Rigidity of the Spacetime Positive Mass Theorem

We prove that if an asymptotically flat initial data set satisfies the dominant energy condition and has $E=|P|$, then $E=|P|=0$, where $(E, P)$ is the ADM energy-momentum vector. Previously the result was only known for spin manifolds [8, 14]. We consider a variational approach to the Regge-Teitelboim Hamiltonian, except that we use a modified constraint operator introduced by the first named author and J. Corvino [15] in place of the usual constraint operator. The spacetime positive mass inequality implies that an initial data set satisfying $E=|P|$ must locally minimize the modified Regge-Teitelboim Hamiltonian among initial data sets with the same modified constraints. The Lagrange multipliers corresponding to this constrained minimizer give rise to asymptotically vacuum Killing initial data which is also asymptotically translational. Earlier work of R. Beig and P. Chru\'{s}ciel [8] then implies that the ADM energy-momentum vector must be zero. Since the variational formalism takes place in the space of initial data sets of low regularity, we prove a spacetime positive mass inequality $E\ge |P|$ in the setting of weighted Sobolev spaces, which may be of independent interest.