I've really missed writing for several days. I was off studying about the Gudermannian yet again in hopes that I could tie up a research project over spring break. It turns out, (as pointed out by Shahin), that the launch angle that gives the maximum range for the catenary trajectory of a relativistic projectile in a gravitational field is the same launch angle that will maximize the arc length along the parabolic trajectory of a classical (non-relativistic) projectile. The derivatives that have to be solved in both cases contain Gudermannian functions, hence all the recent study of hyperbolic math at the expense of my writing. What follows is a free-writing dump capturing everything I’ve been up to for the last several days. This relates to our poster from December that I have yet to fully describe and more about Karapetoff the subject of my history of physics presentation that will come up in a few weeks.
Karapetoff began writing about special relativity in the 1920s. His first several articles treated special relativity using what he called an oblique angle method. As it turns out special relativity can be modeled as a rotation in hyperbolic geometry. Physicists like rotations; we’re familiar with them as they turn up all over our work in things like angular momentum calculations. What physicists in the 1920s weren't as comfortable with was hyperbolic geometry. This was a trend that had already been going on for almost twenty years when Karapetoff began to write his articles.
After the announcement of Einstein’s special theory of relativity, Minkowski, (one of Einstein’s math professors), made a presentation describing the hyperbolic spacetime of the theory. However, Minkowski pretty much stopped emphasizing the hyperbolic geometry part of his special relativity work thereafter. Some historians think it was the general level of discomfort with hyperbolic geometry among physicists that caused this move. In any event, it would be almost 20 years between Karapetoff’s first special relativity papers that emphasized the oblique angle treatment, (making hyperbolic geometry unnecessary), and his papers that emphasized rapidity, ( a concept that only makes sense in the context of hyperbolic geometry).
By using oblique angles, Karapetoff was able to write down geometric diagrams like the following one from his 1926 Journal of the Optical Society of America paper which used all real valued angles. The expense paid for having real valued angles in a non-hyperbolic coordinate system was that the coordinate axes of space and time, (denoted by x , x’, and t, t’ respectively in the diagram below), were no longer at right angles to each other.
Where does the Gudermannian enter into all of this? With Karapetoff’s oblique angle treatment, he was only one step away from the full-blown hyperbolic treatment that he adopted in the 1940s and that we made profitable use of last year. As it turns out, the angle can be turned into a hyperbolic angle as follows, (picture 5),
The second time the Gudermannian was used was to patch Copernicus’ almost correct circular orbits of the planets into Kepler’s elliptical orbits of the planets.
1. History post on Karapetoff
2. Post on hyperbolic geometry and special relativity
3. Karapetoff on special relativity in 1926
4. Alfred Robb's pamphlet on rapidity, (open access)
5. Enders A. Robinson's book on Google Books
6. Mercator projection on Wikipedia
7. The rapidity poster
8. Shahin on relativisitc projectiles
9. MacColl's paper on the relativistic projectile in a uniform gravitational field