Karapetoff, Oblique Angle Special Relativity, and the Gudermannian

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[7] 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 [1]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[2]).

There were other proponents of the special relativity as rotation interpretation such as Sommerfeld, but they tended not to use the oblique angles of Karapetoff favoring instead a treatment that allowed a graphical display of a right angle, (orthogonal), coordinate system of space and time at the expense of using angles with imaginary number, (a number times the square root of negative one), angular values.

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[3] 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.

Karapetoff’s oblique angle treatment begins with defining an angle a as, (picture 2),

where v is the velocity between the laboratory frame and the frame of the moving particle being observed.  Once this angle is defined, Karapetoff shows that the Lorentz, (Karapetoff used the other name, the Fitzgerald contraction), contraction can be written down in terms of the angle as, (picture 3),

and the expression for time dilation can be written down as, (picture 4),

Folks in physics classes, or who have a really good command of trigonometry, will see how the two expressions can be derived from the diagram shown in the first picture.

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),

where w is the hyperbolic velocity angle of the particle known as the rapidity, an expression that Karapetoff attributes to Alfred Robb[4].  The hyperbolic arctangent, (atanh), of the sin of a is a not-so-well known operation called the anti-Gudermannian.

OK, so far this might seem like a somewhat abstract and possibly useless collection of facts about special relativity and Karapetoff’s research, but there’s a cocktail party bit of useful information to all of this though.  As Enders A. Robinson[5] very aptly pointed out, the Gudermannian shows up in three different places throughout history.  First it was used to transfer maps of the Earth plotted on globes to flat pieces of paper.  It’ a more complex operation than it sounds like and is known as the Mercator projection[6].

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.

The third time it was used was to describe the effects of special relativity as shown above.  While Karapetoff never goes into any detail about the anti-Gudermannian, his writings indicate that he was well aware of it and its significance since he mentions it in relation to MacColl’s[9] paper that winds up presenting an expression for the angle that will maximize the relativistic projectile's range and which contains a Gudermannian.  Interestingly, MacColl seems not to have been aware of the anti-Gudermannian lurking in his own expression.  Furthermore it wasn't until 2006 that Shahin[8] recognized that the angle that maximized the relativistic range also maximized the parabolic path length of the classical particle in the same gravitational field.  Shahin didn't spot the anti-Gudermannian lurking in his equations either.  More on how to spot anti-Gudermannians coming soon!

References:
1.  History post on Karapetoff
http://copaseticflow.blogspot.com/2014/03/rapidity-history-and-vladimir-karapetoff.html

2.  Post on hyperbolic geometry and special relativity
http://copaseticflow.blogspot.com/2014/03/rapidity-as-decomposition-and-writing.html

3.  Karapetoff on special relativity in 1926
http://www.opticsinfobase.org/josa/abstract.cfm?uri=josa-13-2-155

4.  Alfred Robb's pamphlet on rapidity, (open access)
https://archive.org/details/opticalgeometryo00robbrich

5.  Enders A. Robinson's book on Google Books
http://books.google.com/books?id=55PuAAAAMAAJ&q=enders+robinson+relativity&dq=enders+robinson+relativity&hl=en&sa=X&ei=1UUkU7y4IYac2QXU14HYDw&ved=0CD4Q6AEwAA

6.  Mercator projection on Wikipedia
https://en.wikipedia.org/wiki/Mercator_projection

7.  The rapidity poster
https://plus.google.com/108242372478733707643/posts/3nY5Ng4knC7

8.  Shahin on relativisitc projectiles
http://iopscience.iop.org/0143-0807/27/1/017

9.  MacColl's paper on the relativistic projectile in a uniform gravitational field
http://www.jstor.org/discover/10.2307/2302436?uid=2&uid=4&sid=21103757754563