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In this paper, we define an energy-momentum vector at the spatial infinity ofeither asymptotically flat or asymptotically hyperbolic initial data setscarrying a non-compact boundary. Under suitable dominant energy conditions(DECs) imposed both on the interior and along the boundary, we prove thecorresponding positive mass inequalities under the assumption that theunderlying manifold is spin. In the asymptotically flat case, we also prove arigidity statement when the energy-momentum vector is lightlike. Our treatmentaims to underline both the common features and the differences between theasymptotically Euclidean and hyperbolic settings, especially regarding theboundary DECs.

Spacetime positive mass theorems for initial data sets with noncompact boundary