Discussion Overview
The discussion centers on the relationship between attractors in general relativity (GR) and Liouville's theorem from classical Hamiltonian mechanics. Participants explore whether GR possesses attractors and if a version of Liouville's theorem applies within the context of GR, particularly considering its Hamiltonian formulation.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- One participant asserts that Liouville's theorem implies there are no attractors in classical Hamiltonian mechanics due to the preservation of phase space volume.
- Another participant suggests that the question of attractors in GR is complex, given the limited understanding of the global properties of solutions to the initial value problem.
- A participant claims that every Hamiltonian system, including GR, obeys Liouville's theorem, but notes that the phase space in field theories is infinite dimensional, complicating the relationship with 3-dimensional volumes of matter.
- One participant discusses the Raychaudhuri equation, suggesting it indicates shrinking volume but acknowledges that GR is invariant under time reversal, allowing for both expansion and contraction scenarios.
- Concerns are raised about the applicability of Liouville's theorem in field theories, with one participant questioning its rigor due to the infinite degrees of freedom involved.
- A later reply confirms that while Liouville's theorem formally holds in field theories, its application may not be fully rigorous.
Areas of Agreement / Disagreement
Participants express differing views on the existence of attractors in GR and the applicability of Liouville's theorem in the context of field theories. The discussion remains unresolved, with multiple competing perspectives presented.
Contextual Notes
Participants highlight limitations in understanding the global properties of solutions in GR and the implications of infinite dimensional phase spaces, which may affect the interpretation of Liouville's theorem.