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

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.

💡 Research Summary

The paper introduces a broad generalisation of the classic Rindler coordinate transformation, showing that the familiar hyperbolic map is just one member of an infinite family of possible mappings. Starting from the two‑dimensional metric
 ds² = –ρ² dT² + dρ²,
the author derives the most general transformation to the flat Minkowski line element by imposing the tensor transformation law (2.2)–(2.4). Introducing two constants α and β, the analysis splits into three qualitatively different regimes depending on the sign of the product αβ.

• αβ > 0 (standard Rindler‑type) – By defining η = √(αβ) the author obtains a one‑parameter family of hyperbolic transformations. The textbook case η = 1 reproduces the usual Rindler wedge, while any η > 0 yields a conformally flat metric with the same causal structure.

• αβ = 0 (tortoise‑like) – Choosing α = 1, β = 0 leads to the simple logarithmic spatial coordinate x = ln ρ and a time coordinate t = T. This is precisely the tortoise coordinate that appears in the near‑horizon analysis of the Schwarzschild solution.

• αβ < 0 (cyclic or “compact” coordinates) – The novel sector is obtained by setting α = 1, β = –1. Solving the resulting differential equations yields
   t = A sin T cos (ln ρ), x = A cos T sin (ln ρ).
  For fixed ρ the trajectory in the (t,x) plane is an ellipse; the allowed range 1 < ρ < e^{π/2} covers only a compact rhombus‑shaped region of Minkowski space (Fig. 1). The author calls the coordinate ρ “cyclic” because it repeats after a shift of π in ln ρ. The associated observers have constant proper acceleration a = 1/ρ, reproducing the familiar Rindler acceleration law while remaining confined to a bounded patch of spacetime.

Having established the 1+1‑dimensional family, the paper lifts the construction to spherically symmetric 1+3 dimensions. Section 3 shows how the near‑horizon limit of the Schwarzschild metric can be written in the form
 ds² ≈ –ρ² dT² + dρ² + r² dΩ²,
with r ≈ 2GM. By treating this expression as an exact global metric rather than an approximation, the author defines a “Spherical‑Rindler black‑hole” solution. This solution differs from the standard Schwarzschild geometry in that the radial coordinate ρ is bounded, so the spacetime consists of a compact region surrounding the horizon rather than an infinite exterior. The paper computes the effective potential for test particles, the location of the innermost stable circular orbit (ISCO), and demonstrates that the geometry admits a flat cylindrical embedding analogous to Flamm’s paraboloid.

Section 4 explores two cosmological applications. First, the Milne universe is recovered as a Rindler‑type patch of Minkowski space, illustrating that the same cyclic construction can describe an expanding cosmology. Second, a Spherical‑Rindler metric with a non‑zero cosmological constant is compared to de Sitter space; the resulting solution retains the cyclic radial coordinate while exhibiting accelerated expansion, suggesting a new class of cosmological models where acceleration and compact radial topology coexist.

The author concludes that the cyclic (αβ < 0) sector provides a previously unnoticed way to chart a finite, conformally flat region of flat spacetime, and that extending this idea to curved backgrounds yields novel black‑hole and cosmological solutions. The mathematical derivations are thorough, the physical interpretation of constant‑ρ observers is clear, and the connection to familiar near‑horizon coordinates is well‑motivated.

Strengths:

1. Systematic derivation of the full family of Rindler‑type transformations, with clear classification by the sign of αβ.
2. Introduction of compact “cyclic” coordinates that have not been explored in the literature, together with a detailed geometric description (ellipses, rhombus region).
3. Application of the formalism to both black‑hole and cosmological settings, providing explicit expressions for ISCO, effective potentials, and embedding diagrams.

Weaknesses / Open Questions:

• The physical relevance of the compact Spherical‑Rindler black‑hole remains speculative; observable signatures (e.g., lensing, quasinormal modes) are not addressed.
• The bounded radial coordinate may lead to non‑trivial matching conditions at the edge of the compact patch; the paper does not discuss how to glue the interior to an exterior asymptotically flat region.
• Stability analysis of the de Sitter‑like solution with a cyclic radius is absent; linear perturbation theory would be needed to assess viability.
• The treatment of the full 1+3‑dimensional metric assumes spherical symmetry but does not explore possible extensions to rotating (Kerr‑type) backgrounds.

Overall, the work opens an interesting avenue by revealing that the familiar Rindler transformation is just one point in a much richer landscape of coordinate maps. The cyclic sector, in particular, may find applications in holographic setups, compactified models, or as a pedagogical tool for illustrating how accelerated frames can be confined to finite regions of flat spacetime. Future work should focus on physical implications, matching to realistic spacetimes, and stability under perturbations.