First law of entanglement entropy in flat-space holography

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 $\mathrm{AdS}/\mathrm{CFT}$ 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 a BMSFT interval and also null and spacelike geodesics in the bulk gravitational theory. The exterior derivative of this form is 0 when it is calculated for the flat-space cosmology. However, for a generic perturbation of three-dimensional global Minkowski spacetime, the exterior derivative of the one-form yields the Einstein equation. This is the first step for constructing bulk geometry by using FLEE in the flat/BMSFT correspondence.