# First Law of Entanglement Entropy in Flat-Space Holography

## TL;DR

This paper establishes a holographic first law of entanglement entropy in flat-space holography, linking BMSFT perturbations to bulk Einstein equations via a geometric integral approach.

## Contribution

It introduces a holographic formulation of the first law of entanglement entropy in flat-space holography, connecting BMSFT perturbations with bulk Einstein equations.

## Key findings

- FLEE for BMSFT perturbed states corresponds to a geometric integral in the bulk.
- Exterior derivative of the one-form vanishes for flat-space cosmology.
- Generic perturbations yield Einstein equations from the FLEE framework.

## Abstract

According to flat/Bondi-Metzner-Sachs invariant field theories (BMSFT) correspondence, asymptotically flat spacetimes in $(d+1)$-dimensions are dual to $d$-dimensional BMSFTs. In this duality, similar to the Ryu-Takayanagi proposal in the AdS/CFT correspondence, the entanglement entropy of subsystems in the field theory side is given by the area of some particular surfaces in the gravity side. In this paper we find the holographic counterpart of the first law of entanglement entropy (FLEE) in a two-dimensional BMSFT. We show that FLEE for the BMSFT perturbed states which are descried by three-dimensional flat-space cosmology, corresponds to the integral of a particular one-form on a closed curve. This curve consists of BMSFT interval and also null and spacelike geodesics in the bulk gravitational theory. Exterior derivative of this form is zero when it is calculated for the flat-space cosmology. However, for a generic perturbation of three-dimensional global Minkowski spacetime, the exterior derivative of one-form yields Einstein equation. This is the first step for constructing bulk geometry by using FLEE in the flat/BMSFT correspondence.