To get someone notified about my message, should I use @reply syntax in chat rooms? Who will get notifications when I send a message to a chat room without @reply?
Is it valid to combine the equations in this answer by @JohnRennie physics.stackexchange.com/a/69048/123208 to calculate the gravitational time dilation at the centre of a uniform sphere relative to its surface? Like this:
I noticed that there's a nice simplification for small $u=r_s/r$. Multiply top & bottom inside the sqrt by $1-u$. Then the numerator becomes $1-5/2u+3/2u^2$, which is almost $(1-5/4u)^2$
I thought so, but I just wanted to check it. :) I'd like to calculate the gravitational potential at the centre of the Earth assuming spherical symmetry, but non-uniform radial density...
I have to admit that the anims in that answer by Prallax are rather impressive.
In that article, he terminates photon trajectories that hit the event horizon. I'm wondering if it's valid to continue them across the horizon. The trajectory uses the $\phi$ angle, and the inverse radius, $u = r_s/r$. We've eliminated, the Schwarzschild $t$ coord, and aren't using the proper time (or an affine parameter).
@JohnRennie I assume the calculations were done using his own code. He might've used ImageMagick to combine the GIFS into an anim, or the old classic, gifsicle.
It's possible to make simple animated vector graphics with SVG. And you can do fancier stuff (including interactive diagrams) by combining JavaScript with SVG. Stack Exchange will display plain SVG, but not if it's combined with JavaScript.
Yes it's perfect reasonable to continue the trajectories across the horizon. You're just solving the null geodesic equation and that can be done both inside and outside the horizon. You need to choose suitable coordinates though for it to make sense. The animated gifs presumably use the Schwarzschild time coordinate so the trajectories would stop at the horizon.
If you're just drawing the trajectory as a line, not as an animation in time, then of course this problem doesn't arise.
I'm currently comparing the standard 2nd order Leapfrog integrator with Yoshida's 4th order variation. The 2nd order plots in blue, the 4th order in red.
I'm basically using the same convention as that Nature article. The starting point is at (x0, b0 + delta), where b0 is the critical impact parameter. If delta>0, the photon is deflected, if delta<0, it hits the EH. The step parameter is the phi angle increment, in degrees.
Don't expect accurate results for delta=0 :)
You can type expressions into the input fields. Eg, set delta to 0.03*exp(-2*pi) and it will give a similar trajectory to delta = 0.03, except that it will do 1 orbit around the photon sphere.