Algorithms for the explicit computation of Penrose diagrams

TL;DR

This paper presents an algorithm for explicitly computing Penrose diagrams for a class of spacetimes, including dynamic cases, ensuring continuous extension across horizons, with practical examples and comparisons to existing methods.

Contribution

The paper introduces a new algorithm for explicit Penrose diagram computation that handles dynamic spacetimes and horizon crossings, extending previous methods.

Findings

Algorithm successfully computes diagrams for static and dynamic spacetimes.

Diagrams extend the metric continuously across horizons.

Implementation demonstrates the method's effectiveness with various examples.

Abstract

An algorithm is given for explicitly computing Penrose diagrams for spacetimes of the form $ds^2 = -f(r)\, dt^2 + f(r)^{-1} \, dr^2 + r^2 \, d\Omega^2$. The resulting diagram coordinates are shown to extend the metric continuously and nondegenerately across an arbitrary number of horizons. The method is extended to include piecewise approximations to dynamically evolving spacetimes using a standard hypersurface junction procedure. Examples generated by an implementation of the algorithm are shown for standard and new cases. In the appendix, this algorithm is compared to existing methods.