- #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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SUMMARY
The discussion centers on a theorem in Minkowski space regarding the existence of a unique time τ such that the radius four-vector r(τ) is orthogonal to a timelike, C² world-line W. The original proposition is deemed false, with participants providing counterexamples, particularly involving non-inertial world lines and the Rindler horizon. A revised conjecture is proposed, asserting that within a neighborhood A of W, for any point P, there exists a unique point Q on W such that the line PQ is orthogonal to W. This conjecture is considered plausible and potentially easier to prove than initially thought.
PREREQUISITES- Minkowski space concepts
- C² differentiability in the context of world-lines
- Understanding of Rindler horizons
- Fermi normal coordinates in general relativity
- Study the properties of C² world-lines in Minkowski space
- Research the implications of Rindler horizons on simultaneity
- Examine the Herglotz-Noether theorem in the context of general relativity
- Explore mathematical treatments of Fermi normal coordinates for rigorous proofs
Mathematicians, physicists, and students of general relativity interested in the properties of spacetime, world-lines, and the implications of simultaneity in Minkowski space.
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