Three dimensional black holes, microstates & the Heisenberg algebra

Three dimensional black holes, microstates

& the Heisenberg algebra

Wout Merbis

Institute for Theoretical Physics, TU Wien

September 28, 2016

Based on [1603.04824]and work to appear with Hamid Afshar, Stephane Detournay, Daniel Grumiller, Alfredo Perez, David Tempo & Ricardo Troncoso

Introduction

Introduction: Black Hole information loss

I⁺

I−

r=0

Black holes emit Hawking radiation

| i

(black body radiation)

Introduction

Introduction: Black Hole information loss

I⁺

I−

r=0

Black holes emit Hawking radiation

| i

(black body radiation) If a pure state

is thrown into the black hole, it cannot be retrieved from the outgoing radiation

| i 6= U| i

Introduction

Introduction: Black Hole information loss

I⁺

I−

r=0

Black holes emit Hawking radiation

| i

(black body radiation) If a pure state

is thrown into the black hole, it cannot be retrieved from the outgoing radiation

| i 6= U| i

Assumptions

I

Initial and final vacuum state is

Introduction

Introduction: Black Hole information loss

I⁺

I−

r=0

Black holes emit Hawking radiation

| i

(black body radiation) If a pure state

is thrown into the black hole, it cannot be retrieved from the outgoing radiation

| i 6= U| i

Assumptions

I

Initial and final vacuum state is unique

I

Black hole have (nearly) no hair

[Hawking, Perry, Strominger, ’16]

Introduction

Introduction: Asymptotic Symmetries

ConformalS²’s I⁺

I⁻

i⁰

i⁺ I

Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS)

I

Leads to infinitely many conservation laws (Weinberg’s soft theorems)

Infinitely many “soft” charges

defined at asymptotic infinity

[Strominger, ’13]

Introduction

Introduction: Asymptotic Symmetries

ConformalS²’s I⁺

I⁻

i−

i⁰

i⁺ I

Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS)

I

Leads to infinitely many conservation laws (Weinberg’s soft theorems)

I

Infinitely many “soft” charges defined at asymptotic infinity

[Strominger, ’13]

Introduction

Introduction: Asymptotic Symmetries

ConformalS²’s I⁺

I⁻

i⁰

i⁺ I

Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS)

I

Leads to infinitely many conservation laws (Weinberg’s soft theorems)

I

Infinitely many “soft” charges defined at asymptotic infinity

[Strominger, ’13]

Introduction

Introduction: Asymptotic Symmetries

ConformalS²’s I⁺

I⁻

i−

i⁰

i⁺ I

Infinite dimensional symmetry group for asymptotically flat spacetimes (BMS)

I

Leads to infinitely many conservation laws (Weinberg’s soft theorems)

I

Infinitely many “soft” charges defined at asymptotic infinity

[Strominger, ’13]

Black holes may carry these soft charges (i.e. soft hair)

Consistency of BH evaporation with the conservation of these symmetries

could correlate early and late time Hawking radiation

Three dimensional gravity

Gravitons in flatland

I

We’ll work in 2+1 dimensions

I

Interesting global solutions

I

Physical state-space is

determined by boundary

conditions

Three dimensional gravity

Gravitons in flatland

I

We’ll work in 2+1 dimensions

I

Interesting global solutions

Flat space cosmologies

[Cornalba, Costa ’02]

I

Physical state-space is

determined by boundary

conditions

Three dimensional gravity

Gravitons in flatland

I

We’ll work in 2+1 dimensions

I

Interesting global solutions

Flat space cosmologies

[Cornalba, Costa ’02]

I

Physical state-space is

determined by boundary

conditions

Three dimensional gravity

Gravitons in flatland

Imposing suitable boundary conditions leads to the asymptotic symmetry algebra of boundary condition preserving gauge transformations. All inequivalent bulk

solutions fall into representations of this asymptotic symmetry algebra.

I

We’ll work in 2+1 dimensions

I

Interesting global solutions

Flat space cosmologies

[Cornalba, Costa ’02]

I

Physical state-space is

determined by boundary

conditions

Three dimensional gravity

Gravitons in flatland

r

t ϕ I

We’ll work in 2+1 dimensions

I

Interesting global solutions

I

Physical state-space is determined by boundary conditions

Asymptotically AdS spacetimes

AdS

Gravity

[Brown, Henneaux ’86]

[Lm,Ln] = (m−n)Lm+n+ c

m(m²−1)δm+n,0

Three dimensional gravity

Gravitons in flatland

ConformalS¹’s I⁺

I−

i⁻

i⁰ i⁺

I

We’ll work in 2+1 dimensions

I

Interesting global solutions

I

Physical state-space is determined by boundary conditions

Asymptotically flat spacetimes

FS

Gravity

[Ashtekar, Biˇc´ak, Schmidt ’96, Barnich, Comp`ere

’06]

[L_m,L_n] = (m−n)L_m+n [L_m,M_n] = (m−n)M_m+n+cM

12m(m²−1)δ_m+n,0

Near horizon boundary conditions

A new set of boundary conditions for 3D gravity

r= 0

r=r^h

r→∞

I Choose boundary conditions such that all solutions have a (regular) horizon (Rindler space)

I Metric to leading order given by ds²=−2aρdv²+ 2dvdρ+γ(ϕ)²dϕ²

(−2ω(ϕ)a⁻¹dϕdρ) +. . . wherea= 2π/T is the Rindler acceleration andγ(ϕ) and ω(ϕ) are arbitrary ‘state-dependent’ functions

Near horizon boundary conditions

A new set of boundary conditions for 3D gravity

v

ρ

= 0

ρ=c

I Choose boundary conditions such that all solutions have a (regular) horizon (Rindler space)

I Metric to leading order given by ds²=−2aρdv²+ 2dvdρ+γ(ϕ)²dϕ²

(−2ω(ϕ)a⁻¹dϕdρ) +. . . wherea= 2π/T is the Rindler acceleration andγ(ϕ) and ω(ϕ) are arbitrary ‘state-dependent’ functions

Near horizon boundary conditions

A new set of boundary conditions for 3D gravity

v

ρ

= 0

ρ=c

I Choose boundary conditions such that all solutions have a (regular) horizon (Rindler space)

I Metric to leading order given by ds²=−2aρdv²+ 2dvdρ+γ(ϕ)²dϕ²

(−2ω(ϕ)a⁻¹dϕdρ) +. . . wherea= 2π/T is the Rindler acceleration andγ(ϕ) and ω(ϕ) are arbitrary ‘state-dependent’ functions The ‘near horizon’ symmetry algebra is spanned by the Fourier modesJ andK

Soft Heisenberg hair

Consequences of the near horizon symmetry algebra

The ‘near horizon’ symmetry algebra is spanned by the Fourier modesJnandKn

ofγ andω[Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso ’16]

[J_n,K_m] =knδ_n+m,0, [J_n,J_m] = 0 = [K_n,K_m]. This is essentially the Heisenberg algebra with two CasimirsJ₀ andK₀.

Soft Heisenberg hair

Consequences of the near horizon symmetry algebra

The ‘near horizon’ symmetry algebra is spanned by the Fourier modesJnandKn

ofγ andω[Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso ’16]

[J_n,K_m] =knδ_n+m,0, [J_n,J_m] = 0 = [K_n,K_m]. This is essentially the Heisenberg algebra with two CasimirsJ₀ andK₀.

Soft Hair

The infinite number of generators J_n andK_n commute with the Hamiltonian and hence all descendents

|ψi=

N

Y(Jn_i)^mi

M

Y(Kk_i)^li|Ei

Soft Heisenberg hair

Consequences of the near horizon symmetry algebra

The ‘near horizon’ symmetry algebra is spanned by the Fourier modesJnandKn

ofγ andω[Afshar, Detournay, Grumiller, WM, Perez, Tempo, Troncoso ’16]

[J_n,K_m] =knδ_n+m,0, [J_n,J_m] = 0 = [K_n,K_m]. This is essentially the Heisenberg algebra with two CasimirsJ₀ andK₀.

Soft Hair

The infinite number of generators J_n andK_n commute with the Hamiltonian and hence all descendents

|ψi=

N

Y

i=1

(Jn_i)^mi

M

Y

i=1

(Kk_i)^li|Ei

have the same energy as|Ei

A puzzle

How could having infinitely many symmetry generators, or black holes with an infinite amount of soft hair ever lead to a finite answer for the black hole entropy?

S_BH

=

4G

Black hole entropy

The BTZ black hole entropy

I The mass of a solution is only well-defined as spatial infinity!

Map near horizon to the asymptotic region and count different microstates there

I Find exact solution which can be extended to the asymptotic region.

I In AdS, the asymptotic Virasoro generatorsL⁺_n andL⁻_n are found to be L^±_n = 1

k X

p

J_n−p^± J_p^±+inJ^± in terms of the near horizon generatorsJ_n^±=¹₂(J_n±K_n).

I Use the Cardy formula, which gives the asymptotic density of states in a (largec) two-dimensional CFT to compute the entropy

S_BH = 2π(J₀⁺+J₀⁻) = 2πJ₀= A 4GN

.

I Similar arguments work for asymptotically flat spacetimes

Black hole entropy

The BTZ black hole entropy

I The mass of a solution is only well-defined as spatial infinity!

Map near horizon to the asymptotic region and count different microstates there

I Find exact solution which can be extended to the asymptotic region.

I In AdS, the asymptotic Virasoro generatorsL⁺_n andL⁻_n are found to be L^±_n = 1

k X

p

J_n−p^± J_p^±+inJ^± in terms of the near horizon generatorsJ_n^±=¹₂(J_n±K_n).

I Use the Cardy formula, which gives the asymptotic density of states in a (largec) two-dimensional CFT to compute the entropy

S_BH = 2π(J₀⁺+J₀⁻) = 2πJ₀= A 4GN

.

I Similar arguments work for asymptotically flat spacetimes

Black hole entropy

The BTZ black hole entropy

I The mass of a solution is only well-defined as spatial infinity!

Map near horizon to the asymptotic region and count different microstates there

I Find exact solution which can be extended to the asymptotic region.

I In AdS, the asymptotic Virasoro generatorsL⁺_n andL⁻_n are found to be L^±_n = 1

k X

p

J_n−p^± J_p^±+inJ^± in terms of the near horizon generatorsJ_n^± =¹₂(J_n±K_n).

I Use the Cardy formula, which gives the asymptotic density of states in a (largec) two-dimensional CFT to compute the entropy

S_BH = 2π(J₀⁺+J₀⁻) = 2πJ₀= A 4GN

.

I Similar arguments work for asymptotically flat spacetimes

Black hole entropy

The BTZ black hole entropy

I The mass of a solution is only well-defined as spatial infinity!

Map near horizon to the asymptotic region and count different microstates there

I Find exact solution which can be extended to the asymptotic region.

I In AdS, the asymptotic Virasoro generatorsL⁺_n andL⁻_n are found to be L^±_n = 1

k X

p

J_n−p^± J_p^±+inJ^± in terms of the near horizon generatorsJ_n^± =¹₂(J_n±K_n).

I Use the Cardy formula, which gives the asymptotic density of states in a (largec) two-dimensional CFT to compute the entropy

S_BH = 2π(J₀⁺+J₀⁻) = 2πJ₀= A 4GN

.

I Similar arguments work for asymptotically flat spacetimes

Black hole entropy

The BTZ black hole entropy

I The mass of a solution is only well-defined as spatial infinity!

Map near horizon to the asymptotic region and count different microstates there

I Find exact solution which can be extended to the asymptotic region.

I In AdS, the asymptotic Virasoro generatorsL⁺_n andL⁻_n are found to be L^±_n = 1

k X

p

J_n−p^± J_p^±+inJ^± in terms of the near horizon generatorsJ_n^± =¹₂(J_n±K_n).

I Use the Cardy formula, which gives the asymptotic density of states in a (largec) two-dimensional CFT to compute the entropy

Summary & Outlook

Summary & Outlook

I

We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of ‘soft hair’ on the black hole horizon

I

The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region

I

Similar constructions work in asymptotically flat spacetimes and the entropy for flat space cosmological solutions can likewise be obtained.

I

Some interesting future research would be

Study dynamics (Could infalling matter excite the soft hairs? Do the soft hairs leave an imprint on outgoing Hawking radiation?)

Summary & Outlook

Summary & Outlook

I

We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of ‘soft hair’ on the black hole horizon

I

The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region

I

Similar constructions work in asymptotically flat spacetimes and the entropy for flat space cosmological solutions can likewise be obtained.

I

Some interesting future research would be

Study dynamics (Could infalling matter excite the soft hairs? Do the soft hairs leave an imprint on outgoing Hawking radiation?)

Summary & Outlook

Summary & Outlook

I

We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of ‘soft hair’ on the black hole horizon

I

The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region

I

Similar constructions work in asymptotically flat spacetimes and the entropy for flat space cosmological solutions can likewise be obtained.

I

Some interesting future research would be

Study dynamics (Could infalling matter excite the soft hairs? Do the soft hairs leave an imprint on outgoing Hawking radiation?)

Summary & Outlook

Summary & Outlook

I

We have discussed new boundary conditions in 3D gravity and shown that they lead to an infinite amount of ‘soft hair’ on the black hole horizon

I

The correct BTZ black hole entropy is recovered after mapping the near horizon region to the asymptotic region

I

Similar constructions work in asymptotically flat spacetimes and the entropy for flat space cosmological solutions can likewise be obtained.

I

Some interesting future research would be

Study dynamics (Could infalling matter excite the soft hairs? Do the

Summary & Outlook

The End

Thank you for your attention