Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass

TL;DR

This work proves a rigidity theorem for $C^{2,a}_{-q}$ asymptotically hyperboloidal spin initial data sets with decay rate $q\in(\tfrac{n}{2},n]$ under the dominant energy condition: if the total mass $m=0$, the data embeds isometrically into Minkowski spacetime. The authors develop a spinorial approach driven by spacetime harmonic functions, constructing mass-minimizing spinors and showing that level-set geometry becomes flat, ultimately yielding a Minkowski embedding and ruling out non-vacuum pp-wave radiation in this setting. The method hinges on precise decay estimates for spinors, an analysis of the asymptotic geometry via a global potential $u$, and a topological argument ensuring a single-end, globally foliated manifold; the result extends the positive mass theorem and rigidity phenomena to the asymptotically hyperboloidal regime in all dimensions. This advances the understanding of rigidity in General Relativity by highlighting a fundamental difference between AH and asymptotically flat spacetimes and constraining the possible massless configurations in the AH context.

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.

Figures (2)

• Figure 1: On the left, an example of a pp-wave is illustrated. The majority of the spacetime is vacuum and coincides with Minkowski space, with the exception of the wave itself, which is highlighted as an orange beam traveling at the speed of light. This beam extends in the $x_1, \dots, x_{n-1}$ directions with an appropriate decay profile. Any asymptotically flat initial data set contained within this spacetime satisfies the dominant energy condition; it has zero mass but non-zero energy. To illustrate the properties of a pp-wave, consider the effect it has on an observer, represented by the red line. As the pp-wave passes through the observer (around time $t=1$), a notable elongation occurs in the $x_n$ direction. This stretching effect is most pronounced near the center of the wave and diminishes as $x_1, \dots, x_{n-1}$ increase. After the wave has passed through the observer, everything returns to its original state. This phenomenon is visually represented on the right.
• Figure 2: Schematic depiction of a pp-wave spacetime. The wave is traveling at the speed of light along the region shaded in orange. Moreover, the spacetime contains an initial data set $(M^n,g,k)$, highlighted in blue, which fails to be asymptotically hyperboloidal

Theorems & Definitions (41)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1: Weighted function spaces
• Definition 2.2: Asymptotically hyperboloidal initial data sets
• Definition 2.3: Dominant energy condition
• Definition 2.4: The energy-momentum vector
• Definition 2.5
• Remark 2.6
• Proposition 2.7
• Proposition 2.8