Black HoleEntropy and SU(2) Chern

Simons Theory

Alejandro Perez Centre de Physique Théorique, Marseille, France

Based on joint work with Jonathan Engle and Karim Noui

The plan

• Brief motivation and introduction

• The main idea in Ashtekar-Krasnov-Corichi treatment of the problem

• Isolated horizons (in the ACK formulation)

• Connection variables (the SU(2) invariant canonical formulation)

• Quantization

• Relation between the U(1) and SU(2)

• Counting

(1)

Black Hole Thermodynamics

stationarystate (1)M, J, Q

stationary state (2)M , J , Q 0th law: the surface gravity κ

is constant on the horizon.

1st law:δM = κ

8π δA + ΩδJ + ΦδQ work terms

2nd law:δA ≥ 0

3rd law: the surface gravity value κ = 0

(extremal BH) cannot be reached by any

physical process.

Ω ≡ horizon angular velocityκ ≡surface gravity (‘grav. force’ at horizon)If a =killing generator, then a∇ab = κb.

Φ ≡electromagnetic potential.

Some definitions

Introduction and motivation

(3)

Hawking Radiation

0th law: the surface gravity κis constant on the horizon.

1st law:δM = κ

8π δA + ΩδJ + ΦδQ work terms

2nd law:δA ≥ 0

3rd law: the surface gravity value κ = 0

(extremal BH) cannot be reached by any

physical process.

T =κ

2π

S = 14A

dynamicalphase

(4)

Black Holes EvaporateQuestions for Quantum GravityT =

κ

2π

S = 14A

dynamicalphase

?

1) Microscopic origing of the entropy.

2) Dynamics of black hole evaporation.

3) What replaces the classical singularity.

4) What happens when a BH completely

evaporates.

Black Hole ThermodynamicsIntroduction and motivation

(5)

Quasilocal definition of BHFrom the Characteristic formulation of GR to isolated

horizons

D(M)

ioMFree Char

acteri

stic D

ataFree Characteristic Data

+

Initial Cauchy Data

D(M)

ioMFree Char

acteri

stic D

ataFree Characteristic Data

+

Initial Cauchy Data

o

Free data

M1

2

IH bo

undar

y con

dition

iradiat

ion

M

(6)

ACK IH boundary condition

• ∆ = S2 ×R

• Einstein’s equations hold on ∆

• For a the null normal ρ = 0 and σ = 0.

• For na (such that · n = 1) ρ = 0 and σ = 0.• For the null vector field na, such that n · = 1,σ = 0 and ρ = −f(v)

ACK isolated horizonsImplications in terms of connection variables

The main equations:equations below are valid on the appropriate

thermal slicingwhere 0th and 1st laws hold.

Σi = −a

πF i(Γ) Ki = −

2π

aei

Ai = Γi + γKi

Fabi(A) = −π(1− γ2)

aΣab

i

Va = 2Γ1, dV = −2π

aΣ1 = −2π

a2

Σ2 = Σ3 = 0

Free data

ioM1

2

Schwarz

shild

data

M

radiat

ionr =

a

4π

In the gauge normal gauge (where is normal to H) the

main equation can be written in AKC form, plus two additional

conditions on Σi

e1

=⇒

(7)

Surface gravity and the value of !0

In IH there is no unique notion of !"a!a"b = ! "b

Lorentz Transf. above"a " "a = exp(#(#(x)))"a

na " na = exp((#(x)))na

normal " normal

M1

2

H

!

M

Can fix freedom by demanding ! to take the Schwarzshild value

! =1

4M=

!$

aH

fixes %0 = #1$2.

Indeed this choice is the one that makes the zero, and first lawof IH look just as the corresponding laws of stationary black holemechanics [ACK, Ashtekar-Beetle-Fairhurst-Krishnan].

qabrarb = 1

A preferred normal direction to H: the allowed histories must satisfy (Ashtekar)

⇐⇒

δijEai Eb

jrarb =√

q

Fixing of the null generator(8)

Kia = −f(v)√

2eia → Ki

a = −

2π

aeia

(a)

(b)

Isolated horizon dynamical systemCovariant phase space formulation

o

Free data

M1

2

IH bo

undar

y con

dition

iradiat

ion

M

In Ashtekar variables, due to the IH boundary condition, no additional boundary term need to be included.

S[e, A+] = − i

κ

MΣi(e) ∧ F i(A+) +

i

κ

τ∞

Σi(e) ∧Ai+

J(δ1, δ2) = −2i

κδ[1Σi ∧ δ2]A

i+ ∀ δ1, δ2 ∈ TpΓ

dJ(δ1, δ2) = 0

The presymplectic current:

Einsteins Equations imply:

0 =iκ

2

∂BJ(δ1, δ2) =

∆δ[1Σi ∧ δ2]A

i+

+

M1

δ[1Σi ∧ δ2]Ai+ −

M2

δ[1Σi ∧ δ2]Ai+

aH

4π

H2

δ[1A+i ∧ δ2]Ai

+ −aH

4π

H1

δ[1A+i ∧ δ2]Ai

+

(a)

(b)

(c)

(9)

o

Free data

M1

2

IH bo

undar

y con

dition

iradiat

ion

M

The conserved presymplectic

form:iκ ΩM (δ1, δ2) =

M

2δ[1Σi ∧ δ2]Ai

+ −aH

2π

H

δ[1A+i ∧ δ2]Ai

+

ΩM1(δ1, δ2) = ΩM2(δ1, δ2)

Reality conditions:

The conserved presymplectic form:

(a)

(b)

Ai+ = Γi + iKi =⇒ Im[ΩM (δ1, δ2)] = 0

κΩM (δ1, δ2) = κRe[ΩM (δ1, δ2)] =

M2δ[1Σi ∧ δ2]K

i

(c)

(d)

Isolated horizon dynamical systemCovariant phase space formulation

(10)

Connection variables for GR

S[e, A] =1κ

M

√−g R[g]The Einstein

Hilbert Action

The (pre) Symplectic Form

Ω(δ1, δ2) =1κ

M

(δ1qab δ2Πab − δ2qab δ1Πab) ∀ δ1, δ2 ∈ T (ΓIH)

qab = eiaej

bδij

Introduce a triad (SU(2) gauge symm.)

New Canonical Variables

Σi = ijk(ej ∧ ek)Ki

a = ebjKabδ

ij

New ConnectionVariables

Σi = ijk(ej ∧ ek)

Ai = γKi + Γ(e)i

Torsion free connectionSecond fundamental form

Ω(δ1, δ2) =1

8πG

Mδ[1Σi ∧ δ2]K

i =1

8πGγ

Mδ[1Σi ∧ δ2]A

i,

The Symplectic Potential

8πGγΘ(δ) =

MΣi ∧ (γδKi) =

MΣi ∧ δ(Γi + γKi)−

MΣi ∧ δΓi

t

=

ab

abP

q

1/2q (K Kq )ab ab

Σi ∧ δΓi = −d(ei ∧ δei)

M

(a)

(11)

The conserved presymplectic

form:

The Symplectic Potential:

8πγΦ(δ) =

M

Σi ∧ δ(γKi + Γi)−

M

Σi ∧ δΓi

=

M

Σi ∧ δA +

H

ei ∧ δei

ΩM (δ1, δ2) =1κγ

M

[δ1Σi ∧ δ2Ai− δ2Σi ∧ δ1Ai

] +1κγ

H

δ1ei ∧ δ2ei

The presymplectic form in connection

variables:

Rewriting the Symplectic Potential:

8πγΦ(δ) =

MΣi ∧ δ(γKi) +

MΣi ∧ δΓi −

MΣi ∧ δΓi

ΩM (δ1, δ2) =1κ

M[δ1Σi ∧ δ2Ki − δ2Σi ∧ δ1Ki] ∀ δ1, δ2 ∈ T (Γcov)

(a)

(b)

(c)

Isolated horizon dynamical systemCovariant phase space formulation

(12)

Σ(α, S) ≡

S⊂H

Tr[αΣ] Σ(α, S),Σ(β, S) = Σ([α,β], S)

S[e, A] =1κ

M

√−g R[g] Due to the IH boundary condition no additional boundary

term need to be included in the action

No boundary term here either

The Symplectic Potential

ΩM (δ1, δ2) =1κ

M[δ1Σi ∧ δ2Ki − δ2Σi ∧ δ1Ki] ∀ δ1, δ2 ∈ T (Γcov)

8πγΦ(δ) =

M

Σi ∧ δ(γKi + Γi)− a

2π(1− γ2)

H

Ai ∧ δAi

=

M

Σi ∧ δ(γKi)−

H

ei ∧ δei +a

2π(1− γ2)Ai ∧ δAi

closed one form in ΓcovFab

i(A) = −π(1− γ2)a

Σabi

ΩM (δ1, δ2) =1κγ

M

[δ1Σi ∧ δ2Ai− δ2Σi ∧ δ1Ai

] +a

π(1− γ2)κγ

H

δ1Ai ∧ δ2Ai

Isolated horizon dynamical systemCovariant phase space formulation

(13)

The quantum Soup! !

!

!

QuantumSoup

(1) Quantize and solve theIH boundary conditionQuantum SU(2) Chern-Simons theorywith defects[Witten, Smolin]!

aH

!(1!"2) F + !"

|"! = 0

jh

(2) Count states using Witten’s results: S = log(N ) withN =

#

n;(j)n1

dim[H CS(j1· · ·jn)] [Kaul-Majumdar (first), (then)Carlip]

with labels j1 · · · jp of the punctures constrained by the condition#n

p=1

$

jp(jp + 1) " aH

8!#$2p# k0 (precise counting done resently)

[Agullo-Barbero-Borja-Diaz-Polo-Villasenor]

, Krasnov]

(14)

[Noui et al. (new ideas)]

The single intertwiner BH model

dim[H CS(j1· · ·jn)] ! dim[Inv(j1 " · · ·" jn)]

The area constraint!n

p=1

"

jp(jp + 1) ! aH

8!"#2p#

dim[H CS(j1· · ·jn)] = dim[Inv(j1 " · · ·" jn)]

We can model the IH system by a single SU(2)intertwiner [Livine-Terno, Rovelli-Krasnov]

(15)

The single intertwiner BH model

dim[H CS(j1· · ·jn)] ! dim[Inv(j1 " · · ·" jn)]

The area constraint!n

p=1

"

jp(jp + 1) ! aH

8!"#2p#

dim[H CS(j1· · ·jn)] = dim[Inv(j1 " · · ·" jn)]

We can model the IH system by a single SU(2)intertwiner [Livine-Terno, Rovelli-Krasnov]

(16)

Relation to U(1) treatment

Domagala-Lewandowski counting

Ghosh-Mitra counting

jhjh

jhjh

(j,m)

(j, m)(j,m)

(j, m)

≈

≈

Density matrix constructed by tracing out “bulk” degrees of freedom where only the

connection is a surface observable

Density matrix constructed by tracing out “bulk” degrees

of freedom where bothconnection and the area

density are a surface observables

(17)

k

4πF − Σiri

|Ψ = 0 Ψ|Σi(δij − rirj)|Ψ = 0

k

4πF − Σiri

|Ψ = 0 Ψ|Σi(δij − rirj)|Ψ = 0

(see Ashtekar-Engle-Van den Broeck)

k

4πF − Σiri

|Ψ = 0 Ψ|Σi(δij − rirj)|Ψ = 0

The issue of fixing the null generatorin the SU(2) treatment

counting remains unchanged after imposing the condition in the quantum theory

jhjh

(j, m)(j,m)

≈

Counting: Density matrix constructed by tracing out “bulk” degrees of freedom

where bothconnection and the area

density are a surface observables

δijEa

i Ebjrarb =

√q

Assuming there is at least one solution to equation (a) per surface state the counting is

unaffected as (a) only restricts bulk d.o.f.

(a)

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(k

4πF i + Σi)|Ψ >= 0

( √h−√q)|Ψ = 0

Conclusions• IH’s [Ashtekar et al.] admits an SU(2) invariant treatment.

The d.o.f. of the horizon are described in terms of an SU(2)Chern-Simons theory with level k = aH

2!"2p#(1!#2) [meets old ideas

by Smolin].

• For |!| !"

3, k0 ! k: S = #0aH

4"2p# # 32 log(aH) with !0 = 0.274067...

[Agullo-Barbero-Borja-Diaz-Polo-Villasenor, Kaul-Majumdar (for#3/2 log)].

• Leading term in entropy coinsides with the U(1) treatment withGhosh-Mitra counting.

• Can model black hole as a single intertwiner! Vindicates Rovelli-Krasnov’s arguments for computting BH entropy in full theory.

• BH entropy from topological limit of GR [Liu-Montesinos-Perez]

(19)

(number of physical states smaller than U(1) case)

(done in the ACK definition, more work needed for general IH)

Future

• Put our long paper on the arxiv. (ENP).

• Extend to general IHs (avoid the fixing of the null generators).

• Treat distorted and rotating case.

• Dynamics, dynamics.