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Simple stationary plasma flows and their exotic black hole duals - PowerPoint PPT Presentation

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


  1. 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

  2. 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

  3. 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

  4. 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

  5. AdS-CFT • Planar black holes have moduli; temperature and velocity UV Horizon Velocity Temperature Thursday, 27 March 14

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. Stationary black holes (with Killing horizons) Numerical methods Thursday, 27 March 14

  13. 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

  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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. Stationary black holes with non -Killing horizons Numerical methods Thursday, 27 March 14

  22. 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

  23. 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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