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."
1.  Takeno on space-times

2.  Franklin biography

3.  Franklin's transforms

4.  Veblen's book on projective geometry

5.  Article referencing Takeno and Franklin

6.  Takeno line element recalculation begins

7.  Gibbons on Takeno and wave geometry


Popular posts from this blog

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

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 rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr. 

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…