Despite all the differential calculus shown in the board work in the picture, the line element basically describes how the Pythagorean theorem for finding the length of a line works in a given coordinate system. The familiar example from middle/high school is that that length squared equals the sum of the squares of the sides.
For what Takeno was up to and what we're doing in our project, we care about what an infinitesimally small distance behaves like in our coordinate system that describes a disc rotating at relativistic speeds. We start out with the line element in cylindrical coordinates, (see the expression 6.1 picture below from Takeno's paper). We then apply the Lorentz-like transformation defined by Takeno in his paper to the different differentials in the line element of equation 6.1. There's nothing to do for rho, the radius of the disc, since it doesn't change under the Takeno transform, (picture 3). I'm also ignoring z since we're on a disc with no height. That leaves the angular coordinate phi and the time coordinate that need to be transformed. That's what's shown in the boadwork in picture 1. The top half of the page is the calculation for phi and the bottom half of the page is the derivation for the time coordinate.
|The Takeno Line Element I'm Shooting For|
|The Takeno Transform|
In each case, the total differential has been taken for the coordinate. What's a total differential? They come up all the time in two places that I'm aware of. The first is in freshman calculus and the second is, well, seven years later in graduate statistical mechanics. Surprisingly, the concept may not always stick. In the end though, you just take the derivative of an expression with respect to every variable in the expression. You'll notice in the board work that the expression for phi prime depends on the radius rho, the angle phi, and the time t. The expression I've written down for d phi prime is the total derivative and has terms that correspond first to the phi partial derivatives of phi prime, then the rho partial derivatives and finally, the last three terms correspond to the time partial derivatives of phi prime.
Once the primed, (transformed), coordinates are written as total derivatives, the next step is to plug them into the equivalent of the Pythagorean theorem. Yup, just square them up and add them together. Since we don't know if the new coordinates are at right angles to each other, we have to multiply all the terms of the resulting total differentials together. Just multiplying the like terms with each other in the manner that you perform a Cartesian dot product isn't enough. We have to do the cross-multiplies,that's why you see the terms for drho dphi and drho dt in Takenos results.
I did just enough board work on the differentials shown above to find out how I could finagle Mathematica into doing most of the work for me. The most current results of the Mathematica work can be found in pdf from and in notebook form in the references below.
1. Takeno's paper
Takeno H. “On Relativistic Theory of Rotating Disk”, Prog. Theor. Phys. 7, (1952), 367
2. The pdf version of the calculations so far
3. The Mathematica notebook so far