Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass

In Special Relativity, massless objects are characterized as either vacuum states or as radiation propagating at the speed of light. This distinction extends to General Relativity for asymptotically flat initial data sets (IDS) $(M^n, g, k)$, where vacuum is represented by slices of Minkowski space, and radiation is modeled by slices of $pp$-wave spacetimes. In contrast, we demonstrate that asymptotically hyperboloidal IDS with zero mass must embed isometrically into Minkowski space, with no possible IDS configurations modeling radiation in this setting. Our result holds under the most general assumptions. The proof relies on precise decay estimates for spinors on level sets of spacetime harmonic functions and works in all dimensions.