Sunday, March 30, 2014

Hyperbolic Motion, Rindler, Minkowski, Kruskal, and Karapetoff

I found something interesting yesterday, well interesting to me anyway.  What follows is a bit of a historical ramble and reference-fest.  Hyperbolic motion which is usually attributed to Wolfgang Rindler was actually first shown by Hermann Minkowski.  Rindler himself references Minkowski[2] in the first paragraph of his paper where hyperbolic motion under uniform acceleration in terms of special relativity is derived[1].  Not only did Minkowski show hyperbolic motion in his first lecture on spacetime, he also pointed out that you'd have to work kind of hard, actually setting all your acceleration components to zero to not exhibit hyperbolic motion, (picture 1).

Hyperbolic motion, I think, is attributed to Rindler because of his fleshing out and full development of the idea including the concept of event horizons.  Rindler pointed out that when an object undergoes uniform acceleration in spacetime, that a light signal sent out after the object will never be able to catch up.  And that, in a very small nutshell is how an event horizon works.

In picture 2, you might have noticed that what looks like Rindler hyperbolic motion in a Minkowski spacetime is referred to by Rinder as a Kruskal diagram.  Kruskal[4] worked on the hyperbolic nature of the Schwarzschild metric and it was his work that Rindler commented on leading up to what would become known as Rindler coordinates.  In addition to the Schwarzschild metric, Kruskal also discussed Einstein-Rosen wormholes, (picture 3).

There are two more interesting notes about Kruskal's paper.  First, Kruskal was working for project Matterhorn at the time of the publication of the article.  Project Matterhorn was a cold-war project to control thermonuclear reactions.  Second, at the recent Texas Symposium on Relativistic Astrophysics, Charles Misner told the story of how the Kruskal article was actually submitted under Kruskal's name by John Wheeler.  Kruskal had described his work to Wheeler earlier and Wheeler had told him he should write it up. Apparently Wheeler got tired of waiting for Kruskal and just went ahead and wrote it up for him!

Picture of the Day:
Sunrise over Texas A&M

1.  Rindler on hyperbolic motion (not open access)

2.  Minkowski's spacetime lecture, (not open access, not even online, but a cheap book):

3.  Rindler on  Kruskal Space (not open access)

4. Kruskal on the Schwarzild metric

5. Project Matterhorn

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