- #31
RockyMarciano
- 588
- 43
Very benevolent of you.lavinia said:
Curvature of Flat Lorentz manifolds
- Context: Graduate
- Thread starter lavinia
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Discussion Overview
The discussion centers on the curvature properties of flat Lorentz manifolds, particularly in relation to Riemannian metrics. Participants explore the implications of these properties in the context of physics, especially in particle physics, and the definitions of Minkowski space as a flat affine manifold versus a quotient of Minkowski space.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants propose that while Minkowski space and Euclidean space have zero curvature tensors, flat Lorentz manifolds may not admit a flat Riemannian metric due to the nature of their defining groups.
- One idea suggests examining the metric corresponding to the flat Lorentz metric by adjusting the signs of inner products.
- There is a discussion about the complexity of defining Minkowski space as a flat affine 4-manifold versus a quotient, with some participants expressing confusion about its representation in physics.
- Some participants note that flat Lorentz manifolds covered by Minkowski space cannot have a flat Riemannian metric and will exhibit non-zero Riemannian curvature for any Levi-Civita connection.
- There is mention of the Levi-Civita connection and its role in deriving curvatures, with a query about the interest in the physics implications of these manifolds.
- Participants discuss the potential for every Lorentz metric to determine a Riemannian metric through a redefinition of inner products, raising concerns about the loss of physical properties in such transformations.
- Several references to literature on flat Lorentz manifolds are shared, with requests for additional resources on both flat and curved manifolds.
Areas of Agreement / Disagreement
Participants express differing views on the implications of flat Lorentz manifolds and their metrics, with no consensus reached on the nature of their curvature or the definitions of Minkowski space. The discussion remains unresolved regarding the relationship between Lorentz and Riemannian metrics.
Contextual Notes
Participants highlight the complexity of definitions and the potential ambiguities in the relationship between flat Lorentz manifolds and Riemannian metrics, particularly in the context of physics and mathematical literature.
- #32
RockyMarciano
- 588
- 43
@lavinia, would Minkowski space itself, for the timelike vector case(where the Lorentz group acts on the interior of the future timelike oriented cone) not have the same issue with its Lorentz isometries acting properly discontinuously and not being a subgroup of rigid euclidean motions?