Holographic phase space and black holes as Holographic - PowerPoint PPT Presentation

Holographic phase space M. F. Paulos Holographic c-functions Holographic phase space and black holes as Holographic renormalization group flows phase space More on holographic phase space M. F. Paulos Laboratoire de Physique Théorique et Hautes Energies, Université Pierre et Marie Curie Based on arXiv: 1101.5993 DAMTP , University of Cambridge, 12/05/2011

Outline Holographic phase space M. F. Paulos Holographic c-functions Holographic Holographic c-functions 1 phase space More on holographic phase space 2 Holographic phase space More on holographic phase space 3

Outline Holographic phase space M. F. Paulos Holographic c-functions Holographic Holographic c-functions 1 phase space More on holographic phase space 2 Holographic phase space More on holographic phase space 3

c-Functions Holographic phase space M. F. Paulos c-functions measure degrees of freedom along RG flows Zamolodchikov ’86, Holographic c-functions Cappelli, Friedan Latorre ’90 Holographic phase space (2 π ) 2 e ipx ( g µν p 2 − p µ p ν )( g ρσ p 2 − p ρ p σ ) � ∞ d 2 p � T µν ( x ) T ρσ (0) � = π � More on d µc ( µ ) holographic p 2 + µ 2 3 phase 0 space Positivity of spectral measure implies � ǫ � ∞ c UV ≡ d µ c ( µ ) ≥ c IR ≡ lim d µ c ( µ ) ǫ → 0 0 0 At fixed points c ( µ ) = cδ ( µ ) : the c-function matches the CFT central charge. Monotonicity ⇒ irreversibility.

Cardy’s conjecture Holographic phase space M. F. Paulos Holographic c-functions In higher d , more anomaly coefficients: Holographic phase � T a a � ≃ ( − 1) d/ 2 A × ( Euler density )+ � space B i × ( Weyl contractions )+ ∇ ( . . . ) More on holographic phase space Conjecture: c-function involves Euler anomaly A Cardy ’88 . No known counter-example, non-trivial evidence available. No proof. Can holography help?

Holography and RG flow Holographic phase space M. F. Paulos Holographic AdS/CFT: a geometrization of RG flow. c-functions Holographic Scale → extra dimension r (non-trivial). phase space More on Example: relevant flows in N = 4 SYM ⇒ domain wall backgrounds holographic phase ds 2 = d r 2 + e 2 A ( r ) ( − d t 2 + d x 2 ) space with A (+ ∞ ) = r/L 1 , A ( −∞ ) = r/L 2 , L 1 > L 2 Shrinking of AdS radius corresponds to loss of degrees of freedom. Can we make this precise?

Holographic c-functions Holographic phase space M. F. Paulos Einstein equations determine: Girardello et al ’98, Freedman et al ’99 . Holographic c-functions ( d − 1) A ′′ ( r ) = ( T t t − T r r ) = − ( ρ + p r ) Holographic phase space Null energy condition ⇒ A ′′ ( r ) ≤ 0 . Define the c -function More on holographic c 0 c AdS ∝ ( L AdS /l P ) d − 2 . phase ⇒ c ( r ) = space l d − 2 ( A ′ ) d − 2 P Radial dependence related to field theory cut-off. Simple argument suggests: Polchinski, Heemskerk ’10 Λ ≃ e A A ′ ( r ) Exact relation unknown.

Lovelock theories of gravity Holographic phase space M. F. Paulos Holographic To distinguish different anomalies need more general gravity theory - c-functions higher derivatives! Holographic phase space Generically introduces ghosts, complicated equations. Special choice: More on Lovelock theory. holographic phase L ≃ R − 2Λ + E 2 k space with E 2 k the (2 k ) dimensional Euler densities. Nice properties! Non-ghosty vacua. Linearized EOM are 2-derivative. Exact black hole solutions exist.

Plan of attack 1. Holographic phase space M. F. Paulos Holographic c-functions Holographic phase What is the plan? space Construct a c -function for Lovelock theories of gravity. More on holographic phase Extra parameters will allow us to determine what the c stands for (i.e. space the Euler anomaly). Strategy is to consider equations of motion and reconstruct c -function from there.

Plan of attack 2. Holographic phase space M. F. Paulos Holographic c-functions Construct a “c-function” for black hole backgrounds: Holographic � dt 2 phase κ + r 2 L 2 dr 2 � ds 2 = − space + r 2 (dΣ d − 2 ) 2 , L 2 f ( r ) f ∞ + κ κ + r 2 L 2 g ( r ) More on holographic phase space Domain wall solutions are special case. Motivation: black hole horizons appear to have “emergent” conformal symmetry, as in extremal black hole geometries containing an AdS 2 factor. Carlip Suggests such geometries describe intriguing RG flows between CFT’s of different dimensionality.

Equations of motion Holographic phase space Action for Lovelock theories: M. F. Paulos Holographic � K � d d x √− g 1 � c-functions � n k c k E 2 k S = R − 2Λ + + S matter l d − 2 Holographic P phase k =2 space tt equation: More on = − L 2 l d − 2 holographic d � � phase r d − 1 Υ[ g ] r d − 2 T t P space t d r d − 2 with c k g k = 1 − g + c 2 g 2 + . . . � Υ[ g ] ≡ In the absence of matter, f = g and exact black hole solutions can be found by solving a polynomial equation! Υ[ g ] = m 0 r d − 1 m 0 ≃ mass.

Equations of motion 2. Holographic phase space M. F. Paulos Holographic Integrating the equation we find c-functions � r Holographic r 0 d r ′ ( r ′ /L ) d − 2 ρ ( r ′ ) Υ[ g ] = L d l d − 2 L d phase M ( r ) ( d − 2) V d − 2 l d − 2 P space ≡ r d − 1 . P d − 2 r d − 1 More on holographic phase Υ[ g ] ≃ ψ , the “Newtonian” potential. space The tt equation can be rewritten d r = 2 L d � d g dΨ � − Υ ′ [ g ] d r . d − 2 The gravitational field tells us about the direction in which g is decreasing.

The N -function Holographic phase space Start from the flow equation: M. F. Paulos � d g d r = 2 L d Holographic dΨ � − Υ ′ [ g ] d r . c-functions d − 2 Holographic phase space Define the N function: More on � K holographic � phase 1 ( d − 2) k � d − 2 k c k ( − g ) k − 1 N ( r ) = space . d − 2 g 2 k =1 The flow equation becomes � L � d � d N − dΨ � d r = √ g d r Important result : it describes the flow of N , in terms of local gravitational field.

Interlude: Euler anomaly Holographic phase space M. F. Paulos Holographic c-functions Conformal anomaly in even-dimensional CFT’s: Holographic phase space � T a a � ≃ ( − 1) d/ 2 A × ( Euler density )+ � B i × ( Weyl contractions )+ ∇ ( . . . ) More on holographic phase Computed holographically by Skenderis, Henningson ’98. space The A coefficient can be extracted from on-shell Lagrangian. Imbimbo,Schwimmer,Theisen ’99 How does A relate to the N function?

Interlude: Euler anomaly Holographic phase space M. F. Paulos Start with action: Holographic c-functions d d x √− g ( L g + L matter ) , � Holographic S = phase space ( L ( k ) has k curvatures) � L ( k ) , More on L g = holographic phase k space and the equation of motion aef + 1 2 g cd L g + ∂ L − 2 ∇ a ∇ b X acbd + X aecf R d ∂g cd = T cd matter with δS X abcd = δR abcd

Interlude: Euler anomaly Holographic phase space M. F. Paulos Holographic 2 g cd L g + ∂ L aef + 1 c-functions − 2 ∇ a ∇ b X acbd + X aecf R d ∂g cd = T cd matter Holographic phase space More on Assume we have an AdS background. In these circumstances we must holographic have T cd = − 1 2 g cd L m , and all covariant derivatives vanish. Taking the phase space trace of the equation of motion above we find � kL k + d � + d � 2 L k − 2 kL k ) 2 L m = 0 k kL k = d X abcd R abcd = � ⇒ 2 ( L g + L m )

Interlude: Euler anomaly Holographic phase space In AdS M. F. Paulos X abcd = ( g ac g bd − g bc g ad ) X rt rt , Holographic c-functions and therefore we conclude Holographic phase space 4 δS L g + L m = dR More on δR rt rt holographic phase space The Euler anomaly is given by: � A = 1 ∂ L g � ∂R abcd ǫ ab ǫ cd . � 2 � boundary where ǫ rt = √− g rr g tt with all other components zero is a spacelike surface binormal. This is very similar to Wald’s black hole entropy formula: √ � h ∂ L g S BH = − 2 π ∂R abcd ǫ ab ǫ cd horizon

Euler anomaly for Lovelock theories Holographic phase space M. F. Paulos For Lovelock theories of gravity, we get Holographic c-functions � L � K � � d − 2 1 ( d − 2) k Holographic � d − 2 k c k ( − f ∞ ) k − 1 A = phase l P d − 2 space f 2 k =1 ∞ More on holographic This should be compared to the N function phase space � K � 1 ( d − 2) k � d − 2 k c k ( − g ) k − 1 N ( r ) = . d − 2 g 2 k =1 Clearly then we have � L � d − 2 N ( ∞ ) = A l P

N as a c -function Holographic phase space More generally, N correctly captures the Euler anomaly in AdS M. F. Paulos background. The flow equation sets its monotonicity: Holographic � L � d � c-functions � d N − dΨ d r = √ g Holographic d r phase space We impose the null energy condition, ρ + p ≥ 0 . However in RG flow More on holographic backgrounds we have phase space p r = − M ( r ) /V d − 2 . r d − 1 This implies − dΨ ρ + p ≥ 0 ⇒ M ( r ) ≤ 0 ⇒ d r ≥ 0 Therefore N ( r ) monotonously decreases from UV to IR in RG flow backgrounds, and equals the Euler anomaly at AdS fixed points: it is a holographic c -function.