I Solve General Relativity "Paradox" w/ Automatic Diff. ODE Solver

[m4r35n357]

After making a couple of comments on this StackExchange question, and pointing yet again to this article, a thought occurred to me.

I have been working on an Automatic Differentiation based ODE solver and equation analyzer, mentioned in this thread. Why not use it to solve equation (7) in the article, and plug in my answer to equation (8)? All I have to do is write a test client containing the appropriate equations ;)

Well, I had to do a bit of work to extract the equation solver results programmatically rather than just printing them out, but I did get it working! Here's what it looks like in the console (the ratio is of the proper times of the radial to circular paths):

$ ./grtwins.py 2 -2 2 1001 0 1e-9 1e-9 | ./plotMany.py 2 300 >/dev/null
taylor module loaded
series module loaded
playground module loaded
Newton's method
ResultType(count=4, sense='-', mode='ROOT', x=1.5843306502838275, f=-7.105427357601002e-15, dx=-1.0310205845716116e-16)
(4.932330448527358, 1.3413587820012287)
n = 1
alpha = 1.5843306502838275
r1 = 4.932330448527358
r2 = 10.0
ratio = 1.3413587820012287

A plot of the alpha function in the article is attached (function value is black, the rest are the first six derivatives). I am still in the process of testing for accuracy, but thought I'd post while I was motivated ;) If anyone wants to join in the fun, the software (which uses Python3 and matplotlib) is here.

 

[A.T.]

After making a couple of comments on this StackExchange question, and pointing yet again to this article, a thought occurred to me.

Here an old thread where various clocks were discussed, including the two mentioned above:
https://www.physicsforums.com/threads/tossing-a-clock.249722/page-3#post-1840160

 

[Dale]

I had to do a bit of work to extract the equation solver results programmatically rather than just printing them out, but I did get it working

Excellent!

 

[m4r35n357]

Excellent!

Cheers ;)

I'm now scratching my head wondering how I can check against the table following eq(8). I am starting with r2, working out q, then alpha, then r1. How to go the other way round is not particularly obvious . . . I can only think of brute-force evaluating lots of r2 and alpha values, inverting using eq(6), and then interpolating (not nice at all!).