![Page 1: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/1.jpg)
Hawking radiation in non-equilibrium SYM plasmas
Derek Teaney
SUNY Stony Brook and RBRC Fellow
• Heavy quarks: Jorge Casalderrey-Solana, DT; hep-th/0701123
• Dam T. Son, DT; JHEP. arXiv:0901.2338
• Simon Caron-Huot, DT, Paul Chesler; PRD, arXiv:1102.1073
• Paul Chesler and DT; arXiv:1112.6196
• Paul Chesler and DT; arXiv:1211.0343
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Brownian Motion and Equilibrium
Md2x
dt2= −η︸︷︷︸
Drag
x + ξ︸︷︷︸Noise
of Brownian Motion
“Artist’s” conception
1. Equilibrium is a state constant fluctuations
2. Equilibrium is a perpetual competition between drag and noise⟨ξ(t)ξ(t′)
⟩= 2Tη δ(t− t′) to reach equilibrium P (p) ∝ e− p2
2MT
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AdS/CFT
• Classical solutions in curved spacetime = CFT for nonzero temperature
ds2 = (πT )2r2[−f(r)dt2 + dx2
]+
dr2
r2f(r)f(r) = 1− 1
r4
Gravity
“Our”world r = ∞
Black Hole r = 1
How can a static metric be dual to equilibrium=constant fluctuations ?
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A heavy quark in AdS/CFT
• Solve classical string (Nambu-Goto) EOM and find:
Gravity
Stretched horizon
r = rm
r = 1
rh = 1 + ǫ
Not the dual of an equilibrated quark!
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Dissipation in classical black hole dynamics Herzog et al; DT J. Casalderrey-Solana; Gubser
Md2xo
dt2= −η︸︷︷︸
Drag
xo η =
√λ
2πgxx(rh) =
√λ
2π(πT )2︸ ︷︷ ︸
Coupling of string to near horizon metric
Classical dissipation determines drag
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Detailed Balance and Hawking Radiation:
Md2xo
dt2= −η︸︷︷︸
Drag
xo + ξ︸︷︷︸Noise
Gravity
UV Quant Flucts
xo
x(t, r)
Evolves to Classical
Prob. Dist :
P [x, πx] ∝ e−βH[x,πx]
Classical Dissipation Balanced by Hawking Radiation. Find in equilibrium:⟨ξ(t)ξ(t′)
⟩= 2Tηδ(t− t′)
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How to generalize to non-equilrium?
![Page 8: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/8.jpg)
Non-equilibrium setup in 4D: (Chesler-Yaffe)
1. Chesler and Yaffe turn on a strong gravitational pulse in “our” world
ds2 = −dt2 + eBo(t)dx2⊥ + e−2Bo(t)dx2
‖
where
Bo(t) ∝ e−t2/∆t2
Vacuum or
Low T plasma
GravitationalPulse
EquilibratedPlasma
Beginning Middle End
Non Equilbrium
plasma
time
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Non-equilibrium setup in 5D Chesler-Yaffe
1. Corresponds to non-equilibrium geometry with BH formation in AdS5
ds2 = −Adv2 + Σ2[eBdx2
⊥ + e−2Bdx2||]
+ 2dr dv ,
Geodesics falling into hole
(Time)
Bndry Pulse
apparent horizon
(hol
ogra
phic
coo
rd)
Even
t Hor
izon
Diverging Geodesics
Solve for A(v, r), B(v, r) and Σ(v, r) with Einstein eqs with B(v, r)→ Bo(t) on bndry.
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The boundary stress tensor
• The energy density increases by 50 times for a gaussian pulse with ∆t = 1/πTf
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-6 -4 -2 0 2 4 6
Tµ
ν / ε
final
v- πTfinal
ε/εf
PL/εf
Pulse On
ǫ = energy density
(time)
PL = 13ǫ
= Longitudinal pressure
Define an effective temperature:
1
Teff(v)= βeff(v) ∝ ε(v)−1/4
![Page 11: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/11.jpg)
Hawking emission and 2pnt functions in this geometry:
Vacuum or
Low T plasma
GravitationalPulse
EquilibratedPlasma
Beginning Middle End
Non Equilbrium
plasma
time
I want to compute the "photon" emission rate in the non-equilibrium plasma.
1. Study the equilibration of 2pnt functions in the plasma.
2. Study the non-equilibrium emission of quanta from the black brane
Emission from CFT is dual to emission from black brane
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Emission of dilatons weakly interacting with equilibrium strongly coupled SYM plasma
Equilibrated Plasma+ 4D Dilaton Field iSint = i
∫d4xφ(x)J(x)
• Emission:
(2π)32kdΓ<
d3k= G<(K) G<(K) =
⟨J(0)J(K)
⟩• Absorption: The absorption rate of Dilatons is
(2π)32kdΓ>
d3k= G>(K) G>(K) =
⟨J(K)J(0)
⟩• FDT: The Fluctuation Dissipation Relation reads[
G<(K)︸ ︷︷ ︸emission
]/[G>(K)︸ ︷︷ ︸
absorption
]= e−ω/T
We will compute the emission and absorption rates and check for detailed balance
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What the classical AdS/CFT usually computes:
Equilibrated Plasma+ 4D Dilaton Field nk = Dilaton occupation number
∂tnk = −nk Γ>︸︷︷︸absorb
+ (1 + nk) Γ<︸︷︷︸emit
• For a classical dilaton field nk � 1 the damping is
∂tnk = −nk × (Γ> − Γ<)︸ ︷︷ ︸classical absorption rate
• The classical absorption rate
G>(K)−G<(K) = −2 ImGR(K) = ρ(K)
Without assuming FDT, only the classical absorption rate is computable
with the classical black brane response.
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Summary: spectral density and statistical fluctuations
1. Spectral Density (commutator or G> −G<)
ρ(t1|t2) = 〈[φ(t1), φ(t2)]〉
• Records the dissipation of classical waves
2. Statistical fluctuations (anti-commutator or 12(G> +G<))
Grr(t1|t2) = 12 〈{φ(t1), φ(t2)}〉
• Invariably suppressed at large N and only due to Hawking radiation.
In non-equilibrium systems these correlators determine the emission/abs rates
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A non-equilibrium definition of the Emission and Absorption Rates
Want to know the rate to emit and absorb in a frequency band ω at time t
1. Wigner Transforms – perfect frequency resolution, but no time resolution
G<(t, ω) =
∫ ∞−∞
d∆te+iω∆t 〈J(t−∆t)J(t+ ∆t)〉
2. Gabor Transform – Wigner smeared with a minimum uncertainty wave packet
G<(to, ωo)︸ ︷︷ ︸Gabor
=
∫dtdω
2π2e−(ω−ωo)2σ2
e−(t−to)2/σ2︸ ︷︷ ︸minimum wave packet
G<(t, ω)︸ ︷︷ ︸Wigner Trans
G<(to, ωo) determines for the local emission rate for a given temporal resolution
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-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-6 -4 -2 0 2 4 6
Tµ
ν / ε
final
v- πTfinal
ε/εf
PL/εf
Resolution
Pulse On
• Temporal Resolution
σvπTf =1√2' 0.7
• Frequency Resolution
σωω' 1/
√2
8' 10%
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Equilibration and the coarse-grained FDT
1. If the FDT is satisfied
G<(K) = e−ω/TG>(K)
then, the coarse-grained quantities satisfy
G<(to, ωo, q)︸ ︷︷ ︸emission
= e−ωoβeff
[eβ
2eff/4σ
2G>(to, ωo − βeff/2σ
2, q)]
︸ ︷︷ ︸absorption
We will monitor this “FDT” as a function of time to quantify equilibrium
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Hawking Radiation in and out of equilibrium
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Equilbrium:
Md2xo
dt2= −η︸︷︷︸
Drag
xo + ξ︸︷︷︸Noise
Gravity
UV Quant Flucts
xo
x(t, r)
Goals:
1. Will show that Hawking radiation is balanced by gravity in equilbrium
2. Generalize to non-equilibrium
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Detailed Balance and Hawking Radiation (Technical Discussion)
Gravity
UV Quant Flucts
xo
x(t, r)
1. Fluctuations:
Grr ≡1
2〈{x(t1, r1), x(t2, r2)}〉 ,
2. Dissipation (Spectral Density)
ρ ≡ 〈[x(t1, r1), x(t2, r2)]〉 .
• Equilibrium≡ Fluctuation Dissipation Theorem
Grr(ω, r1, r2) =
(1
2+ nB(ω)
)ρ(ω, r1, r2) n(ω) ≡ 1
eω/T − 1
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Formulas
• Action for string fluctuations, hµν = string metric
S =
√λ
2π
∫dtdr gxx
[−√hhµν∂µx∂νx
],
• hµν is the string metric
hµνdσµdσν = −(πT )2r2f(r)dt2 +dr2
f(r)r2,
• Retarded Green Function
iGR(t1r1|t2r2) ≡ θ(t− t′) 〈[x(t1, r1), x(t2, r2)]〉 ,
GR(t1r1|t2r2) is the classical response to a force at t2r2√λ
2π
[∂µ gxx
√hhµν∂ν
]GR(t1r1|t2r2) = δ(t1 − t2)δ(r1 − r2) ,
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The classical Green Function or response to a force:√λ
2π
[∂µ gxx
√hhµν∂ν
]GR = F δ(t1 − t2)δ(r1 − r2) ,
Upward wave
Downward wave
External Force
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Outgoing Geo
desic
(Infalling Time)
Ingoing Wave
Outgoing Wave
Reflected Wave
v =Eddington time
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Statistical Fluctuations
Gravity
UV Quant Flucts
xo
x(t, r)
Grr =1
2〈{x(t1, r1), x(t2, r2)}〉
• The statistical correlator obeys the homogeneous EOM
√λ
2π
[∂µ gxx
√hhµν∂ν
]Grr(t1r1|t2r2) = 0
• So:
1. Specify the correlations (or density matrix) in the past
2. Final state fluctuations are correlated only through initial conditions
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Correlations through Initial conditions
Out
goin
g ge
odes
ics
Time
Spec
ify
Initi
al D
ata
-6 -4 -2 0 2 4
0.5
1
1.5
2
2.5
3
3.5
r
v (2πT)
Correlated through
Initial conditions
Horizon characterized by inflating outgoing geodesics:
r(v)− 1 = (ro − 1) eκ (v−vo) with κ ≡ 2πT
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Correlations through Initial conditions
Time
Consider Init
Data HerePoints uncorrelated
by this Init data
-6 -4 -2 0 2 4
0.5
1
1.5
2
2.5
3
3.5
r
v (2πT)
Init data falls in
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Correlations through Initial conditions
-6 -4 -2 0 2 4
0.5
1
1.5
2
2.5
3
3.5
r
v (2πT)
At late times
This is the only
initial data that mattersCorrelated through
Initial conditions
Time
1. Final correlation come from correlated initial data very near horizon
• Short Wavelength
2. Initial data is inflated by near horizon geometry
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Initial Data from Quantum Fluctuations:
1. Initial data is determined at short distance = Flat Space Physics
2. Scalar Field in 1+1D vacuum flat space
1
2〈{φ(X1), φ(X2)}〉 = − 1
4πKlog |µ
∆s2︷ ︸︸ ︷ηµν∆Xµ∆Xν | K=norm of action
3. String flucts in near horizon geometry
Snear−horizon = η
∫dtdr
[−1
2
√hhµν∂µx∂νx
]η =
√λ
2πgxx(rh)︸ ︷︷ ︸
norm of near horizon-action
The near horizon initial condition is:
Grr(v1r1|v2r2)→ − 1
4πηlog
∣∣∣∣∣∣∣µlocal ∆s2︷ ︸︸ ︷2∆v∆r
∣∣∣∣∣∣∣
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Summary: Specify IC and Solve Equations of Motion√λ
2π
[∂µ gxx
√hhµν∂ν
]Grr(t1r1|t2r2) = 0
-6 -4 -2 0 2 4
0.5
1
1.5
2
2.5
3
3.5
r
v (2πT)
log Init Cond
Correlated via
Time
Init. Cond
1′
1
2
2′
∝ log(∆r)
Inflationary near horizon geometry
(r − 1) =⇒ (r − 1)eκt
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From initial data to final correlations in two steps:
t0
r = 1
r = 1 + εr
t
1′
1
2
Wednesday, January 26, 2011
Use Boltzmann approxhere Full wave eqn here
GR(1|1′) =
∫dt2GR(1|2)
[η√hhrr(r2)
←→∂r2
]r2=1+ε
GR(2|1′) ,
(a) From horizon to stretched horizon – Waves are very short wavelength
– Use collisionless Boltzmann approximation (geodesic/WKB/eikonal approx)
(b) The stretched horizon to boundary – Waves are longer wavelength
– Use full wave equation
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Fluctuations from Equations of Motion
Grr(1|2)︸ ︷︷ ︸bulk flucts
=
∫dt1hdt2h GR(1|1h) GR(2|2h)︸ ︷︷ ︸
outgoing Green fcns
Ghrr(1h|2h)︸ ︷︷ ︸horizon flucts
,
-4 -3 -2 -1 0 1 2 3 4 5 0.8
1
1.2
1.4
1.6
1.8
2
r
v (2πT) (Time)
2h
12
Init conditions∝ log(r)
1hGhrr
The fluctuations on the stretched horizon are from UV vacuum flucts in past
Ghrr(t1|t2) = Blow-up of initial data∝ log(r)
=− η
π∂t1∂t2 log |eκ t1 − eκ t2 | .
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The horizon fluctuations and the Lyapunov exponent
-4 -3 -2 -1 0 1 2 3 4 5 0.8
1
1.2
1.4
1.6
1.8
2
r
v (2πT) (Time)
2h
12
Init conditions∝ log(r)
1hGhrr
1. Thermal looking:
Ghrr(ω) =Fourier-Trans of − η
π∂t1∂t2 log |eκ t1 − eκ t2 |
=(
12 + n(ω)
)2ωη n(ω) ≡ 1
e2πω/κ − 1
2. Temperature∝ inflation rate
κ = 2πT = Lyapunov exponent of diverging geodesics
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Dissipation - Spectral Density
Gravity
UV Quant Flucts
xo
x(t, r)
ρ = 〈[x(t1, r1), x(t2, r2)]〉
• The spectral density also obeys the EOM√λ
2π
[∂µ gxx
√hhµν∂ν
]ρ(t1r1|t2r2) = 0
• But initial conditions are given by the canonical commutation relations
η√hhtt(r1) lim
t2→t1
∂t1ρ(t1r1|t2r2) = iδ(r1 − r2) .
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Spectral Density
ρ(1|2)︸ ︷︷ ︸bulk spectral fcn
=
∫dt1hdt2h GR(1|1h) GR(2|2h)︸ ︷︷ ︸
outgoing Green fcns
ρh(1h|2h)︸ ︷︷ ︸horizon spectral fcn
,
-4 -3 -2 -1 0 1 2 3 4 5 0.8
1
1.2
1.4
1.6
1.8
2
r
v (2πT) (Time)
δ(r)
Init conditions
2 1
ρhra−ar
Where the horizon spectral density
ρh(t1, t2) = local due to canonical commutation relations
=2η[−iδ′(t1 − t2)
](2ωη in Fourier space)
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Detailed Balance
Grr(ω, r1, r2) =(
12 + n(ω)
)ρ(ω, r1, r2)
Gravity
UV Quant Flucts
xo
x(t, r)
1. Fluctuations (Anti-commutator)
Grr(ω, r1, r2)︸ ︷︷ ︸bulk flucts
= GR(ω, r1|rh) GR(ω, r2|rh)︸ ︷︷ ︸outgoing Green fcns
(12 + n(ω)
)2ωη︸ ︷︷ ︸
Horizon-flucts
2. Dissipation: (Commutator)
ρ(ω, r1, r2)︸ ︷︷ ︸bulk spec dense
= GR(ω, r1|rh) GR(ω, r2|rh)︸ ︷︷ ︸outgoing Green fcns
2ωη︸︷︷︸Horizon spec dense
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Non-equilibrium
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Fluctuations in non-equilibrium
Event Horizon
log correlation
here
Becomes stat correl
here
• Surface Properties – on event horizon
κ(v)︸︷︷︸time dep. Lyapunov exponent
≡
Metric−coeff︷ ︸︸ ︷1
2
∂A(r, v)
∂r
∣∣∣∣∣∣∣∣∣r=rh(v)
![Page 39: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/39.jpg)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
-10 -5 0 5 10
κ(t
) /
(2πT
f)
t(πTf)
low temperature
hight temperature
(time)
inflation rate κ(t)
![Page 40: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/40.jpg)
Result:
• General form of near horizon fluctuations in non-equilibrium
Ghrr(v1|v2) = −√η(v1)η(v2)
π∂v1∂v2 log |e
∫ v1 κ(v′)dv′ − e∫ v2 κ(v′)dv′ | .
• Can map the near horizon fluctuations up to boundary by finding GR numerically
Event Horizon
Ghrr
GR
GR
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Results for non-equilbrium emission
![Page 42: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/42.jpg)
Emission&Absorption rates and the FDT:
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Tµ
ν /
ε final
(a)
ε/εfPL/εf
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3 4 5 6
g-< /
g-<fin
al
v- πTfinal
FDT-expectation Emission rate
(b)
Lightlike
Timelike
FDT satisfied
Stress Tensor
Emission &Absorption
Timelike: ω ' 8πTf and q = 0 Lightlike: ω ' 8πTf and qT = qL = ω/√
2
![Page 43: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/43.jpg)
Pattern of equilibration:
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Tµ
ν /
εfin
al
(a)
ε/εfPL/εf
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3 4 5 6
g-< /
g-<fin
al
v- πTfinal
FDT-expectation Emission rate
(b)
Lightlike
Timelike
First the stress/geometryequilibrates
then
the emission rateequilibrates
![Page 44: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/44.jpg)
Thermalization of timelike modes q = 0:
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3 4 5 6
g-< / g-
<final
v- πTfinal
Timelike
(c)
ωo/πTf
2
4
8
12
Find that massive timkelike modes thermalize in a finite time:
![Page 45: Derek Teaney SUNY Stony Brook and RBRC Fellow](https://reader036.vdocuments.site/reader036/viewer/2022072502/62dd0c55515c0d474d56f9de/html5/thumbnails/45.jpg)
τthermalize ∼ const ω →∞
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Thermalization of approx lightlike modes (ω ' |q|) Chesler et al, Arnold&Vaman
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3 4 5 6
g-< /
g-<
final
v- πTfinal
Lightlike
(d)
ωo/πTf
2
4
8
12
The harder the lightlike mode, the longer it takes to equilibrate – find that
τthermalize ∼ (ωσ)1/4 for ω →∞ where Q2 = (ω2−q2) ∼ ωσ−1︸ ︷︷ ︸virtuality
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Summary:
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3 4 5 6
g-< / g-
<final
v- πTfinal
Timelike
(c)
ωo/πTf
2
4
8
12
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2 3 4 5 6
g-< / g-
<final
v- πTfinal
Lightlike
(d)
ωo/πTf
2
4
8
12
1. Find that massive timkelike modes thermalize in a finite time:
τthermalize ∼ const ω →∞
2. The harder the lightlike mode, the longer it takes to equilibrate – expect that:
τthermalize ∼ (ωσ)1/4 for ω →∞
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Conclusions
• Derived Hawking Radiation for non-equilibrium geometries
– Hawking radiation produces statistical fluctuations in strongly coupled plasma
• Used this setup to calculate emission rates in far from equilibrium plasma
• Find a distinct pattern of thermalziation (similar to weak coupling):
1. First the stress tensor equilibrates and then the 2pnt funcs equilibrate
2. Highly offshell modes (ω →∞ with k fixed) thermalize first.
3. High momentum onshell modes (ω ' k →∞) thermalize last.