Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass

TL;DR

This paper proves that asymptotically hyperboloidal initial data sets with zero mass are rigidly embedded into Minkowski space, showing no radiation configurations are possible in this setting under general conditions.

Contribution

It establishes a rigidity result for zero-mass asymptotically hyperboloidal initial data sets, extending the understanding of their geometric structure in all dimensions.

Findings

Zero mass implies isometric embedding into Minkowski space.

No radiation configurations exist for zero-mass hyperboloidal data.

The proof uses decay estimates for spinors on spacetime harmonic functions.

Abstract

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.