# Symbolic and Numerical Analysis in General Relativity with Open Source Computer Algebra Systems

## TL;DR

This paper compares open source computer algebra systems for tensor calculations in general relativity, demonstrating their capabilities through solving and visualizing the Klein-Gordon equation and geodesic motion.

## Contribution

It provides a benchmark and detailed analysis of SageMath, Maxima, and Python-based GraviPy for tensor and algebraic computations in general relativity, highlighting their practical applications.

## Key findings

- SageMath and SageManifolds effectively derive and visualize solutions.
- Numerical Klein-Gordon solutions agree with analytical asymptotics.
- Benchmark results guide future choice of tools for relativity calculations.

## Abstract

We study three computer algebra systems, namely SageMath (with SageManifolds package), Maxima (with ctensor package) and Python language (with GraviPy module), which allow tensor manipulation for general relativity calculations along with general algebraic calculations. We present a benchmark of these systems using simple examples. After the general analysis, we focus on the SageMath and SageManifolds system to derive, analyze and visualize the solutions of the massless Klein-Gordon equation and geodesic motion with Hamilton-Jacobi formalism. We compare our numerical result of the Klein-Gordon equation with the asymptotic form of the analytical solution to see that they agree.