Gravitational edge modes, coadjoint orbits, and hydrodynamics

Abstract

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

Extended corner symmetry, charge bracket and Einstein’s equations

“…In the first case, we have that the symplectic structure is conserved in time and that the action of ξ admits a canonical representation on the gravity phase space. This is the case studied in [1,10], where the sets of admissible vector fields form a corner symmetry algebra with the semidirect sum structure g S = diff(S) i sl(2, R) S . In the second case, the symplectic form is not conserved in general.…”

“…As shown in [10], this algebra appears as the Automorphism group of the normal bundle associated with the embedding of S into spacetime. The surface translations, which move the surface along the normals create non-zero flux.…”

“…kinematical sector, namely to transformations that do not move the corner [2][3][4][5][6][7][8][9][10]. The natural next step in this program is to consider the so-called extended corner symmetry group which also includes the possibility to translate the corner along its normal directions.…”

“…This subfactor is the extended symmetry group of the Einstein-Hilbert formulation of gravity. Its corner component, studied in [1,10], is given by the semi-direct product of diffeomorphisms tangent to the sphere and surface boosts. The fully extended version also contains sphere's normal translations and it is given by the semi-direct sum 1 g ext S = diff(S) i sl(2, R) S i (R 2 ) S .…”

On deformations and extensions of Diff(S2)

“…More precisely, a one parameter family of algebras, known as hs [λ], reduces to SU(N ) for integer λ and becomes svect(S 2 ) in the limit λ → ∞ [12]. 1 Furthermore, non-central extensions of the algebra of vector fields on the sphere have been recently considered to correspond to asymptotic symmetry algebras of asymptotically JHEP10(2021)133 flat, gbms [16][17][18][19][20][21], and asymptotically decelerating spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW), gbms s [22], spacetimes at future null infinity, and asymptotically (anti) de-Sitter [23] in four spacetime dimensions. Non-central extensions of this algebra have also been discussed in the context of asymptotic symmetries in null hypersurfaces (including event horizons) [24,25].…”