Flat-Space Energy-Momentum Tensor from BMS/GCA Correspondence

TL;DR

The article develops a flat-space holography framework by formulating AdS$_3$ in the BMS gauge, taking a controlled flat limit $G/\ell\to0$, and establishing a corresponding BMS$_3$/GCA boundary structure. By computing the Brown-York quasi-local stress tensor in AdS$_3$ and performing a careful flat-space contraction, the authors define a finite flat-space energy-momentum tensor $\tilde{T}_{ij}$ whose components are $\tilde{T}_{uu}=M/(16\pi G^2)$, $\tilde{T}_{u\phi}=N/(8\pi G^2)$, and $\tilde{T}_{\phi\phi}=M/ (16\pi)$ with $M$ and $N$ fixed in the flat limit. The conservation laws become ultra-relativistic: $\partial_u M=0$ and $\partial_\phi M - 2\partial_u N=0$, mirroring the flat Einstein equations in BMS gauge, and the associated charges reproduce known mass and angular momentum for 3D flat solutions. These results support the proposal that the holographic dual of asymptotically flat spacetimes is an ultra-relativistic CFT with Galilean conformal symmetry and offer a practical route to compute conserved quantities in flat holography, with potential extensions to higher dimensions.

Abstract

Flat-space limit is well-defined for asymptotically AdS spacetimes written in coordinates called the BMS gauge. For the three-dimensional Einstein gravity with a negative cosmological constant, we calculate the quasi-local energy momentum tensor in the BMS gauge and take its flat-space limit. In defining the flat-space limit, we use the BMS/GCA correspondence which is a duality between gravity in flat-spacetime and a field theory with Galilean conformal symmetry. The resulting stress tensor reproduces correct values for conserved charges of three dimensional asymptotically flat solutions. We show that the conservation relation of the flat-space energy-momentum tensor is given by an ultra-relativistic contraction of its relativistic counterpart. The conservation equations correspond to Einstein equation for the flat metric written in the BMS gauge. Our results provide further checks for the proposal that the holographic dual of asymptotically flat spacetimes is a field theory with Galilean conformal symmetry.