2020/11/23 by Andrew Farley, Farley, Andrew
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Bloch equations #Bloch wave #Classical mechanics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #General relativity #Geodesic #Geodesics in general relativity #Mathematical analysis #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum Mechanics and Non-Hermitian Physics #Quantum mechanics #gr-qc
paper · pdf · doi:10.48550/arxiv.2011.12714
23 pages, no figures. This version 2 includes corrections, clarifications and additional non-Hermitian Hamiltonian and underlying Lie group discussions
openalex publication_date 2020/11/23 · arxiv created 2021/04/19 · arxiv updated 2021/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
New equations are derived which describe the evolution in curved spacetime of null geodesics with non-zero (complex) shear σ and twist ω rates resembling Grishchuk's squeezed states evolution equations from inflationary cosmology. A ``squeeze" angle ϕ (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 ϕ as a polar angle and w as a radial distance, we obtain a mapping to points on the upper sheet, H2+ , of a two-sheet hyperboloid, establishing the connection between gravitational optics and hyperbolic geometry. Points on H2+ trace out paths evolving according to hyperbolic Bloch equations, similar to the optical Bloch equations, which can also be represented as a Schrödinger-like equation with a non-Hermitian Hamiltonian. A single vector equation on H2+ describes the precession of hyperbolic Bloch vectors about a rotation or birefringence vector on H2+ , analogous to the precession of Bloch vectors on the Bloch sphere or Stokes vectors on the Poincaré sphere. Tidal gravitational effects and a non-zero twist ω contribute to the precession of hyperbolic Bloch vectors.