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

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

Λ = − 4 l2 Conformal boundary (Minkowski) z = 0 ds2 = l2 z2

• dz2 + ηµνdxµdxν

CFT coordinates x z Bulk radial coordinate UV IR

Thursday, 27 March 14

AdS-CFT

• Finite temperature CFT (ie. plasma) = planar horizon in the bulk

UV Horizon ds2 = l2 z2 − ✓ 1 − z3 z3 ◆ dt2 + ✓ 1 − z3 z3 ◆−1 dz2 + dx2 !

Thursday, 27 March 14

AdS-CFT

• Planar black holes have moduli; temperature and velocity

UV Horizon Temperature Velocity

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;
• Take;
• For small this gives hydrodynamics, for large it is dominated by

microscopic behaviour.

Horizon Fluid flow y Velocity ds2 = −dt2 + dρ2 + σ(ρ)dy2 ρ

ρ

σ(ρ) = 1 + 0.2 (1 + tanh(βρ))

β

Thursday, 27 March 14

Stationary = dynamical time flow direction

• If flow has ingoing Minkowski region can always boost to obtain a time and

space dependent dynamics; deformation moves through a static plasma.

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

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 ]

ρ

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

• 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 ]

• Reference connection -
• Now;
• Analogous to generalized harmonic coordinates;

RH

µν ⇥ Rµν ⇤(µξν) = 0

ξα ≡ gµν Γα

µν − ¯

Γα

µν

⇥ RH

µν ∼ −1

2gαβ∂α∂βgµν ξα = 0 = ⇒ ⇤2

Sxα = Hα ⇥ gµν ¯

Γα

µν

¯ Γα

µν

Characteristic approach; pure static gravity

Thursday, 27 March 14

• In the dynamical context this fixes the gauge.
• Bianchi identity; (*)
• Dynamically in gen. harm. coords may fix and its time derivative on a

Cauchy surface and then (*) implies vanishes to the future.

• Although one solves one can guarantee finding a solution to

in gen. harm. gauge

• Then since have characteristic hyperbolic evolution

2ξµ + R ν

µ ξν = 0

ξα = 0 ξα RH

µν = 0

Rµν = 0 ξα = 0 RH

µν ∼ −1

2gαβ∂α∂βgµν

Dynamical characteristic approach

Thursday, 27 March 14

• Treat directly in lorentzian signature; [ Adams, Kitchen, TW ’11 ]
• First consider globally timelike stationary killing vector (note - we have

excluded black holes!)

• Since riemannian this gives an elliptic system since;

∂/∂t g = −N(x)

• dt + Ai(x)dxi⇥2 + hij(x)dxidxj

RH

µν ∼ −1

2gαβ∂α∂βgµν + . . . = −1 2hij∂i∂jgµν + . . . hij

Stationary case

Thursday, 27 March 14

• 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

vectors which generate closed or open orbits.

• Assume killing horizon with normal;
• May write the metric as;

T = ∂/∂t Ra K = T + ΩaRa ds2 = gµνdXµdXν = GAB(x)

• dyA + AA

i (x)dxi⇥

dyB + AB

j (x)dxj⇥

+ hij(x)dxidxj Ra = ∂/∂ya ya ∼ ya + 2π

where

yA = {t, ya}

and if closed

Stationary case

Thursday, 27 March 14

• Following uniqueness theorem proofs think of lorentzian spacetime as

fibration over the base (the `orbit space’) with metric

• Key point: *assume* base metric is riemannian. Then the equations are

elliptic;

• As for uniqueness thms horizon and axes of symmetries becomes boundaries
• f base.
• At these boundaries, regularity of the spacetime prescribes certain boundary

conditions - in particular the surface gravity and (angular) velocities are fixed

hij RH

AB

= −1 2gαβ∂α∂βgAB + . . . = −1 2hmn∂m∂nGAB + . . . RH

Ai

= −1 2gαβ∂α∂βgAi + . . . = −1 2hmn∂m∂n

• GABAB

i

⇥ + . . . RH

ij

= −1 2gαβ∂α∂βgij + . . . = −1 2hmn∂m∂n

• hij + GABAA

i AB j

⇥ + . . . hij

Stationary case

Thursday, 27 March 14

• Elliptic formulation as a boundary value problem.
• However the ‘DeTurck’ term only fixes the gauge a postiori.
• Expect a solution in gauge
• There may be other solutions, with non-trivial - ‘Ricci solitons’.

RH

µν = 0

Rµν = 0 ξα = 0 = ⇒ Rµν = (µξν) ξα

Gauge `fixing’

Thursday, 27 March 14

• Since 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; then Bianchi implies;
• However, no such arguments for stationary or general matter cases.

RH

µν = 0

Gauge `fixing’

φ = ξµξµ ≥ 0 Λ r2φ + ξµ∂µφ 0

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]

Elliptic problem Boundary conditions Mixed Hyperbolic-Elliptic problem Elliptic Hyperbolic No inner boundary condition

Thursday, 27 March 14

Some results

• Surface gravity and linear velocity of the horizon [ see Visser et al ]

0.4 0.5 ΩH −5 5 1.5 2 κ2 ρ

χ = ∂ ∂t + ΩH(ρ)R rµ(χνχν) = 2κχµ R2 = 1 R tangent to horizon and

• rthogonal to ∂

∂t , ∂ ∂y with

Thursday, 27 March 14

Some results

• The local velocity of the plasma, measured from the CFT stress tensor.

−5 5 0.4 0.45 0.5 v −2 −1 1 2 0.5 1 ρ

β = {0.2, 0.3, 0.5, 0.7} β = {1, 1.5, 2} v 1 + v2 = hT tρi hT tt + T ρρi

Thursday, 27 March 14

Some results

• The local velocity of the plasma, measured from the CFT stress tensor.

−5 5 0.4 0.45 0.5 v −2 −1 1 2 0.5 1 ρ

β = {0.2, 0.3, 0.5, 0.7} β = {1, 1.5, 2} v 1 + v2 = hT tρi hT tt + T ρρi

Thursday, 27 March 14

Some results

• The local velocity of the plasma, measured from the CFT stress tensor.

−5 5 0.4 0.45 0.5 v −2 −1 1 2 0.5 1 ρ

β = {0.2, 0.3, 0.5, 0.7} β = {1, 1.5, 2} v 1 + v2 = hT tρi hT tt + T ρρi

instability?

Thursday, 27 March 14

Some results

• The local velocity of the plasma, measured from the CFT stress tensor.

−5 5 0.4 0.45 0.5 v −2 −1 1 2 0.5 1 ρ

β = {0.2, 0.3, 0.5, 0.7} β = {1, 1.5, 2} v 1 + v2 = hT tρi hT tt + T ρρi

instability?

Thursday, 27 March 14

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