Black Holes 16 Parthasarathi Majumdar, Saha Institute of Nuclear Physics, Kolkata

Rudolf Peierls Centre for Theoretical Physics, Oxford University Michaelmas Term 2011. December 7, 2011

1 Created with pptalk Slide 2 Electrodynamics in Minkowski sptm: Define charge holographically

Q(V ) ≡ E~ · ndˆ 2a ZS=∂V 2 2 But, Hv = (1/8π)(E~ + B~ ) → photons ab ab Vac GR : no T s.t. ∇aT =0 in bulk

Hv = [NH + N · P] ZS ≈ 0 when H≈ 0, P ≈ 0

⇒ no analogue of E2 + B2 in vac GR! Excitations ‘polymeric’ Created with pptalk Slide 3 Grav energy globally defined 1 H = d2σab∇ K Komar 8π a b ZS∞ Classically, bulk ⇒ boundary entirely Holography: 3 dim bulk info encoded on 2 dim bdy Gravitons ?

Weak field approx gab =g ¯ab + hab bkgd graviton 3 2 3 2 Hv|{z}= (1/8π|{z})[( h) + ( π) ] As |h| ր , bkreactn ր approxn. invalid nonperturbatively Created with pptalk Slide 4 Quantum General Relativity In general there are indep qu fluct on bdy : H = Hv ⊗Hb

|Ψi = cvb |ψvi |χbi ∈Hv ⊗Hb Xv,b blk bdy ˆ ˆ |{z}1 |{z}1 ˆ H = Hv ⊗ + ⊗ Hb blk bdy Hamiltonian constraint (bulk)|{z} |{z} ˆ ˆ ˆ Hv |ψvi = [HEH,v + HMAX,v]|ψvi =0 ˆ ˆ 1 1 ˆ Q = Qv ⊗ + ⊗ Qb

Qˆv|ψvi =0 Created with pptalk Slide 5 New Hamiltonian constraint ˆ ′ Hv|ψvi = 0 ˆ ′ ˆ ˆ Hv ≡ Hv − ΦQv

Grand Partition Function Majhi, PM 2011 ˆ ˆ ZG = Tr exp −βHT + βΦQ 2 ˆ ′ = |cvb| hχb|⊗hψv| exp −βH |ψvi⊗|χbi Xv,b ˆ ′ ˆ ˆ H = HT − ΦQ Observe ˆ ′ ˆ ′ 1 1 ˆ ′ H = (Hv ⊗ + ⊗ Hb) ˆ ′ Hv|ψvi = 0 Created with pptalk Slide 6

ZG = ZGb ˆ ˆ ZGb = Trb exp −β(Hb − ΦQb) Bulk states decouple! Boundary states determine bh thermodynamics completely → Thermal holography ! (PM 2001, 2007; Majhi, PM 2011)

Different from strong holography (’t Hooft 1992; Susskind 1993; Bousso 2002) Holographic Hypothesis (HH) ... Given any closed surface, we can represent all that happens (gravita- tionally) inside it by degrees of freedom on this surface itself. This ... sug- gests that quantum gravity should be described by a topological quantum field theory in which all (gravitational) degrees of freedom are projected onto the boundary. In contrast, ours underlines primacy of boundary states for bh thermodyn Created with pptalk Slide 7 Grand Canonical Ensemble of IHs in rad bath : compute Zb → Scan • Assume equil. IH with fixed AIH,QIH and MIH = M(AIH,QIH).

• Keep Gaussian fluct. (Das, Bhaduri, PM 2001; Chatterjee, PM 2003) 2 •An ∼ nlP , n>> 1 (justify later) 1 S (A ) = S (A ) + log ∆(A ) can IH IH IH 2 IH th fluc corr | {z } The main issue :

• Expect Scan + ve real ⇒ C > 0 (th stab). How/when violated (e.g. Schwarzschild, RN)? Created with pptalk Slide 8

Condition for thermal stability (Majhi, PM 2011)

ZG = g(Am,Qn)exp −β[M(Am,Qn) − ΦQn] m,n X 2 2 LQG : Am ∼ mlP , m>> 1 , Ahor >> lP , Qn ∼ n

dA dQ Z = g(A,Q) exp −β[M(A,Q) − ΦQ] G A Q Z x y = dA dQ eS(A)−βM(A,Q)+βΦQ Z S(A) ≡ log g assumed indep of Q. Measure factors do not contribute. Expansion around A,¯ Q¯ ⇒ : Gaussian approx

⇒ conditions on Hessian matrix for convergence of ZG Created with pptalk Slide 9 ¯ SA(A) β ≡ ¯ ¯ > 0 MA(A, Q) ¯ ¯ ¯ βMAA(A, Q) − SAA(A) > 0

¯ ¯ ¯ ¯ ¯ 2 2 ¯ ¯ βMAA(A, Q) − SAA(A) βMQQ(A, Q) − β MAQ(A, Q) > 0 Solve as Partial differential inequality Ansatz:

M(A,¯ Q¯)= µ(A¯) · χ(Q¯) , χ(0) = 1 → µ µ S χQ χQ AA − AA > µ µ S χ χ A  A A  QQ Created with pptalk Slide 10 Solution : 1 χ(Q) = (1 + CQ)κ−1 κ µ(A) > (αS)κ−1 κ Choose constants κ> 1 , (kBα)κ−1 = MP ⇒

1 M S S κ−1 > M k k (1 + CQ) P B  B  S 1 S S = 1+ ln + ··· > k κ − 1 k (1 + CQ) k B   B   B • Checks out with Q =0 case PM 2007, 2009 • No classical metric used in derivation • Necessary and sufficient condition for qbh to be thermally stable Created with pptalk Slide 11

Fiducial checks Majhi, PM 2011 Reissner Nordstrom A 1/2 4πQ2 M = 1+ RN 4π A   " # Violates stability bound → thermally unstable Anti-de Sitter Reissner Nordstrom A 1/2 4πQ2 A M = 1+ + ADSRN 4π A 4πl2   " # Satisfies stability bound for A >> 4πl2 • Without use of classical metric, have derived a criterion of thermal stability for thermally active horizons • Initial fiducial checks against stationary black hole spacetimes all match • Interplay between thermal vs gravitational instability ?