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Non-equilibrium dynamics and theRobinson-Trautman solution
Kostas Skenderis
Southampton Theory Astrophysics andGravity research centre
STAG RESEARCHCENTERSTAG RESEARCH
CENTERSTAG RESEARCHCENTER
New Frontiers in Dynamical GravityCambridge, UK, 28 March 2014
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Introduction
â Gauge/gravity duality offers a new tool to studynon-equilibrium dynamics at strong coupling.
â AdS black holes correspond to thermal states of the CFT.â Black hole formation corresponds to thermalization.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Introduction
â Hydrodynamics capture the dynamics the longwave-length, late time behavior of QFTs close to thermalequilibrium.
â On the gravitational side, one can construct bulk solutionsin a gradient expansion that describe the hydrodynamicregime.
â Global solutions corresponding to non-equilibriumconfigurations should be well-approximated by thesolutions describing the hydrodynamic regime atsufficiently long distances and late times.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Introduction
â Almost all work on global solutions is numerical.â In this work we aim at obtaining analytic solutions
describing out-of-equilibrium dynamics.â We will discuss this in the context AdS4/CFT3.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
References
â This talk is based on work done with I. Bakas, to appear.â Related work appeared very recently in [G. de Freitas, H.
Reall, 1403.3537]
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Equilibrium configuration
â The thermal state corresponds to the AdS Schwarzschildblack hole
ds2 = −f(r)dt2 +dr2
f(r)+ r2
(dθ2 + sin2θdφ2
),
with f(r) = 1− 2mr − Λ
3 r2.
â Linear perturbations around the Schwarzschild solutiondescribe holographically thermal 2-point functions in thedual QFT.
â From those, using linear response theory, one can obtainthe transport coefficients entering the hydrodynamicdescription close to thermal equilibrium.
â To describe out-of-equilibrium dynamics we need to gobeyond linear perturbations.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Strategy
To describe analytically non-equilibrium phenomena and theirapproach to equilibrium we need
à Exact time-dependent solutions of Einstein equations.à These solutions should limit at late times to the
Schwarzschild solution.
â Can we find analytically exact solutions corresponding tolinear perturbations of the Schwarzschild solution?
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Linear perturbations of AdS Schwarzschild
Parity even metric perturbations of Schwarzschild solution areparametrized by
f(r)H0(r) H1(r) 0 0
H1(r) H0(r)/f(r) 0 0
0 0 r2K(r) 0
0 0 0 r2K(r)sin2θ
e−iωtPl(cosθ) ,
where Pl(cosθ) are Legendre polynomials. (For simplicity weonly display axially symmetric perturbations.)
à There are also parity odd perturbations. We will not needtheir explicit form here.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Effective Schödinger problem
â The study of these perturbations can be reduced to aneffective Schrödinger problem [Regge, Wheeler] [Zerilli] ...(
− d2
dr2?
+W 2 ± dW
dr?
)Ψ(r?) = EΨ(r?) .
à The two signs correspond to the parity even and oddcases.
à E = ω2 − ω2s , ωs = − i
12m(l − 1)l(l + 1)(l + 2) .
à Ψeven(r) = r2
(l−1)(l+2)r+6m
(K(r)− if(r)
ωr H1(r))
and there isa similar formula for the odd case.
à r? is the tortoise radial coordinate, dr? = dr/f(r).
à W (r) = 6mf(r)r[(l−1)(l+2)r+6m] + iωs
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Supersymmetric Quantum mechanics
â There is an underlying supersymmetric structure with Wbeing the superpotential,
Heven = Q†Q+ ω2s , Hodd = QQ
†+ ω2
s
with
Q =
(− d
dr?+W (r?)
), Q† =
(d
dr?+W (r?)
)â Forming
H =
(Heven 0
0 Hodd
)Q =
(0 0Q 0
)one finds that they form a SUSY algebra, {Q,Q†} = H etc.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Remarks
â The Hamiltonian is only formally hermitian.
â Boundary condition break supersymmetry.
â E is not bounded from below, it is not even real.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Zero energy solutions
â A special class of solutions are those with zero energy,
E = 0 ⇔ ω = ωs
â These modes satisfy a first order equation
QΨ0 =
(− d
dr?+W (r?)
)Ψ0 = 0
They are the supersymmetric ground states ofsupersymmetric quantum mechanics.
â These are the so-called algebraically special modes[Chandrasekhar].
â It is these modes that we would like to study at thenon-linear level.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Boundary conditions
â Ψ0 vanishes at the horizon.â It is finite and satisfies mixed boundary conditions at the
conformal boundary,
d
dr?Ψ0(r?) |r?=0 =
(iωs −
2mΛ
(l − 1)(l + 2)
)Ψ0(r? = 0) .
â It is normalizable,∫ 0
−∞dr? | Ψ0(r?) |2<∞ .
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Robinson-Trautman spacetimes
The metric is given by
ds2 = 2r2eΦ(z,z;u)dzdz − 2dudr − F (r, u, z, z)du2
The function F is uniquely determined in terms of Φ,
F = r∂uΦ−∆Φ− 2m
r− Λ
3r2
where Λ is related to the cosmological constant and∆ = eΦ∂z∂z.The function Φ(z, z;u) should solve the followingRobinson-Trautman equation,
3m∂uΦ + ∆∆Φ = 0.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Robinson-Trautman equation and the Calabi flow
â The Robinson-Trautman equation coincides with the Calabiflow on S2 that describes a class of deformations of themetric
ds22 = 2eΦ(z,z;u)dzdz .
â The Calabi flow is defined more generally for a metric gabon a Kähler manifold M by the Calabi equation
∂ugab =∂2R
∂za∂zb
where R is the curvature scalar of g.à It provides volume preserving deformations within a given
Kähler class of the metric.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Calabi flow on S2
â The Calabi flow can be regarded as a non-linear diffusionprocess on S2.
â Starting from a general initial metric gab(z, z; 0), the flowmonotonically deforms the metric to the constant curvaturemetric on S2, described by
eΦ0 =1
(1 + zz/2)2 .
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
AdS Schwarzschild as Robinson-Trautman
â Using the fixed point solution of the Robinson-Trautmanequation
eΦ0 =1
(1 + zz/2)2 .
the metric becomes
ds2 =2r2
(1 + zz/2)2dzdz − 2dudr −(
1− 2m
r− Λ
3r2
)du2
which is the Schwarzschild metric in the Eddington -Filkenstein coordinates.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Zero energy solutions as Robinson-Trautman
â Perturbatively solving the Robinson-Trautman equationaround the round sphere
ds22 = [1 + εl(u)Pl(cosθ)]
(dθ2 + sin2θdφ2
)one finds
εl(u) = εl(0)e−iωsu
withωs = −i(l − 1)l(l + 1)(l + 2)
12m
â This is exactly the frequency of the zero energy solutionswe found earlier!
â Inserting in the Robinson-Trautman metric we find the zeroenergy perturbations of AdS Schwarzschild we discussedearlier.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Summary
The Robinson-Trautman solution is a non-linearversion of the algebraically special perturbations ofSchwarzschild.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Late-time behavior of solutions [Chrusciel, Singleton]
â We parametrize the conformal factor of the S2 line elementas
eΦ(z,z;u) =1
σ2(z, z;u) (1 + zz/2)2 .
â σ(z, z;u) has the following asymptotic expansion
1 + σ1,0(z, z)e−2u/m + σ2,0(z, z)e−4u/m + · · ·+ σ14,0(z, z)e−28u/m
+[σ15,0(z, z) + σ15,1(z, z)u]e−30u/m +O(e−32u/m
).
â The terms with σ1,0, σ5,0, σ15,0, . . . are due to the linearalgebraically special modes with l = 2, 3, 4, . . ..
â The other terms are due to non-linear effects.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Global aspects
For large black holes, the solution does not appear to have asmooth extension beyond u→∞ [Bicak, Podolsky].
r = rh
u = ∞
r = 0
r = ∞I
H +
u = u0
1
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Other properties
â There is a past apparent horizon Σ, whose positionr = U(z, z) and area Area(Σ) we determined.
à At late times, Area(Σ) decreases and becomes equal toarea of the Schwarzschild horizon as u→∞.
â One can define a Bondi massMBondi with the properties
MBondi ≥ m,d
duMBondi ≤ 0,
that satisfies a Penrose inequality
16πM2Bondi ≥ Area(Σ)
(1− Λ
3
Area(Σ)
4π
)2
.
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Holography
â The boundary metric is time-dependent and it is notconformally flat
ds2I = −dt2 − 6
ΛeΦ(z,z;t)dzdz .
where Φ(z, z; t) = Φ(z, z;u = t− r?)|r?=0.â The holographic energy momentum tensor is
κ2T rentt = −2mΛ
3, κ2T ren
tz = −1
2∂z(∆Φ)
κ2T renzz = meΦ , κ2T ren
zz = − 3
4Λ∂t
((∂zΦ)2 − 2∂2
z Φ),
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Algebraically special modes
â The holographically energy momentum tensor for the linearalgebraically special modes can be rewritten in a fluid form
T ab = ρuaub + p∆ab − ησab
à 3-velocity
ut = −1, uφ = 0, uθ =1
4mΛ(l−1)(l+2)e−iωst∂θPl(cosθ)
à viscosity
κ2η =1
4l(l + 1)
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Violation of KSS bound
η
s=l(l + 1)
8π
rh
2m− rh
à The bound η/s ≥ 1/4π is violated for large black holes andsmall enough l.
à These modes however do not satisfy Dirichlet boundaryconditions.
à All modes that violate the bound do not extend smoothlybeyond u =∞ (however there are modes that do not havesmooth extension but nevertheless satisfy the bound).
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution
Conclusions/Outlook
â The Robinson-Trautman solution is a non-linear version ofthe algebraically special perturbation of Schwarzschild.
â One can study quantitatively and analytically the approachto equilibrium and the effects of non-linear terms.
â It would be interesting to understand better holography forthese solutions, in particular the implications of the unusualboundary conditions, the holographic meaning of the Bondimass, the Penrose inequality, etc. ...
Kostas Skenderis Non-equilibrium dynamics and the Robinson-Trautman solution