Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Christian Röken

TL;DR

This work delivers a rigorous, pedagogical construction of Penrose diagrams for the maximally analytically extended Schwarzschild spacetime, grounding the procedure in a precise conformal compactification of the Kruskal–Szekeres representation. It articulates the central idea of reducing the 4D spacetime to a 2D conformal submanifold, then compactifying to a finite diagram that preserves causal structure, and provides an explicit coordinate chain from Schwarzschild to EF, KS1, KS2, and Penrose coordinates. The paper then delineates the diagram’s components (infinity, horizons, singularity, and the four regions) and demonstrates how EF and Penrose foliations reflect different global structures, including a clear visual contrast between spacelike EF slices and null Penrose slices. By combining historical context with a detailed construction and comparative analysis, the work offers a valuable, accessible framework for graduate students to visualize and analyze the causal geometry of black-hole spacetimes through Penrose diagrams.

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.

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

TL;DR

This work delivers a rigorous, pedagogical construction of Penrose diagrams for the maximally analytically extended Schwarzschild spacetime, grounding the procedure in a precise conformal compactification of the Kruskal–Szekeres representation. It articulates the central idea of reducing the 4D spacetime to a 2D conformal submanifold, then compactifying to a finite diagram that preserves causal structure, and provides an explicit coordinate chain from Schwarzschild to EF, KS1, KS2, and Penrose coordinates. The paper then delineates the diagram’s components (infinity, horizons, singularity, and the four regions) and demonstrates how EF and Penrose foliations reflect different global structures, including a clear visual contrast between spacelike EF slices and null Penrose slices. By combining historical context with a detailed construction and comparative analysis, the work offers a valuable, accessible framework for graduate students to visualize and analyze the causal geometry of black-hole spacetimes through Penrose diagrams.

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.

Paper Structure

This paper contains 14 sections, 63 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Historical Remarks on the Exterior Schwarzschild Solution
  4. Modern Account of the Exterior Schwarzschild Solution and Kruskal's Maximal Analytic Extension
  5. Central Idea Behind Penrose Diagrams
  6. Construction Procedure of the Penrose Diagram of the Maximally Analytically Extended Schwarzschild Spacetime
  7. Coordinate Representation for the Penrose Diagram
  8. Components of the Penrose Diagram
  9. Application to the Eddington--Finkelstein and Penrose Foliations of the Schwarzschild Spacetime
  10. Historical Remarks on the Eddington--Finkelstein and Penrose Coordinate Representations
  11. Eddington Coordinates
  12. Finkelstein Coordinates
  13. Penrose Coordinates
  14. Comparison of the Eddington--Finkelstein and Penrose Foliations

Figures (2)

  • Figure 1: \ref{['PenroseKruskal']} Penrose diagram of the maximally analytically extended Schwarzschild spacetime with labels of the exterior and interior boundary components and the different interior regions. \ref{['PenroseKruskalCoords']} The same diagram showing the orientations of the coordinate systems paramount for its construction (scalings are suppressed for simplicity) and depicting particular null hypersurfaces with either $u = \textnormal{const.}$ or $v = \textnormal{const.}$ indicated by the dashed diagonal lines.
  • Figure 2: \ref{['PenroseKruskalEF']} Penrose diagram of the maximally analytically extended Schwarzschild spacetime with level sets of the Eddington--Finkelstein time coordinate $v_{\textnormal{EF}}$ (blue curves) and level sets of the Eddington--Finkelstein radial coordinate $r$ (aquamarine curves). \ref{['PenroseKruskalPen']}--\ref{['PenroseKruskalKS']} The same diagram for the Penrose representation, the Schwarzschild representation extended to the region $\textnormal{B}_{\textnormal{I}} \cup \textnormal{B}_{\textnormal{II}}$, and the Kruskal representation, respectively.

Theorems & Definitions (18)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Remark 3.2
  • Definition 3.3
  • Remark 4.1
  • Remark 4.2
  • ...and 8 more