Discussion Overview
The discussion centers on the nature and number of geodesics connecting two points in various spacetimes, exploring both theoretical and practical implications. Participants examine examples from geometry and physics, including the 2D sphere and Schwarzschild spacetime, while considering the effects of spacetime signature and topology on geodesic existence and uniqueness.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- Some participants note that on a 2D sphere, there are two geodesics connecting two points along great circles, one minimizing and the other maximizing distance.
- Others argue that in 3D space, an infinite number of geodesics can connect the north and south poles.
- One participant questions whether the signature of spacetime (-, +, +, +) imposes restrictions on the number of geodesics, suggesting that the sphere does not have such a signature.
- Another participant suggests that in Schwarzschild spacetime, due to its symmetry, there are infinite geodesics connecting antipodal points with different coordinate times.
- It is proposed that generally, there is no limit to the number of geodesics connecting two points, although a theorem exists stating that geodesics are unique in a neighborhood around every point in Minkowski spacetime.
- One participant mentions that infinitely many null geodesic paths can exist between two events in curved Lorentzian manifolds, referencing the Einstein ring configuration in gravitational lensing.
- Another point raised discusses the infinite number of orbits with the same period around a central body, noting complications from pericenter effects.
- A later reply introduces the question of whether at least one geodesic can always be found connecting two spacetime points, regardless of type.
- One participant highlights that topology affects geodesic existence, using the example of a plane with a removed point that eliminates certain geodesics.
- Another participant questions whether a defect in the manifold can be detected through calculations within the manifold itself.
- Additional references are provided for further reading on geodesic incompleteness and observationally indistinguishable spacetimes.
- One participant reiterates the question of whether the usual spacetime signature restricts the number of geodesics, emphasizing that geodesics can exist without a defined metric if an affine connection is present.
Areas of Agreement / Disagreement
Participants express a range of views on the existence and number of geodesics, with no consensus reached on whether a geodesic can always be found between two points or the implications of spacetime signature and topology.
Contextual Notes
Discussions involve assumptions about the nature of geodesics in different spacetimes, the role of symmetry, and the impact of topology on geodesic existence, which remain unresolved.