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Flat-space limit is well-defined for asymptotically AdS spacetimes written in the BMS coordinates. For the three-dimensional Einstein gravity with negative cosmological constant, we calculate the quasi-local energy momentum tensor in the BMS gauge and take its flat-space limit. In defining flat-space limit we use 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 flat-space energy-momentum tensor is given by an ultra-relativistic contraction from the 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
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