2D Black Hole Light Ray Simulation

Python simulation visualizing the behavior of light rays around a black hole in 2D space using the Schwarzschild metric. The user can observe gravitational lensing effects in response to a strong gravitational field.

Features

• Simulates light ray trajectories in Schwarzschild geometry
• Visualizes multiple light rays with different initial positions
• Animated visualization showing the progressive movement of light rays
• Uses proper relativistic calculations for null geodesics
• Includes numerical stability improvements for accurate simulation

Requirements

• Python 3.x
• Pip for Python 3
• NumPy
• Matplotlib
• SciPy

Windows

You can use winget to easily install python and pip.

winget install Python.Python.3

Linux

Python backend tkinter may need to be installed on some systems.

Debian/Ubuntu based:

sudo apt install python3-tk python3-pip

Fedora:

sudo dnf install python3-tkinter python3-pip

If using system-wide python, make sure to use pip3 to install packages and python3 to create the virtual environment or run the script.

Theory

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 which defines the event horizon of a black hole, $M$ is the mass of the black hole, $G$ is Newton's gravitational constant, and $c$ is the speed of light. In the simulation, normalized units $G = c = 1$ 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:

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

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.

Critical Radius

The photon sphere occurs at $r = \frac{3}{2} r_s $, where light rays can orbit the black hole in unstable circular orbits. Light rays with impact parameters less than the critical value $b_{\text{crit}} = \frac{3\sqrt{3}}{2}r_s$ will be captured by the black hole.

Installation

1. Clone this repository:

git clone https://github.com/ZZBaron/2D-Black-Hole-Sim.git
cd 2D-Black-Hole-Sim

2. Create a python virtual environment (Optional)

Windows

# Replace `env` with your desired environment name
python -m venv env
env\Scripts\activate

Linux/macOS

python3 -m venv env
source env/bin/activate

3. Install the required packages:

pip install -r requirements.txt

Usage

Run the simulation by executing:

python "2D Black Hole Light Sim.py"

License

MIT License