Skip to main content

Takeno Scooped by Thirty Years and Keeping the Line Element Simple Special Rel Style

One major mistake was pointed out in my work on the Takeno line element from yesterday[6].  When working on a theory that describes the behavior of physical quantities with respect to velocity, (like special relativity), don't decompose the velocity into its components like distance and time, or in my case into angular displacement, and time.  Just leave omega, (angular velocity), as omega!  Since omega isn't one of the variables that the derivatives are being taken with respect to, the many terms due to chain ruling out the innards of the hyperbolic trig functions disappeared and the phi and time components of the line element as calculated by Takeno fell out pretty simply.


I also found out last night while reading through one of the Takeno referencing articles[5] I mentioned a few days ago that Takeno was scooped on his transform by about thirty years.  As it turns out, Phillip Franklin[2], (picture 2), a then recent PhD from Princeton, beat Takeno to the punch in 1922[3, open access!].  Franklin was a mathematician by training and did his PhD thesis under Veblen on the four color problem.  Veblen is also interesting because he wrote a book on projective geometry[4, open access], a topic which turns up more or less subtly all over relativity.

As it turns out Takeno wrote more about the relativity than just the Takeno transforms.  Here's an open access example in which he wrote about spherically symmetric space-times[1].  He also helped to develop something called Wave Geometry.  I haven't been able to get my hands on any of the sources that contain Takeno writing on wave geometry yet, but here's what Gibbons had to say about it[7]

"Conformal Killing spinors of course arise naturally in conformal supergravity [16]. As a further illustration  of historical antecedents, it is interesting to recall that the existence of solutions to an equation of the form (2) was the basic assumption behind the theory of “Wave Geometry” which was extensively developed in Hiroshima in the ’30.s.  The introduction to [38] describing the history of these ideas and the fate of those working on them seems to me to be one the most poignant in the physics literature."
References:
1.  Takeno on space-times
http://projecteuclid.org/euclid.jmsj/1261734863

2.  Franklin biography
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Franklin.html

3.  Franklin's transforms
http://www.jstor.org/stable/pdfplus/84534.pdf?acceptTC=true

4.  Veblen's book on projective geometry
https://archive.org/details/117714799_001

5.  Article referencing Takeno and Franklin
http://arxiv.org/pdf/1208.1913.pdf

6.  Takeno line element recalculation begins
http://copaseticflow.blogspot.com/2014/03/re-calculating-takeno-line-element.html

7.  Gibbons on Takeno and wave geometry
http://arxiv.org/pdf/1110.1206.pdf?origin=publication_detail

Labels:

Comments

Post a Comment

Please leave your comments on this topic:

Popular posts from this blog

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though! Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very sim
1 comment
Read more

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems , there seem to be fewer explanations of the procedure for deriving the conversion, so here goes! What do we actually want? To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. What we’ll need: 1. The Cartesian Nabla: 2. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system: How to do it: Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables. The chain
11 comments
Read more

More Cowbell! Record Production using Google Forms and Charts

First, the what : This article shows how to embed a new Google Form into any web page. To demonstrate ths, a chart and form that allow blog readers to control the recording levels of each instrument in Blue Oyster Cult's "(Don't Fear) The Reaper" is used. HTML code from the Google version of the form included on this page is shown and the parts that need to be modified are highlighted. Next, the why : Google recently released an e-mail form feature that allows users of Google Documents to create an e-mail a form that automatically places each user's input into an associated spreadsheet. As it turns out, with a little bit of work, the forms that are created by Google Docs can be embedded into any web page. Now, The Goods: Click on the instrument you want turned up, click the submit button and then refresh the page. Through the magic of Google Forms as soon as you click on submit and refresh this web page, the data chart will update immediately. Turn up the:
Post a Comment
Read more