geodpy

This code aims to simulate geodesics given an arbitrary space-time metric.

Capabilities

• Calculate symbolic geodesics
• Solve the trajectory of a body given an arbitrary metric and symbolic geodesics.
• Calculate the norm of the velocity of a body.
• Plot the trajectories on matplotlib in both 2D and 3D.

Installation using pip

To install the package using pip, first create a virtual python environment in your working directory and activate the environment:

python -m venv venv
source venv/bin/activate

Then, simply install the package using:

pip install geodpy

Depending on your system, you might also need to install manually PyQt5 or PyQt6, which are needed by matplotlib to render the plots dynamically. Also note that by using this method of installation, you do not get access to the hardcoded examples and the documentation. Both of these are currently only available through the GitHub repository. The pip method of installation only grants you access to all the classes and functions of the library.

Installation using git

To install the package from git, you will need to download these dependencies:

• matplotlib
• sympy
• scipy
• numpy

Make sure these modules are correctly installed. More specifically, the package matplotlib might require the installation of PyQt5 or PyQt6 depending on your system. You can install these dependencies using:

pip install matplotlib numpy scipy sympy

Clone the repository and go to the main folder geodpy. You can use these commands:

git clone https://github.com/Sneaker679/geodpy
cd geodpy

In order to be able to import the classes and functions of this library, you will need to add the ./src folder to your PYTHON PATH. Move to this folder using:

cd src

Then, run the following command to add it to your PYTHON PATH:

export PYTHONPATH="$(pwd):$PYTHONPATH"

To add the folder permanently to your PYTHON PATH, run this command instead:

echo 'export PYTHONPATH="$HOME'${PWD/#$HOME/}':$PYTHONPATH"' >> $HOME/.bashrc

Or, for zsh terminals, run this:

echo 'export PYTHONPATH="$HOME'${PWD/#$HOME/}':$PYTHONPATH"' >> $HOME/.zshrc

Basic Usage -> Using the examples

Example with the Schwarzschild metric

In the ./examples/1_schwarzschild folder, run any of the python files except for schwarzschild.py (since it acts as a module for the other files). For instance, you could run:

python3 2_crashing_traj.py

which will run a configuration of the Schwarzschild metric where a body is falling into a Schwarzschild black hole. A plot should be displayed on the screen using matplotlib.

Structure of the examples

With the last example in mind, here is a more detailed overlook of how the examples are structured. This section will make it easier to understand how you can customize the orbits to your liking and experiment with the code.

In the examples folder are many examples that simulate geodesics with varied metrics, including the Schwarzschild, Kerr and Schwarzschild de Sitter metric. All these folders work in the same way. Taking the schwarzschild folder as an example, inside it are many files, the main one being schwarzschild.py. This script acts as a python module for the other files. Inside this code, there is a schwarzschild() function, which you can call with varied parameters to generate different geodesics. The other files in the schwarzschild folder are simple sripts who call the schwarzschild() function with different initial values. As it was shown in the previous section, the 2_crashing_traj.py script, for example, yields a geodesic simulated near the event horizon of a non rotating Schwarzschild blackhole. You can run the other examples in the other folders the same way.

The README inside each example folders specifies the meaning of the constants and quantities used.

Advanced Usage

For more advance usages, refer to ./docs/1_HowTo_trajectories.md, ./docs/2_HowTo_coordinates.md and ./docs/3_HowTo_plotters.md.

Interface

If you wish to see a more detailed overview of all the classes and functions at your disposition, be sure to checkout the ./docs/class_documentation/. There are many Markdown files there explaining how all the classes and functions work. The REAMDEs in this folder also specify how you can import them.

Numerical errors

Since this code uses a generalized approach to solving geodesics, it is prone to significant numerical errors, especially for metrics with complicated algebraic geodesics expressions. For instance, the 3_kerr example has a lot of numerical errors because the metric is not diagonal and its geodesics are very complex. Be mindful of this when you want accurate results.