The Hyperbolic Bloch Equations of General Relativity

TL;DR

This paper introduces hyperbolic Bloch equations derived from general relativity to describe the evolution of null geodesics with shear and twist, linking gravitational optics to hyperbolic geometry and quantum-like equations.

Contribution

It develops a novel set of equations connecting gravitational optics with hyperbolic geometry and Schrödinger-like dynamics, extending the Bloch equations to curved spacetime contexts.

Findings

Establishes a mapping between gravitational optics variables and hyperbolic geometry.

Derives hyperbolic Bloch equations analogous to optical Bloch equations.

Shows gravitational effects induce precession of hyperbolic Bloch vectors.

Abstract

New equations are derived which describe the evolution in curved spacetime of null geodesics with non-zero (complex) shear $\sigma$ and twist $\omega$ rates resembling Grishchuk's squeezed states evolution equations from inflationary cosmology. A ``squeeze" angle $\phi$ (obtained from the direction of the major axis of the elliptical cross section of the congruence and the direction of the shear rate), an ellipse axis ratio parameter $w$ and a rotation angle $v$ are the primary variables. Interpreting $\phi$ as a polar angle and $w$ as a radial distance, we obtain a mapping to points on the upper sheet, $H_{2}^{+}\,,$ of a two-sheet hyperboloid, establishing the connection between gravitational optics and hyperbolic geometry. Points on $H_{2}^{+}$ trace out paths evolving according to hyperbolic Bloch equations, similar to the optical Bloch equations, which can also be represented as a Schr\"{o}dinger-like equation with a non-Hermitian Hamiltonian. A single vector equation on $H_{2}^{+}$ describes the precession of hyperbolic Bloch vectors about a rotation or birefringence vector on $H_{2}^{+}\,,$ analogous to the precession of Bloch vectors on the Bloch sphere or Stokes vectors on the Poincar\'{e} sphere. Tidal gravitational effects and a non-zero twist $\omega$ contribute to the precession of hyperbolic Bloch vectors.