The Construction and Application of Penrose Diagrams, with a Focus on the Maximally Analytically Extended Schwarzschild Spacetime

TL;DR

This paper provides a rigorous construction and analysis of Penrose diagrams for the maximally extended Schwarzschild spacetime, comparing coordinate representations and serving as an educational resource for graduate students.

Contribution

It offers a detailed, mathematically rigorous method for constructing Penrose diagrams of Schwarzschild spacetime and compares different coordinate systems within this framework.

Findings

Constructed the Penrose diagram of the maximally extended Schwarzschild spacetime.

Compared Eddington-Finkelstein and Penrose coordinate representations visually.

Provided pedagogical explanations of Schwarzschild extensions and coordinate systems.

Abstract

We present a detailed, mathematically rigorous description of the construction procedure of Penrose diagrams for the example of the maximal analytic extension of the exterior Schwarzschild spacetime. To this end, we first outline the central idea underlying Penrose diagrams, state the general requirements on the spacetimes to be visualized, and give a definition of Penrose diagrams. We then construct the Penrose diagram of the maximally analytically extended Schwarzschild spacetime and discuss its components and characteristics. As an application, we work out the differences between the Eddington-Finkelstein and Penrose coordinate representations of the Schwarzschild spacetime by visually analyzing-and comparing-the Penrose diagram of the maximally analytically extended Schwarzschild spacetime equipped with, on the one hand, a foliation by the level sets of the Eddington-Finkelstein time coordinate and, on the other hand, a foliation by the level sets of the Penrose null coordinate. Through the whole of the paper, we provide explanatory accounts of the relevant parts of the seminal research papers on the exterior Schwarzschild solution pertaining to coordinatizations and possible extensions by Schwarzschild himself, Kruskal, Eddington, Finkelstein, and Penrose. This paper is primarily of pedagogical nature aimed at graduate students in physics and applied mathematics, serving mainly as an introduction to Penrose diagrams along with descriptions and analyses of coordinate representations and extensions of the exterior Schwarzschild spacetime.