- #31
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Actually the bound in #29 doesn't just apply in 1+1 dimensions. Since both r and a are perpendicular to W, they lie in a spacelike plane whose geometry is Euclidean, so you can apply the Euclidean Cauchy-Schwarz to them.
Every event is simultaneous with you at some time: proof?
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Discussion Overview
The discussion revolves around the concept of simultaneity in Minkowski space, specifically addressing the assertion that every event is simultaneous with an observer at some uniquely defined time. Participants explore the implications of this assertion, propose proofs, and examine counterexamples, focusing on the mathematical and conceptual underpinnings of the claim.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- Some participants propose a theorem regarding the existence of a unique time where a radius four-vector is orthogonal to a timelike world-line in Minkowski space.
- Others argue that the initial formulation of the theorem is false, citing counterexamples involving non-inertial world lines and uniform proper acceleration.
- A later reply questions the uniqueness of the time defined in the theorem, suggesting that multiple events could be simultaneous under certain conditions.
- Participants discuss the necessity of formulating an axiom system for Minkowski space to avoid coordinate-based approaches.
- Some participants highlight the importance of the ##C^2## property of world lines, suggesting it is necessary for certain proofs and theorems related to simultaneity.
- There are mentions of the need for a neighborhood around the world line to ensure the uniqueness of points that are orthogonal to it.
- Concerns are raised about the implications of sharp bends in world lines as potential counterexamples to the proposed theorems.
Areas of Agreement / Disagreement
Participants express multiple competing views on the validity of the original theorem and its implications. The discussion remains unresolved, with no consensus on the correctness of the claims or the proposed proofs.
Contextual Notes
Some participants note limitations in the original formulation, including the dependence on the completeness of the world line and the potential for counterexamples in non-standard topologies.
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