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<p>In Special Relativity, the vanishing of mass indicates either the absence of energy (vacuum) or the presence of massless radiation moving at the speed of light. In General Relativity, the same principle governs the interpretation of zero total mass: asymptotically flat initial data sets (IDS) <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((M^n, g, k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mi>n</mi> </msup> <mo>,</mo> <mi>g</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with vanishing mass correspond either to slices of Minkowski spacetime or to slices of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(pp\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">pp</mi> </mrow> </math></EquationSource> </InlineEquation>-wave spacetimes that model radiation. In contrast, we demonstrate that asymptotically hyperboloidal spin IDS with zero mass must embed isometrically into Minkowski space, with no possible IDS configurations that model radiation in this setting. Our result holds under general decay assumptions where the total mass is well-defined. The proof relies on precise decay estimates for spinors on level sets of spacetime harmonic functions and works in all dimensions.</p>

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Rigidity of Asymptotically Hyperboloidal Initial Data Sets with Vanishing Mass

  • 期刊论文

  • 发表日期：

    2025年10月30日

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  • Sven Hirsch,
  • Hyun Chul Jang,
  • Yiyue Zhang

摘要

In Special Relativity, the vanishing of mass indicates either the absence of energy (vacuum) or the presence of massless radiation moving at the speed of light. In General Relativity, the same principle governs the interpretation of zero total mass: asymptotically flat initial data sets (IDS) \((M^n, g, k)\) ( M n , g , k ) with vanishing mass correspond either to slices of Minkowski spacetime or to slices of \(pp\) pp -wave spacetimes that model radiation. In contrast, we demonstrate that asymptotically hyperboloidal spin IDS with zero mass must embed isometrically into Minkowski space, with no possible IDS configurations that model radiation in this setting. Our result holds under general decay assumptions where the total mass is well-defined. The proof relies on precise decay estimates for spinors on level sets of spacetime harmonic functions and works in all dimensions.