Hamilton's equations of motion from the SU(1,1)/U(1) symmetry of Schwarzschild radial dynamics

TL;DR

The paper demonstrates that particles moving radially in Schwarzschild spacetime exhibit SU(1,1)/U(1) symmetry, enabling a unified time variable and Hamiltonian formulation akin to harmonic oscillators, aiding in quantization.

Contribution

It identifies a global time variable and Hamiltonian structure for radial Schwarzschild dynamics using SU(1,1)/U(1) symmetry, facilitating quantization.

Findings

Particles exhibit SU(1,1)/U(1) symmetry in Schwarzschild spacetime.

A shared global time variable is established for radial motion.

Hamilton's equations resemble harmonic oscillators, aiding quantization.

Abstract

Particles moving on a radial ray with respect to a Schwarzschild mass are shown to have SU(1,1)/U(1) dynamical symmetry. This symmetry is used to identify a global time variable shared by all test particles moving on a radial ray. With this time variable, Hamilton's equations for particles on a radial ray have the form of the harmonic oscillator. The SU(1,1)/U(1) symmetry, in conjunction with the well defined time variable and observer, assists in determining the quantization of the motion of the test particles.