The Hyperbolic Bloch Equations of General Relativity

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.