A note on the Gannon-Lee theorem.

Area of Science:

• General Relativity
• Differential Geometry
• Mathematical Physics

Background:

• Classical singularity theorems in general relativity typically require higher regularity for spacetime metrics.
• Non-globally hyperbolic spacetimes present unique challenges for causal structure analysis.
• The regularity of spacetime metrics is crucial for the validity of singularity theorems.

Purpose of the Study:

• To extend the Gannon-Lee theorem to the lowest regularity class of Lorentzian metrics ($C^1$).
• To investigate the properties of maximizing causal curves in $C^1$-regular spacetimes.
• To bridge the gap between classical singularity theorems and more general spacetime structures.

Main Methods:

• Development of novel techniques to handle lower regularity metrics in Lorentzian geometry.
• Analysis of causal curves and their properties within the framework of $C^1$-Lorentzian manifolds.
• Application of geometric analysis to prove geodesic properties of maximizing causal curves.

Main Results:

• A Gannon-Lee theorem is proven for non-globally hyperbolic Lorentzian metrics with $C^1$ regularity.
• It is demonstrated that maximizing causal curves in $C^1$-spacetimes are geodesics.
• This establishes $C^2$-regularity for these specific curves.

Conclusions:

• The Gannon-Lee theorem is now applicable to a broader class of spacetimes, including those with minimal regularity.
• The regularity of maximizing causal curves is established as $C^2$, providing a more robust understanding of causal structure.
• This work advances the study of singularities in general relativity by accommodating less smooth spacetime structures.