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Hawking's Singularity Theorem for Lipschitz Lorentzian Metrics.

Matteo Calisti1, Melanie Graf2, Eduardo Hafemann2

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Communications in Mathematical Physics
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This study proves Hawking's singularity theorem for less regular spacetime metrics. It introduces novel Ricci curvature estimates and a new method to control geodesic focusing, advancing general relativity.

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Area of Science:

  • General Relativity
  • Differential Geometry
  • Mathematical Physics

Background:

  • Hawking's singularity theorem is a cornerstone of general relativity.
  • The theorem traditionally requires smooth spacetime metrics.
  • Extending the theorem to less regular metrics is crucial for a complete understanding of spacetime.

Purpose of the Study:

  • To prove Hawking's singularity theorem for spacetime metrics with local Lipschitz regularity.
  • To develop new mathematical techniques applicable to singular spacetimes.

Main Methods:

  • Derivation of new estimates for Ricci curvature of regularizing smooth metrics using a Friedrichs-type lemma.
  • Replacement of classical focusing techniques for timelike geodesics with a worldvolume estimate and segment-type inequality.
  • Control of the volume of points in spacelike surfaces with long maximisers.

Main Results:

  • Successful proof of Hawking's singularity theorem for local Lipschitz regular spacetime metrics.
  • Novel Ricci curvature estimates providing deeper insights into metric regularity.
  • A new worldvolume-based method to analyze geodesic behavior in non-smooth spacetimes.

Conclusions:

  • The singularity theorem holds even for metrics with reduced regularity.
  • The developed techniques offer a robust framework for studying singularities in general relativity.
  • This work expands the applicability of singularity theorems in theoretical physics.