Asymptotic symmetries of three dimensional gravity and the membrane paradigm

The asymptotic symmetry group of three-dimensional (anti) de Sitter space is the two dimensional conformal group with central charge $c=3\ell/2G$. Usually the asymptotic charge algebra is derived using the symplectic structure of the bulk Einstein equations. Here, we derive the asymptotic charge algebra by a different route. First, we formulate the dynamics of the boundary as a 1+1-dimensional dynamical system. Then we realize the boundary equations of motion as a Hamiltonian system on the dual Lie algebra, $\mathfrak{g}^*$, of the two-dimensional conformal group. Finally, we use the Lie-Poisson bracket on $\mathfrak{g}^*$ to compute the asymptotic charge algebra. This streamlines the derivation of the asymptotic charge algebra because the Lie-Poisson bracket on the boundary is significantly simpler than the symplectic structure derived from the bulk Einstein equations. It also clarifies the analogy between the infinite dimensional symmetries of gravity and fluid dynamics.


Introduction
The space of solutions of the Einstein equations has a huge gauge symmetry: metrics differing by small diffeomorphisms 1 are physically equivalent. This introduces a great deal of complexity into the construction of a symplectic structure on this space [1,2]. However, when the phase space of a dynamical system carries redundant information, it is often possible to "quotient out" the redundancy through the process of symplectic reduction and obtain a simpler description of the same dynamics. An extremely simple example appears in rigid body dynamics. In this case, the configuration space is the group G = SO(3) and the canonical phase space is the cotangent bundle, T * G. However, the Euler equations of rigid body dynamics are invariant under the left action of G on T * G ∼ = G × g * , so we can quotient out by G and obtain an equivalent Hamiltonian system on g * , the dual of the Lie algebra 2 . In this process, the canonical Poisson bracket on T * G descends to the Lie-Poisson bracket on g * and the dimension of phase space is reduced by half [3].
In this paper, we consider similar examples in general relativity. Our examples are three dimensional gravity with asymptotically de Sitter (dS) and anti de Sitter (AdS) boundary conditions. In either case, the space of physically distinct solutions of the Einstein equations is parametrized by boundary data supported at asymptotic infinity. The projection of the Einstein equations onto asymptotic infinity gives a set of constraints on the boundary data. We regard the constraints as equations of motion for a dynamical system supported on the 1+1-dimensional boundary. This viewpoint is closely related to the black hole membrane paradigm, which treats the projection of Einstein's equation onto a black hole event horizon as a lower dimensional dynamical system [4][5][6]. Physically, the dynamics is equivalent to conservation of the boundary's Brown-York stress tensor.
We show that the dynamics so obtained can be regarded as Hamiltonian flow on g * , where now G is the asymptotic symmetry group of the family of spacetimes under consideration and g * is the dual of its Lie algebra. In both of our examples, G is two copies of the Virasoro group with central charge fixed by the boundary equations of motion and the normalization of the Brown-York stress energy tensor. As in the rigid body example, the Poisson bracket is the Lie-Poisson bracket and the simplicity of the Lie-Poisson bracket streamlines the computations. The resulting formalism resembles the Hamiltonian description of the compressible Euler equations [7], and we use this analogy to clarify the relationship between conservation laws in fluid dynamics and gravity (see also [8-11]).
The examples in this paper are closely related to the description of three dimensional asymptotically flat gravity in [12]. In the future, it would be interesting to derive the Lie-Poisson brackets appearing in all these examples from the Poisson bracket on covariant phase space [1,2] via symplectic reduction. In the meantime, the subject of asymptotic symmetries remains of great current interest for its wide array of applications, ranging from consistency relations in cosmology [13][14][15] to soft theorems in quantum field theory and memory effects in gravitational wave experiments (for a recent review see [16]). The perspective we develop here can, in some cases, allow one to identify asymptotic symmetry groups and compute their charge algebras with relative ease, tools which should prove useful in this rapidly expanding subject.

Virasoro algebra
In this section, we collect some standard facts about the Virasoro algebra (see, e.g., [3,17]) that will be needed in the following sections. This establishes notation and keeps the present paper self-contained.
The Virasoro group is a central extension of Diff(S 1 ), the diffeomorphism group of the circle. So we start by describing Diff(S 1 ). The elements of Diff(S 1 ) are smooth invertible maps S 1 → S 1 , the group multiplication is composition, and the Lie algebra is Vect(S 1 ), the algebra of vector fields on the circle. The Lie bracket is the usual vector field commutator. As a vector space, Vect(S 1 ) has a dual space, the space of linear functionals on Vect(S 1 ). It is convenient to identify the dual space of Vect(S 1 ) with the space of quadratic differentials on S 1 . Given a quadratic differential, u(θ)dθ 2 , and a vector field, f (θ)∂ θ , we form the pairing, (2.1) For each quadratic differential, this pairing defines a linear map Vect(S 1 ) → R. So the dual of Vect(S 1 ) becomes identified with the space of quadratic differentials. As vector spaces, Vect(S 1 ) and its dual are isomorphic. However, vector fields and quadratic differentials transform differently under diffeomorphisms. This will be important in what follows.
The Virasoro algebra is a central extension of Vect(S 1 ). As a vector space it is vir = Vect(S 1 ) × iR (the factor of i and similar factors below are conventional and simplify some formulas). So an element of vir is a pair, (f (θ)∂ θ , −ia), where a ∈ R. Elements of vir are added and subtracted in the obvious way. What is nontrivial is the Lie bracket on vir. It has the form 2) The first entry on the rhs is the usual vector field commutator. The function appearing in the second entry, ω(f (θ)∂ θ , g(θ)∂ θ ), is a bilinear function of vector fields. The form of ω is constrained by the requirement that the bracket (2.2) is indeed a Lie bracket, and in particular that it satisfies the Jacobi identity. This requirement is so severe that there is in fact a unique solution (up to rescalings) , Any element of vir can be expressed as a linear combination of v m . The second entry on the rhs, −i/24δ 0 m , is conventional and we will comment on its significance in a moment. For now, observe that plugging the basis elements into the Lie bracket gives where Z ≡ (0, −i). These commutation relations are often presented as the definition of the Virasoro algebra. The advantage of starting from the basis-independent presentation (2.2)-(2.3) is that we are free to switch to bases other than the v m . This will become useful below.
The central extension appears in (2.5) as a term proportional to Z, with contributions linear and cubic in m. The linear in m contribution could have been eliminated by setting the second entry of v m to zero in (2.4). However, it is standard to include this term. It has the convenient effect of making the central term vanish for commutators involving only v 0 and v ±1 . This will not be important for us, but we follow convention. Note that the m 3 term cannot be eliminated by changing the basis elements. Non-trivial central extensions are classified by 2-cocycles on vir, which are elements of the second cohomology H 2 (vir, R). This is one-dimensional and generated by ω(f (θ)∂ θ , g(θ)∂ θ ) [17]. The term linear in m is a 2-coboundary and can be removed by performing a change of basis.
Returning to the general theory, consider the dual space, vir * , the space of linear functionals on vir. Its elements are pairs, (u(θ)dθ 2 , ic). The pairing between vir and vir * is For each element of vir * , this pairing gives a linear map vir → R. This justifies our identification of vir * with the dual space of vir. As before, note that vir and vir * are isomorphic as vector spaces but they transform differently under diffeomorphisms. Elements of vir are called adjoint vectors and elements of vir * are called coadjoint vectors. The adjoint action of vir on itself is just the Lie bracket (2.2): Note that a and b do not enter the rhs. This reflects the fact that the extension is central, i.e., that R is in the center of vir. To streamline notation we will sometimes not write a and b explicitly on the lhs either. The adjoint action can be transported to vir * using the pairing (2.6). This defines the coadjoint action, The first thing to note is that the coadjoint action is different from the adjoint action (2.7). This is why the distinction between vir and vir * is important: adjoint vectors and coadjoint vectors transform differently under infinitesimal diffeomorphisms (an infinitesimal diffeomorphism is a vector field). In physical problems that arise "in the wild," it is not always obvious at first glance that a field is an element of vir * . The way to check this is to see if the field transforms under diffeomorphisms according to the Virasoro coadjoint action (2.8). In problems with hidden Virasoro symmetry, the third derivative in (2.8) often manifests as a signal of the underlying symmetry. The infinitesimal coadjoint action (2.8) can be integrated to get an action of Diff(S 1 ) on vir * . The result is where η ∈ Diff(S 1 ) and Sch(η) = (η η − 3 2 (η ) 2 )/(η ) 2 is its Schwarzian derivative. Note that the central charge, c, is invariant under the coadjoint action. In other words, the central charge is constant on orbits of the coadjoint action. When we speak of "the" central charge of a physical system, we mean the phase space of the system is a Virasoro coadjoint orbit with central charge c.
The dual of the Lie algebra, vir * , has a Poisson bracket called the Lie-Poisson bracket. A Poisson bracket is an antisymmetric bilinear map satisfying Leibniz's rule and the Jacobi identity. Let F and G be functions on vir * . At a fixed point µ ≡ (u(θ)dθ 2 , ic) ∈ vir * , the Lie-Poisson bracket is f , (2.14) where (abusing notation slightly) f ≡ δH/δu. In applications, one often "works backwards:" given an equation of motion of the form (2.14), one finds a Hamiltonian for which the equation can be realized as Hamiltonian flow on vir * . For example, the Korteweg-de Vries (KdV) equation was discovered long ago as a model for one-dimensional fluid flow and only much later realized as a Hamiltonian flow on vir * [17]. In the next section, we will find equations governing the evolution of the boundary of three-dimensional asymptotically (A)dS spacetimes. We will realize the boundary dynamics as Hamiltonian flow on vir * by comparing the boundary's equations of motion with (2.14) and choosing H appropriately.

Anti-de Sitter
The metric of three-dimensional anti-de Sitter (AdS 3 ) spacetime is where x ± = t/ ± φ are null coordinates. To study fluctuations about (3.1), we introduce a family of metrics which approaches (3.1) asymptotically, as r → ∞. A seemingly reasonable ansatz for this family of metrics is where ϕ(x + , x − ), γ ++ (x + , x − ), and γ −− (x + , x − ) are independent of r and (. . . ) indicates subleading terms in r −1 . This ansatz is not quite right because the curvature of (3.2) falls off too slowly 3 as r → ∞. To fix this, take instead 3) is known to be the most general asymptotically-AdS metric in three dimensions [18]. 3 Let hµν be the induced metric on the boundary (see below for formula). The rhs of Einstein's equation gives 8πG √ −hT r r = O(1) as r → ∞, violating the requirement that the metric approaches vacuum AdS3 asymptotically.
The metric functions ϕ, γ ++ , and γ −− are not entirely arbitrary. They satisfy two constraints coming from the r → ∞ limit of the the r±-components of Einstein's equation. Following the membrane paradigm, we regard these constraints as equations of motion for a dynamical system in 1+1 dimensions. Comparing the constraints with (2.14) will allow us to realize this 1+1-dimensional dynamics as a Hamiltonian system on a coadjoint orbit of vir * × vir * with c = 3 /2G.

Boundary dynamics
We fix a cutoff surface at large but finite r = r c ; this will serve as a proxy for the boundary of spacetime. Ultimately we are interested in the r c → ∞ limit. Let n ≡ (r/ )∂ r be the unit normal and h µν = g µν − n µ n ν be the induced metric on the cutoff surface. At the cutoff, the Einstein equations enforce the constraints 4 1 8πG where terms subleading in 1/r have been dropped. In section 3.3, we will explain the fluid interpretation of (3.4)-(3.5), following the membrane paradigm. The goal for the remainder of this section is to realize these equations as a Hamiltonian flow on vir * × vir * . The last two terms on the rhs's of (3.4)-(3.5) are reminiscent of the Schwarzian derivatives in the transformation law (2.9) for Virasoro coadjoint vectors. To make this correspondence precise, let ∂ + η + = e 2ϕ and ∂ − η − = e 2ϕ . The Schwarzian derivative of η + is A similar equation holds for Sch(η − ). Note that we always take "spatial derivatives" of η + with respect to x + and "spatial derivatives" of η − with respect to x − . The constraint equations (3.4)-(3.5) can be written compactly as Sch(η ± ) = 0, where Γ ±± ≡ γ ±± /(8πG ) and c = 3 /2G. Comparing with (2.9), we see that Γ ++ and Γ −− transform as Virasoro coadjoint vectors with identical central charges, c = 3 /2G. The constraint equations become (3.8) Using the "product rule" for the coadjoint action 5 , this becomes where f ± ≡ ∂ ∓ η ± ∈ g are the adjoint vectors corresponding to the diffeomorphisms η ± . They can be thought of as the generators of an infinite dimensional generalization of translations along x ± . From this, one can read off the equation of motion for Γ ++ which is Notice that the dynamics of Γ ++ is precisely of the form (2.14), with x − playing the role of "time" and x + playing the role of "space." So the dynamics of Γ ++ can be realized as a Hamiltonian flow on a coadjoint orbit of vir * with central charge c = 3 /2G; all that remains is to specify a Hamiltonian, H + , with δH + /δΓ ++ = f + = ∂ − η + . This can be achieved by defining In the same way, we may regard the dynamics of Γ −− as a Hamiltonian flow on a second copy of vir * , albeit with the roles of x + and x − interchanged. Since the dynamics of Γ ++ and Γ −− are decoupled, one can slightly abuse notation and consider a Hamiltonian, H : vir * × vir * → R, for the combined system given by H = H + + H − where the first term evolves in "time" x − and the second one in "time" x + . At this point, we have constructed a Hamiltonian flow on vir * × vir * which may be studied as an interesting dynamical system in its own right, independent of its origins in AdS 3 gravity. The equation of motion (3.10) is reminiscent of the Korteweg-de Vries equation, another example of a Hamiltonian flow on vir * [17]. The unusual feature of our Hamiltonian is the arbitrary function, f ± (x + , x − ). In some sense, we really have a family of Hamiltonian systems parametrized by f ± (x + , x − ).

Charge algebra
Returning to the boundary equations of motion (3.8), we have the conservation law Projecting onto this basis using (2.6) gives an infinite set of conserved charges, (3.14) The Lie-Poisson bracket (2.10) of the charges is We get a second copy of this algebra by considering the boundary equation of motion ∂ + Ξ −− = 0 and regarding x + as "time." In this case, the Lie-Poisson brackets of the corresponding charges, say Q − m , are This is the asymptotic charge algebra of AdS 3 gravity derived long ago by Brown and Henneaux [22], with central charge c = 3 /2G. We have arrived at this result in a new way: by regarding the constraints on the boundary data as a 1+1-dimensional Hamiltonian system and using the corresponding Lie-Poisson bracket (2.10) on vir * × vir * to compute the charge algebra. The simplicity of the Lie-Poisson bracket on vir * × vir * has helped to streamline the derivation.

Membrane paradigm
The idea of regarding the constraints on the boundary data as a dynamical system in 1+1 dimensions is inspired by the black hole membrane paradigm. In this section, we will use the relationship with the membrane paradigm to clarify the connection between gravity and fluid dynamics. Previously we introduced a cutoff surface at large but finite r = r c , with induced metric h µν and unit normal n. This is the analogue of the "stretched horizon" of the black hole membrane paradigm. Following the membrane paradigm, we now assign the cutoff surface a stress-energy tensor, t µν . The stress-energy tensor is defined such that it terminates the gravitational field at the cutoff. The Israel junction condition gives where K µ ν = h δ ν ∇ δ n µ is the extrinsic curvature of the cutoff surface and K is its trace. The term proportional to 1/ is a regularization term added in order to have a finite stress tensor as we approach the boundary 6 [23]. Plugging in the metric (3.3) gives for the diagonal components, while the off-diagonal ones are proportional to the Ricci scalar of the cutoff surface. Now, the variables Γ ±± introduced earlier may be interpreted as energy densities. They are the energy densities of the boundary stress-energy tensor as viewed by null observers with four-momenta k ± = e −2ϕ ∂ ± . It is interesting to look at the trace of the stress-energy tensor where we already see a hint of conformal invariance [23]: where R 2d = 8e −2ϕ γ +− /(r ) 2 is the Ricci scalar of the cutoff surface. This corresponds to the trace anomaly of a conformal field theory with central charge 7 c = 3 /2G. This anomaly only exists for odd bulk dimensions. In these cases, a logarithmic divergence in the action appears and cannot be canceled by a polynomial counterterm without including a cutoff dependence [24][25][26][27][28]. Given that the stress tensor measures the change in the action due to perturbations of the boundary, one finds this anomaly by looking at the trace of (3.19). Now, we move on to analyze the dynamics. Earlier we obtained equations of motion (3.4)-(3.5) for Γ ±± = e −4ϕ t ±± by projecting Einstein's equations onto the cutoff surface. We obtain the same conservation laws by imposing conservation of t µν : √ −hh aµ t µν |ν = 0, (3.21) where the covariant derivative is t µν |ν = h δ ν ∇ δ t µν and the index a corresponds to x ± . This means that the constraint equations (3.4)-(3.5) may be interpreted as energy conservation laws for the boundary stress-energy tensor.
Ordinary (inviscid, nonrelativistic) fluids are governed by the compressible Euler equations. Mass conservation is described by the continuity equation ∂ρ ∂t + ∇ · (ρv) = 0, (3.22) where ρ is the fluid's mass density and v is its velocity. Let X and x denote the positions of a fluid parcel at t = 0 and time t, respectively. These are called Lagrangian and Eulerian coordinates. Consider the diffeomorphism η : X → x. It is related to the fluid velocity by ∂η(X, t) ∂t = v(η(X, t), t). (3.23) In these variables, the continuity equation (3.22) is Recall that the dynamics governing the AdS 3 boundary was described by Comparing (3.24) and (3.25), we see that the maps η ± are analogous to the map η from Lagrangian to Eulerian coordinates. In fact, the gravity equations (3.25) are almost equivalent to two copies of the usual fluid continuity equation, with the only difference being that the dynamics takes place on vir * rather than Ω n (M ).

de Sitter
The metric of three-dimensional de Sitter space (dS 3 ) is where t ± = t/ ± iφ. This is related to the AdS 3 metric (3.1) by the substitution → −i , in which case t ± → ix ± . In this section, we extend our earlier discussion to asymptotically dS 3 spacetimes. The calculations largely resemble the AdS 3 case, so we will be concise. To begin, we introduce a family of metrics which approaches (4.1) asymptotically, as r → ∞: where L +− ≡ 2 ∂ + ∂ − ϕ. This is the dS 3 analogue of (3.3). The metric functions ϕ(t + , t − ), L ++ (t + , t − ), and L −− (t + , t − ) are independent of r. Note that outside the cosmological horizon, r is a timelike coordinate and t is a spacelike coordinate.
As before, we fix a cutoff surface at large but finite r = r c . This is a proxy for the "boundary of spacetime" at r → ∞. In the AdS 3 case, the cutoff was a timelike surface but now it is spacelike. Let n ≡ (r/ )∂ r be the outward pointing unit normal and h µν = g µν + n µ n ν be the induced metric on the cutoff surface. The Einstein equations enforce the constraints 1 8πG Following the AdS 3 example, we set L ±± ≡ L ±± /(8πG ), ∂ + η + = e 2ϕ , and ∂ − η − = e 2ϕ (in this section, ∂ ± = ∂ t ± ). Further define The constraint equations (4.3)-(4.4) become simply So we are in the same situation as before. As in the AdS 3 case, the constraints give a Hamiltonian system on vir * × vir * with identical central charges c = 3 /2G.

Conserved quantities
The conserved quantities, Θ ±± , are elements of vir * × vir * . As before, we may project the conserved quantities onto a basis of vir × vir. In the present case, a natural basis for the left Virasoro is where Z = (0, 1). The conserved quantities are 9) and the Lie-Poisson bracket is The renormalized Brown-York stress energy tensor is now whose diagonal components are t dS ±± = −e 4ϕ L ±± . It is a quotient of dS 3 by a discrete group [28] similar to the BTZ black hole [32]. Rewriting this metric in t ± coordinates and taking t ± → (1 + 2GM 2 /r 2 )t ± we find that at large r, ds 2 = − 2 r 2 dr 2 + r 2 dt + dt − − 2G ( M + iJ)(dt + ) 2 − 2G ( M − iJ)(dt − ) 2 + (. . . ). (4.14) Comparing with (4.2), we find that the mass and angular momentum are Note that the L ±± are complex valued.

Discussion
We analyzed the dynamics of the asymptotic data of dS 3 and AdS 3 by realizing Einstein's equations as a 1+1-dimensional Hamiltonian system. We constructed an infinite set of conserved charges by projecting the conservation laws onto a basis for vir and we computed the asymptotic charge algebra using the Lie-Poisson bracket. The Hamiltonian flow on g * is restricted to coadjoint orbits (c.f. eq. 3.9), which are symplectic manifolds, so they are natural phase spaces for classical dynamics. Moreover, coadjoint orbits of g * can be identified with representations of the group G according to Kirillov's orbit method [33]. So this formulation of the classical theory can give insights into its quantization, see [34][35][36][37][38][39] for some approaches.
As we mentioned at the outset, the subject of asymptotic symmetries has a wide range of applications, and our results give a new perspective on these applications. For example, in cosmology, the change in the curvature perturbation, ζ, produced by an asymptotic symmetry transformation of dS 3 is identified with adiabatic modes of ζ [40]. The relation to the membrane paradigm described in the present paper allows one to interpret the perturbations produced by adiabatic modes as changes in the energy density of the boundary fluid as observed by a null observer (c.f. eq. 4.12). Mathematically, they are paths along coadjoint orbits of vir * × vir * .