Black hole thermodynamics - uni-bielefeld.de · 2014. 5. 12. · Black hole thermodynamics Daniel...

Transcript of Black hole thermodynamics - uni-bielefeld.de · 2014. 5. 12. · Black hole thermodynamics Daniel...

Black hole thermodynamics

Daniel Grumiller

Institute for Theoretical PhysicsVienna University of Technology

Spring workshop/Kosmologietag, Bielefeld, May 2014

with R. McNees and J. Salzer: 1402.5127
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

Daniel Grumiller — Black hole thermodynamics 2/19


Main statement 1


Main statement 2

I quantum gravity entropy matching with semi-classical prediction

SBH =kBc

3

~GN︸︷︷︸=1 in this talk

Ah4

+O(lnAh)

I semi-classical log corrections of entropy


Main statement 2



Main statement 1


Main statement 2


Main statement 2

I information loss, fuzzballs, firewalls, ...

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

Everything is geometry?

Gravity at low energies:I described by general relativity

I basic field: metric gµµI gauge symmetry: diffeomorphisms

δξgµν = ∇µξν +∇νξµI low energy action: Einstein–Hilbert

SEH ∼1

κ

∫d4x√|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 dΩ2S2

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


Gravity at low energies:I described by general relativityI basic field: metric gµµ

I gauge symmetry: diffeomorphisms


SEH ∼1

κ



Rµν = 0


ds2 = −(1− 2M/r) dt2 + dr2/(1− 2M/r) + r2 dΩ2S2



Gravity at low energies:I described by general relativityI basic field: metric gµµI gauge symmetry: diffeomorphisms

δξgµν = ∇µξν +∇νξµ

I low energy action: Einstein–Hilbert

SEH ∼1

κ



Rµν = 0


ds2 = −(1− 2M/r) dt2 + dr2/(1− 2M/r) + r2 dΩ2S2





SEH ∼1

κ



Rµν = 0


ds2 = −(1− 2M/r) dt2 + dr2/(1− 2M/r) + r2 dΩ2S2





SEH ∼1

κ



Rµν = 0


ds2 = −(1− 2M/r) dt2 + dr2/(1− 2M/r) + r2 dΩ2S2





SEH ∼1

κ



Rµν = 0


ds2 = −(1− 2M/r) dt2 + dr2/(1− 2M/r) + r2 dΩ2S2





SEH ∼1

κ



Rµν = 0


ds2 = −(1− 2M/r) dt2 + dr2/(1− 2M/r) + r2 dΩ2S2

Everything phrased in terms of geometry! Classical gravity = geometry!Daniel Grumiller — Black hole thermodynamics 3/19
Prehistory

I 1930ies, TOV: star in hydrostatic equilibriumstar = spherically symmetric, self-gravitating perfect fluid with linearequation of state p ∝ ρ

I 1960ies, Zel’dovich: bound from causality

p ≤ ρ

causal limit p = ρ: speed of sound = speed of lightI with hindsight: interesting thermodynamical properties!I in particular, entropy is not extensive, S(R) 6= R3 (R = size)I e.g. for p = ρ/3 we get S(R) ∝ R3/2I for causal limit p = ρ:

S(R) ∝ R2 ∝ area

Even before Bekenstein–Hawking:

Non-extensive entropy expected from/predicted by GR!

Prehistory



p ≤ ρ

causal limit p = ρ: speed of sound = speed of lightI with hindsight: interesting thermodynamical properties!

I in particular, entropy is not extensive, S(R) 6= R3 (R = size)I e.g. for p = ρ/3 we get S(R) ∝ R3/2I for causal limit p = ρ:




Prehistory



p ≤ ρ

causal limit p = ρ: speed of sound = speed of lightI with hindsight: interesting thermodynamical properties!I in particular, entropy is not extensive, S(R) 6= R3 (R = size)I e.g. for p = ρ/3 we get S(R) ∝ R3/2

I for causal limit p = ρ:




Prehistory



p ≤ ρ





Prehistory



p ≤ ρ





Thermodynamics and black holes — black hole thermodynamics?

Thermodynamics

Zeroth law:T = const. in equilibrium

First law:dE ∼ TdS+ work terms

Second law:dS ≥ 0

Third law:T → 0 impossible

T : temperature

E: energyS: 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 : massA: area (of event horizon)

Formal analogy or actual physics?


Thermodynamics



Second law:dS ≥ 0


T : temperatureE: energyS: entropy




Second law:dA ≥ 0


κ: surface gravityM : massA: area (of event horizon)



Thermodynamics



Second law:dS ≥ 0


T : temperatureE: energyS: entropy




Second law:dA ≥ 0


κ: surface gravityM : massA: area (of event horizon)



Thermodynamics



Second law:dS ≥ 0

Third law:T → 0 impossibleT : temperatureE: energyS: entropy




Second law:dA ≥ 0

Third law:κ→ 0 impossibleκ: surface gravityM : massA: area (of event horizon)


Bekenstein–Hawking entropy

I Gedankenexperiment by Wheeler: throw cup of lukewarm tea intoBH!

δSUniverse < 0

I violates second law of thermodynamics!

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

SBH ∝ Ah

I generalized second law holds:

δStotal = δSUniverse + δSBH ≥ 0

I Hawking: indeed!

SBH =1

4Ah

using semi-classical gravity



δSUniverse < 0



SBH ∝ Ah



I Hawking: indeed!

SBH =1

4Ah




δSUniverse < 0



SBH ∝ Ah



I Hawking: indeed!

SBH =1

4Ah




δSUniverse < 0



SBH ∝ Ah



I Hawking: indeed!

SBH =1

4Ah




δSUniverse < 0



SBH ∝ Ah



I Hawking: indeed!

SBH =1

4Ah




δSUniverse < 0



SBH ∝ Ah



I Hawking: indeed!

SBH =1

4Ah


Hawking temperature

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

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

ds2 = (1− 2M/r) dt2E + dr2/(1− 2M/r) + irrelevantI near horizon approximation: r = 2M + x2/(8M)

ds2 = x2dt2E

16M2+ dx2 + irrelevant

I conical singularity at x = 0, unless

tE ∼ tE + 8πM = tE + βI periodicity in Euclidean time = inverse temperatureI Result: Hawking temperature!

TH =1

8πM

Hawking temperature



ds2 = (1− 2M/r) dt2E + dr2/(1− 2M/r) + irrelevant

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

ds2 = x2dt2E




TH =1

8πM

Hawking temperature




ds2 = x2dt2E




TH =1

8πM

Hawking temperature




ds2 = x2dt2E



tE ∼ tE + 8πM = tE + β

I periodicity in Euclidean time = inverse temperatureI Result: Hawking temperature!

TH =1

8πM

Hawking temperature




ds2 = x2dt2E



tE ∼ tE + 8πM = tE + βI periodicity in Euclidean time = inverse temperature

I Result: Hawking temperature!

TH =1

8πM

Hawking temperature




ds2 = x2dt2E




TH =1

8πM

Free energy from Euclidean path integralMain 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−βΩ

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−βΩ






Z =∫

DgDX exp

(−1~IE [g,X]

)


I Semiclassical limit (~→ 0): dominated by classical solutions (?)

I Exploit relationship between Z and Euclidean partition function

Z ∼ e−βΩ






Z =∫

DgDX exp

(−1~IE [g,X]

)



Z ∼ e−βΩ






Z =∫

DgDX exp

(−1~IE [g,X]

)



Z ∼ e−βΩ




Free energy from Euclidean path integralSemiclassical Approximation

Consider small perturbation around classical solution

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

+1

2δ2IE [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

)× . . .




+1







) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .




+1







) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .




+1







) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .




+1







) ∫DδgDδX exp

(− 1

2~δ2IE

)× . . .

Free energy from Euclidean path integralWhat 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] ≥ 0Typical 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

Holographic renormalization resolves first and second problem!


...everything!


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:



δIE∣∣EOM


]6= 0




...everything!







δIE∣∣EOM


]6= 0




...everything!







δIE∣∣EOM


]6= 0



Free energy from Euclidean path integralWhat could go Wrong? ...everything!






δIE∣∣EOM


]6= 0



Free energy from Euclidean path integralWhat could go Wrong? ...everything!






δIE∣∣EOM


]6= 0



Free energy from Euclidean path integralHolographic renormalization

Subtract suitable boundary terms from the action

Γ = IE − ICTsuch that second problem resolved; typically also resolves first problem

Z ∼∑gcl

exp

(−1~

Γ[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2Γ

)× . . .

I Leading term is finiteI Linear term vanishesI Quadratic term ok in AdS

Leading order (set ~ = 1):

Z(T, X) = e−Γ(T,X) = e−βF (T,X)

Here F is the Helmholtz free energy




Z ∼∑gcl

exp

(−1~

Γ[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2Γ

)× . . .



Z(T, X) = e−Γ(T,X) = e−βF (T,X)





Z ∼∑gcl

exp

(−1~

Γ[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2Γ

)× . . .

I Leading term is finite

I Linear term vanishesI Quadratic term ok in AdS


Z(T, X) = e−Γ(T,X) = e−βF (T,X)





Z ∼∑gcl

exp

(−1~

Γ[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2Γ

)× . . .

I Leading term is finiteI Linear term vanishes

I Quadratic term ok in AdS


Z(T, X) = e−Γ(T,X) = e−βF (T,X)





Z ∼∑gcl

exp

(−1~

Γ[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2Γ

)× . . .



Z(T, X) = e−Γ(T,X) = e−βF (T,X)





Z ∼∑gcl

exp

(−1~

Γ[gcl, Xcl]

) ∫DδgDδX exp

(− 1

2~δ2Γ

)× . . .



Z(T, X) = e−Γ(T,X) = e−βF (T,X)

Here F is the Helmholtz free energyDaniel Grumiller — Black hole thermodynamics 11/19
Example: Hawking–Page phase transition of AdS3/BTZ

I Euclidean action: IE =∫

d3x√g(R+ 2

)+ 2

∫d2x√γ K

I Holographic counterterm: ICT = 2∫

d2x√γ

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

ds2 = cosh2 ρ dt2E + sinh2 ρ dϕ2 + dρ2

yields free energy

FAdS = −1

8I (non-rotating) BTZ BH

ds2 = −(r2 − r2+) dt2 +dr2

r2 − r2++ r2 dϕ2

yields free energy (T = r+/(2π))

FBTZ = −π2T 2

2= −1

8

T 2

T 2crit.

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



d3x√g(R+ 2

)+ 2

∫d2x√γ K


d2x√γ



yields free energy

FAdS = −1


ds2 = −(r2 − r2+) dt2 +dr2

r2 − r2++ r2 dϕ2


FBTZ = −π2T 2

2= −1

8

T 2

T 2crit.




d3x√g(R+ 2

)+ 2

∫d2x√γ K


d2x√γ



yields free energy

FAdS = −1

8

I (non-rotating) BTZ BH

ds2 = −(r2 − r2+) dt2 +dr2

r2 − r2++ r2 dϕ2


FBTZ = −π2T 2

2= −1

8

T 2

T 2crit.




d3x√g(R+ 2

)+ 2

∫d2x√γ K


d2x√γ



yields free energy

FAdS = −1


ds2 = −(r2 − r2+) dt2 +dr2

r2 − r2++ r2 dϕ2


FBTZ = −π2T 2

2= −1

8

T 2

T 2crit.




d3x√g(R+ 2

)+ 2

∫d2x√γ K


d2x√γ



yields free energy

FAdS = −1


ds2 = −(r2 − r2+) dt2 +dr2

r2 − r2++ r2 dϕ2


FBTZ = −π2T 2

2= −1

8

T 2

T 2crit.


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)2dx)2

Flat space cosmology (y ∼ y + 2πr0)

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)2dx)2

Flat space cosmology (y ∼ y + 2πr0)Daniel Grumiller — Black hole thermodynamics 13/19
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 usefulfor quantum gravity checks/concepts

I Information loss?

II Microscopic entropy matching via Cardy formula

I Black hole complementarity and firewalls

I Everything is information?


Summary:






I Information loss?

II Microscopic entropy matching via Cardy formula




Summary:






I Information loss?

I Black hole holography

I Microscopic entropy matching via Cardy formula




Summary:






I Information loss?






Summary:






I Information loss?






Summary:






I Information loss?






Summary:






I Information loss?





Holography — Main ideaaka 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 twodifferent formulations in two different dimensions

I The formulation in higher dimensions is a theory with gravityI The formulation in lower dimensions is a theory without gravity


One of the most fruitful ideas in contemporary theoretical physics:I The number of dimensions is a matter of perspectiveI 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 gravityI The formulation in lower dimensions is a theory without gravity


One of the most fruitful ideas in contemporary theoretical physics:I The number of dimensions is a matter of perspectiveI We can choose to describe the same physical situation using two

different formulations in two different dimensionsI The formulation in higher dimensions is a theory with gravityI The formulation in lower dimensions is a theory without gravity

Why gravity?The holographic principle in black hole physics

Boltzmann/Planck: entropy of photon gas in d spatial dimensions

Sgauge ∝ volume ∝ Ld

Bekenstein/Hawking: entropy of black hole in d spatial dimensions

Sgravity ∝ area ∝ Ld−1

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

Any consistent quantum theory of gravity could/should have a holo-graphic 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. 〈Tµν〉gauge = TBYµν δ(gravity action) =∫

ddx√|h|TBYµν δhµν

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

I Many applications!

I Tool for calculations

I Strongly coupled gauge theories (difficult) mapped to semi-cassicalgravity (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, high Tc superconductors (?), cold atoms (?), strangemetals (?), ...

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

We can expect many new applications in the next decade!

Thanks for your attention!

References

D. Grumiller, R. McNees and J. Salzer,“Black hole thermodynamics — The first half century,”[arXiv:1402.5127 [gr-qc]].

H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel,“Higher spin theory in 3-dimensional flat space,”Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768 [hep-th]].

A. Bagchi, S. Detournay, D. Grumiller and J. Simon,“Cosmic evolution from phase transition of 3-dimensional flat space,”Phys. Rev. Lett. 111 (2013) 181301 [arXiv:1305.2919 [hep-th]].

A. Bagchi, S. Detournay and D. Grumiller,“Flat-Space Chiral Gravity,”Phys. Rev. Lett. 109 (2012) 151301 [arXiv:1208.1658 [hep-th]].

Thanks to Bob McNees for providing the LATEX beamerclass!

Black hole thermodynamics - uni-bielefeld.de · 2014. 5. 12. · Black hole thermodynamics Daniel...

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Transcript of Black hole thermodynamics - uni-bielefeld.de · 2014. 5. 12. · Black hole thermodynamics Daniel...