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The Spherical-Rindler framework: From compact Minkowski regions to black-Hole and cosmological Solutions

Edgar Alejandro León

TL;DR

The paper develops a generalized Rindler framework in Minkowski space, including compact (cyclic) coordinates, and extends these ideas to generate Spherical-Rindler metrics that yield both black-hole and cosmological solutions. It connects these representations to established spacetime constructions by deriving near-horizon Schwarzschild coordinates and reproducing Kruskal–Szekeres through a Rindler-like mapping, while introducing a Spherical-Rindler black-hole type metric with detailed horizon and geodesic structure. In cosmology, the Milne universe is reinterpreted as a Rindler-type patch of Minkowski space, and a spherical-Rindler static cosmological solution with a horizon is analyzed and contrasted with De Sitter geometry. Collectively, the work clarifies how accelerated-frame coordinates encode global curvature and horizon physics, offering a geometrical lens for horizon thermodynamics and embedding properties with potential applications in exact solutions and horizon studies.

Abstract

In this article we first develop novel Rindler-type representations of flat spacetime by demonstrating that the standard hyperbolic transformation is a member of an infinite family of coordinate mappings. We specifically introduce cyclic coordinates, which, in contrast to the conventional Rindler wedge, delineate a compact region of Minkowski spacetime. By extending this framework, and motivated by near horizon coordinates in Schwarzschild metric, we propose a class of Spherical Rindler metrics. We demonstrate the utility of this approach by deriving and analyzing a black hole solution and a cosmological metric, both emerging naturally from a Spherical Rindler origin. Our results highlight unique geometric properties of these solutions, providing new insights into the relationship between accelerated frames and global spacetime curvature.

The Spherical-Rindler framework: From compact Minkowski regions to black-Hole and cosmological Solutions

TL;DR

The paper develops a generalized Rindler framework in Minkowski space, including compact (cyclic) coordinates, and extends these ideas to generate Spherical-Rindler metrics that yield both black-hole and cosmological solutions. It connects these representations to established spacetime constructions by deriving near-horizon Schwarzschild coordinates and reproducing Kruskal–Szekeres through a Rindler-like mapping, while introducing a Spherical-Rindler black-hole type metric with detailed horizon and geodesic structure. In cosmology, the Milne universe is reinterpreted as a Rindler-type patch of Minkowski space, and a spherical-Rindler static cosmological solution with a horizon is analyzed and contrasted with De Sitter geometry. Collectively, the work clarifies how accelerated-frame coordinates encode global curvature and horizon physics, offering a geometrical lens for horizon thermodynamics and embedding properties with potential applications in exact solutions and horizon studies.

Abstract

In this article we first develop novel Rindler-type representations of flat spacetime by demonstrating that the standard hyperbolic transformation is a member of an infinite family of coordinate mappings. We specifically introduce cyclic coordinates, which, in contrast to the conventional Rindler wedge, delineate a compact region of Minkowski spacetime. By extending this framework, and motivated by near horizon coordinates in Schwarzschild metric, we propose a class of Spherical Rindler metrics. We demonstrate the utility of this approach by deriving and analyzing a black hole solution and a cosmological metric, both emerging naturally from a Spherical Rindler origin. Our results highlight unique geometric properties of these solutions, providing new insights into the relationship between accelerated frames and global spacetime curvature.
Paper Structure (13 sections, 50 equations, 3 figures)

This paper contains 13 sections, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rindler in Minkowski Spacetime
  3. Transforming the Rindler metric
  4. A family of Rindler-type transformations
  5. Accelerated observers and conformal flatness
  6. Rindler in Schwarzschild and the Spherical-Rindler Black Hole
  7. From Rindler to Schwarzschild: near horizon coordinates
  8. From Schwarzschild to Kruskal-Szekeres
  9. Spherical-Rindler Black Hole type solution
  10. Rindler representations in cosmology
  11. Milne universe
  12. Spherically symmetric static cosmological solution
  13. Summary and final comments

Figures (3)

  • Figure 1: Pulsating coordinates described above. The light region is interior of the light-cone respect to the event $(t,x)=(0.0)$.
  • Figure 2: Region I is the Rindler wedge. The shaded region is inaccessible to the $\rho =const.$ observers, hidden by the horizon $t=x$.
  • Figure 3: Effective potentials for spacetime (3.15). The lower curve is the only one with $l^2<27r_h^2$.