Hawking's Singularity Theorem for Lipschitz Lorentzian Metrics

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.