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Black Hole Thermodynamics
and Hamilton-Jacobi CountertermBased upon work with Bob McNees
Daniel Grumiller
Center for Theoretical PhysicsMassachusetts Institute of Technology
Penn State University, May 2007
hep-th/0703230
Outline
Introduction
Euclidean Path Integral
Dilaton Gravity in 2D
Free Energy
Applications
D. Grumiller — Black Hole Thermodynamics 2/32
Outline
Introduction
Euclidean Path Integral
Dilaton Gravity in 2D
Free Energy
Applications
D. Grumiller — Black Hole Thermodynamics Introduction 3/32
Black Hole Thermodynamics - Why?
B-H: S = A4GN
, 1st: dE = T dS + work, 2nd: dS ≥ 0
Classical General Relativity
I Four Laws (Bardeen, Carter, Hawking, 1973)
I Gedankenexperiments with entropy(Bekenstein, 1973)
Black Hole Analogues
I Sonic Black Holes (Unruh, 1981)
I Hawking effect in condensed matter?
Black Hole Thermodynamics
Quantum Gravity
I Semiclassical approximation?
I Microstate counting (Strominger, Vafa, 1996;
Ashtekar, Corichi, Baez, Krasnov, 1997)
Dual Formulations
I AdS/CFT (Maldacena 1997, Gubser, Klebanov,
Polyakov 1998, Witten 1998)
I Hawking-Page transition
D. Grumiller — Black Hole Thermodynamics Introduction 4/32
Black Hole Thermodynamics - Why?
B-H: S = A4GN
, 1st: dE = T dS + work, 2nd: dS ≥ 0
Classical General Relativity
I Four Laws (Bardeen, Carter, Hawking, 1973)
I Gedankenexperiments with entropy(Bekenstein, 1973)
Black Hole Analogues
I Sonic Black Holes (Unruh, 1981)
I Hawking effect in condensed matter?
Black Hole Thermodynamics
Quantum Gravity
I Semiclassical approximation?
I Microstate counting (Strominger, Vafa, 1996;
Ashtekar, Corichi, Baez, Krasnov, 1997)
Dual Formulations
I AdS/CFT (Maldacena 1997, Gubser, Klebanov,
Polyakov 1998, Witten 1998)
I Hawking-Page transition
D. Grumiller — Black Hole Thermodynamics Introduction 4/32
Black Hole Thermodynamics - Why?
B-H: S = A4GN
, 1st: dE = T dS + work, 2nd: dS ≥ 0
Classical General Relativity
I Four Laws (Bardeen, Carter, Hawking, 1973)
I Gedankenexperiments with entropy(Bekenstein, 1973)
Black Hole Analogues
I Sonic Black Holes (Unruh, 1981)
I Hawking effect in condensed matter?
Black Hole Thermodynamics
Quantum Gravity
I Semiclassical approximation?
I Microstate counting (Strominger, Vafa, 1996;
Ashtekar, Corichi, Baez, Krasnov, 1997)
Dual Formulations
I AdS/CFT (Maldacena 1997, Gubser, Klebanov,
Polyakov 1998, Witten 1998)
I Hawking-Page transition
D. Grumiller — Black Hole Thermodynamics Introduction 4/32
Black Hole Thermodynamics - Why?
B-H: S = A4GN
, 1st: dE = T dS + work, 2nd: dS ≥ 0
Classical General Relativity
I Four Laws (Bardeen, Carter, Hawking, 1973)
I Gedankenexperiments with entropy(Bekenstein, 1973)
Black Hole Analogues
I Sonic Black Holes (Unruh, 1981)
I Hawking effect in condensed matter?
Black Hole Thermodynamics
Quantum Gravity
I Semiclassical approximation?
I Microstate counting (Strominger, Vafa, 1996;
Ashtekar, Corichi, Baez, Krasnov, 1997)
Dual Formulations
I AdS/CFT (Maldacena 1997, Gubser, Klebanov,
Polyakov 1998, Witten 1998)
I Hawking-Page transition
D. Grumiller — Black Hole Thermodynamics Introduction 4/32
Black Hole Thermodynamics - Why?
B-H: S = A4GN
, 1st: dE = T dS + work, 2nd: dS ≥ 0
Classical General Relativity
I Four Laws (Bardeen, Carter, Hawking, 1973)
I Gedankenexperiments with entropy(Bekenstein, 1973)
Black Hole Analogues
I Sonic Black Holes (Unruh, 1981)
I Hawking effect in condensed matter?
Black Hole Thermodynamics
Quantum Gravity
I Semiclassical approximation?
I Microstate counting (Strominger, Vafa, 1996;
Ashtekar, Corichi, Baez, Krasnov, 1997)
Dual Formulations
I AdS/CFT (Maldacena 1997, Gubser, Klebanov,
Polyakov 1998, Witten 1998)
I Hawking-Page transition
D. Grumiller — Black Hole Thermodynamics Introduction 4/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Black Hole Thermodynamics - How?Many different approaches available...
Approach:
I Physical arguments
I QFT on fixed BG
I Conformal anomaly
I Gravitational anomaly
I Euclidean path integral
Advantage:
I Very simple
I Rigorous, plausible
I Rigorous, simple
I Plausible, simple
I Very simple
Drawback:
I ad-hoc!
I lengthy
I too special?
I additional input?
I physical?
Employ Euclidean Path Integral Approach
I Not convincing “first time”-derivation of Hawking effect
I Convenient short-cut to obtain thermodynamical partition function
I Rather insensitive to matter coupling
I Useful insights about gravitational actions!
D. Grumiller — Black Hole Thermodynamics Introduction 5/32
Outline
Introduction
Euclidean Path Integral
Dilaton Gravity in 2D
Free Energy
Applications
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 6/32
Main Idea
Consider Euclidean path integral (Gibbons, Hawking, 1977)
Z =∫
DgDX exp(−1
~IE [g,X]
)
I g: metric, X: scalar field
I Semiclassical limit (~ → 0): dominated by classical solutions (?)
I Exploit relationship between Z and Euclidean partition function
Z ∼ e−β Ω
I Ω: thermodynamic potential for appropriate ensemble
I β: periodicity in Euclidean time
Requires periodicity in Euclidean time andaccessibility of semi-classical approximation
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 7/32
Main Idea
Consider Euclidean path integral (Gibbons, Hawking, 1977)
Z =∫
DgDX exp(−1
~IE [g,X]
)
I g: metric, X: scalar field
I Semiclassical limit (~ → 0): dominated by classical solutions (?)
I Exploit relationship between Z and Euclidean partition function
Z ∼ e−β Ω
I Ω: thermodynamic potential for appropriate ensemble
I β: periodicity in Euclidean time
Requires periodicity in Euclidean time andaccessibility of semi-classical approximation
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 7/32
Main Idea
Consider Euclidean path integral (Gibbons, Hawking, 1977)
Z =∫
DgDX exp(−1
~IE [g,X]
)
I g: metric, X: scalar field
I Semiclassical limit (~ → 0): dominated by classical solutions (?)
I Exploit relationship between Z and Euclidean partition function
Z ∼ e−β Ω
I Ω: thermodynamic potential for appropriate ensemble
I β: periodicity in Euclidean time
Requires periodicity in Euclidean time andaccessibility of semi-classical approximation
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 7/32
Main Idea
Consider Euclidean path integral (Gibbons, Hawking, 1977)
Z =∫
DgDX exp(−1
~IE [g,X]
)
I g: metric, X: scalar field
I Semiclassical limit (~ → 0): dominated by classical solutions (?)
I Exploit relationship between Z and Euclidean partition function
Z ∼ e−β Ω
I Ω: thermodynamic potential for appropriate ensemble
I β: periodicity in Euclidean time
Requires periodicity in Euclidean time andaccessibility of semi-classical approximation
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 7/32
Main Idea
Consider Euclidean path integral (Gibbons, Hawking, 1977)
Z =∫
DgDX exp(−1
~IE [g,X]
)
I g: metric, X: scalar field
I Semiclassical limit (~ → 0): dominated by classical solutions (?)
I Exploit relationship between Z and Euclidean partition function
Z ∼ e−β Ω
I Ω: thermodynamic potential for appropriate ensemble
I β: periodicity in Euclidean time
Requires periodicity in Euclidean time andaccessibility of semi-classical approximation
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 7/32
Semiclassical Approximation
Consider small perturbation around classical solution
IE [gcl + δg,Xcl + δX] =IE [gcl, Xcl] + δIE [gcl, Xcl; δg, δX]
+12δ2IE [gcl, Xcl; δg, δX] + . . .
I The leading term is the ‘on-shell’ action.
I The linear term should vanish on solutions gcl and Xcl.
I The quadratic term represents the first corrections.
If nothing goes wrong:
Z ∼ exp(−1
~IE [gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2IE
)× . . .
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 8/32
Semiclassical Approximation
Consider small perturbation around classical solution
IE [gcl + δg,Xcl + δX] =IE [gcl, Xcl] + δIE [gcl, Xcl; δg, δX]
+12δ2IE [gcl, Xcl; δg, δX] + . . .
I The leading term is the ‘on-shell’ action.
I The linear term should vanish on solutions gcl and Xcl.
I The quadratic term represents the first corrections.
If nothing goes wrong:
Z ∼ exp(−1
~IE [gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2IE
)× . . .
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 8/32
Semiclassical Approximation
Consider small perturbation around classical solution
IE [gcl + δg,Xcl + δX] =IE [gcl, Xcl] + δIE [gcl, Xcl; δg, δX]
+12δ2IE [gcl, Xcl; δg, δX] + . . .
I The leading term is the ‘on-shell’ action.
I The linear term should vanish on solutions gcl and Xcl.
I The quadratic term represents the first corrections.
If nothing goes wrong:
Z ∼ exp(−1
~IE [gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2IE
)× . . .
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 8/32
Semiclassical Approximation
Consider small perturbation around classical solution
IE [gcl + δg,Xcl + δX] =IE [gcl, Xcl] + δIE [gcl, Xcl; δg, δX]
+12δ2IE [gcl, Xcl; δg, δX] + . . .
I The leading term is the ‘on-shell’ action.
I The linear term should vanish on solutions gcl and Xcl.
I The quadratic term represents the first corrections.
If nothing goes wrong:
Z ∼ exp(−1
~IE [gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2IE
)× . . .
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 8/32
Semiclassical Approximation
Consider small perturbation around classical solution
IE [gcl + δg,Xcl + δX] =IE [gcl, Xcl] + δIE [gcl, Xcl; δg, δX]
+12δ2IE [gcl, Xcl; δg, δX] + . . .
I The leading term is the ‘on-shell’ action.
I The linear term should vanish on solutions gcl and Xcl.
I The quadratic term represents the first corrections.
If nothing goes wrong:
Z ∼ exp(−1
~IE [gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2IE
)× . . .
D. Grumiller — Black Hole Thermodynamics Euclidean Path Integral 8/32
What could go Wrong?
...everything!
Accessibility of the semiclassical approximation requires
1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 03. δ2IE [gcl, Xcl; 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 configurationscontributing to path integral due to boundary terms
δIE∣∣EOM
∼∫
∂Mdx√γ[πab δγab + πX δX
]6= 0
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
What could go Wrong?
...everything!
Accessibility of the semiclassical approximation requires
1. IE [gcl, Xcl] > −∞
2. δIE [gcl, Xcl; δg, δX] = 03. δ2IE [gcl, Xcl; 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 configurationscontributing to path integral due to boundary terms
δIE∣∣EOM
∼∫
∂Mdx√γ[πab δγab + πX δX
]6= 0
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
What could go Wrong?
...everything!
Accessibility of the semiclassical approximation requires
1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 0
3. δ2IE [gcl, Xcl; 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 configurationscontributing to path integral due to boundary terms
δIE∣∣EOM
∼∫
∂Mdx√γ[πab δγab + πX δX
]6= 0
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
What could go Wrong?
...everything!
Accessibility of the semiclassical approximation requires
1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 0
3. δ2IE [gcl, Xcl; 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 configurationscontributing to path integral due to boundary terms
δIE∣∣EOM
∼∫
∂Mdx√γ[πab δγab + πX δX
]6= 0
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
What could go Wrong?...everything!
Accessibility of the semiclassical approximation requires
1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 03. δ2IE [gcl, Xcl; 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 configurationscontributing to path integral due to boundary terms
δIE∣∣EOM
∼∫
∂Mdx√γ[πab δγab + πX δX
]6= 0
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
What could go Wrong?...everything!
Accessibility of the semiclassical approximation requires
1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 03. δ2IE [gcl, Xcl; 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 configurationscontributing to path integral due to boundary terms
δIE∣∣EOM
∼∫
∂Mdx√γ[πab δγab + πX δX
]6= 0
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
Outline
Introduction
Euclidean Path Integral
Dilaton Gravity in 2D
Free Energy
Applications
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 10/32
The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/0204253
Standard form of the action:
IE =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]
−∫
∂Mdx√γ X K−
∫∂Mdx√γL(X)
I Dilaton X defined via coupling to Ricci scalar
I Model specified by kinetic and potential functions for dilaton
I 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
The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/0204253
Standard form of the action:
IE =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]
−∫
∂Mdx√γ X K−
∫∂Mdx√γL(X)
I Dilaton X defined via coupling to Ricci scalar
I Model specified by kinetic and potential functions for dilaton
I 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
The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/0204253
Standard form of the action:
IE =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]
−∫
∂Mdx√γ X K−
∫∂Mdx√γL(X)
I Dilaton X defined via coupling to Ricci scalar
I Model specified by kinetic and potential functions for dilaton
I 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
The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/0204253
Standard form of the action:
IE =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K
−∫
∂Mdx√γL(X)
I Dilaton X defined via coupling to Ricci scalar
I Model specified by kinetic and potential functions for dilaton
I 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
The Action...for a review cf. e.g. DG, W. Kummer and D. Vassilevich, hep-th/0204253
Standard form of the action:
IE =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K−
∫∂Mdx√γL(X)
I Dilaton X defined via coupling to Ricci scalar
I Model specified by kinetic and potential functions for dilaton
I 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
Selected List of ModelsBlack holes in (A)dS, asymptotically flat or arbitrary spaces
Model U(X) V (X)
1. Schwarzschild (1916) − 12X
−λ2
2. Jackiw-Teitelboim (1984) 0 ΛX3. Witten Black Hole (1991) − 1
X−2b2X
4. CGHS (1992) 0 −2b2
5. (A)dS2 ground state (1994) − aX
BX6. Rindler ground state (1996) − a
XBXa
7. Black Hole attractor (2003) 0 BX−1
8. Spherically reduced gravity (N > 3) − N−3(N−2)X
−λ2X(N−4)/(N−2)
9. All above: ab-family (1997) − aX
BXa+b
10. Liouville gravity a beαX
11. Reissner-Nordstrom (1916) − 12X
−λ2 + Q2
X
12. Schwarzschild-(A)dS − 12X
−λ2 − `X13. Katanaev-Volovich (1986) α βX2 − Λ
14. BTZ/Achucarro-Ortiz (1993) 0 Q2
X− J
4X3 − ΛX15. KK reduced CS (2003) 0 1
2X(c−X2)
16. KK red. conf. flat (2006) − 12
tanh (X/2) A sinh X
17. 2D type 0A string Black Hole − 1X
−2b2X + b2q2
8π
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 12/32
Selected List of ModelsBlack holes in (A)dS, asymptotically flat or arbitrary spaces
Model U(X) V (X)
1. Schwarzschild (1916) − 12X
−λ2
2. Jackiw-Teitelboim (1984) 0 ΛX3. Witten Black Hole (1991) − 1
X−2b2X
4. CGHS (1992) 0 −2b2
5. (A)dS2 ground state (1994) − aX
BX6. Rindler ground state (1996) − a
XBXa
7. Black Hole attractor (2003) 0 BX−1
8. Spherically reduced gravity (N > 3) − N−3(N−2)X
−λ2X(N−4)/(N−2)
9. All above: ab-family (1997) − aX
BXa+b
10. Liouville gravity a beαX
11. Reissner-Nordstrom (1916) − 12X
−λ2 + Q2
X
12. Schwarzschild-(A)dS − 12X
−λ2 − `X13. Katanaev-Volovich (1986) α βX2 − Λ
14. BTZ/Achucarro-Ortiz (1993) 0 Q2
X− J
4X3 − ΛX15. KK reduced CS (2003) 0 1
2X(c−X2)
16. KK red. conf. flat (2006) − 12
tanh (X/2) A sinh X
17. 2D type 0A string Black Hole − 1X
−2b2X + b2q2
8π
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 12/32
Equations of Motion (EOM)
Extremize the action: δIE = 0
U(X)∇µX∇νX − 1
2gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0
R +∂U(X)
∂X(∇X)2 + 2 U(X)∇2X − 2
∂V (X)
∂X= 0
I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]I Generalized Birkhoff theorem: at least one Killing vectorI Orbits of this vector are isosurfaces of the dilaton
LkX = kµ∂µX = 0
I Choose henceforth ∂M as X = const. hypersurface
Adapted coordinate system (Lapse and Shift for radial evolution)
X = X(r) ds2 = N(r)2︸ ︷︷ ︸:=ξ(r)−1
dr2 + ξ(r)︸︷︷︸=kµkµ
(dτ + N τ (r)︸ ︷︷ ︸:=0
dr)2
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
Equations of Motion (EOM)
Extremize the action: δIE = 0
U(X)∇µX∇νX − 1
2gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0
R +∂U(X)
∂X(∇X)2 + 2 U(X)∇2X − 2
∂V (X)
∂X= 0
I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]
I Generalized Birkhoff theorem: at least one Killing vectorI Orbits of this vector are isosurfaces of the dilaton
LkX = kµ∂µX = 0
I Choose henceforth ∂M as X = const. hypersurface
Adapted coordinate system (Lapse and Shift for radial evolution)
X = X(r) ds2 = N(r)2︸ ︷︷ ︸:=ξ(r)−1
dr2 + ξ(r)︸︷︷︸=kµkµ
(dτ + N τ (r)︸ ︷︷ ︸:=0
dr)2
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
Equations of Motion (EOM)
Extremize the action: δIE = 0
U(X)∇µX∇νX − 1
2gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0
R +∂U(X)
∂X(∇X)2 + 2 U(X)∇2X − 2
∂V (X)
∂X= 0
I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]I Generalized Birkhoff theorem: at least one Killing vector
I Orbits of this vector are isosurfaces of the dilaton
LkX = kµ∂µX = 0
I Choose henceforth ∂M as X = const. hypersurface
Adapted coordinate system (Lapse and Shift for radial evolution)
X = X(r) ds2 = N(r)2︸ ︷︷ ︸:=ξ(r)−1
dr2 + ξ(r)︸︷︷︸=kµkµ
(dτ + N τ (r)︸ ︷︷ ︸:=0
dr)2
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
Equations of Motion (EOM)
Extremize the action: δIE = 0
U(X)∇µX∇νX − 1
2gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0
R +∂U(X)
∂X(∇X)2 + 2 U(X)∇2X − 2
∂V (X)
∂X= 0
I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]I Generalized Birkhoff theorem: at least one Killing vectorI Orbits of this vector are isosurfaces of the dilaton
LkX = kµ∂µX = 0
I Choose henceforth ∂M as X = const. hypersurface
Adapted coordinate system (Lapse and Shift for radial evolution)
X = X(r) ds2 = N(r)2︸ ︷︷ ︸:=ξ(r)−1
dr2 + ξ(r)︸︷︷︸=kµkµ
(dτ + N τ (r)︸ ︷︷ ︸:=0
dr)2
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
Equations of Motion (EOM)
Extremize the action: δIE = 0
U(X)∇µX∇νX − 1
2gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0
R +∂U(X)
∂X(∇X)2 + 2 U(X)∇2X − 2
∂V (X)
∂X= 0
I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]I Generalized Birkhoff theorem: at least one Killing vectorI Orbits of this vector are isosurfaces of the dilaton
LkX = kµ∂µX = 0
I Choose henceforth ∂M as X = const. hypersurface
Adapted coordinate system (Lapse and Shift for radial evolution)
X = X(r) ds2 = N(r)2︸ ︷︷ ︸:=ξ(r)−1
dr2 + ξ(r)︸︷︷︸=kµkµ
(dτ + N τ (r)︸ ︷︷ ︸:=0
dr)2
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
Equations of Motion (EOM)
Extremize the action: δIE = 0
U(X)∇µX∇νX − 1
2gµνU(X)(∇X)2 − gµνV (X) +∇µ∇νX − gµν∇2X = 0
R +∂U(X)
∂X(∇X)2 + 2 U(X)∇2X − 2
∂V (X)
∂X= 0
I Integrable! [Easier: first order formulation (Ikeda 1993, Schaller, Strobl 1994)]I Generalized Birkhoff theorem: at least one Killing vectorI Orbits of this vector are isosurfaces of the dilaton
LkX = kµ∂µX = 0
I Choose henceforth ∂M as X = const. hypersurface
Adapted coordinate system (Lapse and Shift for radial evolution)
X = X(r) ds2 = N(r)2︸ ︷︷ ︸:=ξ(r)−1
dr2 + ξ(r)︸︷︷︸=kµkµ
(dτ + N τ (r)︸ ︷︷ ︸:=0
dr)2
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 13/32
Solutions
I Define two model-dependent functions
Q(X) := Q0 +∫ X
dX U(X)
w(X) := w0 − 2∫ X
dX V (X)eQ(X)
I Q0 and w0 are arbitrary constants (essentially irrelevant)
I Construct all classical solutions
∂rX = e−Q(X) ξ(X) = eQ(X)(w(X)− 2M
)Constant of motion M (“mass”) characterizes classical solutions
I Absorb Q0 into rescaling of length units
I Shift w0 such that M = 0 ground state solution
I Restrict to positive mass sector M ≥ 0
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
Solutions
I Define two model-dependent functions
Q(X) := Q0 +∫ X
dX U(X)
w(X) := w0 − 2∫ X
dX V (X)eQ(X)
I Q0 and w0 are arbitrary constants (essentially irrelevant)
I Construct all classical solutions
∂rX = e−Q(X) ξ(X) = eQ(X)(w(X)− 2M
)Constant of motion M (“mass”) characterizes classical solutions
I Absorb Q0 into rescaling of length units
I Shift w0 such that M = 0 ground state solution
I Restrict to positive mass sector M ≥ 0
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
Solutions
I Define two model-dependent functions
Q(X) := Q0 +∫ X
dX U(X)
w(X) := w0 − 2∫ X
dX V (X)eQ(X)
I Q0 and w0 are arbitrary constants (essentially irrelevant)
I Construct all classical solutions
∂rX = e−Q(X) ξ(X) = eQ(X)(w(X)− 2M
)
Constant of motion M (“mass”) characterizes classical solutions
I Absorb Q0 into rescaling of length units
I Shift w0 such that M = 0 ground state solution
I Restrict to positive mass sector M ≥ 0
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
Solutions
I Define two model-dependent functions
Q(X) := Q0 +∫ X
dX U(X)
w(X) := w0 − 2∫ X
dX V (X)eQ(X)
I Q0 and w0 are arbitrary constants (essentially irrelevant)
I Construct all classical solutions
∂rX = e−Q(X) ξ(X) = eQ(X)(w(X)− 2M
)Constant of motion M (“mass”) characterizes classical solutions
I Absorb Q0 into rescaling of length units
I Shift w0 such that M = 0 ground state solution
I Restrict to positive mass sector M ≥ 0
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 14/32
Black Holes
HorizonsSolutions with M ≥ 0 exhibit (Killing) horizons for each solution of
w(Xh) = 2M
Killing norm2 ξ(X) = eQ(X)(w(X)− 2M
)≥ 0 on Xh ≤ X <∞
Assumption 1
If there are multiple horizons we take the outermost one
Asymptotics
X →∞: asymptotic region of spacetime; most models: w(X)→∞
Consider only models where w(X)→∞ as X →∞
Assumption 2
Consequence: ξ(X) ∼ eQw as X →∞, i.e., ξ asymptotes to ground state
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
Black Holes
HorizonsSolutions with M ≥ 0 exhibit (Killing) horizons for each solution of
w(Xh) = 2M
Killing norm2 ξ(X) = eQ(X)(w(X)− 2M
)≥ 0 on Xh ≤ X <∞
Assumption 1
If there are multiple horizons we take the outermost one
Asymptotics
X →∞: asymptotic region of spacetime; most models: w(X)→∞
Consider only models where w(X)→∞ as X →∞
Assumption 2
Consequence: ξ(X) ∼ eQw as X →∞, i.e., ξ asymptotes to ground state
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
Black Holes
HorizonsSolutions with M ≥ 0 exhibit (Killing) horizons for each solution of
w(Xh) = 2M
Killing norm2 ξ(X) = eQ(X)(w(X)− 2M
)≥ 0 on Xh ≤ X <∞
Assumption 1
If there are multiple horizons we take the outermost one
Asymptotics
X →∞: asymptotic region of spacetime; most models: w(X)→∞
Consider only models where w(X)→∞ as X →∞
Assumption 2
Consequence: ξ(X) ∼ eQw as X →∞, i.e., ξ asymptotes to ground state
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
Black Holes
HorizonsSolutions with M ≥ 0 exhibit (Killing) horizons for each solution of
w(Xh) = 2M
Killing norm2 ξ(X) = eQ(X)(w(X)− 2M
)≥ 0 on Xh ≤ X <∞
Assumption 1
If there are multiple horizons we take the outermost one
Asymptotics
X →∞: asymptotic region of spacetime; most models: w(X)→∞
Consider only models where w(X)→∞ as X →∞
Assumption 2
Consequence: ξ(X) ∼ eQw as X →∞, i.e., ξ asymptotes to ground state
D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 15/32
Black Hole TemperatureStandard argument: absence of conical singularity requires periodicity in Euclidean time
The gττ component of the metric vanishes at the horizon Xh
Regularity of the metric requires τ ∼ τ + β with periodicity
β =4π∂rξ
∣∣∣∣rh
=4π
w′(X)
∣∣∣∣Xh
I If ξ → 1 at X →∞: β−1 is temperature measured ’at infinity’
I Denote inverse periodicity by T := β−1 = w′(X)4π
∣∣∣Xh
I 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
Outline
Introduction
Euclidean Path Integral
Dilaton Gravity in 2D
Free Energy
Applications
D. Grumiller — Black Hole Thermodynamics Free Energy 17/32
Free Energy?
Not yet!
Given the black hole solution, can we calculate the free energy?
Z ∼ exp(−1
~IE [gcl, Xcl]
)∼ e−β F
Need a limiting procedure to calculate the action. Implement this in acoordinate-independent way by putting a regulator on the dilaton.
X ≤ Xreg
Evaluating the on-shell action leads to three problems
1. On-shell action unbounded from below (cf. second assumption)
I regE = β
(2M − w(Xreg)− 2πXh T
)→ −∞
2. First variation of action not zero for all field configurationscontributing 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
Free Energy? Not yet!
Given the black hole solution, can we calculate the free energy?
Z exp(−1
~IE [gcl, Xcl]
) e−β F
Need a limiting procedure to calculate the action. Implement this in acoordinate-independent way by putting a regulator on the dilaton.
X ≤ Xreg
Evaluating the on-shell action leads to three problems
1. On-shell action unbounded from below (cf. second assumption)
I regE = β
(2M − w(Xreg)− 2πXh T
)→ −∞
2. First variation of action not zero for all field configurationscontributing 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
Free Energy? Not yet!
Given the black hole solution, can we calculate the free energy?
Z exp(−1
~IE [gcl, Xcl]
) e−β F
Need a limiting procedure to calculate the action. Implement this in acoordinate-independent way by putting a regulator on the dilaton.
X ≤ Xreg
Evaluating the on-shell action leads to three problems
1. On-shell action unbounded from below (cf. second assumption)
I regE = β
(2M − w(Xreg)− 2πXh T
)→ −∞
2. First variation of action not zero for all field configurationscontributing 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
Free Energy? Not yet!
Given the black hole solution, can we calculate the free energy?
Z exp(−1
~IE [gcl, Xcl]
) e−β F
Need a limiting procedure to calculate the action. Implement this in acoordinate-independent way by putting a regulator on the dilaton.
X ≤ Xreg
Evaluating the on-shell action leads to three problems
1. On-shell action unbounded from below (cf. second assumption)
I regE = β
(2M − w(Xreg)− 2πXh T
)→ −∞
2. First variation of action not zero for all field configurationscontributing 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
Free Energy? Not yet!
Given the black hole solution, can we calculate the free energy?
Z exp(−1
~IE [gcl, Xcl]
) e−β F
Need a limiting procedure to calculate the action. Implement this in acoordinate-independent way by putting a regulator on the dilaton.
X ≤ Xreg
Evaluating the on-shell action leads to three problems
1. On-shell action unbounded from below (cf. second assumption)
I regE = β
(2M − w(Xreg)− 2πXh T
)→ −∞
2. First variation of action not zero for all field configurationscontributing 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
Variational Properties of the Action
δIE =
ZMd2x
√g
hEµνδgµν + EXδX
i| z
=0(EOM)
+
Z∂M
dx√
γhπabδγab + πXδX
i| z
=0?
6=0
Does this vanish on-shell? Ignore πXδX and focus on πabδγab
δIE =
Zdτ
»−1
2∂rX δξ + . . .
–Recall ξ(X) = w(X)eQ(X) − 2MeQ(X)
Assume that boundary conditions preserved by variations
δξ ∼ δM eQ(X)
Recalling ∂rX = e−Q we get
δIE =∫dτδM 6= 0
D. Grumiller — Black Hole Thermodynamics Free Energy 19/32
Variational Properties of the Action
δIE =
ZMd2x
√g
hEµνδgµν + EXδX
i| z
=0(EOM)
+
Z∂M
dx√
γhπabδγab + πXδX
i| z
=0?
6=0
Does this vanish on-shell? Ignore πXδX and focus on πabδγab
δIE =
Zdτ
»−1
2∂rX δξ + . . .
–
Recall ξ(X) = w(X)eQ(X) − 2MeQ(X)
Assume that boundary conditions preserved by variations
δξ ∼ δM eQ(X)
Recalling ∂rX = e−Q we get
δIE =∫dτδM 6= 0
D. Grumiller — Black Hole Thermodynamics Free Energy 19/32
Variational Properties of the Action
δIE =
ZMd2x
√g
hEµνδgµν + EXδX
i| z
=0(EOM)
+
Z∂M
dx√
γhπabδγab + πXδX
i| z
=0?
6=0
Does this vanish on-shell? Ignore πXδX and focus on πabδγab
δIE =
Zdτ
»−1
2∂rX δξ + . . .
–Recall ξ(X) = w(X)eQ(X) − 2MeQ(X)
Assume that boundary conditions preserved by variations
δξ ∼ δM eQ(X)
Recalling ∂rX = e−Q we get
δIE =∫dτδM 6= 0
D. Grumiller — Black Hole Thermodynamics Free Energy 19/32
Variational Properties of the Action
δIE =
ZMd2x
√g
hEµνδgµν + EXδX
i| z
=0(EOM)
+
Z∂M
dx√
γhπabδγab + πXδX
i| z
=0?
6=0
Does this vanish on-shell? Ignore πXδX and focus on πabδγab
δIE =
Zdτ
»−1
2∂rX δξ + . . .
–Recall ξ(X) = w(X)eQ(X) − 2MeQ(X)
Assume that boundary conditions preserved by variations
δξ ∼ δM eQ(X)
Recalling ∂rX = e−Q we get
δIE =∫dτδM 6= 0
D. Grumiller — Black Hole Thermodynamics Free Energy 19/32
Variational Properties of the Action
δIE =
ZMd2x
√g
hEµνδgµν + EXδX
i| z
=0(EOM)
+
Z∂M
dx√
γhπabδγab + πXδX
i| z
=0?
6=0
Does this vanish on-shell? Ignore πXδX and focus on πabδγab
δIE =
Zdτ
»−1
2∂rX δξ + . . .
–Recall ξ(X) = w(X)eQ(X) − 2MeQ(X)
Assume that boundary conditions preserved by variations
δξ ∼ δM eQ(X)
Recalling ∂rX = e−Q we get
δIE =∫dτδM 6= 0
D. Grumiller — Black Hole Thermodynamics Free Energy 19/32
Boundary Counterterms
I Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999;
Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998)
I More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann,
Marolf 2006)
I Covariant version of surface terms in 3 + 1 gravity (ADM 1962; Regge, Teitelboim
1974)
I Black Holes in 2D: IE = Γ + ICT
1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/0411121)
2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/0703230)
Γ =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K −
∫∂Mdx√γL(X)︸ ︷︷ ︸
ICT
How to determine the boundary counterterm?
D. Grumiller — Black Hole Thermodynamics Free Energy 20/32
Boundary Counterterms
I Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999;
Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998)
I More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann,
Marolf 2006)
I Covariant version of surface terms in 3 + 1 gravity (ADM 1962; Regge, Teitelboim
1974)
I Black Holes in 2D: IE = Γ + ICT
1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/0411121)
2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/0703230)
Γ =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K −
∫∂Mdx√γL(X)︸ ︷︷ ︸
ICT
How to determine the boundary counterterm?
D. Grumiller — Black Hole Thermodynamics Free Energy 20/32
Boundary Counterterms
I Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999;
Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998)
I More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann,
Marolf 2006)
I Covariant version of surface terms in 3 + 1 gravity (ADM 1962; Regge, Teitelboim
1974)
I Black Holes in 2D: IE = Γ + ICT
1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/0411121)
2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/0703230)
Γ =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K −
∫∂Mdx√γL(X)︸ ︷︷ ︸
ICT
How to determine the boundary counterterm?
D. Grumiller — Black Hole Thermodynamics Free Energy 20/32
Boundary Counterterms
I Same idea as boundary counterterms in AdS/CFT (Balasubramanian, Kraus 1999;
Emparan, Johnson, Myers 1999; Henningson, Skenderis 1998)
I More recently in asymptotically flat spacetimes (Kraus, Larsen, Siebelink 1999; Mann,
Marolf 2006)
I Covariant version of surface terms in 3 + 1 gravity (ADM 1962; Regge, Teitelboim
1974)
I Black Holes in 2D: IE = Γ + ICT
1. Witten Black Hole/2D type 0A strings (J. Davis, R. McNees, hep-th/0411121)
2. Generic 2D dilaton gravity (DG, R. McNees, hep-th/0703230)
Γ =− 12
∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K −
∫∂Mdx√γL(X)︸ ︷︷ ︸
ICT
How to determine the boundary counterterm?
D. Grumiller — Black Hole Thermodynamics Free Energy 20/32
Hamilton-Jacobi Equation
Boundary counterterm ICT is solution of the Hamilton-Jacobi equation
1. Begin with ‘Hamiltonian’ associated with radial evolution.
H = 2πXγab πab + 2U(X)
(γab π
ab)2
+ V (X) = 0
2. Momenta are functional derivatives of the on-shell action
πab =1√γ
δ
δ γabIE
∣∣∣EOM
πX =1√γ
δ
δ XIE
∣∣∣EOM
3. Obtain non-linear functional differential equation for on-shell action
4. 2D: simplifies to first order ODE – can solve (essentially uniquely) forICT !
ICT = −∫
∂Mdx√γ√w(X) e−Q(X)
D. Grumiller — Black Hole Thermodynamics Free Energy 21/32
Hamilton-Jacobi Equation
Boundary counterterm ICT is solution of the Hamilton-Jacobi equation
1. Begin with ‘Hamiltonian’ associated with radial evolution.
H = 2πXγab πab + 2U(X)
(γab π
ab)2
+ V (X) = 0
2. Momenta are functional derivatives of the on-shell action
πab =1√γ
δ
δ γabIE
∣∣∣EOM
πX =1√γ
δ
δ XIE
∣∣∣EOM
3. Obtain non-linear functional differential equation for on-shell action
4. 2D: simplifies to first order ODE – can solve (essentially uniquely) forICT !
ICT = −∫
∂Mdx√γ√w(X) e−Q(X)
D. Grumiller — Black Hole Thermodynamics Free Energy 21/32
Hamilton-Jacobi Equation
Boundary counterterm ICT is solution of the Hamilton-Jacobi equation
1. Begin with ‘Hamiltonian’ associated with radial evolution.
H = 2πXγab πab + 2U(X)
(γab π
ab)2
+ V (X) = 0
2. Momenta are functional derivatives of the on-shell action
πab =1√γ
δ
δ γabIE
∣∣∣EOM
πX =1√γ
δ
δ XIE
∣∣∣EOM
3. Obtain non-linear functional differential equation for on-shell action
4. 2D: simplifies to first order ODE – can solve (essentially uniquely) forICT !
ICT = −∫
∂Mdx√γ√w(X) e−Q(X)
D. Grumiller — Black Hole Thermodynamics Free Energy 21/32
Hamilton-Jacobi Equation
Boundary counterterm ICT is solution of the Hamilton-Jacobi equation
1. Begin with ‘Hamiltonian’ associated with radial evolution.
H = 2πXγab πab + 2U(X)
(γab π
ab)2
+ V (X) = 0
2. Momenta are functional derivatives of the on-shell action
πab =1√γ
δ
δ γabIE
∣∣∣EOM
πX =1√γ
δ
δ XIE
∣∣∣EOM
3. Obtain non-linear functional differential equation for on-shell action
4. 2D: simplifies to first order ODE – can solve (essentially uniquely) forICT !
ICT = −∫
∂Mdx√γ√w(X) e−Q(X)
D. Grumiller — Black Hole Thermodynamics Free Energy 21/32
Hamilton-Jacobi Equation
Boundary counterterm ICT is solution of the Hamilton-Jacobi equation
1. Begin with ‘Hamiltonian’ associated with radial evolution.
H = 2πXγab πab + 2U(X)
(γab π
ab)2
+ V (X) = 0
2. Momenta are functional derivatives of the on-shell action
πab =1√γ
δ
δ γabIE
∣∣∣EOM
πX =1√γ
δ
δ XIE
∣∣∣EOM
3. Obtain non-linear functional differential equation for on-shell action
4. 2D: simplifies to first order ODE – can solve (essentially uniquely) forICT !
ICT = −∫
∂Mdx√γ√w(X) e−Q(X)
D. Grumiller — Black Hole Thermodynamics Free Energy 21/32
Hamilton-Jacobi Equation
Boundary counterterm ICT is solution of the Hamilton-Jacobi equation
1. Begin with ‘Hamiltonian’ associated with radial evolution.
H = 2πXγab πab + 2U(X)
(γab π
ab)2
+ V (X) = 0
2. Momenta are functional derivatives of the on-shell action
πab =1√γ
δ
δ γabIE
∣∣∣EOM
πX =1√γ
δ
δ XIE
∣∣∣EOM
3. Obtain non-linear functional differential equation for on-shell action
4. 2D: simplifies to first order ODE – can solve (essentially uniquely) forICT !
ICT = −∫
∂Mdx√γ√w(X) e−Q(X)
D. Grumiller — Black Hole Thermodynamics Free Energy 21/32
The Improved Action
The correct action for 2D dilaton gravity is
Γ =− 12
∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K +
∫∂Mdx√γ√w(X) e−Q(X)
Properties:
1. Yields the same EOM as IE2. Finite on-shell (solves first problem)
Γ∣∣EOM
= β (M − 2πXh T )
3. First variation δΓ vanishes on-shell ∀ δgµν and δX that preserve theboundary conditions (solves second problem)
δΓ∣∣EOM
= 0
Note: counterterm requires specification of integration constant w0, i.e., choice of ground state, but is independent from Q0
D. Grumiller — Black Hole Thermodynamics Free Energy 22/32
The Improved Action
The correct action for 2D dilaton gravity is
Γ =− 12
∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K +
∫∂Mdx√γ√w(X) e−Q(X)
Properties:
1. Yields the same EOM as IE
2. Finite on-shell (solves first problem)
Γ∣∣EOM
= β (M − 2πXh T )
3. First variation δΓ vanishes on-shell ∀ δgµν and δX that preserve theboundary conditions (solves second problem)
δΓ∣∣EOM
= 0
Note: counterterm requires specification of integration constant w0, i.e., choice of ground state, but is independent from Q0
D. Grumiller — Black Hole Thermodynamics Free Energy 22/32
The Improved Action
The correct action for 2D dilaton gravity is
Γ =− 12
∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K +
∫∂Mdx√γ√w(X) e−Q(X)
Properties:
1. Yields the same EOM as IE2. Finite on-shell (solves first problem)
Γ∣∣EOM
= β (M − 2πXh T )
3. First variation δΓ vanishes on-shell ∀ δgµν and δX that preserve theboundary conditions (solves second problem)
δΓ∣∣EOM
= 0
Note: counterterm requires specification of integration constant w0, i.e., choice of ground state, but is independent from Q0
D. Grumiller — Black Hole Thermodynamics Free Energy 22/32
The Improved Action
The correct action for 2D dilaton gravity is
Γ =− 12
∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)
]−∫
∂Mdx√γ X K +
∫∂Mdx√γ√w(X) e−Q(X)
Properties:
1. Yields the same EOM as IE2. Finite on-shell (solves first problem)
Γ∣∣EOM
= β (M − 2πXh T )
3. First variation δΓ vanishes on-shell ∀ δgµν and δX that preserve theboundary conditions (solves second problem)
δΓ∣∣EOM
= 0
Note: counterterm requires specification of integration constant w0, i.e., choice of ground state, but is independent from Q0
D. Grumiller — Black Hole Thermodynamics Free Energy 22/32
Reconsider Semiclassical Approximation
Z ∼ exp(−1
~Γ[gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2Γ)× . . .
I Leading term is finite
I Linear term vanishes
I 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 = Xc
Well-defined canonical ensemble by specifying Xc and Tc = 1/βc
Leading order (set ~ = 1):
Z(Tc, Xc) = e−Γ(Tc,Xc) = e−βcFc(Tc,Xc)
Here Fc is the Helmholtz free energy
D. Grumiller — Black Hole Thermodynamics Free Energy 23/32
Reconsider Semiclassical Approximation
Z ∼ exp(−1
~Γ[gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2Γ)× . . .
I Leading term is finite
I Linear term vanishes
I 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 = Xc
Well-defined canonical ensemble by specifying Xc and Tc = 1/βc
Leading order (set ~ = 1):
Z(Tc, Xc) = e−Γ(Tc,Xc) = e−βcFc(Tc,Xc)
Here Fc is the Helmholtz free energy
D. Grumiller — Black Hole Thermodynamics Free Energy 23/32
Reconsider Semiclassical Approximation
Z ∼ exp(−1
~Γ[gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2Γ)× . . .
I Leading term is finite
I Linear term vanishes
I 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 = Xc
Well-defined canonical ensemble by specifying Xc and Tc = 1/βc
Leading order (set ~ = 1):
Z(Tc, Xc) = e−Γ(Tc,Xc) = e−βcFc(Tc,Xc)
Here Fc is the Helmholtz free energy
D. Grumiller — Black Hole Thermodynamics Free Energy 23/32
Reconsider Semiclassical Approximation
Z ∼ exp(−1
~Γ[gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2Γ)× . . .
I Leading term is finite
I Linear term vanishes
I 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 = Xc
Well-defined canonical ensemble by specifying Xc and Tc = 1/βc
Leading order (set ~ = 1):
Z(Tc, Xc) = e−Γ(Tc,Xc) = e−βcFc(Tc,Xc)
Here Fc is the Helmholtz free energy
D. Grumiller — Black Hole Thermodynamics Free Energy 23/32
Reconsider Semiclassical Approximation
Z ∼ exp(−1
~Γ[gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2Γ)× . . .
I Leading term is finite
I Linear term vanishes
I 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 = Xc
Well-defined canonical ensemble by specifying Xc and Tc = 1/βc
Leading order (set ~ = 1):
Z(Tc, Xc) = e−Γ(Tc,Xc) = e−βcFc(Tc,Xc)
Here Fc is the Helmholtz free energy
D. Grumiller — Black Hole Thermodynamics Free Energy 23/32
Reconsider Semiclassical Approximation
Z ∼ exp(−1
~Γ[gcl, Xcl]
) ∫DδgDδX exp
(− 1
2~δ2Γ)× . . .
I Leading term is finite
I Linear term vanishes
I 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 = Xc
Well-defined canonical ensemble by specifying Xc and Tc = 1/βc
Leading order (set ~ = 1):
Z(Tc, Xc) = e−Γ(Tc,Xc) = e−βcFc(Tc,Xc)
Here Fc is the Helmholtz free energyD. Grumiller — Black Hole Thermodynamics Free Energy 23/32
Free Energy
Γ(Tc, Xc) = βc Fc(Tc, Xc)
Explicitly: Entropy follows immediately (Bekenstein-Hawking law):
S = − ∂Fc
∂Tc
∣∣∣∣Xc
= 2πXh
Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter,
Louis-Martinez, 1995)
Similarly: dilaton chemical potential (surface pressure) ψc = −∂Fc/∂Xc|Tc
D. Grumiller — Black Hole Thermodynamics Free Energy 24/32
Free Energy
Γ(Tc, Xc) = βc Fc(Tc, Xc)
Explicitly:
Fc(Tc, Xc) =√wc e−Qc
(1−
√1− 2M
wc
)− 2πXhTc
Entropy follows immediately (Bekenstein-Hawking law):
S = − ∂Fc
∂Tc
∣∣∣∣Xc
= 2πXh
Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter,
Louis-Martinez, 1995)
Similarly: dilaton chemical potential (surface pressure) ψc = −∂Fc/∂Xc|Tc
D. Grumiller — Black Hole Thermodynamics Free Energy 24/32
Free Energy
Γ(Tc, Xc) = βc Fc(Tc, Xc)
Explicitly:
Fc(Tc, Xc) =√wc e−Qc
(1−
√1− 2M
wc
)︸ ︷︷ ︸
=Ec(Tc,Xc)
− 2πXhTc︸ ︷︷ ︸=STc
Entropy follows immediately (Bekenstein-Hawking law):
S = − ∂Fc
∂Tc
∣∣∣∣Xc
= 2πXh
Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter,
Louis-Martinez, 1995)
Similarly: dilaton chemical potential (surface pressure) ψc = −∂Fc/∂Xc|Tc
D. Grumiller — Black Hole Thermodynamics Free Energy 24/32
Free Energy
Γ(Tc, Xc) = βc Fc(Tc, Xc)
Explicitly:
Fc(Tc, Xc) =√wc e−Qc
(1−
√1− 2M
wc
)︸ ︷︷ ︸
=Ec(Tc,Xc)
− 2πXhTc︸ ︷︷ ︸=STc
Entropy follows immediately (Bekenstein-Hawking law):
S = − ∂Fc
∂Tc
∣∣∣∣Xc
= 2πXh
Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter,
Louis-Martinez, 1995)
Similarly: dilaton chemical potential (surface pressure) ψc = −∂Fc/∂Xc|Tc
D. Grumiller — Black Hole Thermodynamics Free Energy 24/32
Other Thermodynamical Quantities
Standard thermodynamics in canonical ensemble: internal energy,enthalpy, free enthalpy, specific heats, isothermal compressibility, ...
1. Internal energy
Ec = Fc + Tc S = e−Qc
(√ξgc −
√ξc
)≥ 0
Models with Minkowski ground state (ξgc = 1): M = Ec − E2
c2 wc
2. First lawdEc = Tc dS − ψc dXc
Properly accounts for non-linear effects of gravitational binding energy
3. Specific heat at constant dilaton charge Xc
CD = 2πw′hw′′h
1
1 + (w′h)2
2w′′h(wc−2M)
Allows to check for thermodynamic stability: CD(Xc = Xh + ε) > 0
D. Grumiller — Black Hole Thermodynamics Free Energy 25/32
Other Thermodynamical Quantities
Standard thermodynamics in canonical ensemble: internal energy,enthalpy, free enthalpy, specific heats, isothermal compressibility, ...
1. Internal energy
Ec = Fc + Tc S = e−Qc
(√ξgc −
√ξc
)≥ 0
Models with Minkowski ground state (ξgc = 1): M = Ec − E2
c2 wc
2. First lawdEc = Tc dS − ψc dXc
Properly accounts for non-linear effects of gravitational binding energy
3. Specific heat at constant dilaton charge Xc
CD = 2πw′hw′′h
1
1 + (w′h)2
2w′′h(wc−2M)
Allows to check for thermodynamic stability: CD(Xc = Xh + ε) > 0
D. Grumiller — Black Hole Thermodynamics Free Energy 25/32
Other Thermodynamical Quantities
Standard thermodynamics in canonical ensemble: internal energy,enthalpy, free enthalpy, specific heats, isothermal compressibility, ...
1. Internal energy
Ec = Fc + Tc S = e−Qc
(√ξgc −
√ξc
)≥ 0
Models with Minkowski ground state (ξgc = 1): M = Ec − E2
c2 wc
2. First lawdEc = Tc dS − ψc dXc
Properly accounts for non-linear effects of gravitational binding energy
3. Specific heat at constant dilaton charge Xc
CD = 2πw′hw′′h
1
1 + (w′h)2
2w′′h(wc−2M)
Allows to check for thermodynamic stability: CD(Xc = Xh + ε) > 0
D. Grumiller — Black Hole Thermodynamics Free Energy 25/32
Other Thermodynamical Quantities
Standard thermodynamics in canonical ensemble: internal energy,enthalpy, free enthalpy, specific heats, isothermal compressibility, ...
1. Internal energy
Ec = Fc + Tc S = e−Qc
(√ξgc −
√ξc
)≥ 0
Models with Minkowski ground state (ξgc = 1): M = Ec − E2
c2 wc
2. First lawdEc = Tc dS − ψc dXc
Properly accounts for non-linear effects of gravitational binding energy
3. Specific heat at constant dilaton charge Xc
CD = 2πw′hw′′h
1
1 + (w′h)2
2w′′h(wc−2M)
Allows to check for thermodynamic stability: CD(Xc = Xh + ε) > 0D. Grumiller — Black Hole Thermodynamics Free Energy 25/32
Outline
Introduction
Euclidean Path Integral
Dilaton Gravity in 2D
Free Energy
Applications
D. Grumiller — Black Hole Thermodynamics Applications 26/32
Higher Dimensional Black HolesSchwarzschild, Reissner-Nordstrom, BTZ, Schwarzschild-AdS, ...
EHd+1
DG2
spherical reduction
?
It works, regardless of the asymptotics... But nearly no info about HJd!
Main message
Example: Schwarzschild-AdS in d+ 1 dimensions:
U(X) = −(d− 2d− 1
)1X, V (X) = −(const.)X
d−3d−1 − d(d− 1)
2 `2X
D. Grumiller — Black Hole Thermodynamics Applications 27/32
Higher Dimensional Black HolesSchwarzschild, Reissner-Nordstrom, BTZ, Schwarzschild-AdS, ...
EHd+1bound.
- EHd+1 + GHYd
DG2
spherical reduction
? bound.- DG2 + ′GHY′1
spherical reduction
?
It works, regardless of the asymptotics... But nearly no info about HJd!
Main message
Example: Schwarzschild-AdS in d+ 1 dimensions:
U(X) = −(d− 2d− 1
)1X, V (X) = −(const.)X
d−3d−1 − d(d− 1)
2 `2X
D. Grumiller — Black Hole Thermodynamics Applications 27/32
Higher Dimensional Black HolesSchwarzschild, Reissner-Nordstrom, BTZ, Schwarzschild-AdS, ...
EHd+1bound.
- EHd+1 + GHYd?- EHd+1 + GHYd + HJd
DG2
spherical reduction
? bound.- DG2 + ′GHY′1
spherical reduction
? !- DG2 + ′GHY′1 + HJ1
?
?
It works, regardless of the asymptotics... But nearly no info about HJd!
Main message
Example: Schwarzschild-AdS in d+ 1 dimensions:
U(X) = −(d− 2d− 1
)1X, V (X) = −(const.)X
d−3d−1 − d(d− 1)
2 `2X
D. Grumiller — Black Hole Thermodynamics Applications 27/32
Higher Dimensional Black HolesSchwarzschild, Reissner-Nordstrom, BTZ, Schwarzschild-AdS, ...
EHd+1bound.
- EHd+1 + GHYd?- EHd+1 + GHYd + HJd
DG2
spherical reduction
? bound.- DG2 + ′GHY′1
spherical reduction
? !- DG2 + ′GHY′1 + HJ1
?
?
It works, regardless of the asymptotics... But nearly no info about HJd!
Main message
Example: Schwarzschild-AdS in d+ 1 dimensions:
U(X) = −(d− 2d− 1
)1X, V (X) = −(const.)X
d−3d−1 − d(d− 1)
2 `2X
D. Grumiller — Black Hole Thermodynamics Applications 27/32
Higher Dimensional Black HolesSchwarzschild, Reissner-Nordstrom, BTZ, Schwarzschild-AdS, ...
EHd+1bound.
- EHd+1 + GHYd?- EHd+1 + GHYd + HJd
DG2
spherical reduction
? bound.- DG2 + ′GHY′1
spherical reduction
? !- DG2 + ′GHY′1 + HJ1
?
?
It works, regardless of the asymptotics... But nearly no info about HJd!
Main message
Example: Schwarzschild-AdS in d+ 1 dimensions:
U(X) = −(d− 2d− 1
)1X, V (X) = −(const.)X
d−3d−1 − d(d− 1)
2 `2X
D. Grumiller — Black Hole Thermodynamics Applications 27/32
Hawking-Page TransitionSpherically symmetric AdS Black Holes in d + 1 dimensions
CD: specific heat at constant dilaton; rh: horizon radius; `: AdS radiusD. Grumiller — Black Hole Thermodynamics Applications 28/32
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)I 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
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)I 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
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)I 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
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)I 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
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)
I 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
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)I 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
Black Holes in 2D String Theory
Black holes with exact CFT description (SL(2,R)/U(1) gauged WZWmodel) (Witten 1991)
I Large level k: Witten Black Hole (U = −1/X, V ∝ X)Recover well-known results (Gibbons, Perry 1992; Nappi, Pasquinucci 1992)
I Adding D-branes: 2D type 0A stringsDropping (Coulomb-) divergences is wrong! (Davies, McNees 2004)
Problem: have to move the ’cavity’ to infinity in string theory
I Witten Black Hole: cannot remove cavity! (specific heat diverges)I 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
Thermodynamics of the Exact String Black Hole
Interesting geometry: asymptotically flat, one Killing horizon, nosingularity (dilaton violates energy conditions)Singular limits:
I k →∞: singularity appears (Witten Black Hole)I k → 2: horizon disappears (Jackiw-Teitelboim, AdS2)
Thermodynamical properties
1. Positive specific heat CD = # k2T (like degenerate Fermi gas)
2. Hawking temperature T = TH
√1− 2
k (TH : Hagedorn temperature)
3. Logarithmic α′ corrections to entropy (DG 2005)
S = 2π(√
k(k − 2) + arcsinh (√k(k − 2))
)= 2πk + 2π ln k + . . .
4. Partition function for critical value k = 9/4 (setting GN = 1/4)
Z = earcsinh (3/4) = 2
Relations: α′b2 = 1/(k − 2), Dim− 26 + 6α′b2 = 0
D. Grumiller — Black Hole Thermodynamics Applications 30/32
Thermodynamics of the Exact String Black Hole
Interesting geometry: asymptotically flat, one Killing horizon, nosingularity (dilaton violates energy conditions)Singular limits:
I k →∞: singularity appears (Witten Black Hole)I k → 2: horizon disappears (Jackiw-Teitelboim, AdS2)
Thermodynamical properties
1. Positive specific heat CD = # k2T (like degenerate Fermi gas)
2. Hawking temperature T = TH
√1− 2
k (TH : Hagedorn temperature)
3. Logarithmic α′ corrections to entropy (DG 2005)
S = 2π(√
k(k − 2) + arcsinh (√k(k − 2))
)= 2πk + 2π ln k + . . .
4. Partition function for critical value k = 9/4 (setting GN = 1/4)
Z = earcsinh (3/4) = 2
Relations: α′b2 = 1/(k − 2), Dim− 26 + 6α′b2 = 0
D. Grumiller — Black Hole Thermodynamics Applications 30/32
Thermodynamics of the Exact String Black Hole
Interesting geometry: asymptotically flat, one Killing horizon, nosingularity (dilaton violates energy conditions)Singular limits:
I k →∞: singularity appears (Witten Black Hole)I k → 2: horizon disappears (Jackiw-Teitelboim, AdS2)
Thermodynamical properties
1. Positive specific heat CD = # k2T (like degenerate Fermi gas)
2. Hawking temperature T = TH
√1− 2
k (TH : Hagedorn temperature)
3. Logarithmic α′ corrections to entropy (DG 2005)
S = 2π(√
k(k − 2) + arcsinh (√k(k − 2))
)= 2πk + 2π ln k + . . .
4. Partition function for critical value k = 9/4 (setting GN = 1/4)
Z = earcsinh (3/4) = 2
Relations: α′b2 = 1/(k − 2), Dim− 26 + 6α′b2 = 0
D. Grumiller — Black Hole Thermodynamics Applications 30/32
Thermodynamics of the Exact String Black Hole
Interesting geometry: asymptotically flat, one Killing horizon, nosingularity (dilaton violates energy conditions)Singular limits:
I k →∞: singularity appears (Witten Black Hole)I k → 2: horizon disappears (Jackiw-Teitelboim, AdS2)
Thermodynamical properties
1. Positive specific heat CD = # k2T (like degenerate Fermi gas)
2. Hawking temperature T = TH
√1− 2
k (TH : Hagedorn temperature)
3. Logarithmic α′ corrections to entropy (DG 2005)
S = 2π(√
k(k − 2) + arcsinh (√k(k − 2))
)= 2πk + 2π ln k + . . .
4. Partition function for critical value k = 9/4 (setting GN = 1/4)
Z = earcsinh (3/4) = 2
Relations: α′b2 = 1/(k − 2), Dim− 26 + 6α′b2 = 0
D. Grumiller — Black Hole Thermodynamics Applications 30/32
Thermodynamics of the Exact String Black Hole
Interesting geometry: asymptotically flat, one Killing horizon, nosingularity (dilaton violates energy conditions)Singular limits:
I k →∞: singularity appears (Witten Black Hole)I k → 2: horizon disappears (Jackiw-Teitelboim, AdS2)
Thermodynamical properties
1. Positive specific heat CD = # k2T (like degenerate Fermi gas)
2. Hawking temperature T = TH
√1− 2
k (TH : Hagedorn temperature)
3. Logarithmic α′ corrections to entropy (DG 2005)
S = 2π(√
k(k − 2) + arcsinh (√k(k − 2))
)= 2πk + 2π ln k + . . .
4. Partition function for critical value k = 9/4 (setting GN = 1/4)
Z = earcsinh (3/4) = 2
Relations: α′b2 = 1/(k − 2), Dim− 26 + 6α′b2 = 0
D. Grumiller — Black Hole Thermodynamics Applications 30/32
Conclusions...for more info see DG, R. McNees, hep-th/0703230
Main results presented:
I Constructed Hamilton-Jacobi counterterm for generic 2D dilatongravity (with two working assumptions!)
I Derived free energy and its thermodynamic descendants (entropy,internal energy, specific heat, ...)
I Applied it to numerous black holes in various dimensions
Main results not presented:
I Extensitivity and scaling properties
I Nonperturbative stability analysis (tunneling)
I Inclusion of Maxwell fields (charge, spin, ...)
Next steps envisaged:
I Relax working assumptions (dS!)
I Consider matter fields (reconsider counterterm!)
I Impact on quantum theory?
D. Grumiller — Black Hole Thermodynamics Applications 31/32
Conclusions...for more info see DG, R. McNees, hep-th/0703230
Main results presented:
I Constructed Hamilton-Jacobi counterterm for generic 2D dilatongravity (with two working assumptions!)
I Derived free energy and its thermodynamic descendants (entropy,internal energy, specific heat, ...)
I Applied it to numerous black holes in various dimensions
Main results not presented:
I Extensitivity and scaling properties
I Nonperturbative stability analysis (tunneling)
I Inclusion of Maxwell fields (charge, spin, ...)
Next steps envisaged:
I Relax working assumptions (dS!)
I Consider matter fields (reconsider counterterm!)
I Impact on quantum theory?
D. Grumiller — Black Hole Thermodynamics Applications 31/32
Conclusions...for more info see DG, R. McNees, hep-th/0703230
Main results presented:
I Constructed Hamilton-Jacobi counterterm for generic 2D dilatongravity (with two working assumptions!)
I Derived free energy and its thermodynamic descendants (entropy,internal energy, specific heat, ...)
I Applied it to numerous black holes in various dimensions
Main results not presented:
I Extensitivity and scaling properties
I Nonperturbative stability analysis (tunneling)
I Inclusion of Maxwell fields (charge, spin, ...)
Next steps envisaged:
I Relax working assumptions (dS!)
I Consider matter fields (reconsider counterterm!)
I Impact on quantum theory?
D. Grumiller — Black Hole Thermodynamics Applications 31/32
Conclusions...for more info see DG, R. McNees, hep-th/0703230
Main results presented:
I Constructed Hamilton-Jacobi counterterm for generic 2D dilatongravity (with two working assumptions!)
I Derived free energy and its thermodynamic descendants (entropy,internal energy, specific heat, ...)
I Applied it to numerous black holes in various dimensions
Main results not presented:
I Extensitivity and scaling properties
I Nonperturbative stability analysis (tunneling)
I Inclusion of Maxwell fields (charge, spin, ...)
Next steps envisaged:
I Relax working assumptions (dS!)
I Consider matter fields (reconsider counterterm!)
I Impact on quantum theory?
D. Grumiller — Black Hole Thermodynamics Applications 31/32
Conclusions...for more info see DG, R. McNees, hep-th/0703230
Main results presented:
I Constructed Hamilton-Jacobi counterterm for generic 2D dilatongravity (with two working assumptions!)
I Derived free energy and its thermodynamic descendants (entropy,internal energy, specific heat, ...)
I Applied it to numerous black holes in various dimensions
Main results not presented:
I Extensitivity and scaling properties
I Nonperturbative stability analysis (tunneling)
I Inclusion of Maxwell fields (charge, spin, ...)
Next steps envisaged:
I Relax working assumptions (dS!)
I Consider matter fields (reconsider counterterm!)
I Impact on quantum theory?
D. Grumiller — Black Hole Thermodynamics Applications 31/32
Conclusions...for more info see DG, R. McNees, hep-th/0703230
Main results presented:
I Constructed Hamilton-Jacobi counterterm for generic 2D dilatongravity (with two working assumptions!)
I Derived free energy and its thermodynamic descendants (entropy,internal energy, specific heat, ...)
I Applied it to numerous black holes in various dimensions
Main results not presented:
I Extensitivity and scaling properties
I Nonperturbative stability analysis (tunneling)
I Inclusion of Maxwell fields (charge, spin, ...)
Next steps envisaged:
I Relax working assumptions (dS!)
I Consider matter fields (reconsider counterterm!)
I Impact on quantum theory?
D. Grumiller — Black Hole Thermodynamics Applications 31/32
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