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
AdS 4 -CFT 3 • The vacuum geometry is AdS 4 - the CFT ‘lives’ on the boundary Λ = − 4 l 2 UV IR Conformal boundary (Minkowski) ds 2 = l 2 dz 2 + η µ ν dx µ dx ν � � z 2 CFT coordinates x Bulk radial coordinate z z = 0 Thursday, 27 March 14
AdS-CFT • Finite temperature CFT (ie. plasma) = planar horizon in the bulk UV Horizon ! ◆ − 1 ds 2 = l 2 1 − z 3 1 − z 3 ✓ ◆ ✓ dt 2 + dz 2 + dx 2 − z 3 z 3 z 2 0 0 Thursday, 27 March 14
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; ds 2 = − dt 2 + d ρ 2 + σ ( ρ ) dy 2 ρ Horizon ρ Fluid flow y Velocity • Take; σ ( ρ ) = 1 + 0 . 2 (1 + tanh( βρ )) • For small this gives hydrodynamics, for large it is dominated by β 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
Non-Killing horizon • Stationary compact horizons are Killing horizons. [Hawking; Hollands, Ishibashi, Wald] • In particular the temperature and velocity are constant on a Killing horizon • Since the temperature of the fluid must depend on then the horizon in this ρ case cannot be a Killing horizon! • No contradiction as it is not compact • Other examples; shocks [ Kruzenski ], flowing funnels [ Fischetti, Marolf, Santos ] • Related work; plasma flows non-translationally invariant spaces [ Iizuka, Ishibashi, Maeda ] Thursday, 27 March 14
Stationary black holes (with Killing horizons) Numerical methods Thursday, 27 March 14
Dynamical algorithm? • Could use a full dynamical evolution to find a stationary solution as an end state but... • Too much work • Difficult (impossible?) to find unstable solutions • Require very long time evolution for accurate solutions 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 ] ξ α ≡ g µ ν � R H µ ν − ¯ ⇥ Γ α Γ α µ ν ⇥ R µ ν � ⇤ ( µ ξ ν ) = 0 µ ν ¯ • Reference connection - Γ α µ ν µ ν ∼ − 1 • Now; R H 2 g αβ ∂ α ∂ β g µ ν • Analogous to generalized harmonic coordinates; ξ α = 0 = S x α = H α ⇥ � g µ ν ¯ ⇤ 2 Γ α ⇒ µ ν Thursday, 27 March 14
Dynamical characteristic approach • In the dynamical context this fixes the gauge. • Bianchi identity; (*) � 2 ξ µ + R ν µ ξ ν = 0 ξ α = 0 • Dynamically in gen. harm. coords may fix and its time derivative on a Cauchy surface and then (*) implies vanishes to the future. ξ α R H • Although one solves one can guarantee finding a solution to R µ ν = 0 µ ν = 0 in gen. harm. gauge ξ α = 0 µ ν ∼ − 1 • Then since have characteristic hyperbolic evolution R H 2 g αβ ∂ α ∂ β g µ ν Thursday, 27 March 14
Stationary case • Treat directly in lorentzian signature; [ Adams, Kitchen, TW ’11 ] • First consider globally timelike stationary killing vector (note - we have ∂ / ∂ t excluded black holes!) dt + A i ( x ) dx i ⇥ 2 + h ij ( x ) dx i dx j � g = − N ( x ) • Since riemannian this gives an elliptic system since; h ij µ ν ∼ − 1 2 g αβ ∂ α ∂ β g µ ν + . . . = − 1 R H 2 h ij ∂ i ∂ j g µ ν + . . . Thursday, 27 March 14
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 there are additional commuting killing T = ∂ / ∂ t vectors which generate closed or open orbits. R a • Assume killing horizon with normal; K = T + Ω a R a • May write the metric as; y A = { t, y a } ds 2 = g µ ν dX µ dX ν = G AB ( x ) dy A + A A dy B + A B i ( x ) dx i ⇥ � j ( x ) dx j ⇥ + h ij ( x ) dx i dx j � y a ∼ y a + 2 π where and if closed R a = ∂ / ∂ y a Thursday, 27 March 14
Stationary case • Following uniqueness theorem proofs think of lorentzian spacetime as fibration over the base (the `orbit space’) with metric h ij • Key point: *assume* base metric is riemannian. Then the equations are h ij elliptic; − 1 2 g αβ ∂ α ∂ β g AB + . . . = − 1 R H 2 h mn ∂ m ∂ n G AB + . . . = AB − 1 2 g αβ ∂ α ∂ β g Ai + . . . = − 1 R H 2 h mn ∂ m ∂ n G AB A B � ⇥ + . . . = Ai i − 1 2 g αβ ∂ α ∂ β g ij + . . . = − 1 R H 2 h mn ∂ m ∂ n h ij + G AB A A i A B � ⇥ + . . . = ij j • 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 in gauge ξ α = 0 R H µ ν = 0 R µ ν = 0 ⇒ = • There may be other solutions, with non-trivial - ‘Ricci solitons’. R µ ν = � ( µ ξ ν ) ξ α Thursday, 27 March 14
Gauge `fixing’ • Since is elliptic then a solution should be locally unique. Hence can R H µ ν = 0 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 Λ for asymtotally flat, kk or ads b.c.s, and for (extremal) horizons then no solitons are allowed. [ Figueras, Lucietti, TW ’11] • Define; then Bianchi implies; r 2 φ + ξ µ ∂ µ φ � 0 φ = ξ µ ξ µ ≥ 0 • 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 Mixed Hyperbolic-Elliptic problem Elliptic problem Thursday, 27 March 14
Some results • Surface gravity and linear velocity of the horizon [ see Visser et al ] χ = ∂ 0.5 ∂ t + Ω H ( ρ ) R Ω H R tangent to horizon and 0.4 orthogonal to ∂ ∂ ∂ t , ∂ y R 2 = 1 with 2 κ 2 r µ ( χ ν χ ν ) = � 2 κχ µ 1.5 − 5 0 5 ρ Thursday, 27 March 14
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