holographic perfect uidity, cotton energy-momentum duality
TRANSCRIPT
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Holographic perfect fluidity, Cotton
energy-momentum duality and transport
properties
Valentina Pozzoli
CPHT - Ecole Polytechnique
based on arXiv:1309.2310 and arXiv:1206.4351
with M. Caldarelli, R. Leigh, A. Mukhopadhyay,
A. Petkou, M. Petropoulos, K. Siampos
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Outline
1 Introduction
2 Stationary fluidsFluids on curved backgroundsRanders–Papapetrou geometries
3 Fluids and holographyThe holographic expansionSome examples
4 Exact bulk reconstructionMonopolar geometriesDipolar geometriesExact bulk metric
5 Conclusions
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
The fluid/gravity correspondence
Fluid/gravity correspondence: limit of AdS/CFT when the boundary iswell approximated by its long wavelength description.
Stationary black holesin 3+1 dimensions
2+1-dimensional perfect fluidswith non-trivial vorticity
Motivations
• Applications in AdS/CMT (Bose fast-rotating gases, ... )
• Analogue gravity interpretation (applications to meta-materials)
• Search of new black hole solutions
• Information on transport coefficients
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Outline
1 Introduction
2 Stationary fluidsFluids on curved backgroundsRanders–Papapetrou geometries
3 Fluids and holographyThe holographic expansionSome examples
4 Exact bulk reconstructionMonopolar geometriesDipolar geometriesExact bulk metric
5 Conclusions
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Fluids on curved backgrounds
Fluid dynamics on 2+1 dimensional curved backgrounds
Tµν function of uµ, gµν , and their covariant derivatives satisfies Eulerequations
∇µTµν = 0.
The energy–momentum tensor can be expanded as
Tµν = Tµν(0) + Tµν
(1) + Tµν(2) + · · · ,
where the subscript denotes the number of covariant derivatives and
Tµν(0) = εuµuν + p∆µν , ∆µν = uµuν + gµν .
→ Perfect-fluid energy–momentum tensor.→ Higher-order corrections involve the presence of transport coefficients.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Fluids on curved backgrounds
Transport coefficients
Higher order corrections:
Tµν(1) = −
(2ησµν + ζ∆µνΘ + ζHε
ρλ(µuρσλν)),
aµ = uν∆νuµ acceleration,σµν = ∇(µuν) + a(µuν) − 1
2 ∆µν∇ρuρ shear,
Θ = ∇µuµ expansion.
Transport coefficients
η, ζ, ζH , . . . transport coefficients.
• they give information on the microscopic theory,
• dissipative and non-dissipative.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Fluids on curved backgrounds
Conditions for perfect equilibrium
If we want the fluid on a curved background to be in perfect equilibrium
→ the transport coefficient is vanishing,
→ the corresponding tensor is vanishing.
For example, in Minkowskian backgrounds any inertial fluid hasTµν = T perf
µν .
→ Depending on the geometry, different sets of transport coefficients arerequired to vanish.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Randers–Papapetrou geometries
Randers–Papapetrou backgrounds
We consider stationary metrics with unique normalized time-like Killingvector:
ds2 = −(dt − bidxi )2 + aijdx
idx j .
Velocity one-form u = −dt + bidxi with vorcity ω = 1
2db.
In 2+1 dimensions
- Vorticity ωµν = − q2ηµνρu
ρ, where q(x) is a scalar field.
- Curvature Rµν .
- Cotton–York tensor Cµν = εµρσ∇ρ(Rνσ − 1
4Rδνσ).
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Outline
1 Introduction
2 Stationary fluidsFluids on curved backgroundsRanders–Papapetrou geometries
3 Fluids and holographyThe holographic expansionSome examples
4 Exact bulk reconstructionMonopolar geometriesDipolar geometriesExact bulk metric
5 Conclusions
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
The holographic expansion
The holographic expansion
Bulk solutions of general relativity with Λ = −3k2 can always be taken inthe form
ds2 = ηabθaθb = dr2
k2r2 + k2r2ηµνθµθν ≈ dr2
k2r2 + k2r2g(0)µνdxµdxν
→ The fluid boundary data is read at r →∞.
Expansion for large r of the bulk orthonormal frame:
θµ(r , x) = krEµ(x) + 1kr F
µ[2](x) + 1
k2r2 Fµ(x) + · · ·
→ only two independent data:
- the boundary metric ds2 = ηµνEµEν = g(0)µνdxµdxν
- the boundary energy–momentum tensor T = κFµeµ = TµνE
ν ⊗ eµ.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Some examples
Schwarzschild black hole
Bulk data
ds2 = dr2
V (r) − V (r)dt2 + r2(dθ2 + sin2 θdφ2
),
V (r) = 1 + k2r2 − 2M/r .Holographic coordinate: θr = dr/
√V (r) = dr/kr .
Boundary data
Boundary metric: ds2bdy = −dt2 + 1
k2
(dθ2 + sin2 θdφ2
).
Boundary energy–momentum tensor:T = κMk
3
(2dt2 + 1
k2
(dθ2 + sin2 θdφ2
)).
→ static perfect fluid with no vorticity.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Some examples
Taub-NUT black hole
Bulk data
ds2 = dr2
V (r) − V (r) (dt − 2ν cos θdφ)2 + ρ2(dθ2 + sin2 θdφ2
),
V (ρ) = ∆r/ρ2 with
∆r = (r2 − ν2)(1 + k2(r2 + 3ν2)) + 4k2ν r2 − 2Mr ,ρ2 = r2 + ν2.
Boundary data
Boundary metric:ds2
bdy = − (dt + 2ν(1− cos θ)dφ)2 + 1k2
(dθ2 + sin2 θdφ2
).
Velocity: u = −dt + bidxi = −dt + 2ν(cos θ − 1)dφ.
Boundary energy–momentum tensor:T = κMk
3
(2u2 + 1
k2
(dθ2 + sin2 θdφ2
)).
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Some examples
Taub-NUT boundary
→ Perfect fluid with vorticity:
ω = 12 db = −ν sin θdθ ∧ dφ.
In the bulk: non-rigid rotation with angular momentum distributionalong the Misner string.In the boundary: monopolar-like vorticity.
• North pole: angular velocity Ω∞ = νk2.
• South pole: no angular velocity.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Some examples
Kerr black hole
Bulk data
ds2 = dr2
V (r) − V (r)(dt − a
Ξ sin2 θdφ)2
+ ρ2
∆θdθ2 + sin2 θ∆θ
ρ2
(a dt − r2+a2
Ξ dφ)2
,
V (ρ) = ∆r/ρ2 with
∆r = (r2 + a2)(1 + k2r2)− 2Mr , ρ2 = r2 + a2 cos2 θ,∆θ = 1− k2a2 cos2 θ, Ξ = 1− k2a2.
Boundary data
Boundary metric:
ds2bdy = −
(dt − a sin2 θ
Ξ dφ)2
+ 1k2∆θ
(dθ2 +
(∆θ sin θ
Ξ
)2dφ2).
Velocity: u = −dt + bidxi = −dt + a sin2 θ
Ξ dφ.
Boundary energy–momentum tensor:T = κMk
3
(2u2 + 1
k2
(dθ2 + sin2 θdφ2
)).
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Some examples
Kerr boundary
→ Perfect fluid with vorticity:
ω = 12 db = a cos θ sin θ
Ξ dθ ∧ dφ.
In the bulk: rigid rotation.In the boundary: dipolar-like vorticity.
Constant boundary angular velocity Ω = −ak2.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Outline
1 Introduction
2 Stationary fluidsFluids on curved backgroundsRanders–Papapetrou geometries
3 Fluids and holographyThe holographic expansionSome examples
4 Exact bulk reconstructionMonopolar geometriesDipolar geometriesExact bulk metric
5 Conclusions
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Exact bulk reconstruction
Aim: find boundary geometries such that
g(0)µν , T perfµν
exact 3+1 dimensionalEinstein geometry
Answer: perfect-Cotton geometries Cµν = cT perfµν , with c constant:
Cµνdxµdxν = c
2
(2u2 + d`2
).
Valid in general, but explicit solution when an extra isometry is present:
• Monopolar geometries.
• Dipolar geometries.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Monopolar geometries
Monopolar geometries
→ The vorticity is constant.
Boundary properties
• The boundary is an homogeneous space → the space has thestructure of fibrations over S2,R2 or H2.
• No possible correction to the energy–momentum tensor can be built→ there are no constraint on the transport coefficients.
Bulk properties
Bulk geometries: Taub-NUT black holes with regular horizons.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Dipolar geometries
Dipolar geometries
→ The vorticity is not constant, but the space is conformally flat.
Boundary properties
• Axisymmetric spaces: global rigid rotation.
• The geometry allows for corrections to the energy–momentumtensor → holography puts constraints on an infinite number oftransport coefficients.
Bulk properties
Bulk geometries: Kerr black holes with regular horizons.
It is possible to consider general monopolar-dipolar geometries: thecorresponding bulk is Kerr-Taub-NUT.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Exact bulk metric
Solutions with one isometry
Randers–Papapetrou boundary
ds2bd = −u2 + d`2, R = R + q2
2
Bulk uplift of a perfect–Cotton geometry:
ds2 = −2u
(dr − 1
2dxρuσηρσµ
∇µq
)+ ρ2d`2 −
(r2 +
δ
2− q2
4− 1
ρ2(2Mr +
qc
2)
)u2,
δ = R + 3q2, ρ2 = r2 + q2
4 .
- It is a solution of Einstein’s equations.
- It contains the Kerr-Taub-NUT black holes as well as other solutions.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Outline
1 Introduction
2 Stationary fluidsFluids on curved backgroundsRanders–Papapetrou geometries
3 Fluids and holographyThe holographic expansionSome examples
4 Exact bulk reconstructionMonopolar geometriesDipolar geometriesExact bulk metric
5 Conclusions
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique
Introduction Stationary fluids Fluids and holography Exact bulk reconstruction Conclusions
Conclusions
Holographic fluids
Sufficient conditions for the correspondence to be exact and holographicconstraint of the transport coefficients.
Further directions
• Holography as a bottom-up solution generating technique.
• Probe more transport coefficients: higher multipole and perturbativeapproaches.
Holographic perfect fluidity, Cotton energy-momentum duality and transport properties Valentina Pozzoli CPHT - Ecole Polytechnique