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:

$$\frac{\Delta t_c}{\Delta t_s} = \left(\frac{1-\frac32\frac{r_s}r}{1-\frac{r_s}r}\right)^{1/2}$$

Yes

Thanks. That was quick. :)

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 $(1-5/4u)^2$

If you have two points $1$ and $2$ then the time dilation is just $dt_1/dt_2$ (or it's inverse).

If you introduce a third point $3$, e.g. at infinity, then the time dilations relative to this third point are $dt_1/dt_3$ and $dt_2/dt_3$.

And these are just derivatives so is perfectly proper to write $dt_1/dt_2 = dt_1/dt_3 \times dt_3/dt_2$.

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...

Yep. The good old chain rule.

Everyone knows derivatives are really fractions, despite what those pesky mathematicians say :-)

A few days ago, this question about photon trajectories around black holes astronomy.stackexchange.com/q/45099/16685 led me to discover this recent article nature.com/articles/s41598-021-93595-w which inspired me to write some photon orbit plotting code.

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).

I love the GIFs. I wonder what was used to construct them.

@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.

a researcher told me some computational programs take long to learn.

longer than analytical techniques.

I was looking into how decathlon competitions are scored and it looks like phase transition stuff en.wikipedia.org/wiki/Decathlon#Points_system

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.

@JohnRennie Any comment re: chat.stackexchange.com/transcript/message/58809858#58809858 ? Do I need to kill my photon trajectories if they hit the EH, or can I let them continue towards the singularity?

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.

@JohnRennie Oh good. In my calculation, the phi angle acts as a proxy time coordinate.

That should work fine :-)

Though I guarantee some of your audience will think coordinate time moves evenly along the trajectory :-)

@JohnRennie True. I guess I'll need to mention something about that. :)

I'll try to post my current photon trajectory code here. Give me a minute...

sagecell.sagemath.org/…

^Photon trajectory script.

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.

It works very nicely :-)

Aha! Very close to the photon sphere the two integrators predict very different trajectories:

But then small errors will be magnified this close to the photon sphere ...

It's significantly entertaining though. Thanks :-)

@JohnRennie My pleasure! It's fun to be able to write & run stuff like this on my phone.

@JohnRennie this looks like Hopf bifurcation.