Rindler Holography Daniel Grumiller Institute for Theoretical - - PowerPoint PPT Presentation

Rindler Holography

Daniel Grumiller

Institute for Theoretical Physics TU Wien

Workshop on Topics in Three Dimensional Gravity Trieste, March 2016 based on work w. H. Afshar, S. Detournay, W. Merbis, (B. Oblak), A. Perez, D. Tempo, R. Troncoso

Outline

Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments

Daniel Grumiller — Rindler Holography 2/23

Outline

Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments

Daniel Grumiller — Rindler Holography Motivation 3/23

There is a well-known system with many microstates studied for a long time (recently with help of computers)

Daniel Grumiller — Rindler Holography Motivation 4/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Go: ≈ 10172 microstates

Daniel Grumiller — Rindler Holography Motivation 4/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Go: ≈ 10172 microstates (SGo ≈ 396)

Daniel Grumiller — Rindler Holography Motivation 4/23

There is a well-known system with many microstates studied for a long time (recently with help of computers) Go: ≈ 10172 microstates (SGo ≈ 396) → black holes more complicated!

Daniel Grumiller — Rindler Holography Motivation 4/23

Black hole microstates Bekenstein–Hawking SBH = A 4GN [for M⊙ : eSBH ∼ O(e1076) ∼ echess microstates]

◮ Motivation: microscopic understanding of generic black hole entropy

Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking SBH = A 4GN

◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT2 symmetries (Carlip, Strominger, ...)

using Cardy formula

Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking SBH = A 4GN

◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT2 symmetries (Carlip, Strominger, ...)

using Cardy formula

◮ Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT,

warped CFT, ...) → see talk by Stephane Detournay!

Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking SBH = A 4GN

◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT2 symmetries (Carlip, Strominger, ...)

using Cardy formula

◮ Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT,

warped CFT, ...) → see talk by Stephane Detournay!

◮ Main idea: consider near horizon symmetries for non-extremal

horizons

Daniel Grumiller — Rindler Holography Motivation 5/23

Black hole microstates Bekenstein–Hawking SBH = A 4GN

◮ Motivation: microscopic understanding of generic black hole entropy ◮ Microstate counting from CFT2 symmetries (Carlip, Strominger, ...)

using Cardy formula

◮ Generalizations in 2+1 gravity/gravity-like theories (Galilean CFT,

warped CFT, ...) → see talk by Stephane Detournay!

◮ Main idea: consider near horizon symmetries for non-extremal

horizons

◮ Near horizon line-element with Rindler acceleration a:

ds2 = −2aρ dv2 + 2 dv dρ + γ2 dϕ2 + . . . Meaning of coordinates:

◮ ρ: radial direction (ρ = 0 is horizon) ◮ ϕ ∼ ϕ + 2π: angular direction ◮ v: (advanced) time

Daniel Grumiller — Rindler Holography Motivation 5/23

Choices

◮ Rindler acceleration: state-dependent or chemical potential?

Daniel Grumiller — Rindler Holography Motivation 6/23

Choices

◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale

Recall scale invariance a → λa ρ → λρ v→v/λ

• f Rindler metric

ds2 = −2aρ dv2 + 2 dv dρ + γ2 dϕ2

Daniel Grumiller — Rindler Holography Motivation 6/23

Choices

◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in

1512.08233: v ∼ v + 2πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult Recall scale invariance a → λa ρ → λρ v→v/λ

• f Rindler metric

ds2 = −2aρ dv2 + 2 dv dρ + γ2 dϕ2

Daniel Grumiller — Rindler Holography Motivation 6/23

Choices

◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in

1512.08233: v ∼ v + 2πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult

◮ If chemical potential: all states in theory have same

(Unruh-)temperature (see talk by Miguel Pino) TU = a 2π

Daniel Grumiller — Rindler Holography Motivation 6/23

Choices

◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in

1512.08233: v ∼ v + 2πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult

◮ If chemical potential: all states in theory have same

(Unruh-)temperature (see talk by Miguel Pino) TU = a 2π We make this choice in this talk!

Daniel Grumiller — Rindler Holography Motivation 6/23

Choices

◮ Rindler acceleration: state-dependent or chemical potential? ◮ If state-dependent: need mechanism to fix scale — suggestion in

1512.08233: v ∼ v + 2πL Works technically (see talk by Hamid Afshar), but physical interpretation difficult

◮ If chemical potential: all states in theory have same

(Unruh-)temperature (see talk by Miguel Pino) TU = a 2π

◮ Work in 3d Einstein gravity in Chern–Simons formulation (see talk by

Jorge Zanelli!) ICS = ±

• ±

k 4π

• A± ∧ dA± + 2

3A± ∧ A± ∧ A±

with sl(2) connections A± and k = ℓ/(4GN) with AdS radius ℓ = 1

Daniel Grumiller — Rindler Holography Motivation 6/23

Outline

Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 7/23

Diagonal gauge Standard trick: partially fix gauge A± = b−1

± (ρ)

• d+a±(x0, x1)
• b±(ρ)

with some group element b ∈ SL(2) depending on radius ρ Drop ± decorations in most of talk Manifold topologically a cylinder or torus, with radial coordinate ρ and boundary coordinates (x0, x1) ∼ (v, ϕ)

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Diagonal gauge Standard trick: partially fix gauge A = b−1(ρ)

• d+a(x0, x1)
• b(ρ)

with some group element b ∈ SL(2) depending on radius ρ

◮ Standard AdS3 approach: highest weight gauge

a ∼ L+ + L(x0, x1)L− b(ρ) = exp (ρL0) sl(2): [Ln, Lm] = (n − m)Ln+m, n, m = −1, 0, 1

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Diagonal gauge Standard trick: partially fix gauge A = b−1(ρ)

• d+a(x0, x1)
• b(ρ)

with some group element b ∈ SL(2) depending on radius ρ

◮ Standard AdS3 approach: highest weight gauge

a ∼ L+ + L(x0, x1)L− b(ρ) = exp (ρL0) sl(2): [Ln, Lm] = (n − m)Ln+m, n, m = −1, 0, 1

◮ For near horizon purposes diagonal gauge useful:

a ∼ J (x0, x1) L0

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Diagonal gauge Standard trick: partially fix gauge A = b−1(ρ)

• d+a(x0, x1)
• b(ρ)

with some group element b ∈ SL(2) depending on radius ρ

◮ Standard AdS3 approach: highest weight gauge

a ∼ L+ + L(x0, x1)L− b(ρ) = exp (ρL0) sl(2): [Ln, Lm] = (n − m)Ln+m, n, m = −1, 0, 1

◮ For near horizon purposes diagonal gauge useful:

a ∼ J (x0, x1) L0

◮ Precise boundary conditions (ζ: chemical potential):

a = (J dϕ + ζ dv) L0 and b = exp ( 1

ζ L1) · exp ( ρ 2 L−1). (assume constant ζ for simplicity)

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 8/23

Near horizon metric Using gµν = 1

2

• A+

µ − A− µ

A+

ν − A− ν

• Daniel Grumiller — Rindler Holography

Near horizon boundary conditions 9/23

Near horizon metric Using gµν = 1

2

• A+

µ − A− µ

A+

ν − A− ν

• yields (f := 1 + ρ/(2a))

ds2 = −2aρf dv2 + 2 dv dρ − 2ωa−1 dϕ dρ + 4ωρf dv dϕ +

• γ2 + 2ρ

a f(γ2 − ω2)

• dϕ2

state-dependent functions J ± = γ ± ω, chemical potentials ζ± = −a ± Ω For simplicity set Ω = 0 and a = const. in metric above EOM imply ∂vJ ± = ±∂ϕζ±; in this case ∂vJ ± = 0

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 9/23

Near horizon metric Using gµν = 1

2

• A+

µ − A− µ

A+

ν − A− ν

• yields (f := 1 + ρ/(2a))

ds2 = −2aρf dv2 + 2 dv dρ − 2ωa−1 dϕ dρ + 4ωρf dv dϕ +

• γ2 + 2ρ

a f(γ2 − ω2)

• dϕ2

state-dependent functions J ± = γ ± ω, chemical potentials ζ± = −a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: ds2 = −2aρ dv2 + 2 dv dρ + γ2 dϕ2 + . . . Comments:

◮ Recover desired near horizon metric

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 9/23

Near horizon metric Using gµν = 1

2

• A+

µ − A− µ

A+

ν − A− ν

• yields (f := 1 + ρ/(2a))

ds2 = −2aρf dv2 + 2 dv dρ − 2ωa−1 dϕ dρ + 4ωρf dv dϕ +

• γ2 + 2ρ

a f(γ2 − ω2)

• dϕ2

state-dependent functions J ± = γ ± ω, chemical potentials ζ± = −a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: ds2 = −2aρ dv2 + 2 dv dρ + γ2 dϕ2 + . . . Comments:

◮ Recover desired near horizon metric ◮ Rindler acceleration a indeed state-independent

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 9/23

Near horizon metric Using gµν = 1

2

• A+

µ − A− µ

A+

ν − A− ν

• yields (f := 1 + ρ/(2a))

ds2 = −2aρf dv2 + 2 dv dρ − 2ωa−1 dϕ dρ + 4ωρf dv dϕ +

• γ2 + 2ρ

a f(γ2 − ω2)

• dϕ2

state-dependent functions J ± = γ ± ω, chemical potentials ζ± = −a ± Ω Neglecting rotation terms (ω = 0) yields Rindler plus higher order terms: ds2 = −2aρ dv2 + 2 dv dρ + γ2 dϕ2 + . . . Comments:

◮ Recover desired near horizon metric ◮ Rindler acceleration a indeed state-independent ◮ Two state-dependent functions (γ, ω) as usual in 3d gravity

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 9/23

Canonical boundary charges

◮ Canonical boundary charges non-zero for large trafos that preserve

boundary conditions

◮ Zero mode charges: mass and angular momentum

For covariant approach to boundary charges see e.g. talks by Kamal Hajian, Ali Seraj, Hossein Yavartanoo

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 10/23

Canonical boundary charges

◮ Canonical boundary charges non-zero for large trafos that preserve

boundary conditions

◮ Zero mode charges: mass and angular momentum

Background independent result for Chern–Simons yields Q[η] = k 4π

• dϕ η(ϕ) J (ϕ)

◮ Finite ◮ Integrable ◮ Conserved ◮ Non-trivial

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 10/23

Canonical boundary charges

◮ Canonical boundary charges non-zero for large trafos that preserve

boundary conditions

◮ Zero mode charges: mass and angular momentum

Background independent result for Chern–Simons yields Q[η] = k 4π

• dϕ η(ϕ) J (ϕ)

◮ Finite ◮ Integrable ◮ Conserved ◮ Non-trivial

Meaningful near horizon boundary conditions and non-trivial theory!

Daniel Grumiller — Rindler Holography Near horizon boundary conditions 10/23

Outline

Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 11/23

Near horizon symmetry algebra

◮ Near horizon symmetry algebra = all near horizon boundary

conditions preserving trafos, modulo trivial gauge trafos Most general trafo δǫa = dǫ + [a, ǫ] = O(δa) that preserves our boundary conditions for constant ζ given by ǫ = ǫ+L+ + ηL0 + ǫ−L− with ∂vη = 0 implying δǫJ = ∂ϕη

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 12/23

Near horizon symmetry algebra

◮ Near horizon symmetry algebra = all near horizon boundary

conditions preserving trafos, modulo trivial gauge trafos

◮ Expand charges in Fourier modes

J±

n = k

4π

• dϕ einϕJ ± (ϕ)

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 12/23

Near horizon symmetry algebra

◮ Near horizon symmetry algebra = all near horizon boundary

conditions preserving trafos, modulo trivial gauge trafos

◮ Expand charges in Fourier modes

J±

n = k

4π

• dϕ einϕJ ± (ϕ)

◮ Near horizon symmetry algebra

• J±

n , J± m

• = ± 1

2knδn+m, 0

• J+

n , J− m

• = 0

Two ˆ u(1) current algebras with non-zero levels

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 12/23

Near horizon symmetry algebra

◮ Near horizon symmetry algebra = all near horizon boundary

conditions preserving trafos, modulo trivial gauge trafos

◮ Expand charges in Fourier modes

J±

n = k

4π

• dϕ einϕJ ± (ϕ)

◮ Near horizon symmetry algebra

• J±

n , J± m

• = ± 1

2knδn+m, 0

• J+

n , J− m

• = 0

Two ˆ u(1) current algebras with non-zero levels

◮ Much simpler than CFT2, warped CFT2, Galilean CFT2, etc.

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 12/23

Near horizon symmetry algebra

◮ Near horizon symmetry algebra = all near horizon boundary

conditions preserving trafos, modulo trivial gauge trafos

◮ Expand charges in Fourier modes

J±

n = k

4π

• dϕ einϕJ ± (ϕ)

◮ Near horizon symmetry algebra

• J±

n , J± m

• = ± 1

2knδn+m, 0

• J+

n , J− m

• = 0

Two ˆ u(1) current algebras with non-zero levels

◮ Much simpler than CFT2, warped CFT2, Galilean CFT2, etc. ◮ Map

P0 = J+

0 + J−

Pn =

i kn (J+ −n + J− −n) if n = 0

Xn = J+

n − J− n

yields Heisenberg algebra (with Casimirs X0, P0) [Xn, Xm] = [Pn, Pm] = [X0, Pn] = [P0, Xn] = 0 [Xn, Pm]= iδn,m if n = 0

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 12/23

Soft hair

◮ Vacuum descendants |ψ(q)

|ψ(q) ∼

• (J+

−n+

i )m+ i

(J−

−n−

i )m− i |0 Daniel Grumiller — Rindler Holography Soft Heisenberg hair 13/23

Soft hair

◮ Vacuum descendants |ψ(q)

|ψ(q) ∼

• (J+

−n+

i )m+ i

(J−

−n−

i )m− i |0

◮ Hamiltonian

H := Q[ǫ±|∂v] = aP0 commutes with all generators of algebra

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 13/23

Soft hair

◮ Vacuum descendants |ψ(q)

|ψ(q) ∼

• (J+

−n+

i )m+ i

(J−

−n−

i )m− i |0

◮ Hamiltonian

H := Q[ǫ±|∂v] = aP0 commutes with all generators of algebra

◮ Energy of vacuum descendants

Eψ = ψ(q)|H|ψ(q) = Evacψ(q)|ψ(q) = Evac same as energy of vacuum

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 13/23

Soft hair

◮ Vacuum descendants |ψ(q)

|ψ(q) ∼

• (J+

−n+

i )m+ i

(J−

−n−

i )m− i |0

◮ Hamiltonian

H := Q[ǫ±|∂v] = aP0 commutes with all generators of algebra

◮ Energy of vacuum descendants

Eψ = ψ(q)|H|ψ(q) = Evacψ(q)|ψ(q) = Evac same as energy of vacuum

◮ Same conclusion true for descendants of any state!

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 13/23

Soft hair

◮ Vacuum descendants |ψ(q)

|ψ(q) ∼

• (J+

−n+

i )m+ i

(J−

−n−

i )m− i |0

◮ Hamiltonian

H := Q[ǫ±|∂v] = aP0 commutes with all generators of algebra

◮ Energy of vacuum descendants

Eψ = ψ(q)|H|ψ(q) = Evacψ(q)|ψ(q) = Evac same as energy of vacuum

◮ Same conclusion true for descendants of any state!

Soft hair = zero energy excitations on horizon

Daniel Grumiller — Rindler Holography Soft Heisenberg hair 13/23

Outline

Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 14/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−)

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−) ◮ Generic soft hairy black holes (or “black flowers”) from softly

boosting BTZ

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−) ◮ Generic soft hairy black holes (or “black flowers”) from softly

boosting BTZ

◮ Soft hairy black holes remain regular and have same energy as BTZ

(for other boundary conditions generically not true)

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−) ◮ Generic soft hairy black holes (or “black flowers”) from softly

boosting BTZ

◮ Soft hairy black holes remain regular and have same energy as BTZ

(for other boundary conditions generically not true)

◮ Macroscopic entropy

S = 2π(J+

0 + J− 0 ) =

A 4GN calculated directly in Chern–Simons formulation

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−) ◮ Generic soft hairy black holes (or “black flowers”) from softly

boosting BTZ

◮ Soft hairy black holes remain regular and have same energy as BTZ

(for other boundary conditions generically not true)

◮ Macroscopic entropy

S = 2π(J+

0 + J− 0 ) =

A 4GN

◮ No contribution from soft hair charges

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−) ◮ Generic soft hairy black holes (or “black flowers”) from softly

boosting BTZ

◮ Soft hairy black holes remain regular and have same energy as BTZ

(for other boundary conditions generically not true)

◮ Macroscopic entropy

S = 2π(J+

0 + J− 0 ) =

A 4GN

◮ No contribution from soft hair charges ◮ Suggestive that microstate counting should work

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Macroscopic entropy

◮ Zero-mode solutions with constant chemical potentials: BTZ

J±

0 = k 2(r+ ± r−) ◮ Generic soft hairy black holes (or “black flowers”) from softly

boosting BTZ

◮ Soft hairy black holes remain regular and have same energy as BTZ

(for other boundary conditions generically not true)

◮ Macroscopic entropy

S = 2π(J+

0 + J− 0 ) =

A 4GN

◮ No contribution from soft hair charges ◮ Suggestive that microstate counting should work

Before addressing microstates consider map to aymptotic variables

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 15/23

Map to asymptotic variables

◮ Usual asymptotic AdS3 connection with chemical potential µ:

ˆ A = ˆ b−1 d+ˆ a ˆ b ˆ aϕ = L1 − 1

2 L L−1

ˆ b = eρL0 ˆ at = µL1 − µ′L0 + 1

2 µ′′ − 1 2 Lµ

• L−1

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 16/23

Map to asymptotic variables

◮ Usual asymptotic AdS3 connection with chemical potential µ:

ˆ A = ˆ b−1 d+ˆ a ˆ b ˆ aϕ = L1 − 1

2 L L−1

ˆ b = eρL0 ˆ at = µL1 − µ′L0 + 1

2 µ′′ − 1 2 Lµ

• L−1

◮ Gauge trafo ˆ

a = g−1 (d+a) g with g = exp (xL1) · exp (− 1

2J L−1)

where ∂vx − ζx = µ and x′ − J x = 1

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 16/23

Map to asymptotic variables

◮ Usual asymptotic AdS3 connection with chemical potential µ:

ˆ A = ˆ b−1 d+ˆ a ˆ b ˆ aϕ = L1 − 1

2 L L−1

ˆ b = eρL0 ˆ at = µL1 − µ′L0 + 1

2 µ′′ − 1 2 Lµ

• L−1

◮ Gauge trafo ˆ

a = g−1 (d+a) g with g = exp (xL1) · exp (− 1

2J L−1)

where ∂vx − ζx = µ and x′ − J x = 1

◮ Near horizon chemical potential transforms into combination of

asymptotic charge and chemical potential! µ′ − J µ = −ζ

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 16/23

Map to asymptotic variables

◮ Usual asymptotic AdS3 connection with chemical potential µ:

ˆ A = ˆ b−1 d+ˆ a ˆ b ˆ aϕ = L1 − 1

2 L L−1

ˆ b = eρL0 ˆ at = µL1 − µ′L0 + 1

2 µ′′ − 1 2 Lµ

• L−1

◮ Gauge trafo ˆ

a = g−1 (d+a) g with g = exp (xL1) · exp (− 1

2J L−1)

where ∂vx − ζx = µ and x′ − J x = 1

◮ Near horizon chemical potential transforms into combination of

asymptotic charge and chemical potential! µ′ − J µ = −ζ

◮ Asymptotic charges: twisted Sugawara construction (remember

comment in talk by Gaston Giribet) with near horizon charges L = 1

2J 2 + J ′

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 16/23

Map to asymptotic variables

◮ Usual asymptotic AdS3 connection with chemical potential µ:

ˆ A = ˆ b−1 d+ˆ a ˆ b ˆ aϕ = L1 − 1

2 L L−1

ˆ b = eρL0 ˆ at = µL1 − µ′L0 + 1

2 µ′′ − 1 2 Lµ

• L−1

◮ Gauge trafo ˆ

a = g−1 (d+a) g with g = exp (xL1) · exp (− 1

2J L−1)

where ∂vx − ζx = µ and x′ − J x = 1

◮ Near horizon chemical potential transforms into combination of

asymptotic charge and chemical potential! µ′ − J µ = −ζ

◮ Asymptotic charges: twisted Sugawara construction (remember

comment in talk by Gaston Giribet) with near horizon charges L = 1

2J 2 + J ′ ◮ Get Virasoro with non-zero central charge δL = 2Lε′ + L′ε − ε′′′

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 16/23

Remarks on asymptotic and near horizon variables

◮ Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey

still Heisenberg algebra δQ = − k 4π

• dϕ ε δL = − k

4π

• dϕ η δJ

Reason: asymptotic “chemical potentials” µ depend on near horizon charges J and chemical potentials ζ

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 17/23

Remarks on asymptotic and near horizon variables

◮ Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey

still Heisenberg algebra δQ = − k 4π

• dϕ ε δL = − k

4π

• dϕ η δJ

Reason: asymptotic “chemical potentials” µ depend on near horizon charges J and chemical potentials ζ

◮ Our boundary conditions singled out: whole spectrum compatible

with regularity

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 17/23

Remarks on asymptotic and near horizon variables

◮ Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey

still Heisenberg algebra δQ = − k 4π

• dϕ ε δL = − k

4π

• dϕ η δJ

Reason: asymptotic “chemical potentials” µ depend on near horizon charges J and chemical potentials ζ

◮ Our boundary conditions singled out: whole spectrum compatible

with regularity

◮ For constant chemical potential ζ: regularity = holonomy condition

µµ′′ − 1

2µ′ 2 − µ2L = −2π2/β2

Solved automatically from map to asymptotic observables; reminder: µ′ − J µ = −ζ L = 1

2J 2 + J ′

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 17/23

Remarks on asymptotic and near horizon variables

◮ Asymptotic spin-2 currents fulfill Virasoro algebra, but charges obey

still Heisenberg algebra δQ = − k 4π

• dϕ ε δL = − k

4π

• dϕ η δJ

Reason: asymptotic “chemical potentials” µ depend on near horizon charges J and chemical potentials ζ

◮ Our boundary conditions singled out: whole spectrum compatible

with regularity

◮ For constant chemical potential ζ: regularity = holonomy condition

µµ′′ − 1

2µ′ 2 − µ2L = −2π2/β2

Solved automatically from map to asymptotic observables; reminder: µ′ − J µ = −ζ L = 1

2J 2 + J ′

Near horizon boundary conditions natural for near horizon observer

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 17/23

Cardy counting

◮ Idea: use map to asymptotic observables to do standard Cardy

counting

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 18/23

Cardy counting

◮ Idea: use map to asymptotic observables to do standard Cardy

counting

◮ Twisted Sugawara construction expanded in Fourier modes

kLn =

• p∈Z

Jn−pJp+iknJn

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 18/23

Cardy counting

◮ Idea: use map to asymptotic observables to do standard Cardy

counting

◮ Twisted Sugawara construction expanded in Fourier modes

kLn =

• p∈Z

Jn−pJp+iknJn

◮ Starting from Heisenberg algebra obtain semi-classically Virasoro

algebra [Ln, Lm] = (n − m)Ln+m + 1

2 k n3 δn+m, 0

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 18/23

Cardy counting

◮ Idea: use map to asymptotic observables to do standard Cardy

counting

◮ Twisted Sugawara construction expanded in Fourier modes

kLn =

• p∈Z

Jn−pJp+iknJn

◮ Starting from Heisenberg algebra obtain semi-classically Virasoro

algebra [Ln, Lm] = (n − m)Ln+m + 1

2 k n3 δn+m, 0 ◮ Usual Cardy formula yields Bekenstein–Hawking result

SCardy = 2π

• kL+

0 + 2π

• kL−

0 = 2π(J+ 0 + J− 0 ) =

A 4GN = SBH

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 18/23

Cardy counting

◮ Idea: use map to asymptotic observables to do standard Cardy

counting

◮ Twisted Sugawara construction expanded in Fourier modes

kLn =

• p∈Z

Jn−pJp+iknJn

◮ Starting from Heisenberg algebra obtain semi-classically Virasoro

algebra [Ln, Lm] = (n − m)Ln+m + 1

2 k n3 δn+m, 0 ◮ Usual Cardy formula yields Bekenstein–Hawking result

SCardy = 2π

• kL+

0 + 2π

• kL−

0 = 2π(J+ 0 + J− 0 ) =

A 4GN = SBH Precise numerical factor in twist term crucial for correct results

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 18/23

Warped CFT counting

See talk by Stephane Detournay

◮ Map near horizon algebra J± n = 1 2(Jn ± Kn)

Yn ∼

• Jn−pKp

Tn ∼ Jn to centerless warped conformal algebra [Yn, Ym] = (n − m)Yn+m [Yn, Tm] = −mTn+m [Tn, Tm] = 0

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 19/23

Warped CFT counting

See talk by Stephane Detournay

◮ Map near horizon algebra J± n = 1 2(Jn ± Kn)

Yn ∼

• Jn−pKp

Tn ∼ Jn to centerless warped conformal algebra [Yn, Ym] = (n − m)Yn+m [Yn, Tm] = −mTn+m [Tn, Tm] = 0

◮ Modular property Z(β, θ) = Tr (e−βH+iθJ) = Z(2πβ/θ, −4π2/θ)

(H = Q[∂v], J = Q[∂ϕ]) projects partition function to ground state for small imaginary θ (we need θ → 0)

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 19/23

Warped CFT counting

See talk by Stephane Detournay

◮ Map near horizon algebra J± n = 1 2(Jn ± Kn)

Yn ∼

• Jn−pKp

Tn ∼ Jn to centerless warped conformal algebra [Yn, Ym] = (n − m)Yn+m [Yn, Tm] = −mTn+m [Tn, Tm] = 0

◮ Modular property Z(β, θ) = Tr (e−βH+iθJ) = Z(2πβ/θ, −4π2/θ)

(H = Q[∂v], J = Q[∂ϕ]) projects partition function to ground state for small imaginary θ (we need θ → 0)

◮ Assuming J vac = 0 yields

S = βH = SBH Hamiltonian H is product of BH entropy and Unruh temperature

Daniel Grumiller — Rindler Holography Soft hairy black hole entropy 19/23

Outline

Motivation Near horizon boundary conditions Soft Heisenberg hair Soft hairy black hole entropy Concluding comments

Daniel Grumiller — Rindler Holography Concluding comments 20/23

Comparison to related approaches

◮ Brown, Henneaux ’86

Our boundary conditions differ from Brown–Henneaux — their chemical potentials depend on our charges and chemical potentials! Virasoro composite in terms of Heisenberg algebra

Daniel Grumiller — Rindler Holography Concluding comments 21/23

Comparison to related approaches

◮ Brown, Henneaux ’86 ◮ Donnay, Giribet, Gonz´

alez, Pino 1511.08687 — see talk by Miguel Pino!

◮ Observed already H = TSBH ◮ Changing our bc’s to

ds2 = −2aρ dv2+2 dv dρ−2ωa−1 dϕ dρ+4ωρ dv dϕ+

• γ2+ 2ρ

a (γ2−ω2)

• dϕ2+O(ρ2)

yields AKVs ξ = T(ϕ)∂v + Y (ϕ)∂ϕ + O(ρ3)

◮ Up to subleading terms same AKVs as DGGP

But: T and Y state-dependent for our boundary conditions!

Comment: map to Brown–Henneaux variables requires second chemical potential, not just Rindler acceleration!

Warped CFT algebra composite in terms of Heisenberg algebra

Daniel Grumiller — Rindler Holography Concluding comments 21/23

Comparison to related approaches

◮ Brown, Henneaux ’86 ◮ Donnay, Giribet, Gonz´

alez, Pino 1511.08687 — see talk by Miguel Pino!

◮ Afshar, Detournay, DG, Oblak 1512.08233 — see talk by Hamid

Afshar! Rindler acceleration state-dependent in that approach Twisted warped CFT algebra composite in terms of Heisenberg algebra

Daniel Grumiller — Rindler Holography Concluding comments 21/23

Comparison to related approaches

◮ Brown, Henneaux ’86 ◮ Donnay, Giribet, Gonz´

alez, Pino 1511.08687 — see talk by Miguel Pino!

◮ Afshar, Detournay, DG, Oblak 1512.08233 — see talk by Hamid

Afshar!

◮ Hawking, Perry, Strominger 1601.00921 — see also talk by Geoffrey

Comp` ere!

◮ We constructed explicitly gravitational soft hair ◮ We find no soft hair contribution to black hole entropy∗ ◮ BMS3 follows from Sugawara-like construction from Heisenberg algebra

BMS algebra (supertranslations + superrotation) com- posite in terms of near horizon Heisenberg algebra

∗ See comment by Jan de Boer on Tuesday!

Daniel Grumiller — Rindler Holography Concluding comments 21/23

Comparison to related approaches

◮ Brown, Henneaux ’86 ◮ Donnay, Giribet, Gonz´

alez, Pino 1511.08687 — see talk by Miguel Pino!

◮ Afshar, Detournay, DG, Oblak 1512.08233 — see talk by Hamid

Afshar!

◮ Hawking, Perry, Strominger 1601.00921 — see also talk by Geoffrey

Comp` ere!

◮ Comment on complementarity:

◮ Asymptotic Virasoro algebra composite from near horizon

perspective

◮ Same physics described naturally in different variables for

asymptotic and near horizon observers

◮ In particular, asymptotic chemical potentials depend on near

horizon charges and chemical potentials

Daniel Grumiller — Rindler Holography Concluding comments 21/23

Comparison to related approaches

◮ Brown, Henneaux ’86 ◮ Donnay, Giribet, Gonz´

alez, Pino 1511.08687 — see talk by Miguel Pino!

◮ Afshar, Detournay, DG, Oblak 1512.08233 — see talk by Hamid

Afshar!

◮ Hawking, Perry, Strominger 1601.00921 — see also talk by Geoffrey

Comp` ere!

◮ Comment on complementarity:

◮ Asymptotic Virasoro algebra composite from near horizon

perspective

◮ Same physics described naturally in different variables for

asymptotic and near horizon observers

◮ In particular, asymptotic chemical potentials depend on near

horizon charges and chemical potentials

◮ Li, Lucietti 1312.2626 — 3d black holes and descendants

Daniel Grumiller — Rindler Holography Concluding comments 21/23

Elaborations and generalizations

◮ More on dual field theory — to be done ◮ Flat space

◮ Similar story works! ◮ Get centerless BMS3 as composite algebra from Heisenberg algebra! ◮ Soft hairy flat space cosmologies ◮ Asymptotic chemical potentials again depend on near horizon charges

and chemical potentials

◮ Obtain again Bekenstein–Hawking entropy with no soft hair

contribution

Daniel Grumiller — Rindler Holography Concluding comments 22/23

Elaborations and generalizations

◮ More on dual field theory — to be done ◮ Flat space

◮ Similar story works! ◮ Get centerless BMS3 as composite algebra from Heisenberg algebra! ◮ Soft hairy flat space cosmologies ◮ Asymptotic chemical potentials again depend on near horizon charges

and chemical potentials

◮ Obtain again Bekenstein–Hawking entropy with no soft hair

contribution

◮ Massive gravity — To be done! Doable!

Daniel Grumiller — Rindler Holography Concluding comments 22/23

Elaborations and generalizations

◮ More on dual field theory — to be done ◮ Flat space

◮ Similar story works! ◮ Get centerless BMS3 as composite algebra from Heisenberg algebra! ◮ Soft hairy flat space cosmologies ◮ Asymptotic chemical potentials again depend on near horizon charges

and chemical potentials

◮ Obtain again Bekenstein–Hawking entropy with no soft hair

contribution

◮ Massive gravity — To be done! Doable! ◮ Higher spins — with Stefan Prohazka: similar story works!

Daniel Grumiller — Rindler Holography Concluding comments 22/23

Elaborations and generalizations

◮ More on dual field theory — to be done ◮ Flat space

◮ Similar story works! ◮ Get centerless BMS3 as composite algebra from Heisenberg algebra! ◮ Soft hairy flat space cosmologies ◮ Asymptotic chemical potentials again depend on near horizon charges

and chemical potentials

◮ Obtain again Bekenstein–Hawking entropy with no soft hair

contribution

◮ Massive gravity — To be done! Doable! ◮ Higher spins — with Stefan Prohazka: similar story works! ◮ Lower spins — lowest spin gravity! (see Hofman, Rollier 1411.0672)

Daniel Grumiller — Rindler Holography Concluding comments 22/23

Elaborations and generalizations

◮ More on dual field theory — to be done ◮ Flat space

◮ Similar story works! ◮ Get centerless BMS3 as composite algebra from Heisenberg algebra! ◮ Soft hairy flat space cosmologies ◮ Asymptotic chemical potentials again depend on near horizon charges

and chemical potentials

◮ Obtain again Bekenstein–Hawking entropy with no soft hair

contribution

◮ Massive gravity — To be done! Doable! ◮ Higher spins — with Stefan Prohazka: similar story works! ◮ Lower spins — lowest spin gravity! (see Hofman, Rollier 1411.0672) ◮ 4d — Does it work? Is there soft Heisenberg hair? Is BMS4

composite? What are near horizon symmetries? Near horizon symmetries shed new light on soft hair, microstate counting and complementarity

Daniel Grumiller — Rindler Holography Concluding comments 22/23

Thanks for your attention!

• H. Afshar, S. Detournay, D. Grumiller, W. Merbis, A. Perez,
• D. Tempo and R. Troncoso

“Soft Heisenberg hair on black holes in three dimensions,” 1603.04824

Thanks to Bob McNees for providing the L

A

T EX beamerclass! Daniel Grumiller — Rindler Holography Concluding comments 23/23