Black-Hole-Image

This repository is to put the simlation image/result of Kerr / Schwarzschild Black Hole, using the WebGL or Metal. All the simulation is using the natural unit $(G=c=1)$, and the mass of black hole is $M = 1$. For Schwarzschild metric, we using

$$ d s^2 = \left(1 - \frac{r_s}{r}\right)d t^2 + \left(1 - \frac{r_s}{r}\right)^{-1}d r^2 + d\theta^2 + \sin^2\theta,d\phi^2,, $$

where $r_s$ is the Schwarzschild radius given by $r_s = 2GM/c^2$ and here using spherical coordinate $(t,r,\theta,\phi)$. On the other hand, the Kerr metric is given by

$$ g_{\mu\nu} = \eta_{\mu\nu} + \left(\frac{2GMr^3}{r^4 + az^4}\right)k_{\mu},k_{\nu},, $$

where $k_{\mu}$ is a unit vector given by $\displaystyle \vec{k} = \left(\frac {rx+ay}{r^2+a^2},\frac{ry-ax}{r^2+a^2},\frac{z}{r}\right)$, and here using Kerr–Schild "Cartesian" coordinates, which satisfied

$$ x^2 + y^2 + z^2 = r^2 + a^2\left(1 - \frac{z^2}{r^2}\right),, $$

and $a$ is the rotational parameter related to the angular momentum $J$, which is given by $a = J/Mc$. To see more introduction about Kerr metric, see arXiv paper: 2008, Matt Visser, The Kerr spacetime: A brief introduction.

Example

Click image to see video on YouTube

• Download video: video/kerr-4536x2835-1000fs.mp4
• Download TIFF image (Kerr Black hole):
  • kerr-4536x2835-1000fs/frame_0.tiff
  • kerr-1428x844-100fs/frame_0.tiff

Note of ffmpeg

• frame_n.tiff $\to$ a.mp4

  $ ffmpeg -framerate 100 -i frame_%d.tiff -c:v libx264 -crf 18 -preset slow -tune stillimage -pix_fmt yuv420p output.mp4

  $ ffmpeg -framerate 100 -i frame_%d.tiff -vf "scale=4536:2836" -c:v libx264 -crf 18 -preset slow -pix_fmt yuv420p output.mp4