Relativistic Kinematics of Two-Parametric Riemann Surface in Genus Two

TL;DR

This paper models a genus two Riemann surface using hyperbolic octagons, computes the Weil--Petersson symplectic form, and introduces relativistic kinematics via Lorentz boosts and quantization, within a quantum gravity inspired framework.

Contribution

It introduces a novel approach to relativistic kinematics on a genus two Riemann surface using Fenchel--Nielsen variables and WP-area quantization, inspired by loop quantum gravity.

Findings

Computed WP symplectic form for genus two surface

Derived canonical action--angle variables for isoperimetric orbits

Quantized WP-area within a relativistic framework

Abstract

It is considered a model of compact Riemann surface in genus two, represented geometrically by two-parametric hyperbolic octagon with an order four automorphism and described algebraically by the corresponding Fuchsian group. Introducing the Fenchel--Nielsen variables, we compute the Weil--Petersson (WP) symplectic two-form for parameter space and analyze the closed isoperimetric orbits of octagons. WP-Area in parameter space and the canonical action--angle variables for the orbits are found. Exploiting the ideas from the loop quantum gravity, we generate relativistic kinematics by the Lorentz boost and quantize WP-area. We treat the evolution in terms of global variables within the "big bounce" concept.