Differential Geometry with Python

A Python library for differential geometry, mainly based on SymPy, includes tensor analysis on manifolds, designed for computing fundamental geometric objects such as intrinsic and extrinsic curvatures, geodesics, covariant derivatives, divergences, gradients, Laplacians, and for verifying solutions to Einstein's equations.

Overview

In this repository you can find my Manifold and Submanifold class to handle computations within the framework of Differential Geometry. Throughout my academic education, despite attending several classes on General Relativity and Differential Geometry, I have never had the opportunity to directly compute Einstein's equations, which always felt quite strange to me. For this reason, I decided to develop my own library (based on SymPy) to handle tensorial calculations on manifolds. Additionally, you can find a small notebook where I explicitly verify, step by step, that constant curvature geometries satisfy Einstein's vacuum equations, with a particular focus on the hyperbolic solution. Moreover, a brief insight into the topic of geodesics has also been provided in the Geodesics notebook, where they are first computed symbolically, in the fashion of this library, and then solved numerically using NumPy methods.

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Briefs geo-diff notes

Geodesics of a manifold $(M^n,g)$ are curves of the form $\tau\mapsto\left(x^1(\tau),...,x^n(\tau)\right),$ which are solutions of the following differential equation:

$\sum_{\nu,\lambda=1}^n\frac{d^2x^\mu}{d\tau^2}+\Gamma^\mu_{\nu\lambda}\frac{d x^\nu}{d\tau}\frac{d x^\lambda}{d\tau}=0,$ per ogni $\mu=1,...,n$

Where:

• $x^\mu$ are local coordinates on the manifold.
• $\Gamma^\mu_{\nu\lambda}$ are the Christoffel symbols of the Levi-Civita connection on the manifold.
• $\tau$ is the affine parameter (arc length) along a curve on the manifold.

Geodesics are intrinsic in a certain sense, meaning that they depend on the metric $g\in T^*M\odot T^*M$, which is independent of the coordinates in which it is expressed, given by $g=g_{\mu\nu}dx^\mu\odot dx^\nu$, in Einstein notation.

In general, the components of a metric $g$ on $M^n$ are computed via pullback from an ambient space, usually Euclidean. For instance, if there exists an embedding $F:M^n\hookrightarrow\mathbb{R}^m$, with $\mathbf{\delta}=\delta_{ab}dx^a\odot dx^b$ being the standard Euclidean metric, then $g="J_F^T\cdot J_F"=\delta_{ab}\frac{\partial F^a}{\partial x^\mu}\frac{\partial F^b}{\partial x^\nu} dx^\mu\odot dx^\nu$. In other words, $g$ is the pullback of $\delta$ through the parameterization $F$.

The Christoffel symbols are computed using the formula $\Gamma^\rho_{\mu\nu}=\frac{1}{2}\sum_{\lambda}g^{\rho\lambda}\left(\partial_\nu g_{\mu\lambda}+\partial_\mu g_{\nu\lambda}-\partial_\lambda g_{\mu\nu}\right)$, from which the Riemann curvature tensor can be determined as $\mathbf{Riem}=\sum_{\rho,\sigma,\mu,\nu}R^\rho_{\sigma\mu\nu}\partial_\rho\otimes dx^\sigma\otimes dx^\mu\otimes dx^\nu$ where its components are given by $R^\rho_{\sigma\mu\nu}=\sum_\lambda \partial_\mu\Gamma^\rho_{\nu\sigma}-\partial_\nu\Gamma^\rho_{\mu\sigma}+\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}+\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma}$. This is the most general object describing the curvature of a metric $g$, and it possesses various nontrivial traces, including the Ricci tensor $\mathbf{Ric}=\sum_{\rho,\mu,\nu}\overbrace{R^\rho_{\mu\rho\nu}}^{=:R_{\mu\nu}} dx^\mu\odot dx^\nu$ and the Ricci scalar (or scalar curvature) $\mathbf{R}=\sum_{\mu,\nu}g^{\mu\nu}R_{\mu\nu}$. In the case $n=2$, the Riemann tensor $\mathbf{Riem}$ is entirely determined by the scalar curvature $\mathbf{R}$, while for $n=3$ it is determined by the Ricci tensor $\mathbf{Ric}$. There are also other nontrivial traces of the Riemann tensor that are significant in higher dimensions, one of the most fundamental being the Kretschmann invariant $\mathbf{K}=\sum_{\mu,\nu,\rho,\sigma}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$.