readme.md

Black Hole Raytracer

Warning

This is a legacy project which I wrote in 2015, before the first gravitational waves were detected! The coding standards are not that great. I wrote it to try to do something that was my own rather than following the Interstellar papers. It is stupidly slow and unoptimized, and doesn't exploit any of the symmetries of the problem, and uses a bad integration method, and takes 10 minutes per frame even on 2024 hardware.

2015 compilation instructions

To run it on linux, make sure cimg-dev and libx11-dev are installed ("sudo apt-get install cimg-dev libx11-dev"), make sure the obj directory exists, then type "make" and "./program".

My Approach

The Interstellar papers work with spherical symmetry and exploit Killing vectors and conserved quantities to have some elegant integration code. I had a bunch of rendering code laying around using xyz coordinates, and I didn't want to convert between $(x,y,z)$ and $(r,\theta,\varphi)$ coordinates all the time! Fortunately, the Schwarzschild metric is actually quite nice in xyz coordinates. Here are the first components of $g^{\mu\nu}$ in the (- + + +) convention, where $r=\sqrt{x^2+y^2+z^2}$ and I omitted the z column because you can carry on the pattern yourself. Don't take my word for it, this is derived using computer algebra in the Mathematica notebook blackhole-math.nb which you can view in pdf format blackhole-math.pdf.

$$g^{\mu\nu}=\begin{bmatrix} \frac{1}{-1+1/r} & 0 & 0 \\ 0 & 1-\frac{x^2}{r^3} & -\frac{xy}{r^3}\\ 0 & -\frac{xy}{r^3} & 1-\frac{y^2}{r^3} \end{bmatrix}$$

Using this we can integrate ray trajectories directly in "Cartesian" coordinates. For example:

The formulas for $t''(\tau)$, $x''(\tau)$, $y''(\tau)$, and $z''(\tau)$ are as follows, where I note that $\tau$ is not the proper time but parameterizes the lightlike curve $(t,x,y,z)$. Also: the formulas are presented here as a joke, because they are awful. Feel free to laugh along with me.

double ddt = -((dt*(dx*x + dy*y + dz*z))/((Power(x,2) + Power(y,2) + Power(z,2))*(-1 + Sqrt(Power(x,2) + Power(y,2) + Power(z,2)))));

double ddx = (x*(Power(dx,2)*Power(x,2)*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + 2*dx*dz*x*z*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + Power(dz,2)*Power(z,2)*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + 2*dx*dz*x*z*(-2 + 3*Power(x,2) + 3*Power(y,2) + 3*Power(z,2)) - Power(dz,2)*(2*(-1 + Power(x,2) + Power(y,2))*(Power(x,2) + Power(y,2)) + (Power(x,2) + Power(y,2))*Power(z,2) - Power(z,4)) + Power(dt,2)*(-1 + Power(x,2) + Power(y,2) + Power(z,2))*(-Power(x,2) - Power(y,2) - Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))) + 2*dy*y*(dx*x + dz*z)*(-2 + 3*Power(x,2) + 3*Power(y,2) + 3*Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))) + Power(dx,2)*(2*(Power(y,2) + Power(z,2)) + (Power(x,2) + Power(y,2) + Power(z,2))*(Power(x,2) - 2*(Power(y,2) + Power(z,2)))) + Power(dy,2)*(-2*Power(x,4) + Power(y,4) + 2*Power(z,2) - 2*Power(z,4) - Power(x,2)*(-2 + Power(y,2) + 4*Power(z,2)) + Power(y,2)*(-Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))))))/(2.*(-1 + Power(x,2) + Power(y,2) + Power(z,2))*Power(Power(x,2) + Power(y,2) + Power(z,2),2.5));

double ddy = (y*(Power(dx,2)*Power(x,2)*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + 2*dx*dz*x*z*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + Power(dz,2)*Power(z,2)*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + 2*dx*dz*x*z*(-2 + 3*Power(x,2) + 3*Power(y,2) + 3*Power(z,2)) - Power(dz,2)*(2*(-1 + Power(x,2) + Power(y,2))*(Power(x,2) + Power(y,2)) + (Power(x,2) + Power(y,2))*Power(z,2) - Power(z,4)) + Power(dt,2)*(-1 + Power(x,2) + Power(y,2) + Power(z,2))*(-Power(x,2) - Power(y,2) - Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))) + 2*dy*y*(dx*x + dz*z)*(-2 + 3*Power(x,2) + 3*Power(y,2) + 3*Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))) + Power(dx,2)*(2*(Power(y,2) + Power(z,2)) + (Power(x,2) + Power(y,2) + Power(z,2))*(Power(x,2) - 2*(Power(y,2) + Power(z,2)))) + Power(dy,2)*(-2*Power(x,4) + Power(y,4) + 2*Power(z,2) - 2*Power(z,4) - Power(x,2)*(-2 + Power(y,2) + 4*Power(z,2)) + Power(y,2)*(-Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))))))/(2.*(-1 + Power(x,2) + Power(y,2) + Power(z,2))*Power(Power(x,2) + Power(y,2) + Power(z,2),2.5));

double ddz = (z*(Power(dx,2)*Power(x,2)*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + 2*dx*dz*x*z*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + Power(dz,2)*Power(z,2)*Sqrt(Power(x,2) + Power(y,2) + Power(z,2)) + 2*dx*dz*x*z*(-2 + 3*Power(x,2) + 3*Power(y,2) + 3*Power(z,2)) - Power(dz,2)*(2*(-1 + Power(x,2) + Power(y,2))*(Power(x,2) + Power(y,2)) + (Power(x,2) + Power(y,2))*Power(z,2) - Power(z,4)) + Power(dt,2)*(-1 + Power(x,2) + Power(y,2) + Power(z,2))*(-Power(x,2) - Power(y,2) - Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))) + 2*dy*y*(dx*x + dz*z)*(-2 + 3*Power(x,2) + 3*Power(y,2) + 3*Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))) + Power(dx,2)*(2*(Power(y,2) + Power(z,2)) + (Power(x,2) + Power(y,2) + Power(z,2))*(Power(x,2) - 2*(Power(y,2) + Power(z,2)))) + Power(dy,2)*(-2*Power(x,4) + Power(y,4) + 2*Power(z,2) - 2*Power(z,4) - Power(x,2)*(-2 + Power(y,2) + 4*Power(z,2)) + Power(y,2)*(-Power(z,2) + Sqrt(Power(x,2) + Power(y,2) + Power(z,2))))))/(2.*(-1 + Power(x,2) + Power(y,2) + Power(z,2))*Power(Power(x,2) + Power(y,2) + Power(z,2),2.5));