Quasilocal notions of horizons in the ﬂuid/gravity...

Quasilocal notions of horizons in the fluid/gravity duality

Michał P. HellerInstitute 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 boost-invariant setup) in 0910.0748 [hep-th]

wtorek, 10 sierpnia 2010

2

Motivation

• 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

• Hydrodynamic entropy current is defined phenomenologically

• At the current level of understanding it contains an ambiguityQuestion: what is the dual gravitational interpretation of coefficients appering in the boundary construction of entropy current

Gravity:

Hydrodynamics:


3

CFT side: hydrodynamic entropy currents & their divergences

AdS side: various surfaces for which area increase theorems hold

task:match those guys

using fluid/gravity dualityand quasilocal horizons

Goal


4

Means

• Conformal relativistic fluid dynamics

• AdS/CFT, in particular Fluid/Gravity Duality

• Quasilocal horizons and area increase theorems


5

Tµν = � uµuν + p (ηµν + uµuν)

boost

Tµν = diag (�, p, p, p)µν

1) uniformly boosted plasma

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.

Relativistic hydrodynamics(0712.2451 [hep-th], 0712.2456 [hep-th])


How to express it quantitatively?

6

uµuµ (x)TT (x)

Tµν = � (T ) uµuν + p (T ) (ηµν + uµuν)

EOM:

4 PDEs for 4 functions - closed set !

∂µTµν = 0 for uµ = uµ(x), T = T (x)and

There are corrections to this picture suppressed by powers of

lmicro/Lgradient =1

LgradientT

Tekst


Gradient terms and Weyl-covariance

7

• Not all covariant gradients (scalars, vectors, symmetric two-tensors) terms are allowed!

• Certain gradient terms are equivalent on-shell

• The boundary theory is conformal

• Out of all conformal transformations those par ticularly constraining the form of hydrodynamics are Weyl rescalings

• Conformal hydrodynamics should be Weyl-covariant (up to 4th order in derivatives)

• This highly constrains the number of allowed terms

ηµν → e−2ω(x)ηµν , T → eω(x)T and uµ → eω(x)uµ

(0712.2451 [hep-th], 0801.3701 [hep-th], 0906.4787 [hep-th])


Gradient terms and Weyl-covariance

8

• At first order in gradients there is only single term allowed

• At second order there are 10 terms allowed 0906.4787 [hep-th]

• Those guys are building blocks („LEGO® bricks”) of the boundary energy-momentum tensor and an entropy current, as well as dual gravity background

order in gradients, there are three conformal scalars,

S1 = !µ!!µ! , S2 = !µ!!µ! ,

S3 = c2s!!

µ!µ!

ln s + c4s2!!

µ ln s!µ!

ln s " 12u"u#R"# " 1

4R + 1

6(! · u)2 , (2.8)

two conformal vectors orthogonal to uµ,

Vµ1 = !!

"!"µ + 2c2s!

"µ!!

" ln s " uµ

2!"#!"# , Vµ

2 = !!

" !µ" + uµ!"#!"# , (2.9)

and five conformal symmetric traceless tensors orthogonal to uµ ,

Oµ!1 = R<µ!> " c2

s

!

2!<µ!

!!>! ln s + !µ! (! · u) " 2c2

s!<µ!

ln s!!>! ln s

"

,

Oµ!2 = R<µ!> " 2u"u#R"<µ!># ,

Oµ!3 = !<µ

$!!>$ , Oµ!

4 = !<µ$!

!>$ , Oµ!5 = !<µ

$!!>$ . (2.10)

These will be the building blocks of the energy-momentum tensor and entropy currentfor conformal fluids.

2.2 Non-Conformal Fluids

For more general fluids that do not obey conformal invariance, all possible gradients

can contribute. In particular, to first order in gradients there are one scalar and vector,

(! · u) , !µ ln s , (2.11)

in addition to the tensor already found for the conformal case.

At second order, there are four additional scalars,

S4 = (! · u)2 , S5 = R , S6 = !!

µ ln s!µ!

ln s , S7 = u"u#R"# , (2.12)

four additional vectors,

Vµ3 = "µ#u"R"# , Vµ

4 = !µ"!!

" ln s , Vµ5 = !µ"!!

" ln s , Vµ6 = (! · u)!µ

!ln s ,

(2.13)and three additional tensors,

Oµ!6 = u"u#R

"<µ!># , Oµ!7 =

(! · u)

3!µ! , Oµ!

8 = !<µ!

ln s!!>! ln s . (2.14)

– 5 –


The energy-momentum tensor

9

• The most general energy-momentum tensor of conformal fluid up to second order in gradients reads

• Coefficients appearing at different gradient structures (so 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 )

• Transport properties are derived from underlying microscopic description (kinetic theory for weakly coupled fluids or AdS/CFT for holographic plasmas)

• In particular, has a holographic interpretation in terms of graviton’s absorption cross section of a black brane

Tµν = � u

µu

ν + p∆µν − η σµν + #1O

µν1 + #2O

µν2 + #3O

µν3 + #4O

µν4 + #5O

µν5 + . . .

η/s

η

η/s


• Generalization of the thermodynamic entropy to hydrodynamics

• Der ived not ion - contr ucted order by order in a phenomenological manner out of conformal scalars and vectors

Entropy current

10

• Values of coefficients does not affect directly the dynamics of a theory and are constrained purely by second law of thermodynamics and equilibrium limit

(0803.2526 [hep-th], 0906.4787 [hep-th])

equilibrium entropy current must be of second order in gradients [19]. In the past, theform of the non-equilibrium entropy current has often be postulated [7]. More recently,

a more fundamental approach has been advocated [20, 21] that calls for all structuresin a gradient expansion to be allowed. I will follow this approach here, recovering andextending some of the results in Ref. [21].

4.1 Conformal Fluids

For conformal fluids, the entropy current must be built out of elements that are invariantunder conformal transformations, which are the three scalars (2.8) and two vectors (2.9):

Sµnon!eq = suµ +

A1

4S1u

µ + A2S2uµ + A3

!

4S3 !1

2S1 + 2S2

"

uµ

+B1

!

1

2Vµ

1 +uµ

4S1

"

+ B2 (Vµ2 ! uµS2) , (4.1)

where the five coe!cients A1,2,3 and B1,2 are (mass dimension one) functions of entropyonly and the combinations and prefactors have been chosen such as to facilitate compar-

ison to Ref. [21]. For conformal fluids in three dimensions one has c2s = 1

3and " = 0, and

the absence of a second dimensionful scale leads to the relation Ai, Bi " s1/3. Accord-

ing to Boltzmann’s H-theorem, entropy is never allowed to decrease, so the divergenceof the non-equilibrium entropy current should obey

#µSµnon!eq $ 0 .

The divergence of the entropy current is a physical observable, and as such shouldtransform homogeneously under Weyl rescalings. Explicitly, one can convince oneselfthat this is the case by writing

#µSµ =

1%!g

!µ

#%!gSµ

$

& e4w

%!g

!µ

#

e!4w%!g e4wSµ$

.

Taking the covariant derivative of Eq. (4.1), the result for the equilibrium part

#µ (suµ) can be read o# from Eq. (3.4). A somewhat more lengthy calculation (seeappendix B for some useful identities) gives

#µSµnon!eq =

1

2#"

µ#"

! "µ! (!2A3 + B1) +1

3#"

µ "µ!#"

! ln s (!2A3 + B1)

+"µ!

%

#

2T"µ! + Rµ!

&

! $

2T+ A3

'

+ u"u#R"<µ!>#

!

$ ! #%$

T+ A1 + B1 ! 2A3

"

!1

4"µ

%"!%

!

2&1 ! #%$

T+ A1 + B1 ! 2A3

"

+1

3#<µ

"#!>

" ln s&#%$

T! A1 ! 2A3

'

– 9 –

δS ≥ 0 generalizes to∇µSµnon−eq ≥ 0 with Sµ

non−eq = S uµ + . . .

The purpose of this talk is to provide holographic interpretation of Ai’s and Bj’s using fluid/gravity duality and quasilocal horizons


• (Patch of) AdS-Schwarzschild black hole is described by the metric

Black holes in AdS

11

ds2

BH= 2dtdr − r2

�1− π4T 4

r4

�dt2 + r2d�x2

down tosingularity@ r = 0

horizon

boundary@ r = ∞

eventrEH = π T

Thermal hCFT = Bulk black hole

[hep-th/9803131]

• By analyzing the motion of null geodesics in this static background one concludes that the null surface is an event horizonr = πT

radial null geodesics:

ingoing

outgoing

dt = 0

2dr − r2�1− π4T 4

r4

�dt = 0

• Boundary entropy density = area of spatial section of EH1

4GN


Fluid/gravity duality I 0712.2456 [hep-th]

uµuµ (x)TT (x)

gravity solutionis 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 car toons imply that metric dual to perfect fluid hydrodynamics is that of a boosted AdS-Schwarzschild black brane

ds2 = −2uµdxµdr − r2

�1− π4T 4

r4

�uµuνdx

µdxν + r2 (ηµν + uµuν) dxµdxν

• Now and are slowly varying (in hydrodynamic sense) functions of boundary coordinates

• 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

uµ Txµ


Fluid/gravity duality III 0803.2526 [hep-th]

wise by the metric of a uniform brane with the local value of temperature and velocity. This

feature of the solutions – the fact that they are tube-wise indistinguishable from uniformblack brane solutions – is dual to the fact that the Navier-Stokes equations describe the

dynamics of locally equilibrated lumps of fluid.

Second, the gravitational solutions constructed in [33] are regular everywhere away from

a spacelike surface, and moreover the authors conjectured that this singularity is shielded

from the boundary of AdS space by a regular event horizon. We will prove this conjecture

by explicitly constructing the event horizon of the solutions of [33] order by order in thederivative expansion. It should be possible to carry out a parallel study for the solutions

presented in [37] for four dimensions. We will not carry out such a study here; however,

aspects of our discussion are not specific to AdS5 and can be used to infer the desired

features of 2 + 1 dimensional hydrodynamics. We expect that the results of such a study

would be similar to those presented in this paper.

As we have explained above, we study the causal properties – in particular, the structure

of the event horizon for the solutions presented in [33]. We then proceed to investigatevarious aspects of the dynamics – specifically, the entropy production – at this event

horizon. In the rest of the introduction, we will describe the contents of this paper in some

detail, summarizing the salient points.

Fig. 1: Penrose diagram of the uniform black brane and the causal structure of the spacetimesdual to fluid mechanics illustrating the tube structure. The dashed line in the second figuredenotes the future event horizon, while the shaded tube indicates the region of spacetimeover which the solution is well approximated by a tube of the uniform black brane.

As we have discussed above, [33] provides a map from the space of solutions of fluid

dynamics to a spacetime that solves Einstein’s equations. The geometry we obtain out of

this map depends on the specific solution of fluid dynamics we input. In this paper we

4

In ingoing Eddington-Finkelstein coordinates bulk-to-boundary map is provided by radial ingoing null geodesics

Event horizon

Boundary with coordinates and the metric

Tube-wi se approx imat ion natura l ly impl ies bulk-to-boundary map along ingoing null geodesics ( = const)

This cartoon suggests to map things from horizon’s tube to the boundary along = const. This induces natural coordinate frame on the horizon: coordinates

xµ

hµν

xµ

xµ

xµ

r ≈ πT (x) + . . .


„Area theorem” IBoundary with coord. and the metric (4D)

xµ

hµν

Codimension-1 hypersurface (tube) living in the bulk (4D)

Throat-like geometry of fluid/gravity duality (5D)

Foliation (with foliation parameter ) into spacelike slices with 3D volume element on each ~

λ

λ

√qλ

λi

λi + dλ

V

√qλi+dλ

√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


„Area theorem” II

Natural question is how the infinitesimal area element on a slice evolves with the flow of

λ

λi

λi + dλ

V

√qλi+dλ

√qλi

λ

This is trivially expressed by the equation

Of course, we want

(~ „area theorem”)

LV√qλ = V a∂a

√qλ = ∂λ

√qλ

LV√qλ ≥ 0

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 currentNote 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

and

This implies that the boundary „entropy current” is related to evolution vector on the tube by

and „area theorem” is just a statement about its divergence on the boundary

LV√qλ = V λ∂λ

√qλ = ∂λ

�V λ√qλ

�= ∂µ (V

µ√qλ)

∂µ (Vµ√qλ) =

√−h

1√−h

∂µ

�√−h

� √qλ√−h

V µ

��

How does GR guarantee area theorems?

LV√qλ =

√−h∇µS

µ

Sµ =

√qλ√−h

V µ


Null normals to a sliceλ

λi

λi + dλ

V

√qλi+dλ

√qλi

m

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 and second in the opposite direction denoted

Normalization condition (for convenience):

m2 −V 2=

ln

With the convenient normalization choice null normals are

4

Let’s define a foliation by providing a vector field on the purported horizon, which is normal to the slices of foliation:

v = vµ�eµ (14)

This is convenient for our purposes, since we know all the Weyl-invariant vectors which can be built from b and u and

their (boundary) derivatives. So we let vµ be the general combinations of such allowed terms – order by order in the

gradient expansion. In terms of the basis of the 5d bulk one then has

v = vβ�

∂

∂xβ−�∂S

∂r

�−1 � ∂S

∂xβ

�∂

∂r

�(15)

The null normals are defined as linear combinations of m, v, with coefficients fixed to ensure

• l2 = n2 = 0

• l · n = −1

• l, n are future-pointing

• l is outward-pointing

• n is inward-pointing

Assume for definiteness that m is timelike and v is spacelike, and that the signs are chosen so that m is future pointing

and v is outward pointing. In the cases we will be dealing with we will take

l =1

2(m+ V )

n = σ(m− V ) (16)

where σ is to be fixed so as l · n = −1. For this to work we need to normalize m so that m2 = −v2.

With this choice of scaling the evolution vector

V = �− Cn (17)

comes out to be

V = (1

2− Cσ)m+ (

1

2+ Cσ)v (18)

Thus for V be tangent to the MTT we need

C =1

2σ(19)

so that

V = v (20)

As expected, C > 0 corresponds to spacelike V, C < 0 corresponds to timelike V, and C = 0 corresponds to null V.

might be fixed by some additional condition, the most convenient one beingσ

4

Let’s define a foliation by providing a vector field on the purported horizon, which is normal to the slices of foliation:

v = vµ�eµ (14)

This is convenient for our purposes, since we know all the Weyl-invariant vectors which can be built from b and u and

their (boundary) derivatives. So we let vµ be the general combinations of such allowed terms – order by order in the

gradient expansion. In terms of the basis of the 5d bulk one then has

v = vβ�

∂

∂xβ−�∂S

∂r

�−1 � ∂S

∂xβ

�∂

∂r

�(15)

The null normals are defined as linear combinations of m, v, with coefficients fixed to ensure

• l2 = n2 = 0

• l · n = −1

• l, n are future-pointing

• l is outward-pointing

• n is inward-pointing

Assume for definiteness that m is timelike and v is spacelike, and that the signs are chosen so that m is future pointing

and v is outward pointing. In the cases we will be dealing with we will take

l =1

2(m+ V )

n = σ(m− V ) (16)

where σ is to be fixed so as l · n = −1. For this to work we need to normalize m so that m2 = −v2.

With this choice of scaling the evolution vector

V = �− Cn (17)

comes out to be

V = (1

2− Cσ)m+ (

1

2+ Cσ)v (18)

Thus for V be tangent to the MTT we need

C =1

2σ(19)

so that

V = v (20)

As expected, C > 0 corresponds to spacelike V, C < 0 corresponds to timelike V, and C = 0 corresponds to null V.

In GR area theorems follow from properties of null geodesics propagating in the gravitational background


Expansion scalarsLV

√qλ

LV√qλ = L�

√qλ − V 2

2Ln

√qλ

L�√qλ Ln

√qλ

θl =1

√qλ

L�√qλ θn =

1√qλ

Ln√qλ

• Rewrite in terms of null normals. The result is

• and 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

• Standard behavior is that outgoing light front expands in area and ingoing one shrinks ( and )

• 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

and

θl > 0 θn < 0


Marginally trapped surface

LV√qλ = L�

√qλ − V 2

2Ln

√qλ

• Consider now a slicing our tube into surfaces such that and

• Surfaces with and are called marginally trapped

• It turns out that tube in such case must be spacelike or null ( )

• Such tube is called the apparent horizon

• The apparent horizon is defined by the condition , so it is not unique (different foliations leads to distinct tubes)

• Most importantly, area theorem holds on apparent horizon (AH)

• 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

θl = 0 θn < 0

θl = 0 θn < 0

V 2 ≥ 0

LV

√qλ��AH

= −1

2V 2√qλ θn ≥ 0

θl = 0


Event horizon and black holeLV

√qλ = L�

√qλ − V 2

2Ln

√qλ• Formula holds also on the event horizon

• In that case , but leading to another area theorem

• Note however that the event horizon is not defined locally (contrary to apparent horizon), so that standard black hole is defined teleologically

V 2 = 0 θl ≥ 0

LV

√qλ��EH

=√qλ θl ≥ 0

Black hole = Complement of causal pastof future null infinity

or in [hep-th] language

Black hole =• nothing: tracing all light rays in the geometry

• ever: knowledge of the whole spacetime is required

Region from which nothingcan ever escape


Example I: Static AdS-Schwarzschildds2

BH= 2dtdr − r2

�1− π4T 4

r4

�dt2 + r2d�x2 with the event horizon at

Consider tube defined by

rEH = π T

r = constant

Normal to the tube is given by m = dr or

m = ∂t + r2�1− π4T 4

r4

�∂r

Foliate the tube with vector field V = ∂t

Null normals to slices of foliation are given by

� =1

2(m+ V ) = ∂t +

1

2r2

�1− π4T 4

r4

�∂r

n = σ (m− V ) = −∂rCalculate expansion scalars to locate AH

Out

In

θn = −3πT

r< 0θ� =

3r

2πT

�1− π4T 4

r4

�

θ� = 0 for rAH = πT (here EH = AH)

m V


Example II: Vaidya spacetime

and

ds2 = 2dtdr − r2�1− M (t)

r4

�dt2 + r2d�x2

The event horizon

r a d i a l outgoing geodesics

2r� (t)− r (t)2�1− M (t)

r (t)4

�= 0

Apparent horizonHorizon’s tube:

Normal to the tube is given by

Foliate the tube with vector field V = ∂t

Null normals to foliation slices are given by

Calculate expansion scalars to locate AH

θ� = 0 for (here EH AH)

Assume for definiteness that M (t) → Mf when t → ∞

Then for the event horizon is at

t → ∞rEH (∞) = M1/4

f

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 = rAH (t)

m = ∂t +

�r2

�1− M (t)

r4

�− r�

AH(t)

�∂r

� =1

2(m+ V ) = ∂t +

1

2r2

�1− M (t)

r4

�∂r

n = σ (m− V ) = −∂r

θ� =3r

2

�1− M (t)

r4

�θn = −3

r< 0

rAH (t) = M (t)1/4 �=wtorek, 10 sierpnia 2010

This observation that apparent horizons are marginallytrapped then motivates the various modern notions ofquasilocal horizons such as trapping [15], isolated [16],and dynamical horizons [17] (or see review articles such as[30,31]). Though there are significant technicalities, theidea is that the identification of marginally trapped sur-faces, which under arbitrarily small deformations becomefully trapped, is sufficient to signal the presence of a blackhole boundary—even without going through the process offoliating the spacetime and finding apparent horizons oneach slice. In fact, this idea is so pervasive that in numeri-cal relativity [32], the term ‘‘apparent horizon’’ has beencoopted to refer to the outermost surface !!‘" # 0 on agiven slice of spacetime. The study of these ideas is anactive and developing area of research with a fairly com-plicated system of nomenclature but for the purposes ofthis paper, quasilocal black hole (or brane) horizons in(n$ 1)-spacetime dimensions will be understood asn-dimensional hypersurfaces that are foliated by (n% 1)-dimensional marginally trapped spacelike surfaces. Therequirement that there be fully trapped surfaces ‘‘just in-side’’ the horizon can be mathematically written as

L n!!‘" < 0; (5)

where in this case na is any extension of the null normal na

into a neighborhood of the putative horizon [21]. In whatfollows wewill usually adopt Hayward’s nomenclature andrefer to such structures as future outer trapping horizons(FOTH) [15]. Occasionally however we will refer to time-evolved apparent horizons (which are examples of FOTHs)or dynamical horizons [17] which are almost equivalent.

Mathematically the properties of these surfaces (such asexistence, uniqueness, and evolution) may be studied usingstandard techniques from differential geometry. From aphysical perspective however, the key advantages of thesegeneralized apparent horizons include: (1) they are definedby the existence of strong gravitational fields, (2) as forfully trapped surfaces their existence is sufficient to implythe existence of singularities and event horizons, (3) theymay be identified without reference to the far future (if onethinks of them as time-evolved apparent horizons), and(4) their evolution is similarly local. The local nature of

this evolution is demonstrated in Fig. 3. In that case, theexpansion of the event horizon continues to occur in an-ticipation of the arrival of infalling matter, with the actualarrival of the matter slowing or ending that expansion. Bycontrast the apparent horizon evolves in the expected wayin response to the infalling matter—it expands in and onlyin the presence of actual matter crossing the horizon.On the downside, it is well-known that quasilocal hori-

zons are not uniquely defined. For classically definedapparent horizons this is easily be seen. Given a foliationof spacetime we can define a time-evolved apparent hori-zon ! as the union of the apparent horizons on eachsurface. Then, it is clear that different foliations will sam-ple different subsets of all the possible trapped surfaces.Thus, different foliations will define different !. In theextreme case it is known that certain slicings ofSchwarzschild spacetime contain no trapped surfaces atall and so no apparent horizon [33]. We will return to thislack of uniqueness in later sections.

3. The membrane paradigm: A physical approach toblack holes

A third way of looking at black holes focuses not oncausality or geometry but rather on how black holes inter-act with their environment. By definition event horizonscannot directly affect their surroundings (they are not incausal contact with any point outside themselves) andneither can apparent horizons (they are contained withinevent horizons). All that either can do is impose restrictionson the behavior of surfaces ‘‘near’’ the horizons that are incausal contact with the outside. The membrane paradigm

FIG. 3. A simulation similar to that of Fig. 1 though this timetwo distinct shells fall into the black hole. Both the apparent andevent horizons are plotted.

FIG. 2. An ‘‘instant’’ "t along with some of its trapped sur-faces (small black circles), the associated trapped region (darkgray), and the apparent horizon (thick dashed line).

BLACK BRANE ENTROPY AND HYDRODYNAMICS: THE . . . PHYSICAL REVIEW D 80, 126013 (2009)

126013-5

r1 21/40

t

−∞

10

90

20 40 60 80 100 t

0.5

1.0

1.5

2.0M�t�

�100 �50 50 100 t

0.2

0.4

0.6

0.8

1.0area�t�

Dynamics of horizons in Vaidya

EH area

AH area

�20 �15 �10 �5 0 5 10t

0.02

0.04

0.06

0.08

0.10

0.12

0.14

area�t�

This example shows teleological nature of the event horizon

apparent horizon


Example III: perfect fluid metric


�1− π4T 4

r4

�uµuνdx


The leading order metric of fluid/gravity duality inherits the causal structure from stationary (boosted) black brane metric. Reminder: in the static unboosted case

Horizon’s tube: r = πTFoliating vector: V = ∂t

ds2

BH= 2dtdr − r2

�1− π4T 4

r4

�dt2 + r2d�x2

For a static boosted black brane horizon’s tube does not change and foliating vector is proportional to the velocity of boundary fluid


�1− π4T 4

r4

�uµuνdx


Horizon’s tube: r = πT

Foliating vector:V =1

Tuµ∂µ

Remember that in the fluid/gravity duality velocity and temperature are promoted to slowly-varying functions of boundary position (note Weyl transformations formulas )

Horizon’s tube:

Foliating vector:

r = πT (x)

V =1

T (x)uµ (x) ∂µ

, T → eω(x)T and uµ → eω(x)uµ

(or Weyl-invariantly )r T (x)−1 = π

This implies that horizon’s tube and foliating vector can be written in the leading (zeroth) order of bulk gradient expansion as

(this formula is Weyl-invariant)wtorek, 10 sierpnia 2010

Perfect fluid order of fluid/gravity duality

Sµ =

√qλ√−h

V µ

V =1

T (x)uµ (x) ∂µ

r T (x)−1 = π

λ

λi

λi + dλ

V

√qλi+dλ

√qλi

m

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

matches the one from the perfect fluid hydrodynamics

Sµ = s (T (x))uµ (x) with s (T ) =1

2N2

c π2T 3

∇µSµ ∼ θl −

1

2V 2θn = 0 + . . .

and its divergence evaluated on GR side is 0

0 0

(up to higher order gradient terms)


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

V =1

T (x)uµ (x) ∂µr T (x)−1 = πHorizon’s tube: Foliating vector:

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

r T (x)−1�1 + h1 T (x)−2 S1 + h2 T (x)−2 S2 + h3 T (x)−2 S3

�= π

V µ = T−1 uµ + c1T−2V1 + c2T

−2V2 + {e1S1 + e2S2 + e3S3}T−1uµ

Going to the second order


Horizon’s ingredients in the boundary entropy current

Let’s look more closely at horizon’s tube and foliation vector

r T (x)−1�1 + h1 T (x)−2 S1 + h2 T (x)−2 S2 + h3 T (x)−2 S3

�= π

V µ = T−1 uµ + c1T−2V1 + c2T

−2V2 + {e1S1 + e2S2 + e3S3}T−1uµ

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 , i.e. e1,2,3 = 0

V µ = T−1 uµ + c1T−2V1 + c2T

−2V2

The candidate entropy current dual to a surface parametrized by h1, h2, h3, c1, c2 is

(Foliation part)

(Tube part)

Sµ =1

4l3P

√qλ√−h

V µ =1

2N2

c π2T 3

�uµ +

c1(πT )−1V1 + c2(πT )

−1V2+�3h1 −

1

2− π

8+

5

8log 2

�(πT )−2S1 +

�3h2 +

1

2

�(πT )−2S2 + 3h3(πT )

−2S3

�


Conditions for self-consistency

(Foliation part)

(Tube part)

Sµ =1

4l3P

√qλ√−h

V µ =1

2N2

c π2T 3

�uµ +

c1(πT )−1V1 + c2(πT )

−1V2+�3h1 −

1

2− π

8+

5

8log 2

�(πT )−2S1 +

�3h2 +

1

2

�(πT )−2S2 + 3h3(πT )

−2S3

�

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 LV√qλ = L�

√qλ − V 2

2Ln

√qλ ≥ 0

2) Check for which c1 and c2 the foliating vector is indeed hypersurface orthogonal (work-in-progress).


Imposing area theoremThe event horizon Apparent 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]

θ� = 0Condition fixes uniquely h1, h2, h3 but leaves c1, c2 underterminedTwo 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

(Foliation part)

(Tube part)

Sµ =1

4l3P

√qλ√−h

V µ =1

2N2

c π2T 3

�uµ +

c1(πT )−1V1 + c2(πT )

−1V2+�3h1 −

1

2− π

8+

5

8log 2

�(πT )−2S1 +

�3h2 +

1

2

�(πT )−2S2 + 3h3(πT )

−2S3

�

c1 =? c2 =?c1 =1

4c2 =

1

2

h3 =1

72h2 = −3

8

h1 =1

4+

π

16− 1

8log 2

h3 =1

72h2 = −3

8

h1 =1

6+

π

16− 1

8log 2


More on black brane entropy

LV√qλ = L�

√qλ − V 2

2Ln

√qλ• Formula 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

• Hydrodynamic entropy current is defined phenomenologically by

1) having correct equilibrium limit

2) non-negative divergence

3) satisfying symmetries of an underlying microscopic theory

∇µSµnon−eq ≥ 0

Sµnon−eq = S uµ + . . .

• Inspired by 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

LV√qλ =

√−h∇µS

µ


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-Weyl-covariant 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]


Quasilocal notions of horizons in the ﬂuid/gravity...

Documents

Transcript of Quasilocal notions of horizons in the ﬂuid/gravity...