Hydro & Thermo Dynamics at Causal Boundaries, Examples in 3 d Gravity

: We study 3-dimensional gravity on a spacetime bounded by a generic 2-dimensional causal surface. We review the solution phase space speciﬁed by 4 generic functions over the causal boundary, construct the symplectic form over the solution space and the 4 boundary charges and their algebra. The boundary charges label boundary degrees of freedom. Three of these charges extend and generalize the Brown-York charges to the generic causal boundary, are canonical conjugates of boundary metric components and naturally give rise to a ﬂuid description at the causal boundary. Moreover, we show that the boundary charges besides the causal boundary hydrodynamic description, also admit a thermodynamic description with a natural (geometric) causal boundary temperature and angular velocity. When the causal boundary is the asymptotic boundary of the 3 d AdS or ﬂat space, the hydrodynamic description respectively recovers an extension of the known conformal or conformal-Carrollian asymptotic hydrodynamics. When the causal boundary is a generic null surface, we recover the null surface thermodynamics of [1] which is an extension of the usual black hole thermodynamics description


Introduction
It is now a textbook knowledge that typical physical systems in continuum coarse-grained limit exhibit a fluid description governed by hydrodynamic equations.Landau's seminal work [2] extended this to particle physics, proposing that the collective behavior of colliding high-energy and high-density particles can be described by hydrodynamic equations.In more recent reincarnations, Landau's picture has been reformulated within the AdS/CFT setup [3,4], in fluid/gravity correspondence [5][6][7].In the fluid/gravity correspondence, the hydrodynamic limit of a (strongly interacting) conformal field theory which is residing at the boundary of dual asymptotically AdS geometry is related to a gravity dual picture.
The fluid/gravity correspondence for 1 + 1 dimensional fluids both within AdS3/CFT2 [7][8][9][10][11][12] and in 3d flat space holography contexts [13][14][15][16][17][18] is a well-developed topic.The 3d gravity setting is a particularly interesting playground as there are no bulk graviton modes and we have a much better computational handle on both sides of the duality see e.g.[10,12].For the AdS 3 case, the field theory side is a 1 + 1 dimensional conformal field theory (CFT) at the Brown-Henneaux central charge [19] which admits a relativistic conformal fluid description.For the asymptotic flat case, the dual fluid system is a conformal Carrollian fluid residing at the asymptotic null infinity I. See the above references and references therein for more discussions.
In this work, we reconsider the 1 + 1 dimensional fluids with a 3d dual gravity description and extend the fluid/gravity description in some different ways.We establish that the fluid/gravity description for both AdS 3 or flat space cases can be extended to the bulk, where the boundary of spacetime is an arbitrary 2d causal (timelike or null) surface.To this end, one needs to formulate 3d gravity in spacetime with an arbitrary causal boundary.This problem has been explored and well formulated in [20][21][22].The system is in general described by 4 boundary fields (boundary degrees of freedom) for a time-like boundary and by 3 boundary degrees of freedom for the null case.The description developed in [20,21] compared to the other works in the literature cited above has the advantage that it is not relying on asymptotic expansion or falloff behavior of metric components in a particular coordinate system.
Building on the results of [20,21], we establish that boundary degrees of freedom for a generic timelike boundary on flat or AdS gravity cases admit a natural relativistic, but non-conformal hydrodynamic description.This hydrodynamic description in general involves all 4 boundary degrees of freedom.For the cases where the boundary is the asymptotic boundary of the spacetime, however, we show that the fluid description (a relativistic conformal fluid for the AdS 3 case and a conformal Carrollian fluid for the 3d flat space) naturally arises as the description of a part of the boundary degrees of freedom.Explicitly, we show that 2 of the 4 (or 3) boundary degrees of freedom appear in the fluid description for the timelike (or null) boundary cases.We show that the asymptotic hydrodynamic description can be obtained as a limit of our results for general boundaries, when the boundary is taken to infinity and that in this limit one retains the conformal (for AdS) and conformal Carrollian (for flat) fluid description which had been extensively studied in the literature.Finally, for a generic null boundary case, for both flat or AdS cases, we show that there is a hydrodynamic description, as well as an equivalent thermodynamic description, worked out in [1,23].
Besides the hydrodynamic descriptions, we also revisit possible thermodynamic description at a generic causal boundary.In [1] and as an extension of the usual black hole thermodynamic description, see Chapter 5 of [24] for a review, it has been argued that the boundary degrees of freedom at a generic null boundary (not necessarily a black hole horizon) are governed by a generalization of the first law of thermodynamics (which is a local equation at the null boundary).In this work, we show that the null surface thermodynamics can be extended to a generic causal boundary.
Outline of the paper.In section 2 we review the construction of the solution phase space presented in [21] and introduce and fix our conventions and notations.We compute conserved charges and their corresponding algebra and set the stage for our hydrodynamic and thermodynamic analyses of the following sections.In section 3 we give a hydrodynamics description of AdS 3 or 3d flat space bounded by a generic timelike boundary.In section 4 we explore the hydrodynamic description of three cases.In section 4.1 we work out boundary hydrodynamic description for AdS 3 or 3d flat space bounded by a generic null boundary.In section 4.2 we study the hydrodynamic description of the effective conformal fluid at the causal boundary of AdS 3 where recover an extension of the results of [10][11][12] in that only half of the boundary degrees of freedom in our solution space appear in the asymptotic hydrodynamic description.Moreover, we show the hydrodynamics at the causal boundary of AdS 3 may be recovered from the limit of general results in section 3.1 by taking the timelike boundary to infinity.The details of the limit are presented in appendix A. In section 4.3 we explore how a conformal Carrollian hydrodynamic description appears at the asymptotic null boundary of 3d flat space.In section 5 we construct "thermodynamics at the causal boundary", i.e. we show boundary degrees of freedom at a generic causal boundary satisfy a local version of the first law of thermodynamics.We close by discussions and outlook in section 6.

Solution space, a review
In this section, we present some basic notations and conventions and review the construction of the most general solution space in a Gaussian-null-type coordinate system in the presence of a given generic causal boundary.More details may be found in [21].

Spacetime 2+1 decomposition
Given a causal boundary, it is appropriate to consider a 2+1 decomposition and adopt the radial coordinate r such that the boundary resides at constant r and choose the range r ≥ 0. That is, the coordinate r emanates from the causal boundary C r and foliates spacetime into causal hypersurfaces which are parameterized by the advanced time v and the periodic coordinate φ, φ ∼ φ + 2π.See Figs. 1, 2 for timelike boundaries in AdS and flat space and and Figs. 3, 4 for null boundaries in AdS and flat space.The most general form of the metric in these coordinates takes the form, x µ = {v, r, φ} . (2.1) where V, U and R are functions of v, r, φ while η is a function of v, φ.The induced metric on C r is then, We restrict ourselves to a part of spacetime for which V ≥ 0. V > 0 ensures that the vector field normal to the boundary is outward pointing and is consistent with the choice of v to be the advanced time coordinate on C r and for this choice, η > 0. We raise and lower lowercase Latin indices by γ ab and γ ab respectively, where γ ac γ cb = δ a b .We also note that Let s denote the vector field perpendicular to C r , which is normalized as s • s = 1.Then the induced metric on C r is given by The induced metric γ µν can be written in terms of unit timelike vector field t µ and spacelike vector field where Table 1: Weight w for various quantities defined and used in this section and the following sections. (2.7) The timelike vector field t µ is normalized as t 2 = −1 and the spacelike vector field k µ as k 2 = 1.We also need to define a metric on the transverse surface In the 3d case, the non-zero eigenvalue of q µν is essentially the φφ component of the metric.
The two spacelike and timelike vector fields s, t may be written in terms of linear combinations of two normalized null vector fields l, n, with l 2 = n 2 = 0, and l (2.10) Equation (2.9) also makes it clear that ln( √ V ) may be viewed as a boost speed which acts on l, n like scaling by √ V , 1/ √ V , respectively.
We will also be dealing with the "asymptotic boundary" C ∞ = C r→∞ , which is a null (timelike) cylinder for asymptotic flat (AdS 3 ) cases, see Fig. 5.By D v and D v we denote the derivatives along the v respectively on C r and C ∞ , where L X denotes a differential operators which is defined as where O w is an arbitrary function in spacetime with weight w.Weights of different quantities can be found in Table 1.
Independent components of the covariant derivative of the boundary generating spacelike vector field s along the boundary are given through (2.13) Similarly, for the two null vectors l, n the expansions θ l , θ n , the angular velocity, ω l , and non-affinity parameter, κ, are given by (2.14)

Equations of motion
Field equations for Einstein-Λ theory are Straightforward computations show that one can solve for the r-dependence of the 3 functions in the metric (2.1) (recall that η is r-independent) obtained to be [21]: where Ω, λ, η, Υ, U, M are six functions of v, φ and D v is defined in (2.11).Einstein equations yield 2 more constraints/relations among the 6 codimension one functions of v, φ.To analyze these equations we divide them into three parts, (2.17) Among them, The other equations E ss = 0, E ab = 0 are readily satisfied once (2.16) and (2.18) hold.It should be noted that the equations (2.18) consist of two first-order time (v) derivative equations, which are linear in the variables θ s , ω s , and κ t .These equations are completely defined at the boundary C r .Upon these equations, the solution space is completely specified by 4 functions over C r .

Asymptotic metric, asymptotic boundary
The asymptotic behavior in the absence or presence of cosmological constant Λ is different.So we analyze the two asymptotic flat, Λ = 0, and asymptotic AdS 3 , Λ = −1/ 2 , cases separately.
Asymptotic AdS 3 case.The asymptotic value of metric functions (at large r limit) is given by The asymptotic form of the line element is From this, one can read the asymptotic boundary metric as The above is the metric of 1 + 1 dimensional cylinder of unit radius up to the conformal factor P 2 r 2 .The asymptotic (conformal/causal) boundary C ∞ is specified by 3 functions of v, φ, namely λ, η, U. λ(v, φ) −1 is the local scale of asymptotic time and U(v, φ) specifies the local angular velocity of the frame at the boundary.Moreover, note that P sets a local scale of r.
The asymptotic expression of geometric quantities are where and S[X; φ] is the Schwarzian derivative of the quantity X, (2.24) We note that Einstein equations (2.18) lead to [21] Third derivatives terms in equations (2.25) involve λ −1 , U which are components of metric (2.21).Finally, we also note the asymptotic expression of null geometric quantities

26)
Asymptotic flat Λ = 0 case. 1 For the flat case, we have a Carrollian geometry at the asymptotic null boundary.(See [25][26][27][28][29] for more discussions on Carrollian geometries.)It is described by a degenerate metric, with kernel Recalling (2.16), we see that V = − 2DvP λ r + O(r 0 ).Therefore, in order V > 0 we should have D v P < 0 (Note that we have already assumed η, λ > 0 and hence P > 0.) Moreover, to ensure V > 0 in the region of spacetime of interest, the function P should be monotonically decreasing with respect to advanced time.We assume that this condition is satisfied.
The asymptotic expression of geometric quantities are (2.29) and and M still satisfies (2.25) with Λ = 0.

Surface charges, general analysis
This section summarizes charge analysis associated with a generic causal surface.More detailed analysis and discussion may be found in [21].
Causal boundary symmetry.The metric of the form (2.1), with (2.16), is preserved by the infinitesimal diffeomorphism generated by the vector field [21] where T , Z, W , and Y are generic functions at codimension one constant r surfaces, generic functions of v, φ.T generates supertranslation in v direction, Z supertranslation in r direction, W superscaling, and Y superrotations.
Using the adjusted Lie bracket we have where ) The above is a semi-direct sum of Diff(C r ) and supertranslations along r (generated by Z) and superscalings along r (generated by W ). While the Diff(C r ) has a clear geometric meaning at the boundary C r , the Z, W parts do not.In particular, we note that Weyl scaling on C r , transformations γ ab → Wγ ab with γ ab being the boundary metric (2.2), are not generated by our symmetry generators. 2 Note, however, that as we discuss in section 4.2 Weyl scaling becomes a part of our symmetry algebra in r → ∞ limit.Some of the relevant transformation laws are [21] ) ) ) Surface charge.We start from the three-dimensional gravity action in the presence of the cosmological constant where L µ b is a boundary Lagrangian.The pre-symplectic potential for this Lagrangian can be read as, where is Lee-Wald's pre-symplectic potential for the Einstein-Hilbert term and Y µν [g; δg] is the Y -freedom, a covariant skew-symmetric tensor constructed out of metric and its variation.
The Lee-Wald surface charge variation reads where C r,v denotes a constant v slice at the boundary C r .In the first line µν is binormal to C r,v , and h µν = δg µν , its indices are raised and lowered with the background metric g µν and h is its trace.The second line in (2.38) shows that the r-dependence of the Lee-Wald surface charge arises from the last two terms in the above equation.This r-dependence can be removed by adding the following Y -term This Y -term is covariant and is completely built out of geometric quantities defining the boundary C r .
The charge expression then becomes where and As (2.40a) shows, the charge is not integrable in the slicing that T, Z, W, Y are field independent, i.e. when δT = δZ = δW = δY = 0. Explicitly, while the charges of W, Z, Y are integrable, the charge associated with T is not.However, as is made manifest in (2.40b), the charge expression (2.40b) becomes integrable if we assume the new hatted generators are field-independent.That is, change of slicing to the hatted generators makes the charges integrable. 3Recalling the asymptotic boundary metric (2.21), we observe that Ŷ, T are generators of translations in normalized φ, v directions, respectively.We will explore this point more closely in the next sections when we discuss the hydrodynamic description.
The charge variation (2.40b) shows that our solution phase space is well parametrized with four codimension-one surface charges, { M, Υ, Ω, Π}.We crucially note that while these four charges are rindependent, the surface charge is computed by integrating over an arbitrary constant r, v surfaces.We will show in sections 3 and 3.2 that the first two charges lead to a hydrodynamic description at infinity while Ω, Π do not enter the boundary hydrodynamic description.
Surface charge algebra in the direct-sum slicing.In general the algebra of charges is the same as the algebra of symmetry generators, up to possible central terms, i.e.
In the direct-sum hatted slicing of previous page, The explicit form of non-zero commutation relations of the algebra are given by That is, the algebra is a direct sum of a Heisenberg algebra and a BMS 3 (Λ = 0) or Virasoro⊕Virasoro (Λ < 0) algebra at Brown-Henneaux central charge [19,21].Note that while the charges are formally v-dependent, the algebra of charges is the same at any arbitrary v.
Symplectic form.As pointed out, the solution space is equipped with a symplectic 2-form Ω[δg, δg; g].
The explicit form of the symplectic form depends on the Y -freedom.With the Y -term which removes the r-dependence (2.39), the symplectic form is This symplectic form consists of 2 parts, "codimension 1 modes" given by a codimension 1 integral spanned by M, Υ and their respective canonical conjugates λ −1 , U and "codimension 2 modes" given by a codimension 2 integral spanned by Ω, Π which are canonical conjugates of each other. 4We note that for the generic (e.g. higher dimensional gravity) cases besides the codimension 1 and codimension 2 modes, we also have bulk modes (gravitons in higher dimensional gravity), see e.g.[23].Symplectic form for the bulk modes is also given by a codimension one integral, taken over a (partial) Cauchy surface.Such bulk modes are absent in the 3d example we study here.
We should also point out that the solution space is specified by 4 functions of v, φ, whereas in (2.46) we have 6 functions.This is of course the off-shell symplectic form and the codimension 1 modes and their canonical conjugates, as in any Hamiltonian system, are related to each through evolution equations, which in our case are (2.25).

Hydrodynamics at a generic timelike boundary
In this section, we show that the boundary degrees of freedom at a generic timelike boundary C r in AdS 3 (as depicted in Fig. 1) or the flat spacetimes (as depicted in Fig. 2) admit a 1 + 1 dimensional relativistic (but non-conformal) hydrodynamic description.Since the analysis of flat space case can be obtained by simply setting Λ = 0, we focus on the AdS case and in section 3.2 we discuss the flat case.
Causal boundary Brown-York stress tensor.We start with extrinsic curvature of constant r surfaces K µν , where γ α µ = g αν γ µν .The causal boundary Brown-York energy-momentum tensor [34] is by construction a symmetric tensor whose contraction with the normal vector to the causal boundary s µ vanishes, T µν s ν = 0.It can hence be decomposed as where where T is the trace of the causal boundary Brown-York energy-momentum tensor and θ s , ω s , κ t are defined in (2.13).One can readily check that T µν has only 3 non-zero components.These components are those along the constant r surface and will be denoted by Symplectic potential.The symplectic potential on the causal boundary C r , a constant r hypersurface, is Θ C := Cr Θ µ dx µ , with Θ µ defined in (2.36).Its explicit form is One way to fix the two freedoms L µ b , Y µν is the well-known Gibbons-Hawking-York boundary term [35][36][37][38]] where With these choices the symplectic potential becomes Symplectic form.Using the above we learn that As we can see, in the absence of Y • , the off-shell symplectic form consists of three causal boundary Brown-York charges T ab which are canonically conjugate to the boundary metric γ ab .This decomposition appearing in (2.46).As we will see below this 3 + 3 decomposition provides the setting for a hydrodynamic description at a generic timelike boundary C r .
We stress again that T ab and their canonical conjugates γ ab are r-dependent.Moreover, recalling (2.22), observe that E, J , T at large r behave as ∼ 1/r 2 and therefore, combinations R 2 E, R 2 J which appear in the symplectic form (3.9) remain finite at large r and noting that R √ V ∼ r 2 at large r, we see that the symplectic form remains finite at large r, even without the inclusion of the Y -term.That the symplectic form without Y ra • term is finite at large r, is a result of the addition of Gibbons-Hawking-York term (3.7).This point was also noted in [22,39].In the r-independent symplectic form (2.46) this contribution is removed by This choice replaces Gibbons-Hawking-York Y -term with the one used in [21].The wiggles on Cr are to highlight the boundary degrees of freedom, where the Brown-York-type charges T ab are canonical conjugates of boundary metric components γ ab .

Hydrodynamics description at finite distance: AdS 3 case
One should note that the above symplectic form is off-shell and it is subject to (2.18).These equations take the inspiring form, where D a is metric connection compatible with boundary metric γ ab .Note that the r-dependence of T ab is fixed within the solution space specified by (2.16) and (3.11) holds at C r for any r.These equations relate 2 out of 6 functions and hence, as expected and discussed, the solution phase space is governed by 4 functions over C r .
Recalling (3.9) and (3.11), the above description resembles a hydrodynamic description at the boundary C r with energy-momentum tensor T ab which is canonically conjugate to boundary metric γ ab (2.2).While T ab is divergence-free, it is not traceless and the hydrodynamic system is not a conformal one.To understand this, recall that (3.11) is a manifestation of the fact that our boundary symmetries (2.31) include general 2d diffeomorphisms at C r .Nonetheless, as pointed out, Weyl scaling on C r , γ ab → W 2 γ ab , is not a part of our boundary symmetries at generic r and hence the effective relativistic hydrodynamic description at the boundary is not a conformal one.As we see all 3 + 3 modes in the configuration space appear in our hydrodynamic description on generic C r .As we will show in the next subsection in the r → ∞ limit, where the boundary approaches the causal boundary of spacetime, we recover a conformal hydrodynamic description which only involves 2 + 2 codimension 1 modes (cf.discussions below (2.46).The 1 + 1 codimension 2 modes associated with the r supertranslations and superscalings, respectively generated by Z, W , are not relevant to our hydrodynamic description manifested in (3.11).
We note that the hydrodynamic at generic r is not unique.We start with a general analysis of causal boundary Brown-York charges under a Weyl scaling.Consider two boundary metrics related by a Weyl scaling, where W is a generic function on the spacetime and a scalar in the γ-frame.Then, one can readily verify that, where We raise and lower indices for tilde-quantities by γab and γab respectively, as such T ab = Tab .The divergence-free condition (3.11) can be written as, where ∇a is the covariant derivative w.r.t.γab .That is, in a generic Weyl-frame neither the divergence nor the trace of the energy-momentum tensor is zero.
Divergence-free frames.The above is true for an arbitrary Weyl factor W. One may choose W = f (T ), where f is an arbitrary function of T .Then, one can construct a new divergence-free energymomentum tensor T ab , ∇a T ab = 0, where prime denotes derivative w.r.t. the argument.The above also makes it clear that while T ab is divergence-free, it is not trace-less.One may also show, To obtain the above we used, δ Eq. (3.17a) makes it apparent that the Weyl scaling by W = f (T ), amounts to the addition of √ −γ F (T ) to the boundary Lagrangian and (3.17b) relates two divergence-free energy-momentum tensors which are canonical conjugates to two metrics which are related by the Weyl scaling.In other words and more explicitly, the subclass of Weyl scalings by W = f (T ) (together with (3.16)) is a canonical transformation and hence is a (local) symmetry of our solution space. 5Moreover, dealing with divergencefree stress tensors, one may use either the original or tilde-frames and associated stress tensors, respectively T ab , T ab , for the hydrodynamic description.That is, the hydrodynamic description is not unique and since f (T ) is an arbitrary function, there are infinitely many such descriptions. 6

Hydrodynamics at a generic timelike boundary: 3d flat case
While our general analysis of the previous subsection generically works for Λ < 0 and Λ = 0, we focused more on the AdS 3 , corresponding to Λ = −1/ 2 , case.In this section and for completeness we also present the 3d flat space case and consider a generic causal boundary at a constant finite r surface in 3d flat spacetime.To this end, we need to take → ∞ limit of the results in the previous section.
The only difference between flat and AdS cases at the level of the metric is in the absence of R 2 term in V (2.16), and at the level of the boundary symmetry algebra, we have (2.45) with Λ = 0 which is a direct sum of BMS 3 and Heisenberg algebras.The other geometric quantities at a generic boundary C r discussed in section 2, like the dribeins t, s, k and θ s , ω s , κ t are still given by the same expressions with V given in (3.18).
The first step in our hydrodynamic description is working out the causal boundary Brown-York stress tensor T µν .For the flat spacetime, we need to make two different modifications, one in the equations (3.2)-(3.7)where appears explicitly and we can simply take the flat → ∞ limit and the other one is in the implicit dependence on through V.In the latter case, we only need to replace V with (3.18).Boundary Einstein equations are still given by (3.11), with a hydrodynamic description as the AdS case.In this case, too, the trace of T ab is nonzero and for generic r we deal with a non-conformal fluid.One may readily check that analysis in section 3.1 works verbatim for Λ = 0.In particular, the hydrodynamic symplectic form still takes the same form as in (3.9).

Hydrodynamics at a generic null or asymptotic boundaries
The causal boundary Brown-York stress tensor T ab defined at C r for finite r laid the basis for our hydrodynamic description of the previous section.This stress tensor is constructed out of geometric quantities like θ s , κ and ω s as well as the boundary metric γ ab .These geometric quantities need to be reconsidered and redefined in two special cases, when we consider asymptotic r → ∞ and when the boundary is a null surface.The latter can be at finite r in AdS or flat cases as depicted in Fig. 3 and Fig. 4, like the cases discussed in [20], or can be the asymptotic boundary of 3d flat space I, as depicted in the Right figure in Fig. 5.In this section, we explore hydrodynamic description of the boundary theory in these three different cases.

Hydrodynamics on a null surface at finite distance
In this section, we consider a generic null boundary in a spacetime with Λ ≤ 0.Moreover, we restrict the spacetime to V ≥ 0, which we associate with r ≥ 0 in our adopted coordinate system.Requiring that C r at finite r is a null surface that amounts to having V = 0 at the position of the boundary.Requiring the null boundary N to be located at r = 0 yields V = 0 may be viewed as a second-class constraint on a generic causal boundary solution phase space.This yields the null surface solution space analyzed in detail in [20] which is described by three codimension 1 functions, compared to 4 functions for the cases with a timelike boundary [21].That is, when we take the null limit of the timelike boundary C r , we lose one of the charges and its canonical conjugate.
The 2-dimensional null boundary N is a Carrollian geometry which is described by an enhanced Carrollian structure [40][41][42], a 1-dimensional metric, a kernel, and relevant connections and covariant derivatives.In two subsections 4.1 and 4.3, we introduce the Carrollian structure that emerges at the null limit for a boundary at finite r and asymptotic region of 3d flat spacetime.
Basics of Carrollian geometry.The two-dimensional metric (2.2) becomes degenerate in the null limit where V vanishes.Constraint (4.1) induces the following degenerate metric on the boundary where Ū = U (r = 0).From now on, we use the barred notion for quantity X to indicate it computed at r = 0, namely, X = X(r = 0).The null boundary is a Carrollian geometry which besides the degenerate metric γab := ka kb , is equipped with the kernel la la is a null vector that generates the null boundary from the 3d embedding space viewpoint, cf.discussions in section 2.1.Since the metric is degenerate, we need to track the index placement of the geometric quantities.In this regard, we define the one-form dual of la as follows na dx a := − dv , la na = −1 .
This one-form field allows us to define the Ehreshmann connection [29,43].The one-forms {n a , ka } are linearly independent and they form a basis of the cotangent space.Accordingly, the dual basis of the tangent space is { la , ka } and they fulfill the relation The vector field la and the one-form na are respectively pullbacks of the vector field l µ and the one-form n µ defined in (2.9b) to the null surface r = 0, up to proportionality factors.The identity (4.6) allows us to define a projector onto horizontal forms and γab = ka kb defines a partial inverse of the degenerate metric.Taking the null vector n µ defined in section 2.1 to be a null rigging vector, we can define a projector for spacetime vectors onto the null surface Using this projection operator, we define a rigged connection on the null surface [44] where X a = P a µ X µ and X a = P µ a X µ for any vector X µ .Field equations and solution space.One may rewrite (2.18) in terms of variables more appropriate to the null boundary N .To this end, we note that from (2.13) and (2.14) we learn, Replacing the above into (2.18) and using constraint V = 0 on the null boundary N , we arrive at following equations which yields the desired null field equations where κ, ωl , θl are obtained from κ, ω l , θ l in (2.14) computed at r = 0. 7 In the following, we show that the above equations can be derived from the conservation of an energy-momentum tensor.We also note that (4.11b) and (4.11a) yield some useful relations that will be used in the future, Following the construction presented in [38], we define the null boundary energy-momentum tensor as

.14)
7 Here we extend the results presented in [45], which demonstrated that the geometry of a black hole horizon can be described by a Carrollian geometry emerging from an ultra-relativistic limit, in two ways: First, this applies to a generic null surface in the bulk.Second, the ultra-relativistic limit can be obtained by sending the boost parameter ln √ V to infinity.This result is consistent with the recent observation [46] that any null fluid exhibits Carrollian structure.This is analogous to the Brown-York energy-momentum tensor (3.2) for a null boundary.The explicit form of this null energy-momentum tensor is given by To study its conservation through the rigged connection (4.9), we need the spacetime uplift of this tensor.In this regard, we define the spacetime version of the null energy-momentum tensor (4.15) as follows It is easy to check that T a b = P a µ P ν b T µ ν and 8πG T µ ν l ν = θ l l µ .We now have all materials to consider the conservation equation associated with this energy-momentum tensor, namely, The projection of this equation along la and kb gives Symmetry generators at null boundary.To have a consistent solution space in which the null boundary is fixed at r = 0, we need to impose further δV (r = 0) = 0 = δ ξ V (r = 0).Then from (2.34c) we get where Dv := ∂ v − L Ū .A simple and natural solution to (4. 19) is ξ r 0 = 0.This condition in terms of the causal boundary symmetries (2.31) yields which fixes Z in terms of T and its derivatives.We note that this is a field-dependent condition for the Z generator.So, at the null boundary N we remain with symmetry generators T, W, Y and the associated transformations and charges.It is clear that T, Y do not change N at r = 0 and that W , being proportional to r∂ r does not change r = 0 surface either.See [20,47] for more details.
Hydrodynamic symplectic potential and symplectic forms.It is easy to see that the Gibbons-Hawking-York freedoms (3.7) are not appropriate for the null case.One appropriate choice, as discussed in [1,20,48] is to consider the Lee-Wald symplectic potential for the null case, explicitly, With this choice, the Lee-Wald symplectic potential is given by where v 0 is an arbitrary initial value, while the following definitions and the on-shell relations (4.12) have been used in the third line.In the above [γ] denotes a particular field-dependent path in the v, φ plane with δ[γ] = 0 and d is the length element along the path which is field-dependent.One may argue that there exists a field dependent path [γ] such that the above equality is satisfied.A similar symplectic potential has also been discussed in [18].
The Lee-Wald symplectic form is In view of symplectic potential and symplectic form expressions above, some comments are in order: 1.The last term in the first line of (4.22) is a total variation and hence do not contribute to the symplectic form, as is explicitly seen from the first line in (4.24).This term in ΘN is proportional to the trace of stress tensor T and may be absorbed into a W -freedom, a boundary Lagrangian.
2. The first two terms in the first line of (4.24) are proportional to variations of the trace-free part of stress tensor T a b , namely κ, ωl .These two terms may be compared with the first term in (3.8).
3. The third term in the first line of (4.22) which is a total v derivative, may be absorbed into a Y -freedom.This term yields the third term in the first line of (4.24).
4. The first two terms in the first line of (4.24) are the "thermodynamical symplectic form" discussed in [1,48], once we view κ as the temperature, Ω as the entropy, Ω 2 ωl as angular momentum and Ū as angular velocity.
5. The first line of (4.24) and recalling the above comments, reinforces the fact that null surface thermodynamics [1] and the Carrollian limit of causal surface hydrodynamics are closely related.More precisely, they provide a complementary information about the null surface.
6. Interestingly and importantly, the second line in (4.24) shows that the "on-shell" symplectic form, i.e. when field equations (4.12) are imposed on the first line of (4.24), becomes a total time derivative.This renders the on-shell symplectic form as a surface integral (an integral over a codmiension 2 surface, the integral over φ in our case).This is a 3d realization of the general analysis in [23].
7. The last equality in (4.24) makes it clear that Ω, P and ū, J are canonical conjugate of each other, the last equality in (4.24) is in Darboux basis and yields the following Poisson brackets on the solution phase space:

Hydrodynamics at asymptotic causal boundary of AdS 3 space
The above hydrodynamical description on C r at a generic r, becomes more interesting when we take r → ∞ and take C ∞ to be the usual AdS 3 causal (asymptotic) boundary.This is schematically depicted in Fig. 5, Left figure.While Weyl scaling at C r is not among our symmetry generators (2.31), as we will show below, it becomes one of the boundary symmetries at infinity, i.e. the classical symmetry algebra at C ∞ includes Weyl⊕Diff. 8Among other things, this leads to a conformally invariant hydrodynamical description.As is well known, however, due to anomaly in either of Diff or Weyl parts of the symmetry algebra, the boundary stress tensor can be made either divergence-free or traceless, not both simultaneously.
We will establish below that asymptotically there are a continuum of the effective hydrodynamical descriptions of the dual 2d CFT residing at the asymptotic causal boundary.These continuum of descriptions are specified by choices of boundary Lagrangians (W -freedom).In particular, we discuss two trace-free and divergence-free hydrodynamic slicings of the solution phase space and explicitly show how the anomalies associated with Weyl and diffeomorphisms at the asymptotic 2d cylinder can be transformed between these two slicings through choice of boundary Lagrangians and associated change of slicing.In appendix A we derive the asymptotic hydrodynamic description directly from taking the large r limit of what we have in subsection 3.1.Recovering Weyl symmetry at infinity.Consider the asymptotic boundary metric (2.21).It is clear that a scaling in r yields a Weyl transformation on this metric.Therefore, the W symmetry generator (cf.(2.31)) is generating Weyl transformations and T, Y, W generate Weyl⊕Diff.The Z transformation (r supertranslations) do not change this boundary metric at the leading order.Explicitly, using variations in metric coefficients given in [21], one may show that

C r
where σ ab is the boundary metric (2.21).Therefore, unlike the boundary at generic boundary C r which is only Diff invariant, the asymptotic boundary metric is manifestly Weyl⊕Diff.Some further comments are in order: 1. Weyl⊕Diff is a subset of our algebra generated by W, T, Y at C ∞ .
2. The Z transformation, while not acting on the leading order boundary metric, is not a trivial (pure gauge) transformation, and is still a symmetry generator once one considers the subleading terms in the boundary metric.The charges associated to the Z, W transformations (denoted by Ω, Π respectively, cf.(2.40)) are Heisenberg conjugate of each other, either commutator is (8πG) −1 .
3. Recalling the comment in footnote 2, besides the asymptotic boundary, when the boundary is a null surface is another interesting special case where the boundary symmetries coincides with that of a null string.
4. The "enhanced AdS 3 asymptotic symmetries" considered in [51] is a subset of our symmetry algebra generated by v-independent W, T, Y .Explicitly, our symmetry algebra (2.45) extends the algebra in [51] in two different ways: Our symmetry generators are generic functions of v, φ (and not only φ considered in [51]).Moreover, we also have the Z generator which act at subleading order.The charge associated to Z (denoted by Π) is Heisenberg conjugate to the charge associated with the Weyl scaling (denoted by Ω) generated by W .In this sense, our work at asymptotic infinity recovers [22].
Hydrodynamic descriptions, preliminaries.To uncover the hydrodynamical descriptions, we rewrite the asymptotic geometric quantities of the previous section in a particular conformal frame with metric, which its explicit form is We note that √ −γ = ( λ) −1 .Metric at infinity can be expressed in terms of zwibein, These two vector fields can also be obtained by scaling two vector fields t a and k a (2.6) by R and inducing them to the boundary at infinity C ∞ .The deviation tensors associated with these vectors are ∇a tb = − θk ta kb + θt ka kb , ∇a kb = − θk ta tb + θt ka tb , (4.30a) where ∇a is the covariant derivative with respect to boundary metric γab .
Trace-free hydrodynamic slicing.Motivated by the surface charge expression (2.40b) and the symplectic form (2.46), we suggestively define a symmetric, trace-less energy-momentum tensor suitable for the asymptotically AdS spacetimes with the boundary metric given by (4.28) as where While the proposed energy-momentum tensor is trace-free, it is not divergence-free.The main obstacle in making it divergence-free is the anomalous third derivative terms in the equations of motion (2.25).
To take into account these terms, we gather them in a symmetric "anomalous energy-momentum tensor" A ab .Then equations of motion (2.25) can be recast as follows where the anomalous energy-momentum tensor is Here c = 3 /(2G) is the Brown-Henneaux central charge [19] and is the Ricci scalar of the boundary metric γab .Observe that of the anomalous energy-momentum tensor (4.34) is constructed from the second derivatives of the conformal boundary metric (4.28) and hence its divergence in (4.33) yields the third derivative terms in (2.25).
Recalling (4.38), the above makes it clear that upon change of slicing, the divergence and trace-free slicings are mapped onto each other.Note that to compute the above one should view λ, U as functions of Ĵ , Ê, as specified through (2.25).The above analysis and in particular (4.41) shows that hydrodynamic description at infinity only involves 2 of the 4 charges and more interestingly, the 2 charges appearing in the hydrodynamical analysis are the "codimension 1" modes and the "codimension 2" modes Ω, Π, which are generated by Ẑ, Ŵ , do not enter the hydrodynamical description.These modes do not have a direct appearance in the unit circle cylinder at the boundary (4.28).
We close this subsection by the comment that similar energy-momentum tensors have been proposed in [10,11,52], following the fluid/gravity correspondence through a derivative expansion.However, their construction is more restrictive than ours, as it is limited to specific classes of spacetimes.However, here we bypassed this restriction by basing our construction on the surface charge and the symplectic form analyses of the theory.

Hydrodynamics at asymptotic null boundary of 3d flat space
The asymptotic boundary of 3d flat space I is a null surface and analysis of section 4.1 is expected to extend over to this case.However, as in the AdS 3 case discussed in section 4.2, one may get extra enhancements in the asymptotic region.So, we study this case separately.Let us start with the geometry at I. It is a Carrollian geometry described by (2.27) and (2.28).One can readily observe that both W, Z symmetry generators keep the form of asymptotic metric and the kernel vector and one-form to the leading order in r.We note in particular that for asymptotic flat case, the V = 0 condition (4.1) is not required.Therefore, unlike the generic null boundary which discussed in the previous section and like the asymptotic AdS 3 case, the symmetry group contains 4 generators, explicitly, where • • • denotes the subleading terms.As pointed out around eq. (2.27), in order to asymptote to I, D v P < 0, recalling that δ Z D v Ω = D v P, this means δ Z D v Ω < 0. Note that Ω does not explicitly appear in the geometric information at I.
There are associated 4 charges which in the slicing discussed in section 2.4 may be taken to be M, Υ, Ω, Π which satisfy BMS 3 ⊕Heisenberg algebra, (2.45) with Λ = 0 [21]. 9One may use the same symplectic form (2.46), however, this slicing and the associated Y -term used, obscure the hydrodynamic picture which we would like to uncover below.
Carroll structure at null infinity.The conformal boundary structure is given by where the null surface is described by ka and kernel la as follows This structure has the following Ehresmann connection which is dual to the Carrollian vector la These vectors are related to the normal vector of AdS boundary t, through the following scaling Field equations at null infinity.Our solution space has well-defined null dynamics, which can be derived simply by setting Λ to zero.These are equations of motion (2.25) for Λ = 0, explicitly, As we see the equation for ˆ (4.50a) does not explicitly involve the other charge Ĵ .This is reminiscent of the fact that two ˆ charges commute (cf.(2.45c)), i.e. they are supertranslation charges.
Hydrodynamics at null infinity.As in section 4.2 we construct two energy-momentum tensors, a trace-free and a divergent-free one.We may start with the results in section 4.2 and take the appropriate large limit.This parallels the studies in [26,27,[53][54][55][56] (see also [22] for further discussions) where BMS 3 algebra was obtained as a contraction of Vir⊕Vir algebra, corresponding to taking a flat space → ∞ limit of AdS 3 geometry.We can define a trace-free and a divergence-free energy-momentum tensor at null infinity, respectively TI where more definitions can be found in Eqs.(4.30).It is then straightforward to verify that the divergence-free condition ∇a TI a b = 0, yields the on-shell equations (4.50).Given the expressions above some comments are in order: 1.The expressions for ÊI , ĴI have 2 and 0 contributions.The 2 terms, which are divergent in the flat → ∞ limit, are of the form l • ∂ θl , k • ∂ θl , i.e. they are components of gradient of the expansion at I.These 2 terms may be understood noting that unlike a generic null boundary discussed in previous subsection 4.1, we allow for our null boundary at infinity I to also fluctuate and that θl is the expansion of the null boundary I.
2. The 2 terms in the trace of the divergence-free energy-momentum tensor TI are nothing but the extrinsic curvature of I and represent the anomaly term, which appears as the central extension term in the BMS 3 algebra.
3. One may directly verify that the 2 terms do not contribute to ∇a TI a b = 0 and hence to on-shell equations.
4. The angular momentum ĴI do not have an 0 contribution, reflecting the fact that angular momentum part of the algebra (the Virasoro subalgebra of BMS 3 ) has no anomaly.
5. The energy ÊI has an 0 contribution, a reflection of the fact that ÊI is not a scalar under diffeomorphisms at the null boundary.
We close this part noting that one may carry out the analysis of the "hydrodynamic symplectic form" as was done in section 3 and obtain an analogue of (4.41).In a similar way one may show that the divergence-free and trace-free cases are related by a change of slicing.Since the analysis is very similar to what was done in section 3 we do not repeat it again.

Thermodynamic description
In the previous sections, we developed hydrodynamical descriptions at a generic causal boundary C r , which essentially stems from the diffeomorphism invariance of the effective field theory residing at the boundary.In this section, we show that at each boundary we also have a thermodynamical description which also involves the "codimension 2" modes.To work out the thermodynamical description we follow the ideas for "null surface thermodynamics" discussed in [1,48,57] modulo two important differences: (1) Here we seek a thermodynamical description for a generic causal surface and (2) we address the Y -freedom dependence of the description in [1].
As in [1], our starting point is the expression for charge variation (2.40a), which we rewrite for having it at hand, where with (5. 3) The first line in (5.2) is written in terms of parameters more naturally defined at the asymptotic boundary C ∞ or I whereas the second line is in terms of parameters more natural to the boundary at generic r, C r .

Derivation of the first law, a new viewpoint
The seminal Iyer-Wald paper [58] presents a derivation of the first law of black hole thermodynamics within covariant phase space formalism.This derivation is based on the expression for the symplectic form density ω(δ 1 g, δ 2 g) computed over the field variations generated by symmetry generators ξ, ω(δg, δ ξ g).Since the symplectic density is closed, the charge variation / δQ ξ := Σ ω(δg, δ ξ g) where Σ is a partial Cauchy surface, localizes over the two boundaries of Σ, the bifurcation surface of a black hole with a Killing horizon H and asymptotic spatial infinity i 0 .Since the symplectic form is closed, this yields equality, (5.4) The above is true for any ξ.If ξ is taken to be the Killing vector generating the horizon ζ H , recalling that ζ H = i µ i ζ i where ζ i are asymptotic Killing vectors associated with ADM charges and µ i are some constant (chemical potentials), the above yields the standard first law, once we use Wald's entropy formula [58,59].The above derivation crucially depends on the information on the two boundaries of the Cauchy surface, the T δS term comes from LHS of (5.4) which is computed over H and the RHS of (5.4) yield the usual δM − ΩδJ which is computed at i 0 .Recall that in general charge variation / δQ ξ is ambiguous to the choice of Y -freedom.However, since in (5.4) we equate / δQ ξ at two different boundaries H, i 0 , if the Y -term has the same value at both boundaries, it cancels off in the expression for the first law.Explicitly, Iyer-Wald derivation is invariant under this certain class of Y -terms.
In [60] an alternative derivation of the first law was presented which uses the information over a single boundary, building on the fact that for "exact symmetries" (Killing vectors of the black hole background geometry) the charges are symplectic symmetries and they may be computed over any codimension 2 compact spacelike surface, see Chapter 5 of [24] for a more detailed discussion.This derivation has the advantage over the Iyer-Wald derivation in that it relies only on one boundary, however, it has the disadvantage that in general it depends on the Y -term.
Here we present a derivation of the first law which combines all positive features and advantages of the three derivations: 1. Null surface thermodynamics [1], in that it applies to geometries without exact Killing symmetries and importantly, it is a local first law, a first law which is true for any v, φ and that it applies to any null surface.
2. Iyer-Wald formulation [58], in that it is independent of certain class of Y -terms.
3. The derivation in [60], in that the first law may be computed for any arbitrary boundary C r .
In our new viewpoint, we use the same steps of null surface thermodynamics [1], but to remove the Yterm dependence we equate two equivalent expressions for / δH at the same boundary C r .This extends this advantage of Iyer-Wald derivation to any choice of Y -freedom.In the next subsection, we show how this derivation works for the asymptotically AdS 3 case.

Local first law for causal surface thermodynamics
We first note that the r-superscaling aspect Ω, superrotation aspect Υ and the r-supertranslation aspect 2P are integrable while the charge associated with v-supertranslations / δH is not integrable.The first three integrable charges may hence be respectively viewed as charge densities associated with W = 1, Y = 1 and Z = 1.Explicitly, if we denote then The crucial observation made in [1,48] which constitutes the basis of our local first law derivation here is that / δH, as (5.2) explicitly shows, is written in terms of the other three charges and the associated chemical potentials (canonical conjugates to these charges).This observation replaces the relation between the (Killing) symmetry vectors like ζ H = i µ i ζ i which is the basis of first law derivations in [58] or [60].
To obtain the local first law, we start with the identity, (5.7) The above lays the basis of our local first law.Compared to the analysis in [1] it has the advantage and distinctive feature that the contribution of the Y -term to the charge variations drops out from the two sides of the equation and our final first law becomes Y -freedom invariant.This extends the advantage of the Iyer-Wald derivation discussed above to a generic choice of Y -term.After rearrangement of some terms we obtain, where M, Υ are defined in (2.25) and The difference between the first and second equalities in (5.8) is that we have replaced the covariant derivative appropriate for the boundary at infinity D v with D v appropriate for the boundary C r , cf. (2.11).
Next, we note that (5.8) suggests to define the following thermodynamic variables (5.9) Note that κ, U have geometric meanings (2.14).The factors of λ in T, U, the extra Dvλ λ term in T and the contribution from the Schwarzian derivative may be understood as λ is the asymptotic local time scale. 10e also comment that our viewpoint in deriving the first law through the identity (5.7) removes two other disadvantages of the usual Iyer-Wald derivation [24,60]: The normalization of the Killing vectors ζ H , ζ i 6 Discussion and outlook We have studied 3d gravity in the presence of a generic causal (timelike or null) boundary (C r or N ) and constructed the solution phase space.For the generic case, this solution space has 4 boundary degrees of freedom (4 functions on C r ) labeled by 4 surface charges associated with symmetry generators.Two of these symmetry generators are diffeomorphisms at the boundary and two others are associated with supertranslation and super-scaling in the r direction transverse to C r .In the case with a generic timelike boundary and regardless of spacetime being asymptotically AdS 3 or flat space, we have shown by an explicit construction that the boundary degrees of freedom admit a relativistic, but non-conformal, 1 + 1 dimensional fluid description.The key object in this hydrodynamic description is the causal boundary Brown-York energy-momentum tensor at C r .
In our understanding, this generic hydrodynamic description arises as a result of excising the part of spacetime "behind the boundary" which in our conventions corresponds to r < 0 region.When we restrict the spacetime to r ≥ 0 regions and to compensate for the r < 0 part of spacetime, one should associate appropriate degrees of freedom to the boundary.These boundary degrees of freedom in general interact with the "bulk modes" and these interactions give rise to the effective hydrodynamic description.Of course, in the 3d case, where there are no bulk graviton modes, the latter is only a result of Einstein equations of the bulk projected at the boundary, see section 2 or [21] for more discussions.More explicitly, the boundary modes and associated surface charges are giving an effective, coarse-grained description of the would-be microscopic degrees of freedom residing at the boundary.The hydrodynamic description is then nothing but the requirement that this microscopic boundary system is coupled to the bulk modes in a way consistent with general covariance in the bulk.This hydrodynamic description does not specify what the boundary theory is.This hydrodynamic picture for generic timelike boundary should be revisited for three special cases, where the boundary is approaching the asymptotic boundary of spacetime in AdS or flat spacetimes, as well as the cases with null boundary.We hence studied these three special cases in section 4. For the asymptotic AdS case, we first argued that in the symmetry algebra Diff⊕Heisenberg, the Heisenberg part involves Weyl scaling plus its Heisenberg conjugate.In this case, the hydrodynamic description becomes that of a conformal 1 + 1-dimensional fluid.The Heisenberg part of the algebra does not enter the hydrodynamic description, as the symplectic form (4.41) manifests.For a generic null boundary case, as is manifested in (4.24), we have a hydrodynamic description as well as a complementary thermodynamic description discussed in earlier papers [1,48].The hydrodynamic description of the last special case, the asymptotic boundary of flat space I, was constructed as an infinite AdS radius limit [53,54] of the relativistic conformal hydrodynamic appearing in the asymptotic boundary of AdS 3 .This limit is essentially the Carrollian limit of the conformal hydrodynamics and leads to a conformal Carrollian fluid discussed in [15][16][17][18].Our analysis here also sheds light on the notion of the equilibrium limit of the theory at the boundary.While it has been established that Carroll geometries are natural intrinsic geometries of null surfaces, both at finite and infinite regions [38,62,63], there have been some open questions regarding entropy and the nature of hydrodynamic quantities and our analyses here clarify them.
Our analysis has been performed on an extended solution space compared to earlier analyses.This extension made it possible to uncover a non-conformal fluid picture at a generic boundary.In comparison, the conformal fluid description one finds at the asymptotic AdS boundary involves only 2 out of 4 boundary fields that take part in the hydrodynamic description.We discussed that the 2 hydrodynamic degrees of freedom are "codimension 1 modes" whereas the other 2 modes which do not appear in the hydrodynamic description are "codimension 2 modes", as has been manifested in (2.46) and (4.41).The codimension 2 modes are parameterizing the boundary mode and its canonical conjugate and do not enter into how bulk and boundary modes balance each other (what the hydrodynamic picture is about).This is compatible with the general hydrodynamic picture we discussed above.
Among other things, in appendix A we showed how the hydrodynamic descriptions at generic C r and the conformal fluid at the boundary of AdS are related in an explicit large r limit.In this sense, our construction extends those in [18,38] and shows there is a non-singular limit on the finite r null boundary.
For a generic null surface inside AdS 3 which may be the horizon of a BTZ black hole, our construction here establishes the existence of a "hydrodynamical holographic RG flow" from a conformal fluid to a conformal Carroll fluid: As one moves from the AdS 3 causal boundary inward to a null surface in the bulk, the conformal hydrodynamic description at the AdS 3 boundary is transformed to the thermo/hydro dynamic description at the null surface.The hydrodynamic holographic RG flow can connect the two places where hydrodynamic description of gravity arise: membrane paradigm for the (stretched) horizon of black holes [64][65][66] (see also Chapter 6 of [24]) and the fluid/gravity correspondence in the context of AdS/CFT duality [5,6] and its generalizations [67][68][69], which provide a description of the hydrodynamic limit of a field theory residing at the AdS boundary through its gravitational dual.Our analyses here connect the two through an RG flow on the CFT side.It is desirable to explore this hydrodynamic RG flow in a more general (e.g. higher dimensional) settings.
We discussed at the end of section 3.1 that the non-conformal boundary fluid description for a generic boundary C r is not unique.We constructed an infinite family of such energy-momentum tensors related by the subclass of Weyl transformations which are functions of the trace of the causal boundary Brown-York energy-momentum tensor T .We showed that this subclass does not change the hydrodynamic symplectic form and hence represents canonical transformations over the hydrodynamic phase space; i.e. these different hydrodynamic descriptions are physically equivalent.The same non-uniqueness is expected to be there when we consider asymptotic AdS (or flat) boundary cases in the divergence-free frame.That is, there should be infinitely many such divergence-free frames in section 3.1.It is interesting to explore the family of divergence-free descriptions and the implications for the boundary theory.
We also argued that the symmetry algebra generated by ξ(T, Z, W, Y ) (2.32) at large r for AdS case rearranges such that we get an algebra which is an extension of 2d Diff⊕Weyl symmetry at the asymptotic boundary.On the other hand, consistency of the description requires that the boundary theory (which admits a relativistic conformal fluid description in the continuum limit) should have this symmetry as a gauge (local) symmetry.On the other hand, we know that string worldsheet theory is the most general 2d theory with Diff⊕Weyl symmetry.Therefore, the boundary theory should be a string theory.It is also known that we do not have consistent string theory with only AdS 3 as its target space.So, an indirect consequence of our analyses and discussions here is that to upgrade the effective, coarse-grained description we uncovered here to a fully-fledged microscopic (quantum) level, one should supplement the AdS 3 part with internal spaces (like AdS 3 × S 3 × X 4 spaces).In other words, pure AdS 3 quantum gravity is expected not to be a consistent theory.These results resonate with the conclusions in [70] and the follow-up works.
Besides the hydrodynamic description at a generic timelike boundary C r , we showed that the system also admits a thermodynamic description.We argued for the latter by explicitly constructing a local first law of thermodynamics and Gibbs-Duhem relations at C r .This is an extension of our earlier results for generic null boundaries [1].This result may seem intriguing as the lore is that such a thermodynamic description is a feature of null surfaces, in particular (Killing) horizons of black holes.It is desirable to explore if the thermodynamics at the causal boundary discussed here can be extended to more interesting higher dimensional cases and if yes, how does it extend the picture discussed in Ted Jacobson's seminal paper [71].Moreover, to derive this thermodynamic description we discussed a new viewpoint (cf.section 5.1).It is interesting to explore implications of this new viewpoint for the seminal Iyer-Wald derivation [58].
In the 3d Einstein gravity setup we studied here there are no propagating bulk modes.It is interesting to reconsider our analysis for cases with bulk modes.There are two such examples: (1) How does our analysis here extend to higher dimensions; do we have a hydrodynamic description for a generic timelike boundary in d > 3? (2) How is the hydrodynamic in Topologically Massive Gravity (TMG) [72] which admit bulk (chiral) gravitons?Analysis of the solution phase space for the TMG in the presence of a generic null boundary and its thermodynamical description has been studied [31,57].One may extend these analyses for a generic causal boundary and explore the hydro/thermo dynamical descriptions.

Figure 1 :
Figure 1: Portion of AdS3 bounded by a generic timelike boundary Cr.We formulate physics in the shaded region.

Figure 2 :
Figure 2: A generic time-like boundary Cr in flat space.The wiggles on Cr depict the boundary degrees of freedom which are associated with the boundary metric and its geometry.

Figure 3 :
Figure 3: A generic null boundary in AdS3.The null boundary N may be viewed as future (Left figure) or past (Right figure) Poincaré horizon on AdS.We formulate physics in the shaded region and focus on the boundary degrees of freedom residing at N .

Figure 4 :
Figure 4: Null boundaries N in a flat 3d space.The left and right figures respectively show a future or past null boundary.We are interested in formulating physics in the shaded regions bounded by N .
.18b) The above yield the Raychaudhuri and Damour equations (4.11), once we recall that lµ ∂ µ = Dv and Ω θl = Dv Ω.To summarize this part, D b T b a = 0 and therefore T b a may be viewed as energy-momentum tensor associated with the null surface hydrodynamics.Note also that T b a lb = θl 8πG la , i.e. the expansion of null boundary θl is the eigenvalue of the energy-momentum tensor along the null vector la .This eigenvalue equation and D b T b a = 0 may be viewed as properties defining the energy-momentum tensor T b a .

Figure 5 :
Figure 5: Asymptotic boundary of AdS 3 (Left figure) and 3d flat space (Right figure).These asymptotic boundaries may be viewed as asymptotic limits of respective timelike boundaries.