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Aristotelis Panagiotopoulos1, George Sparling2, Marios Christodoulou3

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Researchers explored complete observables in general relativity using set theory. They found that complete observables are not definable for many spacetimes, posing a fundamental challenge for theoretical physics.

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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.