Incompleteness Theorems for Observables in General Relativity

Area of Science:

• Theoretical Physics
• Mathematical Physics
• Descriptive Set Theory

Background:

• The search for complete observables in general relativity is a persistent challenge.
• Complete observables are crucial for a fully predictive theory of gravity.

Purpose of the Study:

• To investigate the definability of complete observables in general relativity.
• To determine if complete observables can be constructed within the framework of set theory.

Main Methods:

• Employing methods from descriptive set theory.
• Analyzing rich collections of spacetimes, including vacuum solutions.
• Utilizing Zermelo-Fraenkel set theory and the axiom of dependent choice.

Main Results:

• Demonstrated that no complete observable is Borel definable on sufficiently rich spacetime collections.
• Showed that it is consistent with standard set theory axioms that no complete observable exists for such collections.
• Confirmed these results even when restricting to vacuum solutions, highlighting the role of local degrees of freedom.

Conclusions:

• The problem of observables in general relativity is fundamentally limited by mathematical definability, analogous to classical geometry problems.
• The existence of local degrees of freedom is a key factor in the non-existence of complete observables.
• This research opens new avenues connecting theoretical physics and descriptive set theory.