Effective actions for fluids from holography and the membrane - PowerPoint PPT Presentation

Effective actions for fluids from holography and the membrane paradigm Natalia Pinzani Fokeeva University of Amsterdam Oxford, 25th November, 2014 based on hep-th: 1405.4243 and on 1411.xxxx with Jan de Boer and Michal P. Heller

Effective actions for fluids from holography and the membrane paradigm Fluid behavior is ubiquitous in physics Fluid dynamics is the low energy effective description of a system valid when fluctuations around thermal equilibrium are sufficiently long-wavelength L ≫ l mfp Conventional description: Effective field theory description: [Landau et al.] • ∇ µ T µν = 0 [Nicolis, Son et al. 2006] • based on an action principle • constitutive relations • more economic and natural • constraints on the transport coefficients coming from • no systematic inclusion of ∇ µ J µ ≥ 0 dissipation so far

Effective actions for fluids from holography and the membrane paradigm What can holography teach us? Holographic gravity = QFT + its renormalization group flow • one should be able to derive the low energy effective action of the dual field theory from holography • In gravity dissipation is naturally encoded in the one way nature of the event horizon [Nickel and Son 2010] ⇒ one should be able to characterize easier the nature of dissipation

Effective actions for fluids from holography and the membrane paradigm Membrane paradigm: [Damour; Thorne, MacDonald and Price 80´s] approximation scheme in which near horizon details of a black hole are neglected and one retains only the ingoing behavior property of the horizon • In order to include dissipation one can rely on such membrane paradigm approximation • It is important to understand what are the limits of validity of such approximation

Contents Effective actions for fluids Motivation 1) Effective actions for fluids from holography 2) Including dissipation 3) Is the membrane paradigm a good approximation? Conclusions & Outlook

Outline Effective actions for fluids Motivation 1) Effective actions for fluids from holography 2) Including dissipation 3) Is the membrane paradigm a good approximation? Conclusions & Outlook

Effective actions for fluids - The degrees of freedom Consider for simplicity an uncharged fluid Fluid degrees of freedom: φ I = φ I ( t, � x ) as a map between Eulerian and Lagrangian coordinates φ ( t, � x ) x 3 φ 3 φ 2 x 2 x 1 φ 1

Effective actions for fluids - The symmetries Internal symmetries: Spacetime symmetries: φ I → φ I + c I • shifts • Poincar´ e invariance φ I → R I J φ J • rotations • for ideal fluids volume preserving diffs invariance � ∂ξ I � φ I → ξ I ( φ ); det = 1 ∂φ J Goldstones break the global subgroup of the internal and spacetime symmetries down to a diagonal subgroup � φ = � x + � π ( t, � x )

Effective actions for fluids - The leading order Based on those symmetries construct the most general effective action in a derivative expansion S (0) + S (1) + S (2) + . . . The leading order effective action: d d x √− gF ( s ) � S (0) = where s is unique volume preserving diffs invariant object � det( ∂ µ φ I ∂ ν φ J g µν ) s =

Effective actions for fluids - The stress-energy tensor • The conserved stress-energy tensor is the ideal fluid stress tensor T (0) µν = p ( g µν + u µ u ν ) + ρ u µ u ν • provided that ρ = − F , p = − F ′ s + F , T = − F ′ J µ = ∗ ( dφ 1 ∧ dφ 2 ∧ . . . ) J µ = s u µ � − J µ J µ , s = • ∇ µ J µ = 0 identically, hence this construction is dissipationless

Effective actions for fluids - The linearized expansion • Divide into longitudinal and transverse modes � φ = � x + � π π = π T + π L , ∇ ∧ π L = 0 , ∇ · π T = 0 such that • The action up to quadratic order in an amplitude expansion S (0) ∼ π T ) 2 + ( ˙ π L ) 2 − c 2 s ( ∇ · π L ) 2 � � d d x � ( ˙ + . . . • The dispersion relations for the Goldstones are then π T : ω T = 0 π L : ω L = c s k • π T is non propagating, reflecting the volume preserving diffs invariance

Effective actions for fluids - Comments • Proceed at higher orders: S (1) , S (2) , . . . • Application to superfluids, solids, inflationary models, quantum Hall effect etc. [Nicolis et al, Rangamani et al, Son et al] • Dissipative effects? • Is volume preserving symmetry a necessary fundamental symmetry? [Rangamani et al 2012]

Outline Effective actions for fluids Motivation 1) Effective actions for fluids from holography 2) Including dissipation 3) Is the membrane paradigm a good approximation? Conclusions & Outlook

Motivation 1) Is it possible to derive an effective action for conformal fluids from holography? ⇒ Yes! This provides an explicit example of such effective actions constructions 2) Is it possible to decouple the dissipative from the dissipationless sector? ⇒ No! Certain divergent terms are only removed when dissipation is taken into account 3) What are the limits of validity of the membrane paradigm as an approximation scheme? ⇒ OK for hydrodynamic quasi normal modes ∗ ! KO for massive quasi normal modes!

Outline Effective actions for fluids Motivation 1) Effective actions for fluids from holography 2) Including dissipation 3) Is the membrane paradigm a good approximation? Conclusions & Outlook

1) Effective actions from holography - The set-up Integrating out high energy d.o.f. ⇔ ¨Integrating out¨ shells of geometry [Faulkner, Liu and Rangamani; Heemskerk and Polchinski, 2010] Singularity S UV • Divide spacetime in UV and IR u = 1 with a finite cutoff u δ Horizon S tot. = S IR + S UV u δ u = 0 • Solve a double-Dirichlet problem for gravitational perturbations in S IR UV B o u The effective action in holography: n d is the (partially) on-shell UV action a r + near horizon limit u δ → 1 y Integrated out d.o.f.

1) Effective actions from holography - The Goldstones Singularity The Goldstones: u = 1 correspond to a spontaneous symmetry Horizon breaking by the classical solution with double-Dirichlet boundary conditions u δ u = 0 Poincar´ e × Poincar´ e → Diag (Rot. + Transl.) Dirichlet Dirichlet π t • On a finite cutoff: π, � B o • On the horizon: � π u n d • Goldstones correspond to a diffeomorphisms x α → x α + ξ α ( x ) r y from general gauge to radial gauge where ξ α ( x ) = π α ( x )

1) Effective actions from holography - Ex:linearized pert. in AdS 5 • Low energy behavior of thermal N = 4 SYM (conformal fluid) • The action is d 5 x √− g ( R − 2Λ) 1 � S = 2 k 2 5 • Black-brane geometry background in AdS ds 2 = − ( πTL ) 2 L 2 ( − f ( u ) dt 2 + d� x 2 )+ 4 u 2 f ( u ) du 2 , f = 1 − u 2 u � dωdk • Linearized perturbations: δh µν ( t, x, u ) = (2 π ) 2 δh µν ( ω, k, u ) e − iωt + ikx transverse sector: δh tα , δh xα , δh αu with α = y, z longitudinal sector: δh tt , δh tx , δh � x , δh tu , δh xu , δh uu x�

1) Effective actions from holography - Ex: transverse sector Transverse sector: δh tα , δh xα , δh αu with α = y, z • E.o.m: 2 second order + 1 first oder constraint • Since we want to solve a double-Dirichlet problem we leave the constraints unsolved • The (hydrodynamic) solution is not unique since it depends on the arbitrary gauge choice encoded in the fields δh uα • Goldstones are the Wilson line-like objects � u δ π α ∼ δh uα du 0 • Imposing radial gauge δh uα = 0 , the Goldstones are non trivial boundary conditions to be imposed on the second boundary

1) Effective actions from holography - Ex: transverse sector • Imposing vanishing Dirichlet boundary conditions the (partially) on-shell UV action � π α ) 2 − c 2 S (0) d 4 x T ( ∇ ∧ π α ) 2 � � ∼ ( ˙ + . . . T • The dispersion relation is non vanishing on a generic cutoff (Volume preserving diffs breaking?) ω T = ± c T k + O ( k 2 ) where u δ c T = → O (1 − u δ ) � − log(1 − u 2 δ )

1) Effective actions from holography - Ex: Longitudinal sector Longitudinal sector: δh tt , δh tx , δh � x , δh tu , δh xu , δh uu x� • The Goldstone bosons are � u δ � u δ π t ∼ π x ∼ δh tu du, δh xu du, 0 0 • δh uu parametrizes the position of the cutoff. Can be integrated out from the effective action � � π t ) 2 − c 2 π t ∂ x π x + S (0) d 4 x t ( ∂ x π t ) 2 ) + f δ ˙ ∼ f δ (( ˙ L π x ) 2 − c 2 s ( ∂ x π x ) 2 � +( ˙ + . . . � � π x ) 2 − c 2 s ( ∂ x π x ) 2 � d 4 x → + O (1 − u δ ) ( ˙

1) Effective actions from holography - Longitudinal sector • The dispersion relation after the near horizon limit u δ → 1 ω L = ± 1 π x : 3 k + O (1 − u δ ) + O ( k 2 ) √ � 2 π t : 3 k + O (1 − u δ ) + O ( k 2 ) ω = ± • However we see that the timelike Goldstone decouples from the effective action when u δ → 1 and one has to discard such mode on the horizon • At higher order in hydro expansion the dispersion relation is divergent! ω L = ± 1 π x : 3 k +( finite + # log(1 − u δ )) k 3 + O (1 − u δ )+ O ( k 4 ) √