Holographic Reconstruction of 3D Flat Space-Time

TL;DR

This work develops a covariant holographic framework for 3D asymptotically flat spacetimes by identifying Carrollian geometry as the natural boundary structure at future null infinity. It introduces general Carrollian boundary sources in BMS gauge, solves Einstein's equations via a derivative expansion to derive Ward identities for a boundary energy-momentum tensor, and shows a well-posed variational principle without a Gibbons–Hawking term. The analysis reveals a vanishing energy flux Ward identity that drives the enhancement to the full BMS_3 algebra and constructs all conserved BMS_3 currents from boundary data using generalized conformal boundary Killing vectors. It further explores torsion-free boundary conditions, derives matching equations to determine the normal vector, and discusses extensions to higher dimensions and connections to broader flat-space holography frameworks.

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.