Upload
others
View
11
Download
0
Embed Size (px)
Citation preview
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.