Symmetries, charges and conservation laws at causal diamonds in general relativity

TL;DR

The paper develops a covariant phase space analysis for vacuum general relativity at the null boundary of causal diamonds, revealing an infinite-dimensional symmetry algebra built from diffeomorphisms of the 2-sphere and boost-type supertranslations. Using Gaussian null coordinates and the Wald-Zoupas prescription, it proves that the associated fluxes are Hamiltonian generators and that, due to smoothness at the bifurcation edge, fluxes are conserved between the past and future boundaries, producing an infinite set of conservation laws for finite gravitational subregions. It further uncovers a nontrivial central extension in the causal-diamond symmetry algebra, with central charges proportional to the area of the bifurcation edge, akin to a local Wald entropy. The results suggest deep, transferable constraints on gravitational subregion dynamics and offer avenues toward quantum-gravity and holographic applications in finite regions.

Abstract

We study the covariant phase space of vacuum general relativity at the null boundary of causal diamonds. The past and future components of such a null boundary each have an infinite-dimensional symmetry algebra consisting of diffeomorphisms of the $2$-sphere and boost supertranslations corresponding to angle-dependent rescalings of affine parameter along the null generators. Associated to these symmetries are charges and fluxes obtained from the covariant phase space formalism using the prescription of Wald and Zoupas. By analyzing the behavior of the spacetime metric near the corners of the causal diamond, we show that the fluxes are also Hamiltonian generators of the symmetries on the phase space. In particular, the supertranslation fluxes yield an infinite family of boost Hamiltonians acting on the gravitational data of causal diamonds. We show that the smoothness of the vector fields representing such symmetries at the bifurcation edge of the causal diamond implies suitable matching conditions between the symmetries on the past and future components of the null boundary. Similarly, the smoothness of the spacetime metric implies that the fluxes of all such symmetries is conserved between the past and future components of the null boundary. This establishes an infinite set of conservation laws for finite subregions in gravity analogous to those at null infinity. We also show that the symmetry algebra at the causal diamond has a non-trivial center corresponding to constant boosts. The central charges associated to these constant boosts are proportional to the area of the bifurcation edge, for any causal diamond, in analogy with the Wald entropy formula.

Figures (1)

• Figure 1: Diagram of a causal diamond in a spacetime $(M,g)$. The points $p^\pm$ denote the corners of the causal diamond and $B$ is the bifurcation edge while $N^\pm$ denote the future/past null surfaces joining $B$ to $p^\pm$, respectively. The functions $v$ and $u$ are affine coordinates with affine null normals $\ell^a$ and $n^a$ on $N^\pm$.

Theorems & Definitions (4)

• Remark 2.1: Symmetry group at $N$
• Remark 2.2: Fluxes from the charges
• Remark 4.1: Affine supertranslations
• Remark 4.2: Non-affine parametrization of the null generators