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Rapidity, History and Vladimir Karapetoff

This April, I’m giving a presentation at the APS April meeting about Vladimir Karapetoff, an electrical engineering professor who worked at Cornell University in the early 1900s.  He wrote rather prolifically and one of the many subjects he covered in depth was special relativity, specifically the use of hyperbolic geometry in special relativity.

I only have ten minutes for the talk.  There are two ways I can put the talk together.  One would be to use science to call attention to the history.  Although he doesn't mention Karapetoff, this approach has been done really nicely and fairly completely by Scott Walter[pdf][1].  The other approach is to cache science in a little bit of history.  This is the approach I’ll take.  The first little bit of the talk will focus on Karapetoff the person.  He’s actually quite interesting.  He was a member of the socialist party when socialism was cool.  He ran for New York state offices on several occasions although he never won an election.  As I mentioned before, he was a prolific writer.  Not only did he write about electrical engineering and physics, he also wrote about social policies in national magazines like the Saturday Evening Post.  He married Rosalie Margaret Cobb[3].  She was a prominent industrial chemist who won the Coating and Graphic Arts Division Award of the Technical Association of the Pulp and Paper Industry.  She later setup an award that memorializes Karapetoff’s name, the Eta Kappa Nu Vladimir Karapetoff Outstanding Technical Achievement Award[2].

At this point, the presentation will branch away from Karapetoff the person towards the science that will be the bulk of the talk.  I’ll point out that in addition to being a prolific writer he was a prolific builder of machines.  He built analog computers that could be used to calculate various quantities in engineering and physics.  Among the machines he built were the Heavisidion for performing transmission line calculations, the Secomor, the Indumor, the Blondelion, the C.P.S’er named after his friend Dr. C. P. Steinmetz who preceded him as head of engineering at GE.  His penchant for building computing machines resulted in the award of the Franklin Institutes Cresson medal in 1928.

 Karapetoff believed that one of the best ways to teach concepts to applied scientists was through the tactile use of a machine.  Along these lines he also constructed two analog computers for performing special relativity calculations.  One was used to demonstrate the special relativistic doppler effect as well as other relations due to the Lorentz transform and the other was used to calculate the aberration of light (picture 2).  While these machines were designed to work with what Karapetoff called the velocity angle in special relativity, he soon turned his sights away from velocity angles and towards the hyperbolic rapidity angles of Minkowski spacetime.

Now we’ve segued into hyperbolic geometry.  The first trick will be to demonstrate how to arrive at hyperbolic geometry when starting from the four-velocity which most physicists are familiar with in its gamma form.  What needs to be shown is as follows (picture 3):

Looking at the last line, you begin to realize that this looks a whole lot like a horizontal component corresponding to a cosine and a vertical component corresponding to a sine.  There’s only one issue.  What should be the length of the hypotenuse in the denominator is all wrong because of the minus sign.  It’s only wrong for Euclidean geometry involving circles though.  For hyperbolic geometry, it’s just right!  At this point, the components of the four velocity can be shown to be (picture 4):

cosh and sinh are the hyperbolic sine and cosine and w is an angle in the hyperbolic velocity plane.  Hyperbolic velocity angles were dubbed rapidities by Alfred Robb in a pamphlet he printed in 1911.  What the above notation points out is something that Minkowski emphasized only in his first lecture on spacetime.  The velocities of all particles in special relativity are constant at the value of c.  This viewpoint was developed further by Sommerfeld who explicitly pointed out that adding rapidities in more than one dimension was analogous to adding angles on a sphere and amounted to performing spherical hyperbolic trigonometry.  Karapetoff built the same set of hyperbolic formulas, but did not mention the spherical nature of the four space geometry.  To the best of my knowledge the concept of the hyperbolic velocity sphere would go largely unknown and unmentioned until Brehme and/or Adler revived it in the 1960s.

At this point, I suspect I’ve gone well beyond my ten minutes.  Consequently, I’ll stop here for the day and get to work on building slides and trimming material!





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