Gravity in Flatland Black holes

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Gravity in Flatland

Black holes in lower dimensions

Daniel Grumiller

Institute for Theoretical Physics TU Wien

Colloquium, U. W¨urzburg, November 2019

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Outline

Motivation

Gravity in three dimensions

Gravity in two dimensions

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Seeing is believing...

Daniel Grumiller — Gravity in Flatland 4/25

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Outline

Motivation

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Some open issues in gravity I IR (classical gravity)

I asymptotic symmetries I soft physics

I near horizon symmetries I UV (quantum gravity)

I black holeevaporation and unitarity I black holemicrostates

I UV/IR (holography)

I AdS/CFT and applications (seeErdmenger,Meyerand collaborators) I precision holography

I generality of holography

I all issues above can be addressed in lower dimensions I lower dimensions technically simpler

I hope to resolve conceptual problems

Daniel Grumiller — Gravity in Flatland Motivation 6/25

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I near horizon symmetries

Equivalence principle needs modification Take-away slogan

I UV (quantum gravity)

I numerous conceptual issues

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I precision holography I generality of holography

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Find ‘hydrogen-atom’ of quantum gravity Take-away homework

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See book byErdmengeror lecture notes 1807.09872

I precision holography I generality of holography

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(When) is quantum gravity in D+ 1 dimensions equivalent to (which) quantum field theory in Ddimensions?

Take-away question(s)

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I IR (classical gravity) I asymptotic symmetries I soft physics

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Gravity in various dimensions

Riemann-tensor ^D²^(D₁₂²⁻¹⁾ components inD dimensions:

I 11D: 1210 (1144 Weyl and 66 Ricci) I 10D: 825 (770 Weyl and 55 Ricci) I 5D: 50 (35 Weyl and 15 Ricci) I 4D: 20 (10 Weyl and 10 Ricci)

I 3D: 6 (Ricci) I 2D: 1 (Ricci scalar)

I 1D: 0 (space or time but not both⇒ no lightcones)

Caveat: just counting tensor components can be misleading as measure of complexity Example: largeDlimit actually simple for some problems (Emparan et al.)

I 2D: lowest dimension exhibitingblack holes (BHs) I Simplest gravitational theories withBHs in 2D I No Einstein gravity

I 3D: lowest dimension exhibitingBHsand gravitons

I Simplest gravitational theories withBHs and gravitons in 3D I Lowest dimension for Einstein gravity (BHs but no gravitons)

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I 11D: 1210 (1144 Weyl and 66 Ricci) I 10D: 825 (770 Weyl and 55 Ricci) I 5D: 50 (35 Weyl and 15 Ricci) I 4D: 20 (10 Weyl and 10 Ricci) I 3D: 6 (Ricci)

I 2D: 1 (Ricci scalar)

Apply as mantra the slogan “as simple as possible, but not simpler”

I Simplest gravitational theories withBHs in 2D I No Einstein gravity

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Outline

Motivation

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Could thisblack holebe the ‘hydrogen atom’ for quantum gravity?

Daniel Grumiller — Gravity in Flatland Gravity in three dimensions 9/25

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Choice of theory

I Choice of bulk action

Pick Einstein–Hilbert action with negative cc (Λ =−1/`²) IEH[g] =− 1

16πG Z

M

d³x√

−g

R+ 2

`²

Usually choose also topology ofM, e.g. cylinder

I Choice of boundary conditions

Crucial to define theory — yields spectrum of ‘edge states’ Pick whatever suits best to describe relevant physics

I Goal: understand holography beyond AdS/CFT I Explain first in general how edge states emerge

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Choice of theory

16πG Z

M

d³x√

−g

R+ 2

`²

Usually choose also topology ofM, e.g. cylinder Main features:

I no local physical degrees of freedom

I rotating (BTZ)black holesolutions analogous to Kerr ds²=−(r²−r₊²)(r²−r²₋)

`²r² dt²+ `²r²dr²

(r²−r²₊)(r²−r₋²)+r²

dϕ−r+r−

`r² dt² I conserved massM = (r²₊+r₋²)/`² and angular mom.J = 2r+r₋/` I Bekenstein–Hawking entropy

S_BH= A

4G =πr+

2G

Hawking–Unruh temperature: T = (r₊² −r₋²)/(2πr+`²) I Choice of boundary conditions

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Choice of theory

16πG Z

M

d³x√

−g

R+ 2

`²

Usually choose also topology ofM, e.g. cylinder Main features:

I no local physical degrees of freedom I all solutions locally and asymptotically AdS₃

I rotating (BTZ)black holesolutions analogous to Kerr ds²=−(r²−r₊²)(r²−r²₋)

`²r² dt²+ `²r²dr²

(r²−r²₊)(r²−r₋²)+r²

dϕ−r+r−

`r² dt² I conserved massM = (r²₊+r₋²)/`² and angular mom.J = 2r+r₋/` I Bekenstein–Hawking entropy

S_BH= A

4G =πr+

2G

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Choice of theory

Pick Einstein–Hilbert action with negative cc (Λ =−1/`²) Main features:

I no local physical degrees of freedom I all solutions locally and asymptotically AdS3

I rotating (BTZ)black holesolutions analogous to Kerr ds²=−(r²−r₊²)(r²−r₋²)

`²r² dt²+ `²r²dr²

(r²−r²₊)(r²−r²₋)+r²

dϕ−r+r₋

`r² dt2

t: time, ϕ∼ϕ+ 2π: angular coordinate,r: radial coordinate r→ ∞: asymptotic region

r→r+≥r−: black holehorizon r→r₋≥0: inner horizon r₊→r₋ >0: extremal BTZ r₋→0: non-rotating BTZ

+ −

I Bekenstein–Hawking entropy S_BH= A

4G =πr₊ 2G

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Choice of theory

`²r² dt²+ `²r²dr²

(r²−r²₊)(r²−r²₋)+r²

dϕ−r+r₋

`r² dt2

I conserved massM = (r₊² +r₋²)/`² and angular mom.J = 2r₊r₋/`

I Bekenstein–Hawking entropy SBH= A

4G =πr₊ 2G

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Choice of theory

`²r² dt²+ `²r²dr²

(r²−r²₊)(r²−r²₋)+r²

dϕ−r+r₋

`r² dt2

I conserved massM = (r₊² +r₋²)/`² and angular mom.J = 2r₊r₋/`

I Bekenstein–Hawking entropy SBH= A

4G =πr₊ 2G

Hawking–Unruh temperature: T = (r₊² −r₋²)/(2πr+`²)

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Choice of theory

Pick Einstein–Hilbert action with negative cc (Λ =−1/`²) I Choice of boundary conditions

Crucial to define theory — yields spectrum of ‘edge states’

Pick whatever suits best to describe relevant physics

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Pick Einstein–Hilbert action with negative cc (Λ =−1/`²) I Choice of boundary conditions

Crucial to define theory — yields spectrum of ‘edge states’

Pick whatever suits best to describe relevant physics

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Physics with boundaries

Science is a differential equation. Religion is a boundary condition. — Alan Turing

I Many QFT applications employ “natural boundary conditions”:

fields and fluctuations tend to zero asymptotically

I Notable exceptions exist in gauge theories with boundaries: e.g. in Quantum Hall effect

I Natural boundary conditions not applicable in gravity: metric must not vanish asymptotically

I Gauge or gravity theories in presence of (asymptotic) boundaries: asymptotic symmetries

I Choice of boundary conditions determines asymptotic symmetries

All boundary condition preserving gauge transformations (bcpgt’s) modulo trivial gauge transformations

Definition of asymptotic symmetries

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I Notable exceptions exist in gauge theories with boundaries:

e.g. in Quantum Hall effect

metric must not vanish asymptotically

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I Natural boundary conditions not applicable in gravity:

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I Gauge or gravity theories in presence of (asymptotic) boundaries:

asymptotic symmetries

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I Gauge or gravity theories in presence of (asymptotic) boundaries:

asymptotic symmetries

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Asymptotic symmetries in gravity

I Impose some bc’s at (asymptotic or actual) boundary:

r→rlim_bgµν(r, xⁱ) = ¯gµν(r_b, xⁱ) +δgµν(r_b, xⁱ)

L_ξgµν !

=O(δg_µν) I typically, Killing vectors can be expanded radially

ξ^µ(r_b, xⁱ)=ξ₍₀₎^µ (r_b, xⁱ)+

Lie bracket quotient algebra of asymptotic Killing vectors modulo trivial diffeos

Definition of asymptotic symmetry algebra

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r→rlim_bg_µν(r, xⁱ) = ¯g_µν(r_b, xⁱ) +δg_µν(r_b, xⁱ) r: some convenient (“radial”) coordinate

I bcpgt’s generated by asymptotic Killing vectors ξ: L_ξgµν !

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r→rlim_bgµν(r, xⁱ) = ¯gµν(rb, xⁱ) +δgµν(rb, xⁱ)

r: some convenient (“radial”) coordinate r_b: value ofr at boundary (could be∞)

L_ξgµν

=! O(δg_µν) I typically, Killing vectors can be expanded radially

ξ^µ(rb, xⁱ)=ξ₍₀₎^µ (rb, xⁱ)+

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r→rlim_bgµν(r, xⁱ) = ¯gµν(rb,xⁱ) +δgµν(rb,xⁱ)

r: some convenient (“radial”) coordinate r_b: value ofr at boundary (could be∞)

xⁱ: remaining coordinates (“boundary” coordinates)

I bcpgt’s generated by asymptotic Killing vectors ξ: L_ξgµν !

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r→rlim_bgµν(r, xⁱ)= ¯gµν(rb, xⁱ) +δgµν(rb, xⁱ)

r: some convenient (“radial”) coordinate r_b: value ofr at boundary (could be∞) xⁱ: remaining coordinates

g_µν: metric compatible with bc’s

L_ξg_µν =^! O(δg_µν) I typically, Killing vectors can be expanded radially

ξ^µ(rb, xⁱ)=ξ₍₀₎^µ (rb, xⁱ)+

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r→rlim_bgµν(r, xⁱ) =¯gµν(rb, xⁱ)+δgµν(rb, xⁱ)

¯

gµν: (asymptotic) background metric

I bcpgt’s generated by asymptotic Killing vectors ξ: L_ξg_µν =^! O(δg_µν) I typically, Killing vectors can be expanded radially

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r→rlim_bgµν(r, xⁱ) = ¯gµν(rb, xⁱ) +δgµν(rb, xⁱ)

¯

gµν: (asymptotic) background metric δg_µν: fluctuations permitted by bc’s

L_ξgµν !

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¯

I bcpgt’s generated by asymptotic Killing vectors ξ:

L_ξgµν !

=O(δg_µν)

I typically, Killing vectors can be expanded radially ξ^µ(r_b, xⁱ)=ξ₍₀₎^µ (r_b, xⁱ)+

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Asymptotic symmetries in gravity — modification of equivalence principle I Impose some bc’s at (asymptotic or actual) boundary:

¯

L_ξgµν !

ξ^µ(r_b, xⁱ)=ξ^µ₍₀₎(r_b, xⁱ)+ subleading terms

ξ₍₀₎^µ (r_b, xⁱ): generates asymptotic symmetries/changes physical state subleading terms: generate trivial diffeos

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Asymptotic symmetries in gravity — modification of equivalence principle I Impose some bc’s at (asymptotic or actual) boundary:

gµν: metric compatible with bc’s

¯

L_ξgµν !

ξ^µ(rb, xⁱ)=ξ₍₀₎^µ (rb, xⁱ)+ trivial diffeos

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Canonical boundary charges

God made the bulk; surfaces were invented by the devil — Wolfgang Pauli

I changing boundary conditions can change physical spectrum

perform Hamiltonian analysis in presence of boundaries I in Hamiltonian language: gauge generatorG[]varies as

δG[] = Z

Σ

(bulk term) δΦ− Z

∂Σ

(boundary term) δΦ not functionally differentiable in general (Σ: constant time slice) I add boundary term to restore functional differentiability

δΓ[] =δG[] +δQ[]=^! Z

Σ

(bulk term) δΦ I yields (variation of) canonical boundary charges

δQ[] = Z

∂Σ

(boundary term) δΦ

Trivial gauge transformations generated by some with Q[] = 0

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simple example: quantum mechanics of free particle on half-line x≥0

time-independent Schr¨odinger equation:

− d²

dx²ψ(x) =Eψ(x) look for (normalizable) bound state solutions,E <0

I Dirichlet bc’s: no bound states I Neumann bc’s: no bound states I Robin bc’s

ψ+αψ⁰

_x=0₊= 0 α∈R⁺ lead to one bound state

ψ(x) x≥0=

r2

αe^−x/α

with energyE=−1/α², localized exponentially nearx= 0 I to distinguish asymptotic symmetries from trivial gauge trafos:

δG[] = Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

δQ[] = Z

∂Σ

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simple example: quantum mechanics of free particle on half-line x≥0 time-independent Schr¨odinger equation:

− d²

I Dirichlet bc’s: no bound states I Neumann bc’s: no bound states

ψ+αψ⁰

ψ(x) x≥0=

r2

αe^−x/α

with energyE=−1/α², localized exponentially nearx= 0 I to distinguish asymptotic symmetries from trivial gauge trafos:

δG[] = Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

δQ[] = Z

∂Σ

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simple example: quantum mechanics of free particle on half-line x≥0 time-independent Schr¨odinger equation:

− d²

I Dirichlet bc’s: no bound states I Neumann bc’s: no bound states I Robin bc’s

ψ+αψ⁰

ψ(x) x≥0=

r2

αe^−x/α

with energyE=−1/α², localized exponentially nearx= 0

I to distinguish asymptotic symmetries from trivial gauge trafos: perform Hamiltonian analysis in presence of boundaries

I in Hamiltonian language: gauge generatorG[]varies as δG[] =

Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

δQ[] = Z

∂Σ

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I to distinguish asymptotic symmetries from trivial gauge trafos: either use Noether’s second theorem and covariant phase space analysis or perform Hamiltonian analysis in presence of boundaries

Some references:

I covariant phase space: Lee, Wald ’90,Iyer, Wald ’94andBarnich, Brandt ’02

I review: seeComp`ere, Fiorucci ’18 and refs. therein

I canonical analysis: Arnowitt, Deser, Misner ’59, Regge, Teitelboim ’74 andBrown, Henneaux ’86

I review: seeBa˜nados, Reyes ’16and refs. therein

δG[] = Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

δQ[] = Z

∂Σ

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I changing boundary conditions can change physical spectrum I to distinguish asymptotic symmetries from trivial gauge trafos:

δG[] = Z

Σ

∂Σ

(boundary term) δΦ not functionally differentiable in general (Σ: constant time slice) Φ: shorthand for phase space variables

: smearing function/parameter of gauge trafos δ: arbitrary field variation

I add boundary term to restore functional differentiability δΓ[] =δG[] +δQ[]=^!

Z

Σ

δQ[] = Z

∂Σ

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δG[] = Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

(bulk term) δΦ

δQ[] = Z

∂Σ

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δG[] = Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

δQ[] = Z

∂Σ

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I to distinguish asymptotic symmetries from trivial gauge trafos:

δG[] = Z

Σ

∂Σ

δΓ[] =δG[] +δQ[]=^! Z

Σ

δQ[] = Z

∂Σ

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Soap bubble metaphor for AdS3

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Brown–Henneaux example of asymptotically AdS3

I Given some bc’s it is easy to determine asymptotic Killing vectors

ds² = dr²+e^2r/` dx⁺dx⁻+O(1) dx^{+ 2}+O(1) dx⁻²+. . . I Metrics above preserved by asymptotic Killing vectors

ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

I Introducing (Fourier) modes l_n^±∼ξ(ε^±=e^inx^±)yields ASA [l_n^±, l^±_m]Lie= (n−m)l^±_n+m

I Introduce also Fourier modes for charges L^±_n =Q[l^±_n] I Canonical realization of asymptotic symmetries

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

12 (n³−n)δ_n+m,₀ with central charge

cBH= 3` 2G I Dual field theory, if it exists, must be CFT2!

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I Given some bc’s it is easy to determine asymptotic Killing vectors I Brown–Henneaux imposed following bc’s

ds² = dr²+e^2r/` dx⁺dx⁻+O(1) dx^{+ 2}+O(1) dx⁻²+. . .

I Metrics above preserved by asymptotic Killing vectors ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

I Introducing (Fourier) modes l_n^±∼ξ(ε^±=e^inx^±)yields ASA [l_n^±, l^±_m]Lie= (n−m)l^±_n+m

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

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ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

I Introducing (Fourier) modes l_n ∼ξ(ε =e )yields ASA [l_n^±, l^±_m]Lie= (n−m)l^±_n+m

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

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ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

I Introducing (Fourier) modes l_n^±∼ξ(ε^±=e^inx^±) yields ASA [l_n^±, l^±_m]Lie= (n−m)l^±_n+m

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

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ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

I Introduce also Fourier modes for charges L^±_n =Q[l^±_n]

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

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ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

cBH= 3`

2G

I Dual field theory, if it exists, must be CFT2!

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ξ=ε⁺(x⁺)∂₊+ε⁻(x⁻)∂−+. . .

i{L^±_n, L^±_m}= (n−m)L^±_n+m+cBH

cBH= 3`

2G I Dual field theory, if it exists, must be CFT2!

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Some checks of AdS3/CFT2

Every AdS3 gravity observable must correspond to some CFT2observable ok, fine, so what about...

I ...correlation functions? I ...entropy?

I ...entanglement entropy?

I ...boundary conditions different from Brown–Henneaux? Different boundary conditions may lead to other symmetries, hence no AdS₃/CFT₂!

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Some checks of AdS3/CFT2

Every AdS3 gravity observable must correspond to some CFT2observable ok, fine, so what about...

I ...correlation functions?

I ...entanglement entropy?

I ...boundary conditions different from Brown–Henneaux? Different boundary conditions may lead to other symmetries, hence no AdS₃/CFT₂!