Discussion Overview
The discussion centers on the comparison between Milne and Minkowski spacetimes, focusing on their metrics and manifold structures. Participants explore the implications of geodesic completeness, the nature of isometries, and the definitions of metrics in the context of flat spacetimes.
Discussion Character
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants argue that Milne and Minkowski are different instances of manifolds with metrics due to the absence of a global isometry between them, with Milne being geodesically incomplete.
- Others clarify that Milne spacetime represents only a "wedge" of Minkowski spacetime, specifically the interior of the future light cone of a chosen origin event, suggesting that isometries exist within that wedge but not globally.
- There is a discussion on whether the distinction between Milne and Minkowski spacetimes depends on how one defines a "metric," with some suggesting that it could be viewed as a different manifold based on the geodesic completeness of the metrics applied.
- Participants note that both spacetimes are homeomorphic to ##R^4##, but the way metrics are applied leads to different properties regarding geodesic completeness.
- Some participants express uncertainty about whether the differences in metrics constitute different manifolds or merely different coordinate representations of the same underlying topology.
- A later reply introduces the idea that Minkowski spacetime could be seen as the analytic continuation of the geodesically incomplete Milne spacetime.
- Discussion also touches on the FLRW metric and its relation to Minkowski spacetime, with some noting that the standard FLRW metric with constant scale factor is indeed Minkowski coordinates covering the entire Minkowski spacetime.
- Participants mention the Einstein static universe as another special case, contrasting its properties with those of Minkowski spacetime.
Areas of Agreement / Disagreement
Participants express differing views on whether Milne and Minkowski should be considered distinct spacetimes or different representations of the same topology. There is no consensus on the implications of their metrics or the nature of their relationship.
Contextual Notes
Participants highlight the importance of definitions and assumptions regarding metrics and topology, noting that the discussion may reflect varying levels of rigor between physicists and mathematicians.