[Paper Review] Hyperbolic nature of uniformly rotating systems and their relation to gravity

B. H. Lavenda|ArXiv.org|Apr 10, 2008

Geomagnetism and Paleomagnetism Studies2 references3 citations

TL;DR

This paper proposes that uniformly rotating reference frames naturally exhibit hyperbolic geometry with constant negative curvature, linking special and general relativity to Lobachevskian (hyperbolic) space. By modeling spacetime using a hyperbolic metric with a space constant κ, the paper derives relativistic effects—such as Lorentz contraction and angular defects—geometrically, showing that the angle of parallelism explains relativistic distortions without invoking mass. The key result is that physical space may be fundamentally hyperbolic, not Euclidean or spherical, with implications for gravity and cosmology.

ABSTRACT

Special relativity corresponds to hyperbolic geometry at constant velocity while the so-called general relativity corresponds to hyperbolic geometry of uniformly accelerated systems. Generalized expressions for angular momentum, centrifugal and Coriolis forces are found in hyperbolic space, which reduce to the usual expressions of Euclidean space when the absolute constant tends to infinity. Gravity enters only in the specification of the absolute constant. A uniformly rotating disc corresponds exactly to hyperbolic geometry with a constant negative Gaussian curvature. The angle defect is related to Lorentz contraction of objects normal to the radial direction. Lobachevsky's angle of parallelism accounts for the apparent relativistic distortion of moving objects and would provide a testing ground to measure a positive defect by replacing large distances by high speeds that are comparable with that of light.

Motivation & Objective

• To establish a geometric foundation for special and general relativity based on hyperbolic (Lobachevskian) geometry rather than Riemannian curvature.
• To explain relativistic effects—such as time dilation, length contraction, and apparent object distortion—not as kinematic effects but as consequences of hyperbolic geometry.
• To challenge the standard interpretation of Einstein's field equations by showing that the vacuum condition R_ij = 0 does not uniquely determine spacetime geometry.
• To propose that the space constant κ, related to the speed of light, defines hyperbolic geometry and underlies the observed relativistic distortions.
• To demonstrate that the Schwarzschild interior solution leads to finite volume in a way inconsistent with an open, hyperbolic universe, suggesting a physical inconsistency in the standard model.

Proposed method

• Uses the hyperbolic metric with a space constant κ to model uniformly rotating systems, where the line element is derived from a modified Euclidean arc length using a refractive index-like factor.
• Applies Beltrami coordinates to express the hyperbolic differential of arc length, relating it to the logarithmic cross-ratio and Doppler shift via the angle of parallelism.
• Transforms the metric into polar coordinates to show that radial and angular components are affected by the ratio r/κ, with the radial component exhibiting hyperbolic behavior.
• Derives the geodesic equations from the hyperbolic metric and shows that the ratio of Euclidean speed to c corresponds to the cosine of the angle of parallelism.
• Compares the hyperbolic metric to the Schwarzschild solution, showing that the standard inner solution leads to a finite volume as r→κ, contradicting the infinite volume expected in hyperbolic space.
• Uses semi-geodesic coordinates (r, σ) to express the invariant hyperbolic line element, demonstrating that r = κ tanh⁻¹(𝐫/κ) maps Euclidean radius to hyperbolic distance.

Experimental results

Research questions

• RQ1 Can relativistic effects such as time dilation and length contraction be interpreted geometrically as angular defects in hyperbolic space?
• RQ2 Does the uniformly rotating disk naturally give rise to a hyperbolic geometry with constant negative curvature?
• RQ3 Is the angle of parallelism in hyperbolic geometry equivalent to the relativistic distortion of moving objects?
• RQ4 Does the standard vacuum condition R_ij = 0 in general relativity fail to uniquely determine spacetime geometry when applied to hyperbolic models?
• RQ5 Does the Schwarzschild interior solution contradict the expected infinite volume of a hyperbolic universe, suggesting a flaw in its physical interpretation?

Key findings

• The uniformly rotating disk corresponds exactly to hyperbolic geometry with constant negative Gaussian curvature, where the radius r is related to the Euclidean radius 𝐫 by r = κ tanh⁻¹(𝐫/κ).
• The circumference of a rotating disk exceeds 2π𝐫 due to Lorentz contraction in the tangential direction, leading to a hyperbolic geometry with larger circumferences than in Euclidean space.
• The angle of parallelism in Lobachevskian geometry accounts for the apparent relativistic distortion of moving objects, with the angular defect matching the relativistic Doppler shift.
• The hyperbolic metric leads to an infinite volume for a hyperbolic sphere as the radius r increases without bound, in contrast to the finite volume predicted by the Schwarzschild interior solution.
• The condition R_ij − ½Rg_ij = 0 vanishes identically in the hyperbolic model regardless of mass or metric parameters, implying that the vacuum Einstein equation does not uniquely determine geometry.
• The Schwarzschild inner solution gives a maximum volume of π²κ³ as 𝐫→κ, which contradicts the expected infinite volume of a truly open, hyperbolic universe, suggesting a physical inconsistency in the model.