Black Hole Thermodynamics and Hamilton-Jacobi Counterterm

(1)

Black Hole Thermodynamics and Hamilton-Jacobi Counterterm

Based upon work with Bob McNees

Daniel Grumiller

Center for Theoretical Physics Massachusetts Institute of Technology

QFEXT07, Leipzig,September 2007

hep-th/0703230

(2)

Outline

Black Hole Thermodynamics from Euclidean Path Integral

Dilaton Gravity in 2D

Free Energy

Applications

D. Grumiller — Black Hole Thermodynamics 2/25

(3)

Outline

Free Energy

Applications

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

(4)

Black Hole Thermodynamics – Recapitulation

B-H: S= _4G^A

N, 1^st: dE =TdS+ work, 2^nd: 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

(5)

B-H: S= _4G^A

N, 1^st: dE =TdS+ work, 2^nd: dS ≥0

Classical General Relativity

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

I Microstate counting(Strominger, Vafa, 1996; Ashtekar, Corichi, Baez, Krasnov, 1997)

Dual Formulations

(6)

B-H: S= _4G^A

N, 1^st: dE =TdS+ work, 2^nd: dS ≥0 Classical General Relativity

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

I Microstate counting(Strominger, Vafa, 1996;

Ashtekar, Corichi, Baez, Krasnov, 1997)

Dual Formulations

(7)

Black Hole Thermodynamics – Recapitulation B-H: S= _4G^A

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

Dual Formulations

(8)

Black Hole Thermodynamics – Recapitulation B-H: S= _4G^A

(Bekenstein, 1973)

Black Hole Thermodynamics

Quantum Gravity

Dual Formulations

(9)

Euclidean Path Integral – Main Idea

Consider Euclidean path integral (Gibbons, Hawking, 1977)

Z = Z

DgDX exp

−1

~I_E[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 and accessibility of semi-classical approximation

(10)

Z = Z

DgDX exp

−1

~I_E[g, X]

I g: metric, X: scalar field

(11)

Z = Z

DgDX exp

−1

~I_E[g, X]

(12)

Z = Z

DgDX exp

−1

~I_E[g, X]

(13)

Z = Z

DgDX exp

−1

~I_E[g, X]

(14)

Semiclassical Approximation

Consider small perturbation around classical solution

IE[gcl+δg, Xcl+δX] =IE[gcl, Xcl] +δIE[gcl, Xcl;δg, δX]

+1

2δ²I_E[g_cl, X_cl;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

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

(15)

IE[gcl+δg, Xcl+δX] =IE[gcl, Xcl]+δIE[gcl, Xcl;δg, δX]

+1

I Theleading term is the ‘on-shell’ action.

I The linear term should vanish on solutions gcl and Xcl.

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

(16)

+1

I Thelinear term should vanish on solutionsgcl and Xcl.

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

(17)

+1

2δ²I_E[g_cl, X_cl;δg, δX]+. . .

I Thequadratic term represents the first corrections.

If nothing goes wrong:

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

(18)

+1

2δ²I_E[g_cl, X_cl;δg, δX]+. . .

I Thequadratic term represents the first corrections.

If nothing goes wrong:

Z ∼exp

−1

~I_E[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²I_E

×. . .

(19)

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. δ²I_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

δIE

EOM∼

Z

∂M

dx√ γ

h

π^abδγab+π_XδX i

6= 0

3. Frequently violated: Gaussian integral may diverge Focus in this talk on thesecond problem!

(20)

...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. δ²I_E[g_cl, X_cl;δg, δX]≥0

Typical gravitational actions evaluated on black hole solutions:

1. Violated: Action unbounded from below

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

(21)

...everything!

2. δI_E[g_cl, X_cl;δg, δX] = 0

3. δ²I_E[g_cl, X_cl;δg, δX]≥0

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

(22)

...everything!

2. δI_E[g_cl, X_cl;δg, δX] = 0

3. δ²I_E[g_cl, X_cl;δg, δX]≥0

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

(23)

...everything!

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

3. Frequently violated: Gaussian integral may diverge

Focus in this talk on thesecond problem!

(24)

...everything!

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

(25)

Outline

Free Energy

Applications

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

(26)

The Action

...for a review cf. e.g. DG, W. Kummer and D. Vassilevich,hep-th/0204253

Standard form of the action:

I_E =−1 2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γXK− Z

∂M

dx√ 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 curvatureof (∂M, γ)

Variational principle: fix X and induced metricγ at∂M

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

For later: Q(X) :=RX

U, w(X) :=RX

e^QV

(27)

The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γXK− Z

∂M

dx√ γL(X)

I Model specified by kinetic and potential functions for dilaton

U, w(X) :=RX

e^QV

(28)

The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X)(∇X)²−2V(X)

− Z

∂M

dx√

γXK− Z

∂M

dx√ γL(X)

I Model specified by kineticandpotential functions for dilaton

U, w(X) :=RX

e^QV

(29)

The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X)(∇X)²−2V(X)

− Z

∂M

dx√ γXK

− Z

∂M

dx√ γL(X)

I Dilaton gravity analog of Gibbons-Hawking-York boundary term:

coupling of X to extrinsic curvatureof(∂M, γ) Variational principle: fix X and induced metricγ at∂M

U, w(X) :=RX

e^QV

(30)

The Action

I_E =−1 2

Z

M

d²x√ g

XR−U(X)(∇X)²−2V(X)

− Z

∂M

dx√

γXK−

Z

∂M

dx√ γL(X)

I Dilaton gravity analog of Gibbons-Hawking-York boundary term:

coupling of X to extrinsic curvatureof(∂M, γ) Variational principle: fix X and induced metricγ at∂M

U, w(X) :=RX

e^QV

(31)

Selected List of Models

Black holes in(A)dS, asymptotically flat or arbitrary spaces

Model U(X) V(X)

1. Schwarzschild (1916) −_2X¹ −λ²

2. Jackiw-Teitelboim (1984) 0 ΛX

3. Witten Black Hole (1991) −_X¹ −2b²X

4. CGHS (1992) 0 −2b²

5.(A)dS2 ground state (1994) −_X^a BX

6. Rindler ground state (1996) −_X^a BX^a

7. Black Hole attractor (2003) 0 BX⁻¹

8. Spherically reduced gravity (N >3) −_(N−2)X^N−3 −λ²X(N−4)/(N−2)

9. All above: ab-family (1997) −_X^a BX^a+b

10. Liouville gravity a be^αX

11. Reissner-Nordstr¨om (1916) −_2X¹ −λ²+^Q_X²

12. Schwarzschild-(A)dS −_2X¹ −λ²−`X

13. Katanaev-Volovich (1986) α βX²−Λ

14. BTZ/Achucarro-Ortiz (1993) 0 ^Q_X² −_4X^J₃−ΛX

15. KK reduced CS (2003) 0 ¹₂X(c−X²)

16. KK red. conf. flat (2006) −¹₂tanh (X/2) AsinhX 17. 2D type 0A string Black Hole −_X¹ −2b²X+^b²_8π^q²

Models of interest: Boundary atX → ∞, w(X → ∞)→ ∞

(32)

Selected List of Models

Black holes in(A)dS,asymptotically flatorarbitrary spaces

Model U(X) V(X)

1. Schwarzschild (1916) −_2X¹ −λ²

2. Jackiw-Teitelboim (1984) 0 ΛX

3. Witten Black Hole (1991) −_X¹ −2b²X

4. CGHS (1992) 0 −2b²

5.(A)dS2 ground state (1994) −_X^a BX

6. Rindler ground state (1996) −_X^a BX^a

7. Black Hole attractor (2003) 0 BX⁻¹

8. Spherically reduced gravity (N >3) −_(N−2)X^N−3 −λ²X(N−4)/(N−2)

9. All above: ab-family (1997) −_X^a BX^a+b

10. Liouville gravity a be^αX

11. Reissner-Nordstr¨om (1916) −_2X¹ −λ²+^Q_X²

12. Schwarzschild-(A)dS −_2X¹ −λ²−`X

13. Katanaev-Volovich (1986) α βX²−Λ

14. BTZ/Achucarro-Ortiz (1993) 0 ^Q_X² −_4X^J₃−ΛX

15. KK reduced CS (2003) 0 ¹₂X(c−X²)

16. KK red. conf. flat (2006) −¹₂tanh (X/2) AsinhX 17. 2D type 0A string Black Hole −_X¹ −2b²X+^b²_8π^q²

Models of interest: Boundary atX → ∞, w(X → ∞)→ ∞

(33)

Outline

Free Energy

Applications

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

(34)

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 actionunbounded from below (cf. second assumption) I_E^reg =β 2M−w(X_reg)−2π X_hT

→ −∞ 2. First variation of actionnot zero for all field configurations

contributing to path integral due to boundary terms

3. Second variation of actionmay lead to divergent Gaussian integral

(35)

Free Energy? Not yet!

Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞ 2. First variation of actionnot zero for all field configurations

contributing to path integral due to boundary terms

(36)

Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞

2. First variation of actionnot zero for all field configurations contributing to path integral due to boundary terms

(37)

Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞

(38)

Zexp

−1

~I_E[g_cl, X_cl]

e^{−β F}

X≤X_reg

→ −∞

(39)

Variational Properties of the Action

δIE= Z

M

d²x√ gh

E^µνδgµν+EXδXi

| {z }

=0(EOM)

+ Z

∂M

dx√ γ h

π^abδγab+πXδXi

| {z }

=0?

6=0

Does this vanish on-shell? IgnoreπXδX and focus onπ^abδγab δIE =

Z dτ

»

−1

2∂rX δξ+. . . –

Square of Killing norm: ξ(X) =w(X)e^Q(X⁾−2Me^Q(X) Assume that boundary conditions preserved by variations

δξ ∼δMe^Q(X)

Because ∂rX =e^−Q we get δI_E =

Z

dτ δM 6= 0

(40)

δIE= Z

M

d²x√ gh

| {z }

=0(EOM)

+ Z

∂M

dx√ γ h

π^abδγab+πXδXi

| {z }

=0?

6=0

Z dτ

»

−1

2∂rX δξ+. . . –

δξ ∼δMe^Q(X)

Z

dτ δM 6= 0

(41)

δIE= Z

M

d²x√ gh

| {z }

=0(EOM)

+ Z

∂M

dx√ γ h

π^abδγab+πXδXi

| {z }

=0?

6=0

Z dτ

»

−1

2∂rX δξ+. . . –

Square of Killing norm: ξ(X) =w(X)e^Q(X⁾−2Me^Q(X)

Assume that boundary conditions preserved by variations δξ ∼δMe^Q(X)

Z

dτ δM 6= 0

(42)

δIE= Z

M

d²x√ gh

| {z }

=0(EOM)

+ Z

∂M

dx√ γ h

π^abδγab+πXδXi

| {z }

=0?

6=0

Z dτ

»

−1

2∂rX δξ+. . . –

δξ ∼δMe^Q(X)

Z

dτ δM 6= 0

(43)

δIE= Z

M

d²x√ gh

| {z }

=0(EOM)

+ Z

∂M

dx√ γ h

π^abδγab+πXδXi

| {z }

=0?

6=0

Z dτ

»

−1

2∂rX δξ+. . . –

δξ ∼δMe^Q(X)

Because ∂rX=e^−Q we get δI_E =

Z

dτ δM 6= 0

(44)

Boundary Counterterms

I Same idea as boundary counterterms inAdS/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: I_E =Γ+I_CT

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=−1 2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K− Z

∂M

dx√ γL(X)

| {z }

ICT

How to determine theboundary counterterm?

(45)

Γ=−1 2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K− Z

∂M

dx√ γL(X)

| {z }

ICT

(46)

Γ=−1 2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K− Z

∂M

dx√ γL(X)

| {z }

ICT

(47)

Γ=−1 2

Z

M

d²x√ g

X R−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K− Z

∂M

dx√ γL(X)

| {z }

ICT

(48)

Hamilton-Jacobi Equation

Boundary counterterm ICT is solution of the Hamilton-Jacobi equation

1. Begin with ‘Hamiltonian’ associated with radial evolution. H= 2π_Xγ_abπ^ab+ 2U(X)

γ_abπ^ab

2

+V(X) = 0

2. Momenta are functional derivatives of the on-shell action π^ab= 1

√γ δ δ γ_ab IE

EOM π_X = 1

√γ δ δ X IE

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!

ICT =− Z

∂M

dx√ γ

q

w(X)e^−Q(X)

(49)

Boundary counterterm ICT is solution of the Hamilton-Jacobi equation 1. Begin with ‘Hamiltonian’ associated with radial evolution.

H= 2π_Xγ_abπ^ab+ 2U(X)

γ_abπ^ab 2

+V(X) = 0

EOM π_X = 1

√γ δ δ X IE

EOM

I_CT!

ICT =− Z

∂M

dx√ γ

q

w(X)e^−Q(X)

(50)

γ_abπ^ab 2

+V(X) = 0

EOM π_X = 1

√γ δ δ X IE

EOM

I_CT!

ICT =− Z

∂M

dx√ γ

q

w(X)e^−Q(X)

(51)

γ_abπ^ab 2

+V(X) = 0

EOM π_X = 1

√γ δ δ X IE

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!

ICT =− Z

∂M

dx√ γ

q

w(X)e^−Q(X)

(52)

γ_abπ^ab 2

+V(X) = 0

EOM π_X = 1

√γ δ δ X IE

EOM

I_CT!

ICT =− Z

∂M

dx√ γ

q

w(X)e^−Q(X)

(53)

γ_abπ^ab 2

+V(X) = 0

EOM π_X = 1

√γ δ δ X IE

EOM

I_CT!

ICT =− Z

∂M

dx√ γ

q

w(X)e^−Q(X)

(54)

The Improved Action

The correct action for 2D dilaton gravity is Γ =−1

2 Z

M

d²x√ g

XR−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K+ Z

∂M

dx√ γ

q

w(X)e^−Q(X) Properties:

1. Yields the same EOM asI_E

2. Finite on-shell (solves first problem) Γ

EOM=β (M −2πX_hT)

3. First variation δΓvanishes on-shell ∀ δgµν and δX that preserve the boundary conditions (solvessecond problem)

δΓ

EOM= 0

Note: counterterm requires specification of integration constantw₀, i.e., choice of ground state, but is independent fromQ₀

(55)

The Improved Action

2 Z

M

d²x√ g

XR−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K+ Z

∂M

dx√ γ

q

δΓ

EOM= 0

(56)

The Improved Action

2 Z

M

d²x√ g

XR−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K+ Z

∂M

dx√ γ

q

δΓ

EOM= 0

(57)

The Improved Action

2 Z

M

d²x√ g

XR−U(X) (∇X)²−2V(X)

− Z

∂M

dx√

γ X K+ Z

∂M

dx√ γ

q

δΓ

EOM= 0

(58)

Free Energy

Γ(T_c, X_c) =β_cF_c(T_c, X_c) Explicitly:

Fc(Tc, Xc) =p

wce^−Q^c 1− r

1−2M wc

!

| {z }

=Ec(Tc,Xc)

−2πXhTc

| {z }

=STc

Entropy follows immediately (Bekenstein-Hawking law):

S =− ∂Fc

∂T_c Xc

= 2πX_h

Entropy determined by dilaton evaluated at the horizon (Gegenberg, Kunstatter, Louis-Martinez, 1995)

Similarly: dilaton chemical potential (surface pressure) ψc=−∂F_c/∂Xc|_T_c