A visual introduction to curved geometry for physicists

TL;DR

This paper offers a visual, intuitive introduction to differential geometry concepts relevant for physicists, focusing on Riemannian and Lorentzian manifolds, with applications to special relativity and spacetime diagrams.

Contribution

It introduces visual methods for understanding differential geometry, derives Thomas precession visually, and presents new techniques for creating Carter-Penrose diagrams and indicating distortions.

Findings

Visual derivation of Thomas precession

New method for generating Carter-Penrose diagrams

Technique for indicating distortion on spacetime diagrams

Abstract

This article provides a gentle, visual introduction to the basic concepts of differential geometry appropriate for students familiar with special relativity. Visual methods are used to explain basics of differential geometry and build intuition for all types of Riemannian and Lorentzian manifolds of constant curvature. A visual derivation of the Thomas precession is given, showcasing the utility of differential geometry while also pointing a spotlight at certain intricacies of Minkowski space crucial from a pedagogical perspective. In addition, a straightforward method to generate some Carter-Penrose diagrams -- suitable for students with no differential geometry knowledge -- is presented, and a new method of indicating distortion on spacetime diagrams is shown.