Inertial Forces as Viewed from the ADM Slicing and their behaviour for Particles in Non-Circular Geodesics

TL;DR

This paper investigates inertial forces on particles in non-circular geodesics within static and stationary spacetimes, revealing that centrifugal force reversal occurs only for prograde orbits around black holes when azimuthal velocity exceeds radial velocity.

Contribution

It reformulates covariant acceleration expressions using ADM 3+1 splitting and analyzes conditions for centrifugal force reversal in various particle motions around black holes.

Findings

Centrifugal force reversal occurs only in prograde orbits.

Reversal depends on azimuthal velocity surpassing radial velocity.

Reversal is influenced by parameters E, l, and Kerr parameter a.

Abstract

Considering the definition of inertial forces acting on a test particle, following non-circular geodesics, in static and stationary space times we show that the centrifugal force reversal occurs only in the case of particles following prograde orbits around black holes. We first rewrite the covariant expressions for the acceleration components in terms of the lapse function, shift vector and the 3-metric $\gm_{ij}$, using the ADM 3+1 splitting and use these, for different cases as given by pure radial motion, pure azimuthal motion and the general non-circular motion. It is found that the reversal occurs only when the azimuthal angular velocity of the particle supersedes the radial velocity, which indeed depends upon the physical parameters $E$, $l$ and the Kerr parameter $a$.