Spinning massive test particles in cosmological and general static spherically symmetric spacetimes

TL;DR

This paper develops a Lagrangian formalism to analyze the motion of spinning massive particles in various cosmological and static spherically symmetric spacetimes, providing exact solutions and new constants of motion.

Contribution

It introduces a comprehensive method for solving spinning particle trajectories in diverse spacetimes, including a new constant of motion in Friedmann--Robertson--Walker spacetime.

Findings

Exact solutions for particle motion in different spacetimes.

Discovery of a new constant of motion in FRW spacetime.

Particle trajectories can have spacelike velocity segments.

Abstract

A Lagrangian formalism is used to study the motion of a spinning massive particle in Friedmann--Robertson--Walker and G\"odel spacetimes, as well as in a general Schwarzschild-like spacetime and in static spherically symmetric conformally flat spacetimes. Exact solutions for the motion of the particle and general exact expressions for the momenta and velocities are displayed for different cases. In particular, the solution for the motion in spherically symmetric metrics is presented in the equatorial plane. The exact solutions are found using constants of motion of the particle, namely its mass, its spin, its angular momentum, and a fourth constant, which is its energy when the metric is time independent, and a different constant otherwise. These constants are associated to Killing vectors. In the case of the motion on the Friedmann--Robertson--Walker metric, a new constant of motion is found. This is the fourth constant which generalizes previously known results obtained for spinless particles. In the case of general Schwarzschild-like spacetimes, our results allow for the exploration of the case of the Reissner-Nordstrom-(Anti)de Sitter metric. Finally, for the case of the conformally flat spacetimes, the solution is explicitly evaluated for different metric tensors associated to a universe filled with static perfect fluids and electromagnetic radiation. For some combination of the values of the constants of motion the particle trajectories may exhibit spacelike velocity vectors in portions of the trajectories.