Black Hole Light Ray Simulation

About

2D Simulation of light paths bending around a massive object according to Einstein's theory of general relativity, creating visual phenomena such as gravitational lensing and photon spheres.

Schwarzschild Metric

The Schwarzschild metric describes the geometry of spacetime around a non-rotating, chargeless, and spherically symmetric mass:

\[ ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2 + r^2d\Omega^2 \]

where \(r_s = \frac{2GM}{c^2}\) is the Schwarzschild radius, \(M\) is the mass of the black hole, \(G\) is the gravitational constant, and \(c\) is the speed of light.

Schwarzschild Radius

The Schwarzschild radius defines the event horizon of a black hole:

\[ r_s = \frac{2GM}{c^2} \]

In the simulation, normalized units in which \(G = c = 1\) are used, so the Schwarzschild radius becomes \(r_s = 2M\).

Geodesic Equations

Light rays follow null geodesics, which are paths where the spacetime interval is zero (\(ds^2 = 0\)). For motion in the equatorial plane (\(\theta = \pi/2\)), the geodesic equations reduce to

\[\frac{dr}{d\lambda} = f(r) \cdot p_r \] \[\frac{d\phi}{d\lambda} = \frac{L}{r^2} \] \[\frac{dp_r}{d\lambda} = \frac{L^2}{r^3}f(r) - \frac{r_s}{2r^2}f(r) \] \[\frac{dL}{d\lambda} = 0 \]

where \(f(r) = 1 - \frac{r_s}{r}\), \(p_r\) is the radial momentum component, and \(L\) is the angular momentum (conserved).

Effective Potential

The motion of light rays can be understood through an effective potential:

\[ V_{eff}(r) = \frac{L^2}{2r^2}\left(1 - \frac{r_s}{r}\right) \]

The closest approach of a light ray to the black hole is determined by the impact parameter \(b = \frac{L}{E}\), where \(E\) is the energy of the photon.