DynamicsofNear-Extremal BlackHolesinAdS 4 arXiv:1802.09547 with A. - PowerPoint PPT Presentation

DynamicsofNear-Extremal BlackHolesinAdS 4 arXiv:1802.09547 with A. Shukla, R. M. Soni, S. P. Trivedi & M. V. Vishal by Pranjal Nayak Great Lakes Strings 2018 April 15, 2018

Main results The Einstein-Maxwell theory in 4-dimensions doesn’t flow to JT theory in IR limit However, the dynamics, at low energies and to leading order in the parameter L / r h , is well approximated by the Jackiw-Teitelboim theory of gravity The low-energy dynamics is determined by symmetry considerations alone, with the JT theory being the simplest realisation of these symmetries 1/27

Introduction Introduction 2/27

B=0 makes a consistent theory, but it lacks any interesting dynamics! How to regulate the backreaction was studied by [Almehiri-Polchinski] Problems with AdS 2 /CFT 1 Degrees of Freedom counting in a d-dimensional theory of gravity: d ( d − 3)/2 tells us that there ‘-1’ degrees of freedom in 2-dimensions [c.f. Finn’s talk] ! A theory with scaling symmetry in time direction has density of states, ρ ( E ) = A δ ( E ) + B E Introduction 3/27

How to regulate the backreaction was studied by [Almehiri-Polchinski] Problems with AdS 2 /CFT 1 Degrees of Freedom counting in a d-dimensional theory of gravity: d ( d − 3)/2 tells us that there ‘-1’ degrees of freedom in 2-dimensions [c.f. Finn’s talk] ! A theory with scaling symmetry in time direction has density of states, ρ ( E ) = A δ ( E ) + B E B=0 makes a consistent theory, but it lacks any interesting dynamics! Introduction 3/27

Problems with AdS 2 /CFT 1 Degrees of Freedom counting in a d-dimensional theory of gravity: d ( d − 3)/2 tells us that there ‘-1’ degrees of freedom in 2-dimensions [c.f. Finn’s talk] ! A theory with scaling symmetry in time direction has density of states, ρ ( E ) = A δ ( E ) + B E B=0 makes a consistent theory, but it lacks any interesting dynamics! How to regulate the backreaction was studied by [Almehiri-Polchinski] Introduction 3/27

this gives rise to, r h f t t G bdy nAdS 2 /nCFT 1 Recently proposed duality between SYK/tensor model and JT theory [c.f. Sumit’s talk, Kitaev, Maldacena-Stanford] Polyakov induced gravity theory can also be shown to reproduce the same physics [Mandal-Nayak-Wadia] Other models of 2-dimensional gravity can be shown to reproduce the same physics [today’s talk :)] Symmetry breaking structure: Reparametrization → SL (2 , R ) Introduction 4/27

nAdS 2 /nCFT 1 Recently proposed duality between SYK/tensor model and JT theory [c.f. Sumit’s talk, Kitaev, Maldacena-Stanford] Polyakov induced gravity theory can also be shown to reproduce the same physics [Mandal-Nayak-Wadia] Other models of 2-dimensional gravity can be shown to reproduce the same physics [today’s talk :)] Symmetry breaking structure: Reparametrization → SL (2 , R ) this gives rise to, − r 2 ∫ h { f ( t ) , t } G α bdy Introduction 4/27

Space of AAdS 2 Geometries 2-dimensional geometries with constant negative curvature and asymptotic AdS boundary conditions can be generated by applying large difgeomorphisms [Mandal-Nayak-Wadia, Jensen] Introduction 5/27

Space of AAdS 2 Geometries In Fefgerman-Graham gauge, δ g zz = 0 = δ g zt , these geometries are characterized by metric, ( ) 2 ) ds 2 = L 2 1 − z 2 { f ( t ) , t } ( dz 2 + dt 2 2 z 2 2 These modes as the pseudo-Goldstone modes that Sumit talked about yesterday Introduction 5/27

Action on AAdS 2 geometries In models of pure 2-dimensional gravity, these geometries have a trivial action cost associated with them when the backreaction is regulated, and reparametrization symmetry is broken the action on these geometries is given by a Schwarzian action, − r 2 ∫ h { f ( t ) , t } G α bdy Introduction 6/27

S-waveReduction S-wave Reduction 7/27

S-wave Reduction of Einstein-Maxwell Einstein-Maxwell system in 4-dimensions 1 ∫ 1 ∫ ( ) S = − d 4 x g ˆ R − 2ˆ d 3 x γ K (3) √ √ ˆ Λ − ˆ 16 π G 8 π G + 1 ∫ d 4 x √ g F 2 ˆ 4 G can be reduced in the S-wave sector using the following metric ansatz, ds 2 = g αβ ( t , r ) dx α dx β + Φ 2 ( t , r ) d Ω 2 2 S-wave Reduction 8/27

S-wave Reduction of Einstein-Maxwell S = − 1 d 2 x √ g ∫ [ Λ) + 2( ∇ Φ) 2 ] 2 + Φ 2 ( R − 2ˆ 4 G + 2 π Q 2 ∫ d 2 x √ g 1 Φ 2 − 1 ∫ √ γ Φ 2 K . m G 2 G bdy To compare with the JT action, we need to rescale the 2-dimensional metric, g αβ → r h Φ g αβ and redefine, Φ = r h (1 + ϕ ) S-wave Reduction 8/27

Does JT still play a role in higher dimensional low energy computation? S-wave Reduction of Einstein-Maxwell Then the action that one obtains is, S = − r 2 (∫ d 2 x √ g R + 2 √ γ K ) ∫ h 4 G bdy − r 2 d 2 x √ g φ ( R − Λ 2 ) − r 2 ∫ ∫ √ γ φ K h h 2 G G bdy +3 r 2 d 2 x √ g φ 2 − r 2 h κ ∫ ∫ √ γ φ 2 K h 2 G G L 2 bdy 2 S-wave Reduction 9/27

S-wave Reduction of Einstein-Maxwell Then the action that one obtains is, S = − r 2 (∫ d 2 x √ g R + 2 √ γ K ) ∫ h 4 G bdy − r 2 d 2 x √ g φ ( R − Λ 2 ) − r 2 ∫ ∫ √ γ φ K h h 2 G G bdy +3 r 2 d 2 x √ g φ 2 − r 2 h κ ∫ ∫ √ γ φ 2 K h 2 G G L 2 bdy 2 Does JT still play a role in higher dimensional low energy computation? S-wave Reduction 9/27

4DSphericallySymmetric Reissner-NordströmBH 4D Spherically Symmetric Reissner-Nordström BH 10/27

r h r h r h Q ext r h M ext L G L Solution Einstein-Maxwell system has following BH solution: ds 2 = − a ( r ) 2 dt 2 + a ( r ) − 2 dr 2 + b ( r ) 2 ( d θ 2 + sin 2 θ d φ 2 ) a ( r ) 2 = 1 − 2 GM + 4 π Q 2 + r 2 r r 2 L 2 b ( r ) 2 = r 2 4D Spherically Symmetric Reissner-Nordström BH 11/27

Solution Einstein-Maxwell system has following Extremal BH solution: ds 2 = − a ( r ) 2 dt 2 + a ( r ) − 2 dr 2 + b ( r ) 2 ( d θ 2 + sin 2 θ d φ 2 ) a ( r ) 2 = ( r − r h ) 2 L 2 + 3 r 2 h + 2 rr h + r 2 ) ( r 2 L 2 b ( r ) 2 = r 2 h + 3 r 4 M ext = r h 1 + 2 r 2 ext = 1 ( ) ( ) h h Q 2 r 2 , L 2 G L 2 4 π 4D Spherically Symmetric Reissner-Nordström BH 11/27

Near Horizon Limit The extremal solution has a near horizon AdS 2 limit, ( r − r h ) ≪ r h : − ( r − r h ) 2 L 2 [ ] ds 2 = dt 2 + ( r − r h ) 2 dr 2 + r 2 h ( d θ 2 + sin 2 θ d φ 2 ) 2 L 2 2 L L 2 = 6 , is the radius of AdS 2 √ Components of the above AdS 2 metric receive ( r − r h ) corrections @ O r h ‘Boundary’ of AdS 2 is in the region ( r − r h ) ≫ L 2 r h ≫ ( r − r h ) ≫ L 4D Spherically Symmetric Reissner-Nordström BH 12/27

Near Horizon Limit For r → ∞ the geometry is asymptotically AdS 4 . r → r c , where L ≪ r c − r h ≪ r h , is the asymptotic AdS 2 × S 2 region. The horizon at extremality is at r = r h . 4D Spherically Symmetric Reissner-Nordström BH 13/27

Thermodynamics in Near-Extremal BH Extremal Blackholes have 0 temperature Heating the BH slightly gives rise to Near-Extremal BH, the degenerate horizon splits into inner and outer horizons, r ± = r h ± δ r h , δ r h ≪ r h T = L 2 + 6 r 2 δ r h δ r h → 3 h 2 π L 2 r 2 L 2 π h This is achieved by changing the mass of the BH, h ( L 2 + 6 r 2 δ M = δ r 2 h ) 2 GL 2 r h 4D Spherically Symmetric Reissner-Nordström BH 14/27

Thermodynamics in Near-Extremal BH Thermodynamic partition function can be computed by evaluating the on-shell action (with correct holographic counterterms [Skenderis-Solodukhin, Balasubramanian-Krauss] ), Z [ β ] = e − β F = e − S − S count 1 ∫ √ g ( R − 2Λ) − 1 ∫ √ γ K S = − 16 π G 8 π G M ∂ M + 1 ∫ √ g F 2 4 G M 1 + L 2 1 ( ) ∫ √ γ S count = 4 R 3 4 π GL ∂ M 4D Spherically Symmetric Reissner-Nordström BH 14/27

Thermodynamics in Near-Extremal BH In the generic case, we get, β F = β M − S ent = β M − π r 2 + G For the near extremal BH, to the leading order β F = β M ext − βδ M − π r 2 h G Other thermodynamic quantities: Entropy, S ent = π r 2 h G 3 G TL 2 r h Specific heat, C = d δ M dT = 2 π 2 4D Spherically Symmetric Reissner-Nordström BH 15/27

Comparing with the results of JT The action for JT gravity, S JT = − r 2 (∫ d 2 x √ g R + 2 ∫ √ γ K ) h 4 G bdy − r 2 (∫ d 2 x √ g ϕ ( R + 2 ) ∫ √ γ ϕ ( K − 1 )) h + 2 2 G L 2 L 2 bdy 2 Finite temperature solutions of JT theory are given by, ( ( r − r h ) 2 − 2 G δ M dr 2 ) ds 2 = d τ 2 + L 2 r h ( ( r − r h ) 2 ) − 2 G δ M 2 L 2 r h 2 Topological term = 4 π R = − 2/ L 2 2 therefore the bulk integral doesn’t contribute Boundary integral evaluates to − βδ M 4D Spherically Symmetric Reissner-Nordström BH 16/27

Comparing with the results of JT The action for JT gravity, S JT = − r 2 (∫ d 2 x √ g R + 2 ) ∫ √ γ K h 4 G bdy − r 2 (∫ d 2 x √ g ϕ ( ) ( )) R + 2 ∫ √ γ ϕ K − 1 h + 2 2 G L 2 L 2 bdy 2 Topological term = 4 π R = − 2/ L 2 2 therefore the bulk integral doesn’t contribute Boundary integral evaluates to − βδ M S JT = − βδ M − π r 2 h G 4D Spherically Symmetric Reissner-Nordström BH 16/27

Computingthe4-ptFunction Computing the 4-pt Function 17/27