Black Hole Thermodynamics and Hamilton-Jacobi …quark.itp.tuwien.ac.at/~grumil/pdf/leipzig2007.pdfBlack Hole Thermodynamics and Hamilton-Jacobi Counterterm Based upon work with Bob

Black Hole Thermodynamics

and Hamilton-Jacobi CountertermBased upon work with Bob McNees

Daniel Grumiller

Center for Theoretical PhysicsMassachusetts Institute of Technology

QFEXT07, Leipzig, September 2007

hep-th/0703230

Outline

Black Hole Thermodynamics from Euclidean Path Integral

Dilaton Gravity in 2D

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics 2/25

Outline



Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Black Hole Thermodynamics from Euclidean Path Integral 3/25

Black Hole Thermodynamics – Recapitulation

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?


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



B-H: S = A4GN









Quantum Gravity




Dual Formulations






B-H: S = A4GN









Quantum Gravity




Dual Formulations






B-H: S = A4GN









Quantum Gravity




Dual Formulations






B-H: S = A4GN









Quantum Gravity




Dual Formulations





Euclidean Path Integral – 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




Z =∫

DgDX exp(−1

~IE [g,X]

)




Z ∼ e−β Ω







Z =∫

DgDX exp(−1

~IE [g,X]

)




Z ∼ e−β Ω







Z =∫

DgDX exp(−1

~IE [g,X]

)




Z ∼ e−β Ω







Z =∫

DgDX exp(−1

~IE [g,X]

)




Z ∼ e−β Ω





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

)× . . .





+12δ2IE [gcl, Xcl; δg, δX] + . . .





Z ∼ exp(−1

~IE [gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .





+12δ2IE [gcl, Xcl; δg, δX] + . . .





Z ∼ exp(−1

~IE [gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .





+12δ2IE [gcl, Xcl; δg, δX] + . . .





Z ∼ exp(−1

~IE [gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .





+12δ2IE [gcl, Xcl; δg, δX] + . . .





Z ∼ exp(−1

~IE [gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .


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!



...everything!


1. IE [gcl, Xcl] > −∞

2. δIE [gcl, Xcl; δg, δX] = 03. δ2IE [gcl, Xcl; δg, δX] ≥ 0




δIE∣∣EOM

∼∫


]6= 0





...everything!


1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 0

3. δ2IE [gcl, Xcl; δg, δX] ≥ 0




δIE∣∣EOM

∼∫


]6= 0





...everything!


1. IE [gcl, Xcl] > −∞2. δIE [gcl, Xcl; δg, δX] = 0

3. δ2IE [gcl, Xcl; δg, δX] ≥ 0




δIE∣∣EOM

∼∫


]6= 0




What could go Wrong?...everything!






δIE∣∣EOM

∼∫


]6= 0




What could go Wrong?...everything!






δIE∣∣EOM

∼∫


]6= 0




Outline



Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 8/25

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!For later: Q(X) :=

∫ XU , w(X) :=

∫ XeQV




IE =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]

−∫

∂Mdx√γ X K−

∫∂Mdx√γL(X)






∫ XU , w(X) :=

∫ XeQV




IE =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]

−∫

∂Mdx√γ X K−

∫∂Mdx√γL(X)





Note: additional boundary term allowed consistent with classical solutions,variational principle and symmetries!

For later: Q(X) :=∫ X

U , w(X) :=∫ X

eQV




IE =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K

−∫

∂Mdx√γL(X)





Note: additional boundary term allowed consistent with classical solutions,variational principle and symmetries!

For later: Q(X) :=∫ X

U , w(X) :=∫ X

eQV




IE =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K−

∫∂Mdx√γL(X)






∫ XU , w(X) :=

∫ XeQV


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

Models of interest: Boundary at X →∞, w(X →∞)→∞D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 10/25

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

Models of interest: Boundary at X →∞, w(X →∞)→∞D. Grumiller — Black Hole Thermodynamics Dilaton Gravity in 2D 10/25

Outline



Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Free Energy 11/25

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


Free Energy? Not yet!


Z exp(−1

~IE [gcl, Xcl]

) e−β F


X ≤ Xreg



I regE = β


)→ −∞






Z exp(−1

~IE [gcl, Xcl]

) e−β F


X ≤ Xreg



I regE = β


)→ −∞






Z exp(−1

~IE [gcl, Xcl]

) e−β F


X ≤ Xreg



I regE = β


)→ −∞






Z exp(−1

~IE [gcl, Xcl]

) e−β F


X ≤ Xreg



I regE = β


)→ −∞




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 + . . .

–Square of Killing norm: ξ(X) = w(X)eQ(X) − 2MeQ(X)

Assume that boundary conditions preserved by variations

δξ ∼ δM eQ(X)

Because ∂rX = e−Q we get

δIE =∫dτδM 6= 0



δIE =

ZMd2x

√g


i| z

=0(EOM)

+

Z∂M

dx√


i| z

=0?

6=0


δIE =

Zdτ

»−1

2∂rX δξ + . . .

–

Square of Killing norm: ξ(X) = w(X)eQ(X) − 2MeQ(X)


δξ ∼ δM eQ(X)





δIE =

ZMd2x

√g


i| z

=0(EOM)

+

Z∂M

dx√


i| z

=0?

6=0


δIE =

Zdτ

»−1

2∂rX δξ + . . .



δξ ∼ δM eQ(X)





δIE =

ZMd2x

√g


i| z

=0(EOM)

+

Z∂M

dx√


i| z

=0?

6=0


δIE =

Zdτ

»−1

2∂rX δξ + . . .



δξ ∼ δM eQ(X)





δIE =

ZMd2x

√g


i| z

=0(EOM)

+

Z∂M

dx√


i| z

=0?

6=0


δIE =

Zdτ

»−1

2∂rX δξ + . . .



δξ ∼ δM eQ(X)




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?






Marolf 2006)


1974)




Γ =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K −


ICT







Marolf 2006)


1974)




Γ =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K −


ICT







Marolf 2006)


1974)




Γ =− 12

∫Md2x√g[X R− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K −


ICT



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)






(γab π

ab)2

+ V (X) = 0


πab =1√γ

δ

δ γabIE

∣∣∣EOM

πX =1√γ

δ

δ XIE

∣∣∣EOM



ICT = −∫







(γab π

ab)2

+ V (X) = 0


πab =1√γ

δ

δ γabIE

∣∣∣EOM

πX =1√γ

δ

δ XIE

∣∣∣EOM



ICT = −∫







(γab π

ab)2

+ V (X) = 0


πab =1√γ

δ

δ γabIE

∣∣∣EOM

πX =1√γ

δ

δ XIE

∣∣∣EOM



ICT = −∫







(γab π

ab)2

+ V (X) = 0


πab =1√γ

δ

δ γabIE

∣∣∣EOM

πX =1√γ

δ

δ XIE

∣∣∣EOM



ICT = −∫







(γab π

ab)2

+ V (X) = 0


πab =1√γ

δ

δ γabIE

∣∣∣EOM

πX =1√γ

δ

δ XIE

∣∣∣EOM



ICT = −∫



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


The Improved Action


Γ =− 12

∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K +


Properties:

1. Yields the same EOM as IE

2. Finite on-shell (solves first problem)

Γ∣∣EOM

= β (M − 2πXh T )


δΓ∣∣EOM

= 0



The Improved Action


Γ =− 12

∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K +


Properties:


Γ∣∣EOM

= β (M − 2πXh T )


δΓ∣∣EOM

= 0



The Improved Action


Γ =− 12

∫Md2x√g[XR− U(X) (∇X)2 − 2V (X)

]−∫

∂Mdx√γ X K +


Properties:


Γ∣∣EOM

= β (M − 2πXh T )


δΓ∣∣EOM

= 0



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


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




1. Internal energy


(√ξgc −

√ξc

)≥ 0


c2 wc





1

1 + (w′h)2

2w′′h(wc−2M)





1. Internal energy


(√ξgc −

√ξc

)≥ 0


c2 wc





1

1 + (w′h)2

2w′′h(wc−2M)





1. Internal energy


(√ξgc −

√ξc

)≥ 0


c2 wc





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 18/25

Outline



Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics Applications 19/25

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



EHd+1bound.

- EHd+1 + GHYd

DG2

spherical reduction

? bound.- DG2 + ′GHY′1

spherical reduction

?


Main message


U(X) = −(d− 2d− 1

)1X, V (X) = −(const.)X

d−3d−1 − d(d− 1)

2 `2X



EHd+1bound.

- EHd+1 + GHYd?- EHd+1 + GHYd + HJd

DG2

spherical reduction


spherical reduction

? !- DG2 + ′GHY′1 + HJ1

?

?


Main message


U(X) = −(d− 2d− 1

)1X, V (X) = −(const.)X

d−3d−1 − d(d− 1)

2 `2X



EHd+1bound.


DG2

spherical reduction


spherical reduction

? !- DG2 + ′GHY′1 + HJ1

?

?


Main message


U(X) = −(d− 2d− 1

)1X, V (X) = −(const.)X

d−3d−1 − d(d− 1)

2 `2X



EHd+1bound.


DG2

spherical reduction


spherical reduction

? !- DG2 + ′GHY′1 + HJ1

?

?


Main message


U(X) = −(d− 2d− 1

)1X, V (X) = −(const.)X

d−3d−1 − d(d− 1)

2 `2X


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 21/25

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





































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)
























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








√1− 2



S = 2π(√

k(k − 2) + arcsinh (√k(k − 2))

)= 2πk + 2π ln k + . . .











√1− 2



S = 2π(√

k(k − 2) + arcsinh (√k(k − 2))

)= 2πk + 2π ln k + . . .











√1− 2



S = 2π(√

k(k − 2) + arcsinh (√k(k − 2))

)= 2πk + 2π ln k + . . .











√1− 2



S = 2π(√

k(k − 2) + arcsinh (√k(k − 2))

)= 2πk + 2π ln k + . . .





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?








































































Thanks for the attention...

...and thanks to Bob McNees for the style and source files of his talk!D. Grumiller — Black Hole Thermodynamics Applications 25/25

Documents

Black Hole Thermodynamics and Hamilton-Jacobi …quark.itp.tuwien.ac.at/~grumil/pdf/leipzig2007.pdfBlack Hole Thermodynamics and Hamilton-Jacobi Counterterm Based upon work with Bob