Rotating Black Hole State Counting in Loop Quantum Gravity

Rotating black hole state counting in Loop Quantum Gravity

Jibril Ben Achour Center for Field Theory and Particles Physics Fudan University, Shanghai

LeCosPa Workshop December 5th 2016

Based on : arXiv:1607.02380 [gr-qc], JHEP 1608 (2016) 149 [BA, Noui, Perez] arXiv:1406.6021 [gr-qc], JHEP 1506 (2015) 145 [BA, Mouchet, Noui]

0 / 15 Overview

1. LQG: self dual vs real variables

2. Isolated horizon : a brief review

3. State counting with γ = i

1 / 15 LQG: self dual vs real variables Overview

1. LQG: self dual vs real variables

2. Isolated horizon : a brief review

3. State counting with γ = i

2 / 15 LQG: self dual vs real variables

Ashtekar self dual gravity

Reformulation in term of ﬁrst order variables I I J ? A tetrad e related to the metric as: gµν = eµeν ηIJ ? A so(3, 1) spin connection ωIJ related to the Levi Civita connection Gravity in term of the self dual Ashtekar’s variables [Ashtekar (1986)] C i C a ? New complex canonical variables: ( Aa, Ei )

C i C b i b C i i (0)i C a abc j k { Aa, Ej } = iδj δa Aa = ωa − iωa Ei = ijkebec (1)

? Very important improvement: drastic simpliﬁcation of the constraints of GR ! ? Provided a reformulation of GR in term of a SL(2,C) background independent Yang Mills theory ? However, need to impose reality conditions to recover real GR √ ab C a C b C i C ¯i i −qq = Tr( E E ) ∈ R Aa + Aa = 2iωa (2)

? No one knows how to impose those reality conditions at the quantum level... Open problem up to now !

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Real Ashtekar-Barbero gravity

Avoid the reality conditions: turn the complex self dual variables for real ones ! ? Introduction of the real SU(2) Ashtekar-Barbero phase space [Barbero (1994)] i a ? New real canonical variables: (Aa,Ei )

i b i b i i (0)i a abc j k {Aa,Ej } = γδj δa Aa = ωa − γωa Ei = ijkebec (3) One free real parameter: the Barbero-Immirzi parameter γ ? First advantage: no need to impose any reality conditions ? Second advantage: wick rotation of the phase space, turn the non compact gauge group SL(2, C) to a compact one SU(2)

? First drawback: complicated hamiltonian constraint i ? Second drawback: Aa not a space-time connection ( ... anomaly in the quantum theory) ? Third drawback: the quantum Barbero transform is not unitary

4 / 15 Isolated horizon : a brief review Overview

1. LQG: self dual vs real variables

2. Isolated horizon : a brief review

3. State counting with γ = i

5 / 15 Isolated horizon : a brief review

? Quasi-local description (no reference to infinity) : isolated horizon ? Null hypersurface, expansion: θ = 0, defined by a unique equivalence class of null tangent vectors [χ]: χ ∼ cχ0 ? No reference to a metric : only boundary conditions ? Follow the same thermodynamical laws than a black hole ... but with quasi local quantities ! ? However, the first law (and the notion of energy) is not uniquely defined ... Surface gravity depends on the choice of χ inside [χ] ...

6 / 15 Isolated horizon : a brief review Classical description ? Symplectic structure induced on the boundary : Chern Simons theory Z Ω(δ1, δ2) ∝ δ1A ∧ δ2A (4) ∂M ? Gravitational degrees of freedom of freedom are governed by a Chern-Simons theory coupled to punctures k Z 2 SIH = < A ∧ dA + A ∧ A ∧ A > +SPunctures (5) 4π H 3 ? The coupling constant k related to the area and the angular momentum: A A J kstatic = 2 2 krotating = 2 2 − 2 (6) 4πlpγ(1 − γ ) 8πγ(¯γ − 1)lp lp

? Classical solutions: ﬂat connections with singularities (topological defects) at the punctures

kstatic i 1 i krotating i 1 i i i F (A) = 2 Σ F (A) = 2 Σ + pδ1δN + pδ2δS (7) 4π 8πlpγ 4π 8πlpγ

i i j k 2 with Σab = jkeaeb and p = (k + J/lp).

7 / 15 Isolated horizon : a brief review

The quantization in a nutshell ? Quantization of the Chern-Simons theory coupled to point like particles [Perez, Noui, Engle, Pranzetti (2009)] [Perez, Pranzeti, Frodden (2012)] ? Hilbert space : space of (q-deformed) SU(2) intertwiners

H(j1, ...., jn) = Inv(Vj1 ⊗ .... ⊗ Vjn ⊗ VJ ) (8)

? Each punctures (quantum gavitationnal d.o.f) carries an SUq(2) representation Vj : labelled with spin j

? Two punctures associated with the rotational d.o.f are also carrying an SUq(2) representation VJ with spin J ? Degeneracy of the horizon = dimension of the Hilbert space ... given by the Verlinde formula:

k+1 n πddi 2 X 2 πd 2 πdJ Y sin k+2 Nk(J, di, n) = sin sin (9) k + 2 k + 2 k + 2 sin πd d=1 i=1 k+2

with di = 2ji + 1 and dJ = 2J + 1.

8 / 15 Isolated horizon : a brief review

Static state counting for γ ∈ R ? At the leading order, with the one color assumption, the entropy is given by

γ0 A n Sγ (A) = 2 g(n, d) ∝ d (10) γ lp

? Reproduce the Bekenstein-Hawking area law at leading order up to a (rather unatural) ﬁne tuning in the Barbero-Immirzi parameter

log dj γ = γ0 = (11) πdj

where dj = 2j + 1. The degeneracy g(n, d) is not "holographic". Rotating state counting for γ ∈ R ? At the leading order, with the one color assumption, the entropy is given by α log α + (1 − α) log (1 − α) A A √ Sγ (A, J) = − 2 + o( 2 ) (12) πγ 3 4lp lp where α = 1/2 + J/n ? Does not reproduce the Bekenstein-Hawking area law at leading order in this model !

9 / 15 State counting with γ = i Overview

1. LQG: self dual vs real variables

2. Isolated horizon : a brief review

3. State counting with γ = i

10 / 15 State counting with γ = i Deﬁning the self dual static BH in LQG: from γ ∈ R to γ ± i ? First consequence: deﬁnition of the static IH implies that: γ = ±i → k = ±λi 2πγk A = γ ∈ → A = 2πλ γ = i (13) 1 − γ2 R [Frodden, Geiller, Noui, Perez (2012)] [BA, Mouchet, Noui (2014)] ? Have to deal with the analytic continuation of Chern Simons theory [Witten (2010) : Morse theory] ? k enters in the upper bound of the sum in the Verlind formula .... not well suited for analytic continuation purposes ! ? Idea: → need to reformulate the Verlinde formula into an integral on the complex plane .... I n i 2 Y sinh dlz N (n, d ) = dz sinh z coth(k + 2)z (14) k l π sinh z C l=1

iπp ∗ ? For k ∈ R, poles are located at zp = k+2 with p ∈ N ? Contour C encloses the imaginary axis between [0, iπ] ? How the structure of the poles is modiﬁed once k = ±iλ ? [BA, Mouchet, Noui (2014)]

11 / 15 State counting with γ = i The poles structure : deriving a analytical continuation procedure ? First case : (dj , k) ∈ N, poles on the imaginary axis iπp z = p ∈ ∗ (15) k + 2 N

? Second case : (dj , k) ∈ N × iR, poles on the real axis πp z = − p ∈ ∗ (16) λ N Integral vanishes since the countour doesn’t enclose any poles (residus theorem) : inconsistent ! ? Third case : (dj , k) ∈ iR × iR, poles both on the real as well as on the imaginary axis ! πp z = − p ∈ ∗ z = iπm m ∈ ∗ (17) λ N m N The third case is the only possibility to have a mathematically well deﬁned analytical continuation of the Hilbert space dimension. The procedure reads

γ = i =⇒ k ∈ iR =⇒ dj = 2j + 1 ∈ iR (18) meaning 1 γ = ±i j = (is − 1) s ∈ + (19) 2 R ? For rotating BH, same conclusions, but need to generalize to γ = i =⇒ k ∈ C 12 / 15 State counting with γ = i

Static case for γ = i ? Reproduce exactly the Bekenstein-Hawking area law at leading order: no need for ﬁne tuning [BA, Mouchet, Noui (2014)] A A A S(A) = 2 d(n, d) = P ( 2 ) exp ( 2 ) (20) 4lp 4lp 4lp

2 ? Reproduce also the expected quantum corrections ∼ −3/2 log (A/lp) under some additional natural assumptions ? All those results remain if we relax the one color hypothesis Rotating case for γ = i ? The procedure extend to the rotating case [BA, Noui, Perez (2016)]. Reproduce as well the Bekenstein-Hawking area law at leading order: A A A S(A) = 2 d(n, d, J) = P ( 2 ,J) exp ( 2 + 2πiJ) (21) 4lp 4lp 4lp

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More on the analytic continuation procedure ? The prescription trivially solve the reality conditions at the level of the area spectrum

2p 2p 2 A(Σ)|Ψ >= 8πγlp j(j + 1)|Ψ > → A(Σ)|Ψ >= 4πlp s + 1)|Ψ > (22)

? How do the quantum states transform under this mapping ? Non compact spin network [Freidel, Livine (200)]? SL(2, C) non compact spin networks ... ? Several related results in LQG, mostly in the context of black holes thermodynamics [Pranzetti (2013)] [Pranzetti, Sahlmann (2015)] [Geiller, Noui(2014)], 2 + 1 LQG [BA, Geiller, Noui (2013)], spinfoams models [Neiman], LQC [BA, Grain, Noui (2015)], as well as in midi-superspaces loop models of black holes and cosmology [BA, Brahma, Marciano (2016)] ? Main message : self dual variables seems to be more suited with respect to the semi-classical sector of the theory in the sector of black hole quantization, as well as for the implementation of quantum covariance in midi-superspace models in LQG

14 / 15 State counting with γ = i

General considerations on the analytic continuation procedure ? Concrete proposal to extract physical predictions from self dual LQG (and by pass the reality conditions problem) ? Still open problem : what is the target Hilbert space ? Non compact spin network ? ? Relation to the Wick Rotation program in LQG : Thiemann’s program ? Need to be tested on other setting in order to test the robustness of this approach

? Could open a new window towards computing predictions of self dual LQG, and by passing the old reality conditions problem ... which remain out of reach since the very advent of the self dual Ashtekar variables (1986)

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