Representations of the Schwarzschild Geometry: From Classical Coordinates to the Golden Minisuperspace
Year: 2026
Type: article
Abstract: We derive and present visual and geometric representations of the Schwarzschild metric. The analysis spans classical (tortoise, Kruskal) and compact pulsating (diamond) coordinates. Building upon a reduced radial minisuperspace formulation of Schwarzschild dynamics—wherein a Fibonacci matrix naturally yields golden-ratio eigenvalues—we introduce the golden representation and its complex spiral mapping. While the underlying reduced Lagrangian and radial minisuperspace metric are known in the literature, we demonstrate how this flat configuration space provides a useful visual tool for disentangling physical observables. Specifically, we show that the local gravitational redshift, the areal radius and the logarithmic excess in radial Schwarzschild coordinate travel time are mapped onto orthogonal coordinate projections and complex polar components within the radial minisuperspace geometry.
Source: Universe
Author: Edgar Alejandro León
Institution: Universidad Autónoma de Sinaloa
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,930
OpenAlex ID: W7222924190