Spinning gravimagnetic particles in Schwarzschild-like black holes

TL;DR

This paper investigates the dynamics of spinning particles with gravimagnetic moments in various Schwarzschild-like black hole spacetimes, deriving algebraic equations for their motion and providing numerical solutions.

Contribution

It introduces a Hamiltonian framework for gravimagnetic particles, simplifies the equations of motion to algebraic form, and analyzes solution conditions for different gravimagnetic moments.

Findings

Equation for P_0(r) decouples and is solvable as a 6th order polynomial.

Conditions for existence of solutions depend on gravimagnetic moment value.

Numerical algorithms for generating particle momentum solutions are developed.

Abstract

We study the motion of a spinning particle with gravimagnetic moment in Schwarzschild-like spacetimes with a metric $ds^2=-f(r) dt^2 + f^{-1}(r) dr^2 + r^2 d\Omega^2$, specifically we deal with Schwarzschild, Reissner-Nordstrom black holes as well as Ayon-Beato-Garcia and Bardeen regular spacetimes. First, we introduce the Hamiltonian system of equations which describes such kind of particles. In the case of null gravimagnetic moment, the equations are equivalent to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations. Working in the equatorial plane, using the constants of motion generated by the symmetries of the considered spacetimes and the Spin Supplementary Conditions (SSC), we change the problem of solving six differential equations for the momenta and the non-vanishing spin-tensor components to solving six algebraic equations. We show that the equation for the $P_0(r)$ component totally decouples, $P_0(r)$ can be found by solving a 6th order polynomial. We analyze the conditions for existence of solutions of this algebraic system for the relevant cases of gravimagnetic moment equal to unit, which corresponds to a gravimagnetic particle, and zero which corresponds to the MTPD system. A numerical algorithm to generate solutions of the momenta $P_{\mu}$ is provided and some solutions are generated.