The quantum fate of black hole horizons Sergey Solodukhin LMPT (Tours)/CERN Talk at RTG “Models of Gravity” workshop, 19-20 February 2018, Bremen With Cl´ ement Berthiere and Deb Sarkar; arXiv:1712.09914 Sergey Solodukhin The quantum fate of black hole horizons
Outline of the talk Motivations: wormholes as black hole mimickers Properties of classical black hole horizons Semiclassical gravity Black holes or wormholes in semiclassical gravity Conclusions Sergey Solodukhin The quantum fate of black hole horizons
Motivation Perturbations of classical black holes: have continuous spectrum Sergey Solodukhin The quantum fate of black hole horizons
Motivation Perturbations of classical black holes: have continuous spectrum have complex poles (QNM) Sergey Solodukhin The quantum fate of black hole horizons
Motivation Perturbations of classical black holes: have continuous spectrum have complex poles (QNM) relaxation back to equilibrium is due to exponential decay Sergey Solodukhin The quantum fate of black hole horizons
On the other hand, black holes have finite entropy! As any (classical or quantum) system of finite entropy they should show Poincar´ e recurrences t Poincare ∼ e S BH Susskind et al. ’02 Sergey Solodukhin The quantum fate of black hole horizons
The source of this discrepancy is infinite volume in optical metric ds 2 = g ( r ) dt 2 + g ( r ) − 1 dr 2 + r 2 d ω 2 d = g ( r ) ds 2 opt , g ( r ) = 1 − r + / r , 0 ≤ t ≤ β H drr 4 ˆ ( r − r + ) 2 → ∞ if r → r + V opt = 4 πβ H Sergey Solodukhin The quantum fate of black hole horizons
A smooth way to regularize this divergence is to replace black hole with a wormhole g tt → g tt + λ 2 , λ 2 ≪ 1 wh = ( g ( r ) + λ 2 ) dt 2 + g ( r ) − 1 dr 2 + r 2 d ω 2 ds 2 d Sergey Solodukhin The quantum fate of black hole horizons
New properties: there is no event horizon instead there is a throat at r = r + of size L ∼ r + ln 1 /λ t throat ∼ λ t ∞ two new time scales: t Heisenberg ∼ ln 1 /λ t Poincare ∼ 1 /λ If λ ∼ e − S BH one has a realization of Susskind’s ideas Important: during time scales ≪ t Heisenberg , t Poincare no difference with true black holes S.S.’04, ’05; T. Damour and S.S.’07 Sergey Solodukhin The quantum fate of black hole horizons
Applications in astrophysics/ gravitational waves: Wormholes of this type are considered as exotic compact object (ECOs) that may produce same gravitational wave signals as black holes Many papers including Cardoso, Franzin and Pani ’16; Bueno, Cano, Goelen, Hertog and Vernocke ’17 Sergey Solodukhin The quantum fate of black hole horizons
So far our wormhole was considered as a phenomenological metric. We obtain it as a solution to equations of semiclassical gravity. Sergey Solodukhin The quantum fate of black hole horizons
Two aspects of black hole horizons Universality at horizon ds 2 = g ( r ) dt 2 + e 2 φ ( r ) g − 1 ( r ) dr 2 + r 2 d ω 2 d g ( r ) = 4 π β ( r − r + ) + O ( r − r + ) 2 , φ ( r ) = O ( r − r + ) Optical metric ds 2 = g ( z ) ds 2 opt = dt 2 + dz 2 + R 2 ( z ) d ω 2 ds 2 opt , d g ( z ) ∼ e − 4 π z /β + . . . , R 2 ( z ) ∼ e 4 π z /β + . . . z → ∞ Optical spacetime is product space S β 1 × M 3 Near horizon M 3 is hyperbolic space H 3 of radius β/ 2 π It is a solution to GR equations to leading order for any β Sergey Solodukhin The quantum fate of black hole horizons
Horizon as a minimal surface ds 2 = Ω 2 ( ρ ) dt 2 + d ρ 2 + r 2 ( ρ ) d ω 2 d Ω 2 = g and ρ is geodesic radial coordinate Einstein equations: 2 rr ′′ + r ′ 2 − 1 = 0 Ω( r ′ 2 − 1) + 2 rr ′ Ω ′ = 0 If 2-sphere at ρ = ρ + has minimal area r ′ ( ρ + ) = 0 then Ω( ρ + ) = 0 and this sphere is a horizon Sergey Solodukhin The quantum fate of black hole horizons
Goal of this talk Study whether same properties are valid in semiclassical gravity (SG) Claims static spherically symmetric metric with a horizon of finite (non-vanishing) temperature is not a solution to SG in SG a 2-sphere of minimal area embedded in static space-time is not a horizon. Instead it is a throat of a wormhole Ω 2 at throat is bounded by e − S BH (consistent with Susskind’s ideas) Possible temperature is different from Hawking temperature and is exponentially small Sergey Solodukhin The quantum fate of black hole horizons
Before we start: general form of 4-metric we shall consider ds 2 = Ω 2 ( z ) dt 2 + N 2 ( z ) dz 2 + R 2 ( z ) d ω 2 Ω( z ) = e σ ( z ) � � , 2 Useful choices of coordinates (gauge fixing): Black hole horizon R ( z ) ≃ r + e 2 π z /β σ ( z ) = − 2 π z /β + . . . , N ( z ) = 1 , apriori β and r + are not related Minimal sphere N ( z ) = 1 / Ω( ρ ) , z = ρ , R ( ρ ) = r ( ρ ) / Ω( ρ ) r ( ρ ) is geometric radius of sphere Sergey Solodukhin The quantum fate of black hole horizons
Semiclassical Gravity (SG) Consider backreaction from quantized scalar, gauge and fermion fields on non-quantized geometric background. Non-perturbative handle is due to the study of conformal anomaly. Fradkin-Tseytlin ’84, Dowker-Schofield ’90, Mazur-Motola ’01 Two important contributions: due to anomaly and due to optical metric Sergey Solodukhin The quantum fate of black hole horizons
Gravitational action 1 ˆ W grav = − R [ G ] + Γ[ G ] 16 π G N Γ[ G ] is quantum effective action, result of integrating out quantum fields we represent G µν = e 2 σ g µν , quantum effective action transforms as a ˆ b ˆ σ E − 2 b ˆ Γ[ e 2 σ g ] = − σ C 2 + (2 ˜ G µν ∂ µ σ∂ ν σ +2 � σ ( ∇ σ ) 2 +( ∇ σ ) 4 )+Γ 0 [ g ] 16 π 2 16 π 2 16 π 2 G µν = R µν − 1 ˜ 2 g µν R is Einstein tensor C 2 = Riem 2 − 2 Ricci 2 + 1 3 R 2 , E = Riem 2 − 4 Ricci 2 + R 2 1 1 a = 120 ( n 0 + 6 n 1 / 2 + 12 n 1 ) , b = 360 ( n 0 + 11 n 1 / 2 + 62 n 1 ) Sergey Solodukhin The quantum fate of black hole horizons
Effective action on optical metric: 1 × M 3 ] = − π 2 λ H ˆ ˆ Γ 0 = Γ[ S β 90 β 3 c H 1 + R M 3 144 β M 3 M 3 c H = n 0 + 7 2 n 1 / 2 + 2 n 1 , λ H = n 1 / 2 + 4 n 1 - exact result if M 3 = H 3 - we dropped higher curvature (non-local) terms - general structure discussed by Gusev and Zelnikov ’98 Sergey Solodukhin The quantum fate of black hole horizons
Horizons in SG Variations of W grav [ σ ( z ) , N ( z ) , R ( z )] w.r.t. σ ( z ), N ( z ) and R ( z ) give semiclassical gravitational equations Some observations: E ( g optical ) = 0 and C 2 ( g optical ) → C 2 ( S 1 × H 3 ) = 0 as z → ∞ Variation w.r.t. N ( z ) will produce divergent (as z → ∞ ) terms. These terms will come from derivatives of σ in b-anomaly and from Γ 0 , the divergence is due to divergent volume density on M 3 π 2 180 β 4 r 2 + e 4 π z /β (360 b − 2 c H − 10 λ H ) δ N W grav = π 2 + e 4 π z /β . 180 β 4 r 2 = − � n 0 + 6 n 1 / 2 − 18 n 1 � - Curiously, the equations are satisfied for N = 4 SYM theory (have to look at subleading terms) - For generic set of fields divergent term is there so that no static solutions with horizons in SG! Sergey Solodukhin The quantum fate of black hole horizons
Minimal sphere We look for solutions with a turning point: r ′ ( ρ ) = 0 and Ω ′ ( ρ ) = 0 at ρ = ρ + such that r ′′ > 0 and Ω ′′ > 0 Such a solution is parametrized by values of r and Ω at turning point r is the radius of classical horizon Values of second derivatives r ′′ and Ω ′′ are determined by r and Ω via gravitational equations Additionally, there arise consistency conditions on possible values of Ω provided r can be arbitrary Variation w.r.t. N ( z ) takes the form at the turning point a ln Ω − 1 ( γκ r 2 r 2 λκ (Ω rr ′′ − r 2 Ω ′′ ) 2 = y 2 Ω 2 , y 2 = 1 − β 4 Ω 4 − β 2 Ω 2 + 1) κ ¯ a = a / 12 π 2 , γ = c H π 2 / 90, λ = λ H / 72 κ = 8 π G N , ¯ Condition that y 2 ≥ 0 restricts possible values of Ω ! Sergey Solodukhin The quantum fate of black hole horizons
For simplicity consider λ H = 0 (only scalars) Then condition y 2 ≥ 0 is equivalent to condition Ω ≥ γ r 4 Ω 4 ln Ω 0 Ω 0 = e − r 2 a β 4 , κ ¯ a ¯ Notice that r 2 a is proportional to Bekenstein-Hawking entropy S BH = 8 π 2 r 2 /κ of κ ¯ classical black hole It immediately follows that - Ω < Ω 0 = e − r 2 κ ¯ a - T 4 = 1 /β 4 < ¯ 4 γ r 4 Ω 4 a 0 , i.e. temperature is exponentially small! Conditions r ′′ > 0 and Ω ′′ > 0 impose extra constraints on possible values of Ω. In classical limit ¯ a → 0 so that Ω 0 = 0 and the throat becomes horizon! Sergey Solodukhin The quantum fate of black hole horizons
Conclusions Wormhole modification Sergey Solodukhin The quantum fate of black hole horizons
Conclusions Wormhole modification Non-perturbative and exact result Sergey Solodukhin The quantum fate of black hole horizons
Conclusions Wormhole modification Non-perturbative and exact result Experimentally hard to measure such small deviations t distinguish ∼ GM log 1 / Ω 0 ∼ G 2 M 3 Damour, Solodukhin ’07 Sergey Solodukhin The quantum fate of black hole horizons
Conclusions Wormhole modification Non-perturbative and exact result Experimentally hard to measure such small deviations t distinguish ∼ GM log 1 / Ω 0 ∼ G 2 M 3 Damour, Solodukhin ’07 Bound on temperature Sergey Solodukhin The quantum fate of black hole horizons
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