Black hole thermodynamics

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Black hole thermodynamics

Daniel Grumiller

Institute for Theoretical Physics Vienna University of Technology

Spring workshop/Kosmologietag,Bielefeld, May 2014

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Main statements and overview

Black holes are thermal states Main statement 1

I four laws of black hole mechanics/thermodynamics

I phase transitions between black holes and vacuum

Black hole thermodynamics useful for quantum gravity checks Main statement 2

Black hole thermodynamics useful for quantum gravity concepts Main statement 2

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I quantum gravity entropy matching with semi-classical prediction SBH= kBc³

~G_N

| {z }

=1 in this talk

Ah

4 +O(lnAh)

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I information loss, fuzzballs, firewalls, ...

I black hole holography, AdS/CFT, gauge/gravity correspondence, ...

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Everything is geometry?

Gravity at low energies:

I described by general relativity

I basic field: metric g_µµ

I gauge symmetry: diffeomorphisms

δ_ξg_µν =∇_µξ_ν +∇_νξ_µ

I low energy action: Einstein–Hilbert SEH ∼ 1

κ Z

d⁴xp

|g|(Λ +R) + marginal + irrelevant

I basic field equations: Einstein equations Rµν = 0

I simplest solution: spherically symmetric Schwarzschild BH ds² =−(1−2M/r) dt²+ dr²/(1−2M/r) +r² dΩ²_S2

Everything phrased in terms of geometry! Classical gravity = geometry!

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κ Z

d⁴xp

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κ Z

d⁴xp

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κ Z

d⁴xp

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κ Z

d⁴xp

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κ Z

d⁴xp

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κ Z

d⁴xp

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Prehistory

I 1930ies, TOV: star in hydrostatic equilibrium

star = spherically symmetric, self-gravitating perfect fluid with linear equation of statep∝ρ

I 1960ies, Zel’dovich: bound from causality p≤ρ

causal limitp=ρ: speed of sound = speed of light

I with hindsight: interesting thermodynamical properties!

I in particular, entropy is not extensive,S(R)6=R³ (R=size)

I e.g. for p=ρ/3we get S(R)∝R^3/2

I for causal limitp=ρ:

S(R)∝R² ∝area Even before Bekenstein–Hawking:

Non-extensive entropy expected from/predicted by GR!

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Prehistory

I e.g. for p=ρ/3we get S(R)∝R^3/2

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Prehistory

I e.g. for p=ρ/3we getS(R)∝R^3/2

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Prehistory

S(R)∝R² ∝area

Even before Bekenstein–Hawking:

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Prehistory

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Thermodynamics and black holes — black hole thermodynamics?

Thermodynamics Zeroth law:

T =const. in equilibrium

First law:

dE∼T dS+work terms Second law:

dS≥0 Third law:

T →0impossible

T: temperature

E: energy S: entropy

Black hole mechanics Zeroth law:

κ=const. f. stationary black holes

First law:

dM ∼κdA+work terms Second law:

dA≥0 Third law: κ→0 impossible

κ: surface gravity

M: mass

A: area (of event horizon) Formal analogy or actual physics?

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T =const. in equilibrium First law:

dE∼T dS+work terms

Second law: dS≥0 Third law:

T →0impossible

T: temperature E: energy S: entropy

κ=const. f. stationary black holes First law:

dM ∼κdA+work terms

Second law: dA≥0 Third law: κ→0 impossible

κ: surface gravity M: mass

A: area (of event horizon)

Formal analogy or actual physics?

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dS≥0

Third law:

T →0impossible

T: temperature E: energy

dA≥0

Third law: κ→0 impossible

κ: surface gravity M: mass

Formal analogy or actual physics?

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dS≥0 Third law:

T →0 impossible T: temperature E: energy S: entropy

dA≥0 Third law:

κ→0 impossible κ: surface gravity M: mass

A: area (of event horizon) Formal analogy or actual physics?

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Bekenstein–Hawking entropy

I Gedankenexperiment by Wheeler: throw cup of lukewarm tea into BH!

δS_Universe<0

I violates second law of thermodynamics!

I Bekenstein: BHs have entropy proportional to area of event horizon! S_BH ∝A_h

I generalized second law holds:

δS_total =δS_Universe+δS_BH≥0

I Hawking: indeed!

S_BH = 1 4A_h using semi-classical gravity

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δS_Universe<0

I Hawking: indeed!

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δS_Universe<0

I Hawking: indeed!

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δS_Universe<0

I Bekenstein: BHs have entropy proportional to area of event horizon!

S_BH ∝A_h

I Hawking: indeed!

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δS_Universe<0

S_BH ∝A_h

I Hawking: indeed!

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δS_Universe<0

S_BH ∝A_h

I Hawking: indeed!

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Hawking temperature

I Hawking effect from QFT on fixed (curved) background: particle production with thermal spectrum at Hawking temperature

I Slick shortcut: Euclidean BHs! (t_E: Euclidean time)

ds² = (1−2M/r) dt²_E + dr²/(1−2M/r) + irrelevant

I near horizon approximation: r = 2M +x²/(8M) ds² =x² dt²_E

16M² + dx²+ irrelevant

I conical singularity atx= 0, unless

tE ∼tE + 8πM =tE +β

I periodicity in Euclidean time = inverse temperature

I Result: Hawking temperature!

T_H = 1 8πM

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Hawking temperature

T_H = 1 8πM

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Hawking temperature

T_H = 1 8πM

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Hawking temperature

T_H = 1 8πM

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Hawking temperature

T_H = 1 8πM

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Hawking temperature

T_H = 1 8πM

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Free energy from 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

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Main idea

Z = Z

DgDX exp

−1

~I_E[g, X]

I g: metric, X: scalar field

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Main idea

Z = Z

DgDX exp

−1

~I_E[g, X]

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Main idea

Z = Z

DgDX exp

−1

~I_E[g, X]

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Main idea

Z = Z

DgDX exp

−1

~I_E[g, X]

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Semiclassical Approximation

Consider small perturbation around classical solution

I_E[g_cl+δg, X_cl+δX] =I_E[g_cl, X_cl] +δI_E[g_cl, X_cl;δg, δX]

+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 g_cl and X_cl.

I The quadratic term represents the first corrections. If nothing goes wrong:

Z ∼exp

−1

~IE[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²IE

×. . .

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I_E[g_cl+δg, X_cl+δX] =I_E[g_cl, X_cl]+δI_E[g_cl, X_cl;δg, δX]

+1

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

I The linear term should vanish on solutions g_cl and X_cl.

Z ∼exp

−1

~IE[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²IE

×. . .

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+1

I Thelinear term should vanish on solutionsg_cl and X_cl.

Z ∼exp

−1

~IE[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²IE

×. . .

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+1

2δ²I_E[g_cl, X_cl;δg, δX]+. . .

I Thequadratic term represents the first corrections.

If nothing goes wrong:

Z ∼exp

−1

~IE[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²IE

×. . .

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+1

2δ²I_E[g_cl, X_cl;δg, δX]+. . .

I Thequadratic term represents the first corrections.

If nothing goes wrong:

Z ∼exp

−1

~IE[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²IE

×. . .

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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 Holographic renormalization resolvesfirstand second problem!

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...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

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...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

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...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

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What could go Wrong? ...everything!

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

Holographic renormalization resolvesfirstand second problem!

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What could go Wrong? ...everything!

δIE

EOM∼

Z

∂M

dx√ γ

h

6= 0

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Holographic renormalization

Subtract suitable boundary terms from the action Γ =IE−ICT

such that second problemresolved; typically also resolvesfirst problem

Z ∼X

gcl

exp

−1

~Γ[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²Γ

×. . .

I Leading term is finite

I Linear termvanishes

I Quadratic termok in AdS

Leading order (set ~= 1):

Z(T, X) =e^{−Γ(T, X)} =e^{−βF(T, X)} Here F is the Helmholtz free energy

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Z ∼X

gcl

exp

−1

~Γ[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²Γ

×. . .

I Quadratic termok in AdS Leading order (set ~= 1):

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Z ∼X

gcl

exp

−1

~Γ[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²Γ

×. . .

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Z ∼X

gcl

exp

−1

~Γ[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²Γ

×. . .

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Z ∼X

gcl

exp

−1

~Γ[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²Γ

×. . .

I Quadratic termok in AdS

Leading order (set ~= 1):

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Z ∼X

gcl

exp

−1

~Γ[g_cl, X_cl] Z

DδgDδXexp

− 1 2~δ²Γ

×. . .

Z(T, X) =e^{−Γ(T, X)} =e^−βF^{(T, X)} Here F is the Helmholtz free energy

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Example: Hawking–Page phase transition of AdS3/BTZ

I Euclidean action: IE =R d³x√

g R+ 2 + 2R

d²x√ γ K

I Holographic counterterm: ICT= 2R d²x√

γ

I (Euclidean) AdS₃ (t_E ∼t_E +β,ϕ∼ϕ+ 2π)

ds² = cosh²ρ dt²_E+ sinh²ρ dϕ²+ dρ² yields free energy

FAdS=−1 8

I (non-rotating) BTZ BH

ds² =−(r²−r²₊) dt²+ dr²

r²−r²₊ +r²dϕ² yields free energy (T =r₊/(2π))

FBTZ=−π²T² 2 =−1

8 T² T_crit.²

I Critical Hawking–Page temperature: Tcrit.= 1/(2π)

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g R+ 2 + 2R

d²x√ γ K

γ

FAdS=−1 8

ds² =−(r²−r²₊) dt²+ dr²

FBTZ=−π²T² 2 =−1

8 T² T_crit.²

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g R+ 2 + 2R

d²x√ γ K

γ

FAdS=−1 8

ds² =−(r²−r²₊) dt²+ dr²

FBTZ=−π²T² 2 =−1

8 T² T_crit.²

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g R+ 2 + 2R

d²x√ γ K

γ

FAdS=−1 8

ds² =−(r²−r²₊) dt²+ dr²

FBTZ=−π²T² 2 =−1

8 T² T_crit.²

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g R+ 2 + 2R

d²x√ γ K

γ

FAdS=−1 8

ds² =−(r²−r²₊) dt²+ dr²

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Works also for flat space and expanding universe (in 2+1)

Bagchi, Detournay, Grumiller & Simon PRL ’13

Hot flat space (ϕ∼ϕ+ 2π)

ds² = dt²+ dr²+r² dϕ²

ds² = dτ²+(Eτ)² dx²

1 + (Eτ)² + 1 + (Eτ)²

dy+ (Eτ)²

1 + (Eτ)² dx2

Flat space cosmology (y ∼y+ 2πr0)

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Works also for flat space and expanding universe (in 2+1)

Bagchi, Detournay, Grumiller & Simon PRL ’13

Hot flat space (ϕ∼ϕ+ 2π)

ds² = dt²+ dr²+r² dϕ²

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Summary, outlook and rest of the talk (if time permits) Summary:

I Black hole are thermal states

I Calculation of free energy requires holographic renormalization

I Interesting phase transitions possible

I Generalizable to (flat space) cosmologies

Regarding the other two main points: Black hole thermodynamics useful for quantum gravity checks/concepts

I Information loss?

I

I Microscopic entropy matching via Cardy formula

I Black hole complementarity and firewalls

I Everything is information?

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I Information loss?

I

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I Information loss?

I Black hole holography

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I Information loss?

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I Information loss?

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I Information loss?

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I Information loss?

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Holography — Main idea

aka gauge/gravity duality, aka AdS/CFT correspondence

One of the most fruitful ideas in contemporary theoretical physics:

I The number of dimensions is a matter of perspective

I We can choose to describe the same physical situation using two different formulations in two different dimensions

I The formulation in higher dimensions is a theory with gravity

I The formulation in lower dimensions is a theory without gravity

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I We can choose to describe the same physical situation using two different formulations in two different dimensions

I The formulation in higher dimensions is a theory with gravity

I The formulation in lower dimensions is a theory without gravity

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Why gravity?

The holographic principle in black hole physics

Boltzmann/Planck: entropy of photon gas indspatial dimensions S_gauge∝volume∝L^d

Bekenstein/Hawking: entropy of black hole in dspatial dimensions S_gravity∝area∝L^d−1

Daring idea by ’t Hooft/Susskind (1990ies):

Any consistent quantum theory of gravity could/should have a holographic formulation in terms of a field theory in one dimension lower Ground-breaking discovery by Maldacena (1997):

Holographic principle is realized in string theory in specific way e.g. hT_µνi_gauge=T_µν^BY δ(gravity action) =

Z

d^dxp

|h|T_µν^BY δh^µν

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Why gravity?

Any consistent quantum theory of gravity could/should have a holographic formulation in terms of a field theory in one dimension lower

Ground-breaking discovery by Maldacena (1997):

Holographic principle is realized in string theory in specific way e.g. hT_µνi_gauge=T_µν^BY δ(gravity action) =

Z

d^dxp

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Why gravity?

Holographic principle is realized in string theory in specific way

e.g. hT_µνi_gauge=T_µν^BY δ(gravity action) = Z

d^dxp

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Why gravity?

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Why should I care?

...and why were there>9700papers on holography in the past 17 years?

I Many applications!

I Tool for calculations

I Strongly coupled gauge theories (difficult) mapped to semi-cassical gravity (simple)

I Quantum gravity (difficult) mapped to weakly coupled gauge theories (simple)

I Examples of first type: heavy ion collisions at RHIC and LHC, superfluidity, highT_csuperconductors (?), cold atoms (?), strange metals (?), ...

I Examples of the second type: microscopic understanding of black holes, information paradox, 3D quantum gravity, flat space holography, non-AdS holography, higher-spin holography, ...

We can expect many new applications in the next decade!

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Why should I care?

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Why should I care?

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Why should I care?

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Why should I care?

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Why should I care?