A Lean and Mean Introduction to Modern General Relativity

TL;DR

This paper provides a concise, modern introduction to general relativity using differential geometry, covering advanced topics with a focus on physical intuition suitable for upper-year undergraduates.

Contribution

It offers a streamlined, physically motivated presentation of general relativity, including topics not typically covered in standard courses, from a modern geometric perspective.

Findings

Introduces affine spaces and tensor comparison in an educational context

Provides a novel approach to Schwarzschild and Kruskal-Szekeres coordinates

Includes detailed treatment of boundary conditions and manifold integration

Abstract

Notes prepared for the introductory general relativity course PHYSICS 748 at The University of Auckland. They are designed to introduce general relativity to upper-year undergraduate students directly using the modern language of differential geometry but in a physically motivated way, and throughout keeping a logical flow from section to section and chapter to chapter. In doing so, they necessarily cover a number of topics either not normally treated in an introductory course, or from a novel perspective. These include for example: affine spaces, comparing and contrasting rank-2 tensors with matrices, integration on manifolds, the Rindler metric, including a proper near-source boundary condition for the Schwarzschild metric, approaching Kruskal-Szekeres coordinates from a Rindler perspective, and more.