Black Hole Entropy and SU(2) Chern Simons Theoryrelativity.phys.lsu.edu/ilqgs/perez050410.pdfEntropy...
Transcript of Black Hole Entropy and SU(2) Chern Simons Theoryrelativity.phys.lsu.edu/ilqgs/perez050410.pdfEntropy...
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)
(18)
(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.