Holographic Reconstruction of 3D Flat Space-Time

TL;DR

This paper explores the structure of asymptotically flat 3D space-times in Einstein gravity, revealing a boundary description via Carrollian geometry, and derives conserved BMS$_3$ currents from boundary data.

Contribution

It introduces a boundary Carrollian geometric framework for flat space holography and derives a consistent variational principle and conserved currents at null infinity.

Findings

Boundary is described by Carrollian geometry.

A well-posed variational problem at null infinity is established.

Conserved BMS$_3$ currents are constructed from boundary data.

Abstract

We study asymptotically flat space-times in 3 dimensions for Einstein gravity near future null infinity and show that the boundary is described by Carrollian geometry. This is used to add sources to the BMS gauge corresponding to a non-trivial boundary metric in the sense of Carrollian geometry. We then solve the Einstein equations in a derivative expansion and derive a general set of equations that take the form of Ward identities. Next, it is shown that there is a well-posed variational problem at future null infinity without the need to add any boundary term. By varying the on-shell action with respect to the metric data of the boundary Carrollian geometry we are able to define a boundary energy-momentum tensor at future null infinity. We show that its diffeomorphism Ward identity is compatible with Einstein's equations. There is another Ward identity that states that the energy flux vanishes. It is this fact that is responsible for the enhancement of global symmetries to the full BMS$_3$ algebra when we are dealing with constant boundary sources. Using a notion of generalized conformal boundary Killing vector we can construct all conserved BMS$_3$ currents from the boundary energy-momentum tensor.