A. We classify simply-connected homogeneous (D + 1)-dimensional spacetimes for kinematical and aristotelian Lie groups with D-dimensional space isotropy for all D 0. Besides well-known spacetimes like Minkowski and (anti) de Sitter we find several new classes of geometries, some of which exist only for D = 1, 2. These geometries share the same amount of symmetry (spatial rotations, boosts and spatio-temporal translations) as the maximally symmetric spacetimes, but unlike them they do not necessarily admit an invariant metric. We determine the possible limits between the spacetimes and interpret them in terms of contractions of the corresponding transitive Lie algebras. We investigate geometrical properties of the spacetimes such as whether they are reductive or symmetric as well as the existence of invariant structures (riemannian, lorentzian, galilean, carrollian, aristotelian) and, when appropriate, discuss the torsion and curvature of the canonical invariant connection as a means of characterising the different spacetimes.A partial mathematical answer to this question is the classification of kinematical Lie algebras up to isomorphism, which started with the seminal work of Bacry and Lévy-Leblond [1] and of Bacry and Nuyts [2], who classified kinematical algebras (with space isotropy) in the classical case of 3 + 1 dimensions, and culminated recently with a classification in arbitrary dimension using techniques in deformation theory [3,4,5]. The reason this classification is only a partial answer is that the isomorphism type of the Lie algebra is too coarse an invariant: it does not determine uniquely the geometric realisation of the Lie algebra. The ur-example is the Lorentz Lie algebra so(D + 1, 1), which acts transitively and isometrically both on de Sitter spacetime and on hyperbolic space in D + 1 dimensions, and, in what is possibly a new twist on an old tale, we will see that it also acts transitively on a carrollian spacetime of the same dimension.The first step towards a complete answer to the fundamental question was taken already in the original paper [1] of Bacry and Lévy-Leblond. Although restricted to 3 + 1 dimensions and to spacetimes admitting parity and time-reversal transformations, they already distinguish between the abstract Lie algebras and their geometric realisations on homogeneous spacetimes, arriving at a list of eleven possible kinematics. Our more refined analysis in this paper reduces that list to ten, since the para-galilean and static kinematical Lie algebras lead to isomorphic homogeneous aristotelian spacetimes. In addition, we drop the requirement of parity or time-reversal symmetries and we work in arbitrary (positive) dimension D + 1.More precisely, in this paper we give a more complete answer to the fundamental question by classifying the geometric realisations of kinematical Lie algebras on simply-connected homogeneous spacetimes. The classification we present in this paper, while encompassing the classical geometries like (anti) de Sitter, Minkowski, galilean and carrolli...

Abstract:We initiate the study of non-and ultra-relativistic higher spin theories. For sake of simplicity we focus on the spin-3 case in three dimensions. We classify all kinematical algebras that can be obtained by all possible Inönü-Wigner contraction procedures of the kinematical algebra of spin-3 theory in three dimensional (anti-) de Sitter space-time. We demonstrate how to construct associated actions of Chern-Simons type, directly in the ultra-relativistic case and by suitable algebraic extensions in the non-relativistic case. We show how to give these kinematical algebras an infinite-dimensional lift by imposing suitable boundary conditions in a theory we call "Carroll Gravity", whose asymptotic symmetry algebra turns out to be an infinite-dimensional extension of the Carroll algebra.

We construct a new set of boundary conditions for higher spin gravity, inspired by a recent "soft Heisenberg hair"-proposal for General Relativity on three-dimensional Anti-de Sitter space. The asymptotic symmetry algebra consists of a set of affineû(1) current algebras. Its associated canonical charges generate higher spin soft hair. We focus first on the spin-3 case and then extend some of our main results to spin-N , many of which resemble the spin-2 results: the generators of the asymptotic W 3 algebra naturally emerge from composite operators of theû(1) charges through a twisted Sugawara construction; our boundary conditions ensure regularity of the Euclidean solutions space independently of the values of the charges; solutions, which we call "higher spin black flowers", are stationary but not necessarily spherically symmetric. Finally, we derive the entropy of higher spin black flowers, and find that for the branch that is continuously connected to the BTZ black hole, it depends only on the affine purely gravitational zero modes. Using our map to W -algebra currents we recover well-known expressions for higher spin entropy. We also address higher spin black flowers in the metric formalism and achieve full consistency with previous results.

We extend a recent classification of three-dimensional spatially isotropic homogeneous spacetimes to Chern-Simons theories as three-dimensional gravity theories on these spacetimes. By this we find gravitational theories for all carrollian, galilean, and aristotelian counterparts of the lorentzian theories. In order to define a nondegenerate bilinear form for each of the theories, we introduce (not necessarily central) extensions of the original kinematical algebras. Using the structure of so-called double extensions, this can be done systematically. For homogeneous spaces that arise as a limit of (anti-)de Sitter spacetime, we show that it is possible to take the limit on the level of the action, after an appropriate extension. We extend our systematic construction of nondegenerate bilinear forms also to all higher-dimensional kinematical algebras.