Expanding plasmas from Anti de Sitter black holes

Transcription

1 Expanding plasmas from Anti de Sitter black holes (based on [hep-th]) Giancarlo Camilo University of São Paulo Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 1 / 15

2 Objective Long-term goal: Understand the far-from-equilibrium behavior of QFTs. HERE: A good starting point: a QFT plasma undergoing (spatial) expansion. Simplifying assumption #1: Put the QFT in an expanding FLRW background: ds 2 = dt 2 + a(t) 2 dω 2 k Even a locally static fluid in FLRW has a nonzero expansion rate since spacetime itself expands, u µ = (1, 0, 0, 0) µ u µ = 3ȧ(t) a(t) Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 2 / 15

3 Statement of the problem Simplifying assumption #2: Approach the problem using holography, i.e., consider strongly coupled CFT with known AdS gravity dual The Problem: Find slicing of static bulk metrics such that AdS bdry = FLRW Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 3 / 15

4 Begin with generic AdS 5 BH written in standard (t, r) coordinates: ds 2 = f(r)dt 2 + f(r) 1 dr 2 + Σ(r) 2 dω 2 k f and Σ arbitrary except for f(r) r2 L 2, Σ(r) r L for r (AdS asymptotics) f(r h ) = 0 (event horizon) k = +1, 0, 1 all allowed Eventual supporting matter fields are OK (scalar hair, charge, etc) Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 4 / 15

5 ds 2 = f(r)dt 2 + f(r) 1 dr 2 + Σ(r) 2 dω 2 k STEP 1: Go to EF coords (v, r), i.e., dv = dt + f(r) 1 dr STEP 2: Introduce fake time dependence by redefining v as dv = dv a(v ) STEP 3: Redefine radial coordinate as r = Ra(V ) so as to put the metric back in EF-like coordinates (V, R): [ ] f(ra) ds 2 = 2dV dr a 2 2Rȧ dv 2 + Σ(Ra) 2 dω 2 k a Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 5 / 15

6 [ ] f(ra) ds 2 = 2dV dr a 2 2Rȧ dv 2 + Σ(Ra) 2 dω 2 k a At R = : ds 2 = R2 [ dv 2 L 2 + a(v ) 2 dω 2 ] k, so we have achieved our goal of setting the AdS to be FLRW Holographic dual of a CFT in a FLRW background Using it we can now extract info about how the CFT plasma expands, T µν (V ),... Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 6 / 15

7 Example 1: AdS-Schwarzschild Einstein-(Λ < 0) action: S = 1 16πG 5 d 5 x g [ R + 12 ] ( ) L 2 Static solution: f(r) = r2 1 + kl2 M Σ(r) = r L 2 r 2 r 4 L [ R ds 2 2 = 2dV dr (1 L 2 + kl2 R 2 a 2 M ) ] R 4 a 4 2Rȧ dv 2 + R2 a 2 a L 2 dω2 k Holographic dual of N = 4 SYM plasma in FLRW spacetime. The plasma has a local temperature T (V ) = kl2 + 2r 2 h 2πL 2 r h a. Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 7 / 15

8 Use G 5 = πl3 2N 2 c T ττ E = 3(ȧ2 + k) M 64πG 5 a 4 T i i P = (ȧ2 + k) 2 + 4M 4aä(ȧ 2 + k) 64πG 5 a 4 and M = M(T ) to put in pure CFT language. For ex., for k = 0 E = 3N c 2 T 4 + 3N c 2 ȧ π 2 a 4 P = E 3 N c 2 äȧ 2 8π 2 a 3. Nontrivial a(t) breaks conformal symmetry and leads to a conformal anomaly T µ µ = 3P E = 3ä(ȧ2 + k) 16πG 5 a 3 Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 8 / 15

9 Example 2: AdS-Gauss-Bonnet 1 EGB-(Λ < 0) action: S = 16πG 5 d 5 x [ g R + 12 ] L 2 + L2 2 λ (R GB abcdr abcd 4R ab R ab + R 2 ) Holographic dual of a CFT with two central charges c, b c = πl3 AdS 8G 5 1 4λGB b = πl3 AdS 8G 5 ( λGB ) E and P of the expanding plasma: E = 3(ȧ2 + k) M 64πG 5 a 4 3 [ 15(k + ȧ 2 ) 2 + 4M 64aä(k + ȧ 2 aä) ] 128πG 5 a 4 λ GB P = (ȧ2 + k) 2 + 4M 4aä(k + ȧ 2 ) 64πG 5 a 4 15(k + ȧ2 ) 2 + 4M 4aä[31(k + ȧ 2 ) 16aä] 128πG 5 a 4 λ GB ( ) Conformal anomaly T µ µ = 3P E = λ 3ä(k+ȧ 2 ) GB 16πG 5a 3 Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 9 / 15

10 Example 3: AdS-Reissner-Nordström Einstein-Maxwell-(Λ < 0) action: 1 S = d 5 x g 16πG 5 [ R + 12 L 2 1 ] 4 F abf ab Static solution (planar horizon k = 0): ) f(r) = (1 r2 Mr Q2 + 4 A adx a = µ L 2 ( ) 1 r2 h dt r 2 r 6 Σ(r) = r L Dual to a CFT plasma with temperature T and chemical potential µ Interesting also to illustrate how our foliation works in the presence of matter fields coupled to the metric Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 10 / 15

11 FLRW foliation dual to the expanding plasma [ ( ds 2 R 2 = 2dV dr 1 M L 2 R 4 a + Q2 4 R 6 a 6 ( A adx a = µ 1 r2 h R 2 a 2 ) [( 1 a L 2 ȧ/ra 2 1 M ) 2R ȧ a + Q2 R 4 a 4 R 6 a 6 ] dv 2 + R2 a 2 dx2 ) L 2 ] L 2 /R 2 a dv dr 1 M + Q2 R 4 a 4 R 6 a 6 The plasma is subject to a time-dependent chemical potential µ(v ) µ a(v ) : A µ dx µ bdry R= = µ a dv. Cool lesson: Flat space plasma subject to a quench µ(v ) in the chemical potential experiences nonequilibrium dynamics similar to a cosmological evolution with a(v ) µ(v ) 1 Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 11 / 15

12 Energy density E T ττ, pressure P T i i, and charge density Q J τ E = 3ȧ4 + 12r 2 h (r2 h µ2 ) 64πG 5 a 4 P = ȧ4 + 4r 2 h (r2 h µ2 ) 4aäȧ 2 64πG 5 a 4 Q = µ( 2r 2 h + 2ȧ2 + aä ) 16πG 5 a 3. Conformal anomaly is the same as in SAdS: T µν (V ) is conserved while charge density Q is not: T µ µ = 3ä(ȧ2 +k) 16πG 5a 3 µ T µν = 0 µ J µ = µ ( 3ȧä + a... a ) 2a3 Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 12 / 15

13 Entropy production [ ] f(ra) ds 2 = 2dV dr a 2 2Rȧ dv 2 + Σ(Ra) 2 dω 2 k a Apparent horizon R h (V ) defined by f(r h a)σ (R h a) = 0. Nonequilibrium entropy density of the dual expanding plasma can be associated with the BH entropy of the apparent horizon s BH = Σ(R ha) 3 4G 5 ds For conformal plasmas: dv = 0, i.e., no entropy production by the plasma (no dissipation due to ζ = 0) For nonconformal plasmas: ds dv > 0 due to dissipative effects. Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 13 / 15

14 Conclusions and Outlook New slicing of generic AdS BHs such that AdS = FLRW. EF coordinates make the task trivial Applicable to a variety of solutions, supported by external fields or not New arena to explore nonequilibrium dynamics of CFTs analytically Quenches (time-dependent couplings) arise naturally when analyzing BHs supported by matter fields: Gauge field A µ dx µ (r) µdv + A µ dx µ (Ra) µ a dv + Scalar hair φ(r) Jr d + φ(ra) Ja d R d + Cosmology?. Thank you! Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 14 / 15

15 Quantum quench J(t) Ja(t) d of a relevant O : S QFT = S CFT + d d xja(t) d O (flat) (t, x) flat Weyl rescaling ds 2 flat = a(t) 2 ds 2 FLRW : g = a(t) d O (FLRW) (τ, x) = a(t) O (flat) (t, x) S QFT = S CFT + d d x gjo (FLRW) (τ, x) FLRW Constant deformation J but in a FLRW spacetime! Giancarlo Camilo (IFUSP) Expanding plasmas from AdS black holes 15 / 15