches available... Approach: Physical arguments QFT on fixed BG Advantage: Very simple Rigorous, plausible Drawback: ad-hoc! lengthy D. Grumiller Black Hole Thermodynamics Introduction 5/32
12 Black Hole Thermodynamics - How? Many different approaches available... Approach: Physical arguments QFT on fixed BG Conformal anomaly Advantage: Very simple Rigorous, plausible Rigorous, simple Drawback: ad-hoc! lengthy too special? D. Grumiller Black Hole Thermodynamics Introduction 5/32
13 Black Hole Thermodynamics - How? Many different approaches available... Approach: Physical arguments QFT on fixed BG Conformal anomaly Gravitational anomaly Advantage: Very simple Rigorous, plausible Rigorous, simple Plausible, simple Drawback: ad-hoc! lengthy too special? additional input? D. Grumiller Black Hole Thermodynamics Introduction 5/32
14 Black Hole Thermodynamics - How? Many different approaches available... Approach: Physical arguments QFT on fixed BG Conformal anomaly Gravitational anomaly Euclidean path integral Advantage: Very simple Rigorous, plausible Rigorous, simple Plausible, simple Very simple Drawback: ad-hoc! lengthy too special? additional input? physical? D. Grumiller Black Hole Thermodynamics Introduction 5/32
15 Black Hole Thermodynamics - How? Many different approaches available... Approach: Physical arguments QFT on fixed BG Conformal anomaly Gravitational anomaly Euclidean path integral Advantage: Very simple Rigorous, plausible Rigorous, simple Plausible, simple Very simple Drawback: ad-hoc! lengthy too special? additional input? physical? Employ Euclidean Path Integral Approach Not convincing first time -derivation of Hawking effect Convenient short-cut to obtain thermodynamical partition function Rather insensitive to matter coupling Useful insights about gravitational actions! D. Grumiller Black Hole Thermodynamics Introduction 5/32
16 Outline Introduction Euclidean Path Integral Dilaton Gravity in 2D Free Energy Applications D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 6/32
17 Main Idea Consider Euclidean path integral (Gibbons, Hawking, 1977) ( Z = DgDX exp 1 ) I E[g, X] D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 7/32
18 Main Idea Consider Euclidean path integral (Gibbons, Hawking, 1977) ( Z = DgDX exp 1 ) I E[g, X] g: metric, X: scalar field D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 7/32
19 Main Idea Consider Euclidean path integral (Gibbons, Hawking, 1977) ( Z = DgDX exp 1 ) I E[g, X] g: metric, X: scalar field Semiclassical limit ( 0): dominated by classical solutions (?) D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 7/32
20 Main Idea Consider Euclidean path integral (Gibbons, Hawking, 1977) ( Z = DgDX exp 1 ) I E[g, X] g: metric, X: scalar field Semiclassical limit ( 0): dominated by classical solutions (?) Exploit relationship between Z and Euclidean partition function Z e β Ω Ω: thermodynamic potential for appropriate ensemble β: periodicity in Euclidean time D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 7/32
21 Main Idea Consider Euclidean path integral (Gibbons, Hawking, 1977) ( Z = DgDX exp 1 ) I E[g, X] g: metric, X: scalar field Semiclassical limit ( 0): dominated by classical solutions (?) Exploit relationship between Z and Euclidean partition function Z e β Ω Ω: thermodynamic potential for appropriate ensemble β: periodicity in Euclidean time Requires periodicity in Euclidean time and accessibility of semi-classical approximation D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 7/32
22 Semiclassical Approximation Consider small perturbation around classical solution I E [g cl + δg, X cl + δx] =I E [g cl, X cl ] + δi E [g cl, X cl ; δg, δx] δ2 I E [g cl, X cl ; δg, δx] +... D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 8/32
23 Semiclassical Approximation Consider small perturbation around classical solution I E [g cl + δg, X cl + δx] =I E [g cl, X cl ] + δi E [g cl, X cl ; δg, δx] δ2 I E [g cl, X cl ; δg, δx] +... The leading term is the on-shell action. D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 8/32
24 Semiclassical Approximation Consider small perturbation around classical solution I E [g cl + δg, X cl + δx] =I E [g cl, X cl ] + δi E [g cl, X cl ; δg, δx] δ2 I E [g cl, X cl ; δg, δx] +... The leading term is the on-shell action. The linear term should vanish on solutions g cl and X cl. D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 8/32
25 Semiclassical Approximation Consider small perturbation around classical solution I E [g cl + δg, X cl + δx] =I E [g cl, X cl ] + δi E [g cl, X cl ; δg, δx] δ2 I E [g cl, X cl ; δg, δx] +... The leading term is the on-shell action. The linear term should vanish on solutions g cl and X cl. The quadratic term represents the first corrections. D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 8/32
26 Semiclassical Approximation Consider small perturbation around classical solution I E [g cl + δg, X cl + δx] =I E [g cl, X cl ] + δi E [g cl, X cl ; δg, δx] δ2 I E [g cl, X cl ; δg, δx] +... The leading term is the on-shell action. The linear term should vanish on solutions g cl and X cl. The quadratic term represents the first corrections. If nothing goes wrong: ( Z exp 1 ) I E[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 I E... D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 8/32
27 What could go Wrong? Accessibility of the semiclassical approximation requires D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 9/32
28 What could go Wrong? Accessibility of the semiclassical approximation requires 1. I E [g cl, X cl ] > Typical gravitational actions evaluated on black hole solutions: 1. Violated: Action unbounded from below D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 9/32
29 What could go Wrong? Accessibility of the semiclassical approximation requires 1. I E [g cl, X cl ] > 2. δi E [g cl, X cl ; δg, δx] = 0 Typical gravitational actions evaluated on black hole solutions: 1. Violated: Action unbounded from below 2. Violated: First variation of action not zero for all field configurations contributing to path integral due to boundary terms D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 9/32
30 What could go Wrong? Accessibility of the semiclassical approximation requires 1. I E [g cl, X cl ] > 2. δi E [g cl, X cl ; δg, δx] = 0 Typical gravitational actions evaluated on black hole solutions: 1. Violated: Action unbounded from below 2. Violated: First variation of action not zero for all field configurations contributing to path integral due to boundary terms δi EOM E dx [ ] γ π ab δγ ab + π X δx 0 M D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 9/32
31 What could go Wrong?...everything! Accessibility of the semiclassical approximation requires 1. I E [g cl, X cl ] > 2. δi E [g cl, X cl ; δg, δx] = 0 3. δ 2 I E [g cl, X cl ; δg, δx] 0 Typical gravitational actions evaluated on black hole solutions: 1. Violated: Action unbounded from below 2. Violated: First variation of action not zero for all field configurations contributing to path integral due to boundary terms δi EOM E dx [ ] γ π ab δγ ab + π X δx 0 M 3. Frequently violated: Gaussian integral may diverge D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 9/32
32 What could go Wrong?...everything! Accessibility of the semiclassical approximation requires 1. I E [g cl, X cl ] > 2. δi E [g cl, X cl ; δg, δx] = 0 3. δ 2 I E [g cl, X cl ; δg, δx] 0 Typical gravitational actions evaluated on black hole solutions: 1. Violated: Action unbounded from below 2. Violated: First variation of action not zero for all field configurations contributing to path integral due to boundary terms δi EOM E dx [ ] γ π ab δγ ab + π X δx 0 M 3. Frequently violated: Gaussian integral may diverge Focus in this talk on the second problem! D. Grumiller Black Hole Thermodynamics Euclidean Path Integral 9/32
33 Outline Introduction Euclidean Path Integral Dilaton Gravity in 2D Free Energy Applications D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 10/32
34 The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/ Standard form of the action: I E = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 11/32
35 The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/ Standard form of the action: I E = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M Dilaton X defined via coupling to Ricci scalar D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 11/32
36 The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/ Standard form of the action: I E = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M Dilaton X defined via coupling to Ricci scalar Model specified by kinetic and potential functions for dilaton D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 11/32
37 The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/ Standard form of the action: I E = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M dx γ X K M Dilaton X defined via coupling to Ricci scalar Model specified by kinetic and potential functions for dilaton Dilaton gravity analog of Gibbons-Hawking-York boundary term: coupling of X to extrinsic curvature of ( M, γ) Variational principle: fix X and induced metric γ at M D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 11/32
38 The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/ Standard form of the action: I E = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M dx γ X K dx γl(x) M M Dilaton X defined via coupling to Ricci scalar Model specified by kinetic and potential functions for dilaton Dilaton gravity analog of Gibbons-Hawking-York boundary term: coupling of X to extrinsic curvature of ( M, γ) Variational principle: fix X and induced metric γ at M Note: additional boundary term allowed consistent with classical solutions, variational principle and symmetries! D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 11/32
39 Selected List of Models Black holes in (A)dS, asymptotically flat or arbitrary spaces Model U(X) V (X) 1. Schwarzschild (1916) 1 λ 2 2X 2. Jackiw-Teitelboim (1984) 0 ΛX 3. Witten Black Hole (1991) 1 2b 2 X X 4. CGHS (1992) 0 2b 2 5. (A)dS 2 ground state (1994) a BX X 6. Rindler ground state (1996) a BX a X 7. Black Hole attractor (2003) 0 BX 1 8. Spherically reduced gravity (N > 3) N 3 λ 2 X (N 4)/(N 2) (N 2)X 9. All above: ab-family (1997) a BX a+b X 10. Liouville gravity a be αx 11. Reissner-Nordström (1916) 1 2X 12. Schwarzschild-(A)dS 1 2X λ 2 + Q2 X λ 2 lx 13. Katanaev-Volovich (1986) α βx 2 Λ Q 14. BTZ/Achucarro-Ortiz (1993) 0 2 J ΛX X 4X KK reduced CS (2003) 0 X(c 2 X2 ) 16. KK red. conf. flat (2006) 1 tanh (X/2) A sinh X D type 0A string Black Hole 1 2b 2 X + b2 q 2 X 8π D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 12/32
40 Selected List of Models Black holes in (A)dS, asymptotically flat or arbitrary spaces Model U(X) V (X) 1. Schwarzschild (1916) 1 λ 2 2X 2. Jackiw-Teitelboim (1984) 0 ΛX 3. Witten Black Hole (1991) 1 2b 2 X X 4. CGHS (1992) 0 2b 2 5. (A)dS 2 ground state (1994) a BX X 6. Rindler ground state (1996) a BX a X 7. Black Hole attractor (2003) 0 BX 1 8. Spherically reduced gravity (N > 3) N 3 λ 2 X (N 4)/(N 2) (N 2)X 9. All above: ab-family (1997) a BX a+b X 10. Liouville gravity a be αx 11. Reissner-Nordström (1916) 1 2X 12. Schwarzschild-(A)dS 1 2X λ 2 + Q2 X λ 2 lx 13. Katanaev-Volovich (1986) α βx 2 Λ Q 14. BTZ/Achucarro-Ortiz (1993) 0 2 J ΛX X 4X KK reduced CS (2003) 0 X(c 2 X2 ) 16. KK red. conf. flat (2006) 1 tanh (X/2) A sinh X D type 0A string Black Hole 1 2b 2 X + b2 q 2 X 8π D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 12/32
41 Equations of Motion (EOM) Extremize the action: δi E = 0 U(X) µx νx 1 2 gµνu(x)( X)2 g µνv (X) + µ νx g µν 2 X = 0 R + U(X) X ( X)2 + 2 U(X) 2 V (X) X 2 X = 0 D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
42 Equations of Motion (EOM) Extremize the action: δi E = 0 U(X) µx νx 1 2 gµνu(x)( X)2 g µνv (X) + µ νx g µν 2 X = 0 R + U(X) X ( X)2 + 2 U(X) 2 V (X) X 2 X = 0 Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)] D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
43 Equations of Motion (EOM) Extremize the action: δi E = 0 U(X) µx νx 1 2 gµνu(x)( X)2 g µνv (X) + µ νx g µν 2 X = 0 R + U(X) X ( X)2 + 2 U(X) 2 V (X) X 2 X = 0 Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)] Generalized Birkhoff theorem: at least one Killing vector D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
44 Equations of Motion (EOM) Extremize the action: δi E = 0 U(X) µx νx 1 2 gµνu(x)( X)2 g µνv (X) + µ νx g µν 2 X = 0 R + U(X) X ( X)2 + 2 U(X) 2 V (X) X 2 X = 0 Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)] Generalized Birkhoff theorem: at least one Killing vector Orbits of this vector are isosurfaces of the dilaton L k X = k µ µ X = 0 D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
45 Equations of Motion (EOM) Extremize the action: δi E = 0 U(X) µx νx 1 2 gµνu(x)( X)2 g µνv (X) + µ νx g µν 2 X = 0 R + U(X) X ( X)2 + 2 U(X) 2 V (X) X 2 X = 0 Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)] Generalized Birkhoff theorem: at least one Killing vector Orbits of this vector are isosurfaces of the dilaton L k X = k µ µ X = 0 Choose henceforth M as X = const. hypersurface D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
46 Equations of Motion (EOM) Extremize the action: δi E = 0 U(X) µx νx 1 2 gµνu(x)( X)2 g µνv (X) + µ νx g µν 2 X = 0 R + U(X) X ( X)2 + 2 U(X) 2 V (X) X 2 X = 0 Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)] Generalized Birkhoff theorem: at least one Killing vector Orbits of this vector are isosurfaces of the dilaton L k X = k µ µ X = 0 Choose henceforth M as X = const. hypersurface Adapted coordinate system (Lapse and Shift for radial evolution) X = X(r) ds 2 = N(r) 2 }{{} dr2 + ξ(r) (dτ + N τ (r) dr) 2 }{{}}{{} :=ξ(r) 1 =k µ k µ :=0 D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
47 Solutions Define two model-dependent functions Q(X) := Q 0 + X d X U( X) X w(x) := w 0 2 d X Q( X) V ( X)e D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
48 Solutions Define two model-dependent functions Q(X) := Q 0 + X d X U( X) X w(x) := w 0 2 d X Q( X) V ( X)e Q 0 and w 0 are arbitrary constants (essentially irrelevant) D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
49 Solutions Define two model-dependent functions Q(X) := Q 0 + X d X U( X) X w(x) := w 0 2 d X Q( X) V ( X)e Q 0 and w 0 are arbitrary constants (essentially irrelevant) Construct all classical solutions r X = e Q(X) ξ(x) = e Q(X) ( w(x) 2M ) D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
50 Solutions Define two model-dependent functions Q(X) := Q 0 + X d X U( X) X w(x) := w 0 2 d X Q( X) V ( X)e Q 0 and w 0 are arbitrary constants (essentially irrelevant) Construct all classical solutions r X = e Q(X) ξ(x) = e Q(X) ( w(x) 2M ) Constant of motion M ( mass ) characterizes classical solutions Absorb Q 0 into rescaling of length units Shift w 0 such that M = 0 ground state solution Restrict to positive mass sector M 0 D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
51 Black Holes Horizons Solutions with M 0 exhibit (Killing) horizons for each solution of w(x h ) = 2M D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
52 Black Holes Horizons Solutions with M 0 exhibit (Killing) horizons for each solution of Assumption 1 w(x h ) = 2M Killing norm 2 ξ(x) = e Q(X) ( w(x) 2M ) 0 on X h X < If there are multiple horizons we take the outermost one D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
53 Black Holes Horizons Solutions with M 0 exhibit (Killing) horizons for each solution of Assumption 1 w(x h ) = 2M Killing norm 2 ξ(x) = e Q(X) ( w(x) 2M ) 0 on X h X < If there are multiple horizons we take the outermost one Asymptotics X : asymptotic region of spacetime; most models: w(x) D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
54 Black Holes Horizons Solutions with M 0 exhibit (Killing) horizons for each solution of Assumption 1 w(x h ) = 2M Killing norm 2 ξ(x) = e Q(X) ( w(x) 2M ) 0 on X h X < If there are multiple horizons we take the outermost one Asymptotics X : asymptotic region of spacetime; most models: w(x) Assumption 2 Consider only models where w(x) as X Consequence: ξ(x) e Q w as X, i.e., ξ asymptotes to ground state D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
55 Black Hole Temperature Standard argument: absence of conical singularity requires periodicity in Euclidean time The g ττ component of the metric vanishes at the horizon X h Regularity of the metric requires τ τ + β with periodicity β = 4π r ξ = 4π rh w (X) Xh If ξ 1 at X : β 1 is temperature measured at infinity Denote inverse periodicity by T := β 1 = w (X) 4π Xh Proper local temperature related to β 1 by Tolman factor T (X) = 1 ξ(x) β 1 So far no action required but only a line-element D. Grumiller Black Hole Thermodynamics Dilaton Gravity in 2D 16/32
56 Outline Introduction Euclidean Path Integral Dilaton Gravity in 2D Free Energy Applications D. Grumiller Black Hole Thermodynamics Free Energy 17/32
57 Free Energy? Given the black hole solution, can we calculate the free energy? ( Z exp 1 ) I E[g cl, X cl ] e β F D. Grumiller Black Hole Thermodynamics Free Energy 18/32
58 Free Energy? Not yet! Given the black hole solution, can we calculate the free energy? ( Z exp 1 ) I E[g cl, X cl ] e β F Need a limiting procedure to calculate the action. Implement this in a coordinate-independent way by putting a regulator on the dilaton. X X reg Evaluating the on-shell action leads to three problems D. Grumiller Black Hole Thermodynamics Free Energy 18/32
59 Free Energy? Not yet! Given the black hole solution, can we calculate the free energy? ( Z exp 1 ) I E[g cl, X cl ] e β F Need a limiting procedure to calculate the action. Implement this in a coordinate-independent way by putting a regulator on the dilaton. X X reg Evaluating the on-shell action leads to three problems 1. On-shell action unbounded from below (cf. second assumption) I reg E = β ( 2 M w(x reg ) 2π X h T ) D. Grumiller Black Hole Thermodynamics Free Energy 18/32
60 Free Energy? Not yet! Given the black hole solution, can we calculate the free energy? ( Z exp 1 ) I E[g cl, X cl ] e β F Need a limiting procedure to calculate the action. Implement this in a coordinate-independent way by putting a regulator on the dilaton. X X reg Evaluating the on-shell action leads to three problems 1. On-shell action unbounded from below (cf. second assumption) I reg E = β ( 2 M w(x reg ) 2π X h T ) 2. First variation of action not zero for all field configurations contributing to path integral due to boundary terms D. Grumiller Black Hole Thermodynamics Free Energy 18/32
61 Free Energy? Not yet! Given the black hole solution, can we calculate the free energy? ( Z exp 1 ) I E[g cl, X cl ] e β F Need a limiting procedure to calculate the action. Implement this in a coordinate-independent way by putting a regulator on the dilaton. X X reg Evaluating the on-shell action leads to three problems 1. On-shell action unbounded from below (cf. second assumption) I reg E = β ( 2 M w(x reg ) 2π X h T ) 2. First variation of action not zero for all field configurations contributing to path integral due to boundary terms 3. Second variation of action may lead to divergent Gaussian integral D. Grumiller Black Hole Thermodynamics Free Energy 18/32
62 Variational Properties of the Action Z δi E = d 2 x h i Z g E µν δg µν + E XδX + dx h i γ π ab δγ ab + π XδX M {z } M {z } =0(EOM) =0? D. Grumiller Black Hole Thermodynamics Free Energy 19/32
63 Variational Properties of the Action Z δi E = d 2 x h i Z g E µν δg µν + E XδX + dx h i γ π ab δγ ab + π XδX M {z } M {z } =0(EOM) =0? Does this vanish on-shell? Ignore π X δx and focus on π ab δγ ab Z δi E = dτ» 12 rx δξ +... D. Grumiller Black Hole Thermodynamics Free Energy 19/32
64 Variational Properties of the Action Z δi E = d 2 x h i Z g E µν δg µν + E XδX + dx h i γ π ab δγ ab + π XδX M {z } M {z } =0(EOM) =0? Does this vanish on-shell? Ignore π X δx and focus on π ab δγ ab Z δi E = dτ» 12 rx δξ +... Recall ξ(x) = w(x)e Q(X) 2Me Q(X) D. Grumiller Black Hole Thermodynamics Free Energy 19/32
65 Variational Properties of the Action Z δi E = d 2 x h i Z g E µν δg µν + E XδX + dx h i γ π ab δγ ab + π XδX M {z } M {z } =0(EOM) =0? Does this vanish on-shell? Ignore π X δx and focus on π ab δγ ab Z δi E = dτ» 12 rx δξ +... Recall ξ(x) = w(x)e Q(X) 2Me Q(X) Assume that boundary conditions preserved by variations δξ δm e Q(X) D. Grumiller Black Hole Thermodynamics Free Energy 19/32
66 Variational Properties of the Action Z δi E = d 2 x h i Z g E µν δg µν + E XδX + dx h i γ π ab δγ ab + π XδX M {z } M {z } =0(EOM) 0 Does this vanish on-shell? Ignore π X δx and focus on π ab δγ ab Z δi E = dτ» 12 rx δξ +... Recall ξ(x) = w(x)e Q(X) 2Me Q(X) Assume that boundary conditions preserved by variations δξ δm e Q(X) Recalling r X = e Q we get δi E = dτδm 0 D. Grumiller Black Hole Thermodynamics Free Energy 19/32
67 Boundary Counterterms Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999; Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998) More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann, Marolf 2006) Covariant version of surface terms in gravity (ADM 1962; Regge, Teitelboim 1974) D. Grumiller Black Hole Thermodynamics Free Energy 20/32
68 Boundary Counterterms Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999; Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998) More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann, Marolf 2006) Covariant version of surface terms in gravity (ADM 1962; Regge, Teitelboim 1974) Black Holes in 2D: I E = Γ + I CT 1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/ ) 2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/ ) D. Grumiller Black Hole Thermodynamics Free Energy 20/32
69 Boundary Counterterms Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999; Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998) More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann, Marolf 2006) Covariant version of surface terms in gravity (ADM 1962; Regge, Teitelboim 1974) Black Holes in 2D: I E = Γ + I CT 1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/ ) 2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/ ) Γ = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M dx γ X K dx γl(x) M M }{{} I CT D. Grumiller Black Hole Thermodynamics Free Energy 20/32
70 Boundary Counterterms Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999; Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998) More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann, Marolf 2006) Covariant version of surface terms in gravity (ADM 1962; Regge, Teitelboim 1974) Black Holes in 2D: I E = Γ + I CT 1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/ ) 2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/ ) Γ = 1 d 2 x g [ X R U(X) ( X) 2 2 V (X) ] 2 M dx γ X K dx γl(x) M M }{{} I CT How to determine the boundary counterterm? D. Grumiller Black Hole Thermodynamics Free Energy 20/32
71 Hamilton-Jacobi Equation Boundary counterterm I CT is solution of the Hamilton-Jacobi equation D. Grumiller Black Hole Thermodynamics Free Energy 21/32
72 Hamilton-Jacobi Equation Boundary counterterm I CT is solution of the Hamilton-Jacobi equation 1. Begin with Hamiltonian associated with radial evolution. ( H = 2 π X γ ab π ab + 2 U(X) γ ab π ab) 2 + V (X) = 0 D. Grumiller Black Hole Thermodynamics Free Energy 21/32
73 Hamilton-Jacobi Equation Boundary counterterm I CT is solution of the Hamilton-Jacobi equation 1. Begin with Hamiltonian associated with radial evolution. ( H = 2 π X γ ab π ab + 2 U(X) γ ab π ab) 2 + V (X) = 0 2. Momenta are functional derivatives of the on-shell action π ab = 1 γ δ EOM I E π δ γ X = 1 ab γ δ δ X I E EOM D. Grumiller Black Hole Thermodynamics Free Energy 21/32
74 Hamilton-Jacobi Equation Boundary counterterm I CT is solution of the Hamilton-Jacobi equation 1. Begin with Hamiltonian associated with radial evolution. ( H = 2 π X γ ab π ab + 2 U(X) γ ab π ab) 2 + V (X) = 0 2. Momenta are functional derivatives of the on-shell action π ab = 1 γ δ EOM I E π δ γ X = 1 ab γ δ δ X I E EOM 3. Obtain non-linear functional differential equation for on-shell action D. Grumiller Black Hole Thermodynamics Free Energy 21/32
75 Hamilton-Jacobi Equation Boundary counterterm I CT is solution of the Hamilton-Jacobi equation 1. Begin with Hamiltonian associated with radial evolution. ( H = 2 π X γ ab π ab + 2 U(X) γ ab π ab) 2 + V (X) = 0 2. Momenta are functional derivatives of the on-shell action π ab = 1 γ δ EOM I E π δ γ X = 1 ab γ δ δ X I E EOM 3. Obtain non-linear functional differential equation for on-shell action 4. 2D: simplifies to first order ODE can solve (essentially uniquely) for I CT! D. Grumiller Black Hole Thermodynamics Free Energy 21/32
76 Hamilton-Jacobi Equation Boundary counterterm I CT is solution of the Hamilton-Jacobi equation 1. Begin with Hamiltonian associated with radial evolution. ( H = 2 π X γ ab π ab + 2 U(X) γ ab π ab) 2 + V (X) = 0 2. Momenta are functional derivatives of the on-shell action π ab = 1 γ δ EOM I E π δ γ X = 1 ab γ δ δ X I E EOM 3. Obtain non-linear functional differential equation for on-shell action 4. 2D: simplifies to first order ODE can solve (essentially uniquely) for I CT! I CT = dx γ w(x) e Q(X) M D. Grumiller Black Hole Thermodynamics Free Energy 21/32
77 The Improved Action The correct action for 2D dilaton gravity is Γ = 1 d 2 x g [ XR U(X) ( X) 2 2 V (X) ] 2 M dx γ X K + dx γ w(x) e Q(X) Properties: M M D. Grumiller Black Hole Thermodynamics Free Energy 22/32
78 The Improved Action The correct action for 2D dilaton gravity is Γ = 1 d 2 x g [ XR U(X) ( X) 2 2 V (X) ] 2 M dx γ X K + dx γ w(x) e Q(X) M Properties: 1. Yields the same EOM as I E M D. Grumiller Black Hole Thermodynamics Free Energy 22/32
79 The Improved Action The correct action for 2D dilaton gravity is Γ = 1 d 2 x g [ XR U(X) ( X) 2 2 V (X) ] 2 M dx γ X K + dx γ w(x) e Q(X) M M Properties: 1. Yields the same EOM as I E 2. Finite on-shell (solves first problem) Γ EOM = β (M 2πX h T ) D. Grumiller Black Hole Thermodynamics Free Energy 22/32
80 The Improved Action The correct action for 2D dilaton gravity is Γ = 1 d 2 x g [ XR U(X) ( X) 2 2 V (X) ] 2 M dx γ X K + dx γ w(x) e Q(X) M M Properties: 1. Yields the same EOM as I E 2. Finite on-shell (solves first problem) Γ EOM = β (M 2πX h T ) 3. First variation δγ vanishes on-shell δg µν and δx that preserve the boundary conditions (solves second problem) δγ EOM = 0 Note: counterterm requires specification of integration constant w 0, i.e., choice of ground state, but is independent from Q 0 D. Grumiller Black Hole Thermodynamics Free Energy 22/32
81 Reconsider Semiclassical Approximation ( Z exp 1 ) Γ[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 Γ... D. Grumiller Black Hole Thermodynamics Free Energy 23/32
82 Reconsider Semiclassical Approximation ( Z exp 1 ) Γ[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 Γ... Leading term is finite D. Grumiller Black Hole Thermodynamics Free Energy 23/32
83 Reconsider Semiclassical Approximation ( Z exp 1 ) Γ[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 Γ... Leading term is finite Linear term vanishes D. Grumiller Black Hole Thermodynamics Free Energy 23/32
84 Reconsider Semiclassical Approximation ( Z exp 1 ) Γ[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 Γ... Leading term is finite Linear term vanishes Still have to worry about quadratic term D. Grumiller Black Hole Thermodynamics Free Energy 23/32
85 Reconsider Semiclassical Approximation ( Z exp 1 ) Γ[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 Γ... Leading term is finite Linear term vanishes Still have to worry about quadratic term Solved by putting the Black Hole in a box (York, 1986; Gibbons, Perry, 1992) Cavity wall determined by X = X c Well-defined canonical ensemble by specifying X c and T c = 1/β c D. Grumiller Black Hole Thermodynamics Free Energy 23/32
86 Reconsider Semiclassical Approximation ( Z exp 1 ) Γ[g cl, X cl ] ( Dδg DδX exp 1 ) 2 δ2 Γ... Leading term is finite Linear term vanishes Still have to worry about quadratic term Solved by putting the Black Hole in a box (York, 1986; Gibbons, Perry, 1992) Cavity wall determined by X = X c Well-defined canonical ensemble by specifying X c and T c = 1/β c Leading order (set = 1): Z(T c, X c ) = e Γ(Tc,Xc) = e βcfc(tc,xc) Here F c is the Helmholtz free energy D. Grumiller Black Hole Thermodynamics Free Energy 23/32
87 Free Energy Γ(T c, X c ) = β c F c (T c, X c ) D. Grumiller Black Hole Thermodynamics Free Energy 24/32
88 Free Energy Γ(T c, X c ) = β c F c (T c, X c ) Explicitly: F c (T c, X c ) = ) w c e (1 Qc 1 2Mwc 2πX h T c D. Grumiller Black Hole Thermodynamics Free Energy 24/32
89 Free Energy Explicitly: Γ(T c, X c ) = β c F c (T c, X c ) F c (T c, X c ) = w c e Qc (1 1 2Mwc ) } {{ } =E c(t c,x c) Entropy follows immediately (Bekenstein-Hawking law): S = F c T c = 2πX h Xc 2πX h T c }{{} =ST c Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter, Louis-Martinez, 1995) D. Grumiller Black Hole Thermodynamics Free Energy 24/32
90 Free Energy Explicitly: Γ(T c, X c ) = β c F c (T c, X c ) F c (T c, X c ) = w c e Qc (1 1 2Mwc ) } {{ } =E c(t c,x c) Entropy follows immediately (Bekenstein-Hawking law): S = F c T c = 2πX h Xc 2πX h T c }{{} =ST c Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter, Louis-Martinez, 1995) Similarly: dilaton chemical potential (surface pressure) ψ c = F c / X c Tc D. Grumiller Black Hole Thermodynamics Free Energy 24/32
91 Other Thermodynamical Quantities Standard thermodynamics in canonical ensemble: internal energy, enthalpy, free enthalpy, specific heats, isothermal compressibility,... D. Grumiller Black Hole Thermodynamics Free Energy 25/32
92 Other Thermodynamical Quantities Standard thermodynamics in canonical ensemble: internal energy, enthalpy, free enthalpy, specific heats, isothermal compressibility, Internal energy E c = F c + T c S = e Qc ( ξ g c ξ c ) 0 Models with Minkowski ground state (ξ g c = 1): M = E c E2 c 2 w c D. Grumiller Black Hole Thermodynamics Free Energy 25/32
93 Other Thermodynamical Quantities Standard thermodynamics in canonical ensemble: internal energy, enthalpy, free enthalpy, specific heats, isothermal compressibility, Internal energy E c = F c + T c S = e Qc ( ξ g c ξ c ) 0 Models with Minkowski ground state (ξ g c = 1): M = E c E2 c 2 w c 2. First law de c = T c ds ψ c dx c Properly accounts for non-linear effects of gravitational binding energy D. Grumiller Black Hole Thermodynamics Free Energy 25/32
94 Other Thermodynamical Quantities Standard thermodynamics in canonical ensemble: internal energy, enthalpy, free enthalpy, specific heats, isothermal compressibility, Internal energy E c = F c + T c S = e Qc ( ξ g c ξ c ) 0 Models with Minkowski ground state (ξ g c = 1): M = E c E2 c 2 w c 2. First law de c = T c ds ψ c dx c Properly accounts for non-linear effects of gravitational binding energy 3. Specific heat at constant dilaton charge X c C D = 2π w h w h (w h )2 2w h (wc 2M) Allows to check for thermodynamic stability: C D (X c = X h + ε) > 0 D. Grumiller Black Hole Thermodynamics Free Energy 25/32
95 Outline Introduction Euclidean Path Integral Dilaton Gravity in 2D Free Energy Applications D. Grumiller Black Hole Thermodynamics Applications 26/32
96 Higher Dimensional Black Holes Schwarzschild, Reissner-Nordström, BTZ, Schwarzschild-AdS,... EH d+1 DG 2 spherical reduction D. Grumiller Black Hole Thermodynamics Applications 27/32
97 Higher Dimensional Black Holes Schwarzschild, Reissner-Nordström, BTZ, Schwarzschild-AdS,... EH d+1 bound. EH d+1 + GHY d spherical reduction spherical reduction bound. DG 2 DG 2 + GHY 1 D. Grumiller Black Hole Thermodynamics Applications 27/32
98 Higher Dimensional Black Holes Schwarzschild, Reissner-Nordström, BTZ, Schwarzschild-AdS,... EH d+1 bound. EH d+1 + GHY d? EH d+1 + GHY d + HJ d spherical reduction spherical reduction? bound. DG 2 DG 2 + GHY 1! DG 2 + GHY 1 + HJ 1 D. Grumiller Black Hole Thermodynamics Applications 27/32
99 Higher Dimensional Black Holes Schwarzschild, Reissner-Nordström, BTZ, Schwarzschild-AdS,... EH d+1 bound. EH d+1 + GHY d? EH d+1 + GHY d + HJ d spherical reduction spherical reduction? bound. DG 2 DG 2 + GHY! 1 DG 2 + GHY 1 + HJ 1 Main message It works, regardless of the asymptotics... But nearly no info about HJ d! D. Grumiller Black Hole Thermodynamics Applications 27/32
100 Higher Dimensional Black Holes Schwarzschild, Reissner-Nordström, BTZ, Schwarzschild-AdS,... EH d+1 bound. EH d+1 + GHY d? EH d+1 + GHY d + HJ d spherical reduction spherical reduction? bound. DG 2 DG 2 + GHY! 1 DG 2 + GHY 1 + HJ 1 Main message It works, regardless of the asymptotics... But nearly no info about HJ d! Example: Schwarzschild-AdS in d + 1 dimensions: U(X) = ( ) d 2 1 d 1 X, d 3 d(d 1) V (X) = (const.)x d 1 2 l 2 X D. Grumiller Black Hole Thermodynamics Applications 27/32
101 Hawking-Page Transition Spherically symmetric AdS Black Holes in d + 1 dimensions C D : specific heat at constant dilaton; r h : horizon radius; l: AdS radius D. Grumiller Black Hole Thermodynamics Applications 28/32
102 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) D. Grumiller Black Hole Thermodynamics Applications 29/32
103 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) Large level k: Witten Black Hole (U = 1/X, V X) Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992) D. Grumiller Black Hole Thermodynamics Applications 29/32
104 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) Large level k: Witten Black Hole (U = 1/X, V X) Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992) Adding D-branes: 2D type 0A strings Dropping (Coulomb-) divergences is wrong! (Davies, McNees 2004) D. Grumiller Black Hole Thermodynamics Applications 29/32
105 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) Large level k: Witten Black Hole (U = 1/X, V X) Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992) Adding D-branes: 2D type 0A strings Dropping (Coulomb-) divergences is wrong! (Davies, McNees 2004) Problem: have to move the cavity to infinity in string theory D. Grumiller Black Hole Thermodynamics Applications 29/32
106 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) Large level k: Witten Black Hole (U = 1/X, V X) Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992) Adding D-branes: 2D type 0A strings Dropping (Coulomb-) divergences is wrong! (Davies, McNees 2004) Problem: have to move the cavity to infinity in string theory Witten Black Hole: cannot remove cavity! (specific heat diverges) D. Grumiller Black Hole Thermodynamics Applications 29/32
107 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) Large level k: Witten Black Hole (U = 1/X, V X) Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992) Adding D-branes: 2D type 0A strings Dropping (Coulomb-) divergences is wrong! (Davies, McNees 2004) Problem: have to move the cavity to infinity in string theory Witten Black Hole: cannot remove cavity! (specific heat diverges) Need finite k corrections (α corrections): exact string Black Hole (Dijkgraaf, Verlinde, Verlinde, 1992) D. Grumiller Black Hole Thermodynamics Applications 29/32
108 Black Holes in 2D String Theory Black holes with exact CFT description (SL(2, R)/U(1) gauged WZW model) (Witten 1991) Large level k: Witten Black Hole (U = 1/X, V X) Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992) Adding D-branes: 2D type 0A strings Dropping (Coulomb-) divergences is wrong! (Davies, McNees 2004) Problem: have to move the cavity to infinity in string theory Witten Black Hole: cannot remove cavity! (specific heat diverges) Need finite k corrections (α corrections): exact string Black Hole (Dijkgraaf, Verlinde, Verlinde, 1992) Exact string Black Hole allows removal of cavity! String theory is its own reservoir D. Grumiller Black Hole Thermodynamics Applications 29/32
109 Thermodynamics of the Exact String Black Hole Interesting geometry: asymptotically flat, one Killing horizon, no singularity (dilaton violates energy conditions) Singular limits: k : singularity appears (Witten Black Hole) k 2: horizon disappears (Jackiw-Teitelboim, AdS 2 ) Thermodynamical properties D. Grumiller Black Hole Thermodynamics Applications 30/32
110 Thermodynamics of the Exact String Black Hole Interesting geometry: asymptotically flat, one Killing horizon, no singularity (dilaton violates energy conditions) Singular limits: k : singularity appears (Witten Black Hole) k 2: horizon disappears (Jackiw-Teitelboim, AdS 2 ) Thermodynamical properties 1. Positive specific heat C D = # k 2 T (like degenerate Fermi gas) D. Grumiller Black Hole Thermodynamics Applications 30/32
111 Thermodynamics of the Exact String Black Hole Interesting geometry: asymptotically flat, one Killing horizon, no singularity (dilaton violates energy conditions) Singular limits: k : singularity appears (Witten Black Hole) k 2: horizon disappears (Jackiw-Teitelboim, AdS 2 ) Thermodynamical properties 1. Positive specific heat C D = # k 2 T (like degenerate Fermi gas) 2. Hawking temperature T = T H 1 2 k (T H: Hagedorn temperature) D. Grumiller Black Hole Thermodynamics Applications 30/32
112 Thermodynamics of the Exact String Black Hole Interesting geometry: asymptotically flat, one Killing horizon, no singularity (dilaton violates energy conditions) Singular limits: k : singularity appears (Witten Black Hole) k 2: horizon disappears (Jackiw-Teitelboim, AdS 2 ) Thermodynamical properties 1. Positive specific heat C D = # k 2 T (like degenerate Fermi gas) 2. Hawking temperature T = T H 1 2 k (T H: Hagedorn temperature) 3. Logarithmic α corrections to entropy (DG 2005) ( ) S = 2π k(k 2) + arcsinh ( k(k 2)) = 2πk + 2π ln k +... D. Grumiller Black Hole Thermodynamics Applications 30/32
113 Thermodynamics of the Exact String Black Hole Interesting geometry: asymptotically flat, one Killing horizon, no singularity (dilaton violates energy conditions) Singular limits: k : singularity appears (Witten Black Hole) k 2: horizon disappears (Jackiw-Teitelboim, AdS 2 ) Thermodynamical properties 1. Positive specific heat C D = # k 2 T (like degenerate Fermi gas) 2. Hawking temperature T = T H 1 2 k (T H: Hagedorn temperature) 3. Logarithmic α corrections to entropy (DG 2005) ( ) S = 2π k(k 2) + arcsinh ( k(k 2)) = 2πk + 2π ln k Partition function for critical value k = 9/4 (setting G N = 1/4) Z = e arcsinh (3/4) = 2 Relations: α b 2 = 1/(k 2), Dim α b 2 = 0 D. Grumiller Black Hole Thermodynamics Applications 30/32
114 Conclusions...for more info see DG, R. McNees, hep-th/ Main results presented: D. Grumiller Black Hole Thermodynamics Applications 31/32
115 Conclusions...for more info see DG, R. McNees, hep-th/ Main results presented: Constructed Hamilton-Jacobi counterterm for generic 2D dilaton gravity (with two working assumptions!) D. Grumiller Black Hole Thermodynamics Applications 31/32
116 Conclusions...for more info see DG, R. McNees, hep-th/ Main results presented: Constructed Hamilton-Jacobi counterterm for generic 2D dilaton gravity (with two working assumptions!) Derived free energy and its thermodynamic descendants (entropy, internal energy, specific heat,...) D. Grumiller Black Hole Thermodynamics Applications 31/32
117 Conclusions...for more info see DG, R. McNees, hep-th/ Main results presented: Constructed Hamilton-Jacobi counterterm for generic 2D dilaton gravity (with two working assumptions!) Derived free energy and its thermodynamic descendants (entropy, internal energy, specific heat,...) Applied it to numerous black holes in various dimensions D. Grumiller Black Hole Thermodynamics Applications 31/32
118 Conclusions...for more info see DG, R. McNees, hep-th/ Main results presented: Constructed Hamilton-Jacobi counterterm for generic 2D dilaton gravity (with two working assumptions!) Derived free energy and its thermodynamic descendants (entropy, internal energy, specific heat,...) Applied it to numerous black holes in various dimensions Main results not presented: Extensitivity and scaling properties Nonperturbative stability analysis (tunneling) Inclusion of Maxwell fields (charge, spin,...) D. Grumiller Black Hole Thermodynamics Applications 31/32
119 Conclusions...for more info see DG, R. McNees, hep-th/ Main results presented: Constructed Hamilton-Jacobi counterterm for generic 2D dilaton gravity (with two working assumptions!) Derived free energy and its thermodynamic descendants (entropy, internal energy, specific heat,...) Applied it to numerous black holes in various dimensions Main results not presented: Extensitivity and scaling properties Nonperturbative stability analysis (tunneling) Inclusion of Maxwell fields (charge, spin,...) Next steps envisaged: Relax working assumptions (ds!) Consider matter fields (reconsider counterterm!) Impact on quantum theory? D. Grumiller Black Hole Thermodynamics Applications 31/32
120 Thanks for the attention......and thanks to Bob McNees for the style and source files of his talk! D. Grumiller Black Hole Thermodynamics Applications 32/32
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