Schwarzschild Spacetime without Coordinates

TL;DR

This paper presents a coordinate-free method to construct the full Schwarzschild spacetime, including Kruskal-Szekeres extension, using orthonormal Lorentz frames and Einstein equations, avoiding explicit metric forms.

Contribution

It introduces a novel coordinate-free approach to derive the Schwarzschild spacetime and its extension directly from the Einstein equations.

Findings

Successfully constructs the Kruskal-Szekeres spacetime without coordinates

Provides a new perspective on spacetime construction in general relativity

Simplifies the understanding of Schwarzschild spacetime extensions

Abstract

We discuss how to construct the full Schwarzschild (Kruskal-Szekeres) spacetime in one swoop by using the bundle of orthonormal Lorentz frames and the Einstein equation without the use of coordinates. We never have to write down the Kruskal-Szekeres or an equivalent form of the metric.