Simple stationary plasma flows and their

Simple stationary plasma flows and their exotic black hole duals Toby Wiseman (Imperial) Work with Pau Figueras (DAMTP) • arXiv:1212.4498 See also arXiv:1312.0612 with Don Marolf and Mukund Rangamani (Cambridge ’14) Thursday, 27 March 14 Plan for this talk • AdS-CFT and plasma dynamics • Stationary plasma flows = ‘dynamics’ and have non-Killing horizons • Numerical methods for stationary non-Killing horizons • Numerical results for a particular set of stationary flows Thursday, 27 March 14 AdS4-CFT3 • The vacuum geometry is AdS4 - the CFT ‘lives’ on the boundary ⇤= UV Conformal boundary (Minkowski) l2 ds = dz + ⌘µ⌫ dxµ dx⌫ z CFT coordinates x Bulk radial coordinate z z=0 Thursday, 27 March 14 IR l2 AdS-CFT • Finite temperature CFT (ie plasma) = planar horizon in the bulk UV l ds2 = z Thursday, 27 March 14 Horizon ✓ ◆ z z03 ✓ dt + ◆ z z03 dz + dx2 ! AdS-CFT • Planar black holes have moduli; temperature and velocity UV Horizon Velocity Temperature Thursday, 27 March 14 AdS-CFT: Fluid/gravity • Fluid/gravity correspondence [ Bhattacharyya, Hubeny, Minwalla, Rangamani ; Baier et al ’07 ] • The black holes have moduli; the temperature and velocity • In the moduli space approximation to the dynamics these moduli obey the relativistic fluid equations • To next order, there is a viscous correction, and then higher derivative corrections that can be computed that characterize the microphysics of the plasma • Beyond slow variations, the gravity solution computes the plasma behaviour in the dual strongly coupled gauge theory! But obviously it is difficult to find these solutions - typically requiring dynamical numerical GR [Chesler-Yaffe, ] Thursday, 27 March 14 Dynamics • The power of AdS-CFT is that it allows access to this regime beyond hydro, which is very interesting from the perspective of the dual QFT - in particular it allows one to study ‘quench’ behaviour • Focus on homogeneous quenches, but also now inhomogeneous numerical codes available [ Batilan, Gubser, Pretorius ] • Recent work by Balasubramanian and Herzog; implemented a Chesler-Yaffe style code to simulate time and spatial deformations of the boundary for planar bulk horizons Thursday, 27 March 14 Beyond hydro • The key point I wish to emphasize is that; • One can study the ‘beyond hydro’ regime, and quenches, in the context of stationary solutions • All that is required for entropy production (which is typical for departure from hydro) is that the CFT plasma is flowing - however all its stress tensor vev can be time independent • On can use a global Lorentz transformation to map stationary flows into dynamical ‘quench like’ behaviour - yields preferred set of dynamics Thursday, 27 March 14 Stationary plasma flow • Make plasma flow in ⇢ direction in metric; ⇢ ds2 = dt2 + d⇢2 + (⇢)dy Horizon Fluid flow y Velocity • Take; (⇢) = + 0.2 (1 + tanh( ⇢)) • For small this gives hydrodynamics, for large it is dominated by microscopic behaviour Thursday, 27 March 14 Stationary = dynamical • If flow has ingoing Minkowski region can always boost to obtain a time and space dependent dynamics; deformation moves through a static plasma time flow direction Thursday, 27 March 14 Characteristic approach; pure static gravity • Static problem should be elliptic; specify asymptotics and horizon regularity • Use a characteristic version of the einstein eq - `Harmonic einstein eqn’ - to manifest this character - or the DeTurck ‘trick‘ [ Headrick, Kitchen, TW ’09 ] H Rµ⇥ ⇥ Rµ⇥ ⇤(µ • Reference connection • Now; H Rµ⌅ ⇥ g ⇥ ⇥) =0 ⇥g ¯ µ⇤ µ⇤ µ⇤ ¯ µ⇤ ⇥ ⇥ gµ⌅ • Analogous to generalized harmonic coordinates; =0 = Thursday, 27 March 14 ⇤2S x = H ⇥ g µ⇤ ¯ µ⇤ Dynamical characteristic approach • In the dynamical context this fixes the gauge • Bianchi identity; µ + Rµ⇥ ⇥ =0 (*) = and its time derivative on a • Dynamically in gen harm coords may fix Cauchy surface and then (*) implies vanishes to the future H R • Although one solves µ⇥ = one can guarantee finding a solution to Rµ⇥ = in gen harm gauge • Then Thursday, 27 March 14 H R since µ⌅ ⇥ g =0 ⇥ ⇥ gµ⌅ have characteristic hyperbolic evolution Stationary case • Treat directly in lorentzian signature; [ Adams, Kitchen, TW ’11 ] • First consider globally timelike stationary killing vector ⇥/⇥t (note - we have excluded black holes!) ⇥2 g= N (x) dt + Ai (x)dxi + hij (x)dxi dxj • Since hij riemannian this gives an elliptic system since; H Rµ⌅ Thursday, 27 March 14 ⇥ g ⇥ ⇥ ⇥⇥ gµ⌅ + = ij h ⇥i ⇥j gµ⌅ + Stationary case • However for horizons and in particular ergoregions there is no globally timelike stationary killing field • Assume rigidity (Hawking; Ishibashi, Hollands, Wald); • Assume in addition to T = ⇥/⇥t there are additional commuting killing vectors Ra which generate closed or open orbits • Assume killing horizon with normal; • May write the metric as; y A = {t, y a } ds = gµ⇥ dX dX = GAB (x) dy + µ where Thursday, 27 March 14 ⇥ Ra = ⇥/⇥y a K=T + A i AA (x)dx i ⇥ and if closed a Ra dy + B ya j AB (x)dx j ya + ⇥ + hij (x)dxi dxj Stationary case • Following uniqueness theorem proofs think of lorentzian spacetime as fibration over the base (the `orbit space’) with metric hij • Key point: elliptic; *assume* base metric hij is riemannian Then the equations are H RAB = H RAi = H Rij = g g g mn ⇥ ⇥⇥ gAB + = h ⇥m ⇥n GAB + ⇥ mn ⇥ B ⇥ ⇥⇥ gAi + = h ⇥m ⇥n GAB Ai + ⇥ mn ⇥ A B ⇥ ⇥⇥ gij + = h ⇥m ⇥n hij + GAB Ai Aj + ⇥ • As for uniqueness thms horizon and axes of symmetries becomes boundaries of base • At these boundaries, regularity of the spacetime prescribes certain boundary conditions - in particular the surface gravity and (angular) velocities are fixed Thursday, 27 March 14 Gauge `fixing’ • Elliptic formulation as a boundary value problem • However the ‘DeTurck’ term only fixes the gauge a postiori • Expect a solution Rµ⇥ = in gauge • There may be other solutions, Rµ⇥ = Thursday, 27 March 14 =0 (µ ⇥) = H Rµ⇥ =0 with non-trivial - ‘Ricci solitons’ Gauge `fixing’ H R • Since µ⇥ = is elliptic then a solution should be locally unique Hence can always distinguish a soliton from a ricci flat solution • However, there may exist only ricci flat solutions; • Bourguignon (’79) proves on compact manifold no solitons exist • We showed that for static vacuum spacetime with zero or negative ⇤ , then for asymtotally flat, kk or ads b.c.s, and for (extremal) horizons then no solitons are allowed [ Figueras, Lucietti, TW ’11] • Define; = ⇠ µ ⇠µ then Bianchi implies; r2 + ⇠ µ @µ • However, no such arguments for stationary or general matter cases Thursday, 27 March 14 Stationary black holes with non-Killing horizons Numerical methods Thursday, 27 March 14 The ingoing method for black holes • Instead of using coordinates adapted to the stationary Killing horizon, we use ingoing coordinates that extend inside the horizon; [ Figueras, TW ’12] see also [ Fischetti, Marolf, Santos ’12] No inner boundary condition Hyperbolic Elliptic Boundary conditions Elliptic problem Thursday, 27 March 14 Mixed Hyperbolic-Elliptic problem Some results • Surface gravity and linear velocity of the horizon [ see Visser et al ] 0.4 R tangent to horizon and orthogonal to @ , @ @t @y ΩH 0.5 @ = + ⌦H (⇢)R @t with R2 = κ 2 rµ ( 1.5 Thursday, 27 March 14 −5 ρ ⌫ ⌫ )= 2 µ v hT t⇢ i = 1+v hT tt + T ⇢⇢ i Some results • The local velocity of the plasma, measured from the CFT stress tensor 0.5 = {0.2, 0.3, 0.5, 0.7} v 0.45 0.4 −5 = {1, 1.5, 2} 0.5 −2 Thursday, 27 March 14 −1 ρ v hT t⇢ i = 1+v hT tt + T ⇢⇢ i Some results • The local velocity of the plasma, measured from the CFT stress tensor 0.5 = {0.2, 0.3, 0.5, 0.7} v 0.45 0.4 −5 = {1, 1.5, 2} 0.5 −2 Thursday, 27 March 14 −1 ρ v hT t⇢ i = 1+v hT tt + T ⇢⇢ i Some results • The local velocity of the plasma, measured from the CFT stress tensor 0.5 = {0.2, 0.3, 0.5, 0.7} v 0.45 0.4 −5 instability? = {1, 1.5, 2} 0.5 −2 Thursday, 27 March 14 −1 ρ v hT t⇢ i = 1+v hT tt + T ⇢⇢ i Some results • The local velocity of the plasma, measured from the CFT stress tensor 0.5 = {0.2, 0.3, 0.5, 0.7} v 0.45 0.4 −5 instability? = {1, 1.5, 2} 0.5 −2 Thursday, 27 March 14 −1 ρ Summary • Interestingly, in AdS-CFT, the black holes dual to relatively simple gauge theory physics can be very subtle • Duals to stationary plasma flows we expect to be stationary black holes with non-Killing horizons • Stationary flows with an asymptotic Minkowski region can be boosted to ‘dynamical’ flows, with a time and space dependent deformation applied to a static plasma • Numerical methods exist to find such solutions • Interesting to study their dynamics - eg turbulent instabilties? Thursday, 27 March 14 ... AdS-CFT and plasma dynamics • Stationary plasma flows = ‘dynamics’ and have non-Killing horizons • Numerical methods for stationary non-Killing horizons • Numerical results for a particular set of stationary. .. dual to relatively simple gauge theory physics can be very subtle • Duals to stationary plasma flows we expect to be stationary black holes with non-Killing horizons • Stationary flows with an asymptotic... Lorentz transformation to map stationary flows into dynamical ‘quench like’ behaviour - yields preferred set of dynamics Thursday, 27 March 14 Stationary plasma flow • Make plasma flow in ⇢ direction