Quasilocal notions of horizons in the fluid/gravity duality Michał P. Heller Institute of Physics Jagiellonian University, Cracow & Institute for Nuclear Studies, Warsaw based on work-in-progress with Ivan Booth, Grzegorz Plewa and Michał Spaliński some ideas were introduced (in the boostinvariant setup) in 0910.0748 [hep-th]
Motivation Gravity: Area increase theorems hold both for apparent and event horizon Those notions of gravitational horizons differ out of equilibrium Question: what might be the role of all those surfaces in AdS/CFT Hydrodynamics: Hydrodynamic entropy current is defined phenomenologically At the current level of understanding it contains an ambiguity Question: what is the dual gravitational interpretation of coefficients appering in the boundary construction of entropy current 2
Goal CFT side: hydrodynamic entropy currents & their divergences task: match those guys using fluid/gravity duality and quasilocal horizons AdS side: various surfaces for which area increase theorems hold 3
Means Conformal relativistic fluid dynamics AdS/CFT, in particular Fluid/Gravity Duality Quasilocal horizons and area increase theorems 4
Relativistic hydrodynamics (0712.2451 [hep-th], 0712.2456 [hep-th]) 1) uniformly boosted plasma T µν = diag (, p,p,p) µν T µν = u µ u ν + p (η µν + u µ u ν ) boost 2) general flow Providing scales of changes of velocity and temperature are large compared to microscopic scale, patches of fluid can be well-approximated by uniform flow. 5
How to express it quantitatively? T (x) u µ (x) T u µ EOM: µ T µν =0 Tekst for u µ = u µ (x), T = T (x)and T µν = (T ) u µ u ν + p (T )(η µν + u µ u ν ) 4 PDEs for 4 functions - closed set! here are corrections to this picture suppressed by powers of l micro /L gradient = 6 1 L gradient T
Gradient terms and Weyl-covariance Not all covariant gradients (scalars, vectors, symmetric twotensors) terms are allowed! Certain gradient terms are equivalent on-shell The boundary theory is conformal Out of all conformal transformations those particularly constraining the form of hydrodynamics are Weyl rescalings η µν e 2ω(x) η µν,t e ω(x) T and u µ e ω(x) u µ Conformal hydrodynamics should be Weyl-covariant (up to 4th order in derivatives) This highly constrains the number of allowed terms (0712.2451 [hep-th], 0801.3701 [hep-th], 0906.4787 [hep-th]) 7
Gradient terms and Weyl-covariance At first order in gradients there is only single term allowed At second order there are 10 terms allowed 0906.4787 [hep-th] order in gradients, there are three conformal scalars, S 1 = σ µν σ µν, S 2 = Ω µν Ω µν, S 3 = c 2 s µ µ ln s + c4 s 2 µ ln s µ ln s 1 u 2 αu β R αβ 1 R + 1 ( 4 6 u)2, (2.8) two conformal vectors orthogonal to u µ, V µ 1 = α σαµ +2c 2 s σαµ α ln s uµ 2 σ αβσ αβ, V µ 2 = α Ωµα + u µ Ω αβ Ω αβ, (2.9) Those guys are building blocks ( LEGO bricks ) of the boundary energy-momentum tensor and an entropy current, as well as dual and five conformal symmetric traceless tensors orthogonal to u µ, O µν 1 = R <µν> c 2 s O µν 2 = R <µν> 2u α u β R α<µν>β, gravity background ( 2 <µ ν> ln s + σµν ( u) 2c 2 s <µ 8 ln s ν> ln s), O µν 3 = σ <µ λ σν>λ, O µν 4 = σ <µ λ Ων>λ, O µν 5 = Ω <µ λ Ων>λ. (2.10)
The energy-momentum tensor The most general energy-momentum tensor of conformal fluid up to second order in gradients reads T µν = u µ u ν + p µν ησ µν +# 1 O µν 1 +# 2 O µν 2 +# 3 O µν 3 +# 4 O µν 4 +# 5 O µν 5 +... Coefficients appearing at different gradient structures (so η and #i s) are various transport coefficients of the fluid under considerations Their concrete values affect the dynamics of the fluid (e.g. elliptic flow in the case of heavy ion collisions is very sensitive to η/s) Transport properties are derived from underlying microscopic description (kinetic theory for weakly coupled fluids or AdS/CFT for holographic plasmas) In particular, η/s has a holographic interpretation in terms of graviton s absorption cross section of a black brane 9
Entropy current Generalization of the thermodynamic entropy to hydrodynamics Derived notion - contructed order by order in a phenomenological manner out of conformal scalars and vectors S µ non eq = su µ + A 1 4 S 1u µ + A 2 S 2 u µ + A 3 Values of coefficients does not affect directly the dynamics of a theory and are constrained purely by second law of thermodynamics and equilibrium limit 10 (0803.2526 [hep-th], 0906.4787 [hep-th]) δs 0 generalizes to µ S µ non eq 0 with S µ non eq = Su µ +... +B 1 ( 1 2 Vµ 1 + uµ 4 S 1 (4S 3 12 S 1 +2S 2 ) ) u µ + B 2 (V µ 2 u µ S 2 ), (4.1 The purpose of this talk is to provide holographic interpretation of Ai s and Bj s using fluid/gravity duality and quasilocal horizons
Black holes in AdS [hep-th/9803131] Thermal hcft = Bulk black hole (Patch of) AdS-Schwarzschild black hole is described by the metric boundary @ r = r EH = π T down to singularity @ r = 0 event horizon ds 2 BH = 2dtdr r 2 1 π4 T 4 radial null geodesics: ingoing outgoing dt =0 2dr r 2 1 π4 T 4 r 4 dt 2 + r 2 dx 2 By analyzing the motion of null geodesics in this static background one concludes that the null surface r = πt is an event horizon 1 Boundary entropy density = area of spatial section of EH 4G N r 4 dt =0 11
Fluid/gravity duality I 0712.2456 [hep-th] T (x) u µ (x) T u µ gravity solution is patch-wise approximated by boosted black branes Sewing conditions between neighboring patches - gradient expansion in the bulk! Transport coeffs determined by the regularity of the bulk gradient terms.
Fluid/gravity duality II 0712.2456 [hep-th] Previous cartoons imply that metric dual to perfect fluid hydrodynamics is that of a boosted AdS-Schwarzschild black brane ds 2 = 2u µ dx µ dr r 2 1 π4 T 4 r 4 u µ u ν dx µ dx ν + r 2 (η µν + u µ u ν )dx µ dx ν Now and T are slowly varying (in hydrodynamic sense) functions u µ of boundary coordinates x µ This is an approximate solution of EOMs and receives gradient corrections build up from conformal scalars, vectors and tensors Most importantly, perfect fluid metric inherits (part of) the causal structure (future event horizon) of static black brane
Fluid/gravity duality III 0803.2526 [hep-th] In ingoing Eddington-Finkelstein coordinates bulk-to-boundary map is provided by radial ingoing null geodesics Boundary with and the metric x µ h µν coordinates Event horizon r πt (x)+... Tube-wise approximation naturally implies bulk-toboundary map along ingoing null geodesics ( x µ = const) This cartoon suggests to map things from horizon s tube to the boundary along on the horizon: x µ x µ = const. This induces natural coordinate frame coordinates
Area theorem I Boundary with and the metric x µ h µν coord. (4D) Codimension-1 hypersurface (tube) living in the bulk (4D) λ qλi +dλ λ i +dλ Throat-like geometry of fluid/gravity duality (5D) Foliation (with foliation parameter λ ) into spacelike slices with 3D volume element on each ~ qλ V qλi V is a vector field which lives in a tube and is orthogonal to foliation slices. It evolves slices of foliation onto each other pointing along the direction of increasing λ λ i
Area theorem II λ Natural question is how the infinitesimal area element on a slice evolves with the flow of This is trivially expressed by the equation L V qλ = V a a qλ = λ qλ Of course, we want L V qλ 0 λ qλi +dλ V λ i +dλ λ i (~ area theorem ) qλi 1: What is the relation between the evolution vector V on the tube and boundary entropy current? 2: What is the relation between area theorem on the tube and divergence of the entropy current? 3: On which tubes GR guarantees area theorem? 4: How to find those surfaces within the fluid/gravity duality?
Area theorem and entropy current Note now that if the tube is sufficiently close to equilibrated event horizon then ingoing null geodesics will induce on it naturally boundary coordinate system (bulk-to-boundary map) Consider now L V qλ = V λ λ qλ = λ V λ q λ = µ (V µ q λ ) and µ (V µ q λ )= h 1 h µ h qλ h V µ This implies that the boundary entropy current is related to evolution vector on the tube by S µ = qλ h V µ and area theorem is just a statement about its divergence on the boundary How does GR guarantee area theorems? L V qλ = h µ S µ
Null normals to a slice In GR area theorems follow from properties of null geodesics propagating in the gravitational background λ Slices itself are codimension-2 surfaces, so have 2 normal directions. Those are spanned by the evolution vector V and vector normal to the tube itself denoted by m Out of V and m one can form two future-pointing null vectors, one pointing towards the boundary denoted l and second in the opposite direction denoted n With the convenient normalization choice null normals are m qλi +dλ V qλi λ i +dλ λ i l = 1 2 (m + V ) n = σ(m V ) σ might be fixed by some additional condition, the most convenient one being l n = 1 Normalization condition (for convenience): = m 2 V 2
Rewrite L V qλ Expansion scalars in terms of null normals. The result is L V qλ = L qλ V 2 2 L n qλ L qλ and L n qλ denote how the area of the light front emitted from the portion of the slice change in the ingoing and outgoing directions Those guys are related to the expansion scalars Sasha Husa introduced in his talk by θ l = 1 L qλ θ n = 1 and L n qλ qλ qλ Standard behavior is that outgoing light front expands in area and ingoing one shrinks ( θ l > 0 and θ n < 0) However strong gravitational fields (inside black holes) affect this intuition, so that non-standard behavior of outgoing expansion scalar is a local characteristic of being confined to a black hole region Blackboard
Marginally trapped surface Consider now a slicing our tube into surfaces such that L V qλ = L qλ V 2 2 L n qλ θ l =0and θ n < 0 Surfaces with θ l =0and θ n < 0 are called marginally trapped It turns out that tube in such case must be spacelike or null ( V 2 0) Such tube is called the apparent horizon The apparent horizon is defined by the condition θ l =0, so it is not unique (different foliations leads to distinct tubes) Most importantly, area theorem holds on apparent horizon (AH) L V qλ AH = 1 2 V 2 q λ θ n 0 Apparent horizon defines thus an entropy current. If there are different apparent horizons, there will be different entropy currents In stationary solutions AH (if exists) coincide with event horizon
Event horizon and black hole L V qλ = L qλ V 2 qλ Formula 2 L n holds also on the event horizon In that case V 2 =0, but θ l 0 leading to another area theorem L V qλ EH = q λ θ l 0 Note however that the event horizon is not defined locally (contrary to apparent horizon), so that standard black hole is defined teleologically Black hole = or in [hep-th] language Black hole = nothing: tracing all light rays in the geometry Complement of causal past of future null infinity Region from which nothing can ever escape ever: knowledge of the whole spacetime is required
Example I: Static AdS-Schwarzschild ds 2 BH = 2dtdr r 2 1 π4 T 4 r 4 dt 2 + r 2 dx 2 with the event horizon at r EH = π T Consider tube defined by r = constant m V Normal to the tube is given by m =dr or m = t + r 2 1 π4 T 4 Foliate the tube with vector field V = t Null normals to slices of foliation are given by = 1 2 (m + V )= t + 1 2 r2 1 π4 T 4 Out In n = σ (m V )= r Calculate expansion scalars to locate AH θ = 3r 2πT 1 π4 T 4 r 4 θ =0 for r AH = πt θ n = 3πT r r 4 r 4 < 0 (here EH = AH) r r
radial outgoing geodesics Example II: Vaidya spacetime ds 2 = 2dtdr r 2 1 M (t) The event horizon 2r (t) r (t) 2 Assume for definiteness that M (t) M f when t Then for is at t r EH ( ) =M 1/4 f 1 M (t) r (t) 4 =0 the event horizon The position of the event horizon is given by solving the equation for radial outgoing geodesics with initial condition defined in the far future This illustrates teleological nature of the event horizon r 4 Horizon s tube: dt 2 + r 2 dx 2 Apparent horizon r = r AH (t) Normal to the tube is given by m = t + r 2 1 M (t) rah (t) Foliate the tube with vector field V = t Null normals to foliation slices are given by Calculate expansion scalars to locate AH and θ =0 for r AH (t) =M (t) 1/4 (here EH = AH) r 4 = 1 2 (m + V )= t + 1 2 r2 1 M (t) n = σ (m V )= r θ = 3r 2 1 M (t) r 4 r 4 θ n = 3 r < 0 r r
Mt 2.0 Dynamics of horizons in Vaidya t 90 1.5 1.0 10 0.5 EH area AH area 20 40 60 80 100 t areat 1.0 0.8 0.6 0.4 0.2 126013-5 0 FIG. 3. A simulation similar to that of Fig. 1 though this time two distinct shells fall into the black 1 hole. Both 2 1/4 r the apparent and event horizons are plotted. areat This example shows teleological nature of the event horizon apparent horizon 0.14 0.12 0.10 0.08 0.06 0.04 0.02 100 50 50 100 t 20 15 10 5 0 5 10 t
Example III: perfect fluid metric The leading order metric of fluid/gravity duality inherits the causal structure from stationary (boosted) black brane metric. Reminder: in the static unboosted case ds 2 BH = 2dtdr r 2 1 π4 T 4 r 4 dt 2 + r 2 dx 2 Horizon s tube: r = πt Foliating vector: V = t For a static boosted black brane horizon s tube does not change and foliating vector is proportional to the velocity ds 2 = 2u µ dx µ dr r 2 1 π4 T 4 of boundary fluid u µ u ν dx µ dx ν + r 2 (η µν + u µ u ν )dx µ dx ν r 4 Horizon s tube: r = πt µ dx ν + r 2 (η µν + u µ u ν )dx µ dx ν Foliating vector: V = 1 T uµ µ Remember that in the fluid/gravity duality velocity and temperature are promoted to slowly-varying functions of boundary position (note Weyl transformations formulas,t e ω(x) T and u µ e ω(x) u µ ) This implies that horizon s tube and foliating vector can be written in the leading (zeroth) order of bulk gradient expansion as Horizon s tube: Foliating vector: r = πt (x) V = 1 T (x) uµ (x) µ (or Weyl-invariantly rt(x) 1 = π ) (this formula is Weyl-invariant)
Perfect fluid order of fluid/gravity duality In the leading order the event horizon coincides with apparent horizon. This is inherited from the static black brane case. In particular, the foliating vector is null on the horizon. The entropy current defined on those horizons S µ = qλ h V µ matches the one from the perfect fluid hydrodynamics rt(x) 1 = π S µ = s (T (x)) u µ (x) with s (T )= 1 2 N 2 c π 2 T 3 and its divergence evaluated on GR side is 0 m qλi +dλ V qλi λ i +dλ λ i V = 1 T (x) uµ (x) µ λ µ S µ θ l 1 2 V 2 θ n =0+... (up to higher order gradient terms) 0 0
The quasilocal horizons within the fluid/gravity duality need to be constructed in gradient expansion by specifying foliating vector and horizon s tube. Those in the leading order are given by Horizon s tube: Going to the second order rt(x) 1 = π Foliating vector: V = 1 T (x) uµ (x) µ Note that there are no first order Weyl-covariant scalars nor vectors and at second order there are only three conformal (Weyl-covariant) scalars and two vectors, which are orthogonal to fluid velocity 0906.4787 [hep-th] The most general definition of a horizon and a foliation up to 2nd order is given by rt(x) 1 1+h 1 T (x) 2 S 1 + h 2 T (x) 2 S 2 + h 3 T (x) 2 S 3 = π V µ = T 1 u µ + c 1 T 2 V 1 + c 2 T 2 V 2 + {e 1 S 1 + e 2 S 2 + e 3 S 3 } T 1 u µ
Horizon s ingredients in the boundary entropy current Let s look more closely at horizon s tube and foliation vector rt(x) 1 1+h 1 T (x) 2 S 1 + h 2 T (x) 2 S 2 + h 3 T (x) 2 S 3 = π V µ = T 1 u µ + c 1 T 2 V 1 + c 2 T 2 V 2 + {e 1 S 1 + e 2 S 2 + e 3 S 3 } T 1 u µ Up to second order in gradients there are in total 8 c-numbers parametrizing this gravitational construction: 3 in the definition of a tube and 5 in the definition of foliating vector. It turns out however, that one can construct two null vectors out of a vector normal to the tube m and foliating vector V only if e 1,2,3 =0, i.e. V µ = T 1 u µ + c 1 T 2 V 1 + c 2 T 2 V 2 The candidate entropy current dual to a surface parametrized by h1, h2, h3, c1, c2 is S µ = (Foliation part) (Tube part) 1 4l 3 P qλ h V µ = 1 2 N 2 c π 2 T 3 u µ + c 1 (πt ) 1 V 1 + c 2 (πt ) 1 V 2 + 3h 1 12 π8 + 58 log 2 (πt ) 2 S 1 + 3h 2 + 1 (πt ) 2 S 2 +3h 3 (πt ) 2 S 3 2
Conditions for self-consistency S µ = (Foliation part) (Tube part) 1 4l 3 P qλ h V µ = 1 2 N 2 c π 2 T 3 u µ + c 1 (πt ) 1 V 1 + c 2 (πt ) 1 V 2 + 3h 1 12 π8 + 58 log 2 (πt ) 2 S 1 + 3h 2 + 1 (πt ) 2 S 2 +3h 3 (πt ) 2 S 3 2 We considered so far the most general surfaces within Weyl-covariant gradient expansion having correct equilibrium limit. This lead to the most general boundary current sharing those features. We still need to do two things 1) Impose the area theorem L V qλ = L qλ V 2 2 L n qλ 0 2) Check for which c1 and c2 the foliating vector is indeed hypersurface orthogonal (work-in-progress).
Imposing area theorem The event horizon Null surface - vector normal to horizon s tube is null and is proportional to foliating vector This condition fixes h1, h2, h3 and c1, c2 Entropy current agrees 100% with the one obtained in 0809.4272 [hep-th] S µ = (Foliation part) (Tube part) 1 4l 3 P qλ h V µ = 1 2 N 2 c π 2 T 3 u µ + c 1 (πt ) 1 V 1 + c 2 (πt ) 1 V 2 + 3h 1 12 π8 + 58 log 2 (πt ) 2 S 1 + Apparent horizon Condition θ =0fixes uniquely h1, h2, h3 but leaves c1, c2 undertermined Two possibilities: either c1, c2 are free paramters which lead to distinct apparent horizons at quartic order in gradients or are fixed by properly imposing the condition of hypersurface orthogonality 3h 2 + 1 (πt ) 2 S 2 +3h 3 (πt ) 2 S 3 2 h 1 = 1 4 + π h h 3 = 1 2 = 3 16 1 8 log 2 h 1 = 1 6 + π h h3 = 1 2 = 3 16 1 8 log 2 8 c 1 = 1 c c 1 =? c 2 =? 4 2 = 1 72 8 72 2
More on black brane entropy Hydrodynamic entropy current is defined phenomenologically by 1) having correct equilibrium limits µ non eq = Su µ +... 2) non-negative divergence µ S µ non eq 0 3) satisfying symmetries of an underlying microscopic theory L V qλ = L qλ V 2 2 L Formula n qλ holds both on the event horizon and apparent horizon(s) leading to at least 2 physically acceptable (from a point of view of phenomenological definition) boundary entropy currents Proposal Inspired by L V qλ = h µ S µ one can mimic this definition on the GR side, so that phenomenological horizon of near-equilibrium black brane is defined to be a codimension-1 surface, which 1) has correct equilibrium limit (coincides with EH at late time ) 2) has foliation leading to a notion of area theorem 3) satisfies symmetries of dual gauge theory
Summary This talk showed the construction of entropy currents on quasilocal horizons within the fluid/gravity duality background This construction is the most general one assuming correct equilibrium limit, gradient expansion and symmetries of a dual theory (Weyl-covariance) In particular, there are two entropy currents defined by apparent and event horizons Possible ambiguity in the hydrodynamic entropy current can be mimicked in the bulk by adopting a phenomenological definition of horizon and entropy 32
Open questions Give physical argument why foliating vector beyond perfect fluid situation must be transverse Check for which c1 and c2 foliating vector is hypersurface orthogonal Relax the assumption of Weyl-covariance and look for non-weylcovariant surfaces in conformal fluid/gravity background Understand the relation between area increase theorems and entropy current for background dual to charged fluid dynamics Construct non-conformal fluid/gravity background and analyze its horizons and their structure Make contact with Paul Romatschke s argument, which fixes all but a single coefficient in the entropy current 0906.4787 [hep-th]