Time Functions on Lorentzian Length Spaces

Area of Science:

• General Relativity
• Differential Geometry
• Mathematical Physics

Background:

• Time functions are essential in general relativity for understanding spacetime causality and solving Einstein's equations.
• Previous work often assumed manifold structures, limiting applicability to more singular spacetimes.

Purpose of the Study:

• To establish classical existence results for time functions in generalized Lorentzian spaces.
• To characterize the conditions for the existence of time functions without requiring manifold structures.
• To connect time functions to causality and global hyperbolicity in diverse spacetimes.

Main Methods:

• Analysis of Lorentzian (pre-)length spaces, including causally plain continuous spacetimes and closed cone fields.
• Characterization of time functions using K-causality.
• Investigation of modified Geroch volume functions.
• Identification of criteria for global hyperbolicity via Cauchy time functions and sets.

Main Results:

• Fundamental classical existence results for time functions are established for a broad class of spacetimes.
• Time function existence is characterized by K-causality.
• A modified Geroch volume function is shown to be a time function if and only if the spacetime is causally continuous.
• Global hyperbolicity is characterized by the existence of Cauchy time functions and Cauchy sets.

Conclusions:

• Time functions can be defined and studied in highly singular spacetimes, not requiring a manifold structure.
• The results provide a unified framework for understanding time functions and causality across different types of spacetimes.
• This work deepens the understanding of the relationship between causal structure, time functions, and the formulation of gravitational theories.