We study general relativity at a null boundary using the covariant phase space formalism. We define a covariant phase space and compute the algebra of symmetries at the null boundary by considering the boundary-preserving diffeomorphisms that preserve this phase space. This algebra is the semi-direct sum of diffeomorphisms on the two sphere and a nonabelian algebra of supertranslations that has some similarities to supertranslations at null infinity. By using the general prescription developed by Wald and Zoupas, we derive the localized charges of this algebra at cross sections of the null surface as well as the associated fluxes. Our analysis is covariant and applies to general non-stationary null surfaces. We also derive the global charges that generate the symmetries for event horizons, and show that these obey the same algebra as the linearized diffeomorphisms, without any central extension. Our results show that supertranslations play an important role not just at null infinity but at all null boundaries, including non-stationary event horizons. They should facilitate further investigations of whether horizon symmetries and conservation laws in black hole spacetimes play a role in the information loss problem, as suggested by Hawking, Perry, and Strominger.

We derive the first law of black hole mechanics for physical theories based on a local, covariant and gauge-invariant Lagrangian where the dynamical fields transform non-trivially under the action of some internal gauge transformations. The theories of interest include General Relativity formulated in terms of tetrads, Einstein-Yang-Mills theory and Einstein-Dirac theory. Since the dynamical fields of these theories have some internal gauge freedom, we argue that there is no natural group action of diffeomorphisms of spacetime on such dynamical fields. In general, such fields cannot even be represented as smooth, globally well-defined tensor fields on spacetime. Consequently the derivation of the first law by Iyer and Wald cannot be used directly. Nevertheless, we show how such theories can be formulated on a principal bundle and that there is a natural action of automorphisms of the bundle on the fields. These bundle automorphisms encode both spacetime diffeomorphisms and internal gauge transformations. Using this reformulation we define the Noether charge associated to an infinitesimal automorphism and the corresponding notion of stationarity and axisymmetry of the dynamical fields. We first show that we can define certain potentials and charges at the horizon of a black hole so that the potentials are constant on the bifurcate Killing horizon, giving a generalised zeroth law for bifurcate Killing horizons. We further identify the gravitational potential and perturbed charge as the temperature and perturbed entropy of the black hole which gives an explicit formula for the perturbed entropy analogous to the Wald entropy formula. We then obtain a general first law of black hole mechanics for such theories. The first law relates the perturbed Hamiltonians at spatial infinity and the horizon, and

There is significant recent work on coupling matter to Newton-Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non-relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton-Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion paper [1] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether-Ward identities, and transport phenomena and thermodynamics of nonrelativistic fluids.Recently there has been a revival of interest in the Newton-Cartan description of nonrelativistic spacetimes in the condensed matter literature [2][3][4][5][6][7][8][9][10][11][12] where, it has been used with great effect to describe phenomena in the quantum Hall effect and various transport phenomena in condensed matter systems. Newton-Cartan spacetimes are used to describe matter fields and their interaction with general background geometries which are consistent with non-relativistic Galilean invariance. Newton-Cartan geometry also arises in the study of nonrelativistic holographic systems, where the boundary theory realizes a "twistless-torsionful" Newton-Cartan geometry [13][14][15][16][17][18][19]. On the other hand, in the gravitational physics literature (see [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and also Ch.12 of [35] and Ch.4 of [36]) Newton-Cartan geometry has been well studied as a diffeomorphism-covariant, geometric way to describe Newtonian gravity. As such, the Newton-Cartan spacetimes considered there belong to a much more restricted class. This divergence of interests has lead to some conflicts in the construction (or at least in the interpretation) of Newton-Cartan spacetimes. One of the aims of this work is to alleviate these conflicts and set a clear stage for describing both Newtonian gravity and matter R I J := dω I J + ω I K ∧ ω K J (2.13b) 7Here we use the convention that the connection is a 1-form valued in the Lie algebra, instead of viewing it as 1-form components in a set of bases given by the generators of the Lie algebra.

We consider non-relativistic curved geometries and argue that the background structure should be generalized from that considered in previous works. In this approach the derivative operator is defined by a Galilean spin connection valued in the Lie algebra of the Galilean group. This includes the usual spin connection plus an additional "boost connection" which parameterizes the freedom in the derivative operator not fixed by torsion or metric compatibility. As an example we write down the most general theory of dissipative fluids consistent with the second law in curved non-relativistic geometries and find significant differences in the allowed transport coefficients from those found previously. Kubo formulas for all response coefficients are presented. Our approach also immediately generalizes to systems with independent mass and charge currents as would arise in multicomponent fluids. Along the way we also discuss how to write general locally Galilean invariant non-relativistic actions for multiple particle species at any order in derivatives. A detailed review of the geometry and its relation to non-relativistic limits may be found in a companion paper.