bjob/gros

General Relativity Orbit Simulator

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Status

Overview

gros is a python package to numerically calculate and simulate particle trajectories based on the field equations of general relativity. A user needs to define a certain metric by providing the mass of a central gravitational attractor and the start coordinates and velocity of the test particle.

Installation

Clone the repository from https://github.com/BjoB/gros. After this you can install the package via pip in your choosen environment:

pip install .

Theoretical background

To simulate particle trajectories around a spherically symmetric body, we use the Schwarzschild solution of Einstein's field equations, describing the exterior spacetime for our use case. Starting from the field equations

$$\large R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi%20G}{c^4}T_{\mu\nu}$$

with a vanishing energy momentum tensor

$$\large T_{\mu\nu}=0$$

the Schwarzschild metric can be derived as

$$\large ds^2=g_{\mu\nu}dx^\mu dx^\nu=c^2 (1-\frac{r_s}{r})dt^2-\frac{1}{1-r_s/r}dr^2-r^2d\theta^2-r^2sin^2\theta d\phi^2$$

with the Schwarzschild radius $r_s=2GM/c^2$ and the space time coordinates based on spherical coordinates $(x^0,x^1,x^2,x^3) \mapsto (ct, r,\theta, \phi)$.

The intrinsic space time curvature can be derived from the metric by evaluating the Christoffel symbols given with

$$\large \Gamma_{\alpha\nu}^{\beta}=\frac{1}{2}g^{\mu\beta}(\partial_{\alpha}g_{\mu\nu}+\partial_{\nu}g_{\mu\alpha}-\partial_\mu g_{\alpha\nu})$$

After calculating these coefficients and using the proper time as parameter, the motion of of a particle in the gravitational field can be retrieved by solving the system of differential equations given with the geodesic equations

$$\large \frac{d^2 x^\mu}{d\tau^2}+\Gamma_{\alpha\beta}^{\mu}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=0$$

Examples

Some simple simulations can be found in the examples directory.

Particle on Mercury's orbit

This example simulates a particle orbiting the sun with some initial orbital parameters taken from https://nssdc.gsfc.nasa.gov/planetary/factsheet/. After calculating the trajectory, an animation will be generated, which can be used to track the particle with a previously choosen step size.

Earth as a black hole

What if earth was a black hole? The according example shows how a particle would act in short distance of 30m. Especially the perihelion precession is visualized as a direct effect of general relativity. Additionally the gravitational time dilation can be tracked along the animation frames with τ as the proper time of the particle. t is the calculated coordinate time, which can be seen as the measured proper time of a hypothetical observer positioned infinitely far away from the gravitational center.