Schwarzschild Orbits in Cartesian coordinates

[CarlB]

Exiting! When can we see the working applet?

Here it is:
http://www.gaugegravity.com/testapplet/SweetGravity.html

I'd have put it up yesterday, but it still had the debug prints turned on and I didn't have protection against division by zero at the singularity.

The Painleve orbits are labeled as "gauge" because the purpose of the applet is to show off the gauge gravity equations. The Schwarzschild coordinates are labeled "Einstein".

The Schwarzschild and Painleve orbits are similar but not quite identical. I can make them identical by correcting the initial conditions. That is the initial conditions for (coordinate) velocity:

\left. \frac{dx}{dt}\right|_0

means different things for Schwarzschild and Painleve coordinates because t is different between them. I'm contemplating adding a button to the applet that will change the initial conditions for Schwarzschild to match the Painleve or vice versa.

By the way, if you look around in the literature, you will find formulas for approximate relativistic corrections to Newton's equations of motion. What I've computed here are the exact relativistic corrections to Newton's equations of motion. After I do the rotating black hole in Doran coordinates I will publish this.

Here are some papers talking about 1st order corrections to Newton:

http://www.numdam.org/numdam-bin/fitem?id=AIHPA_1985__43_1_107_0
http://www.obs-azur.fr/gemini/pagesperso/pireaux/proc/SCRMI_integrator.pdf
http://syrte.obspm.fr/journees2004/PDF/Pireaux.pdf
http://adsabs.harvard.edu/abs/1986IAUS..114..105N

[Latest Update]: The equations take the form of a ratio of two quantities. In post #53 I factored the denominator factors into:

4 \;I \;\left(4I - \left(\frac{\partial I}{\partial \dot{x}}\frac{\partial I}{\partial \dot{y}}\right)^2\right)

As usual, this is wrong. The correct factorization is:

4 \;I \;\left(4I - \left(\frac{\partial I}{\partial \dot{x}}\right)^2 - \left(\frac{\partial I}{\partial \dot{y}}\right)^2\right)

It turns out that the complicated part of the factorization simplifies to -4. Thus the denominator is:

-16 I

I've now factored the I out of the numerator. Consequently I have a set of equations that have no singularities other than at the origin. The numerator is still messy, but probably can be simplified (which I'm doing).
[/Latest Update]

Carl

 

[CarlB]

From reducing the equations of motion, it's pretty clear that your life is easier if you write your Painleve metric in the following fashion:

ds^2 = \left(dx + \sqrt{2}xr^{-1.5}\;dt\right)^2 +<br /> \left(dy + \sqrt{2}yr^{-1.5}\;dt\right)^2 +<br /> \left(dz + \sqrt{2}zr^{-1.5}\;dt\right)^2 - dt^2

Then the calculations end up using terms like \dot{x}+\sqrt{2}xr^{-1.5}, which in the river vernacular is the velocity relative to the river. For example, if x and \dot{x} are both positive, then you are climbing out of the black hole and swimming against the stream. Therefore the relative velocity is increased.

I've got the terms fairly well reduced and well behaved, but I suspect that if I play with them for a little longer I'll get them much better.

Carl

 

[ZapperZ]

Based on all the reports that I have received, this thread has been going in the wrong path for a long time.

If you wish to continue, please do so in the IR forum.

Zz.