Wednesday, October 25, 2023

New Video on Non-Euclidean Geometry and Relativity

The latest Veritasium video is on:
How One Line in the Oldest Math Text Hinted at Hidden Universes
It is excellent as usual, and concerns the development of non-Euclidean geometry, from Euclid to relativity. It separates special and general relativity, and non-euclidean geometry being essential to general relativity.

In particular it tells how a supernova was seen 4 times in 2014, and predicted to be seen again in 2015, all being an illusion from gravitational lensing.

This is all correct, but special relativity has had a much bigger impact on XX century Physics than general relativity, and special relativity is also founded on non-euclidean geometry.

In the Veritasium model the "one line" is Euclid's Fifth Postulate about the existence of parallel lines. The "hidden universes" are model geometries where the postulate fails. Relativity is non-euclidean because it curves space. But it does not mention the bigger non-euclidean aspect -- light rays have null length. In euclidean geometry, every line segment has a positive length.

Euclidean geometry is based on Euclidean distances. The distance between two points is given by the Pythagorean Theorem. Time is the fourth dimension. The Minkowski metric applies, not a Euclidean one.

The non-euclidean geometry was appreciated early on, at least by Poincare in 1905, Minkowski in 1907, and Varicak in 1910. Einstein disagreed with it.

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