Near-horizon properties of trajectories with finite force relevant for Ba\~{n}ados-Silk-West effect

TL;DR

This paper investigates the near-horizon behavior of particle trajectories with finite forces relevant to the BSW effect, identifying conditions under which unbounded collision energies are possible near black holes.

Contribution

It analyzes the influence of finite forces on particle trajectories near horizons, extending the understanding of conditions for the BSW effect beyond idealized free particles.

Findings

Certain configurations allow the BSW effect with finite forces.

Negative energy particles cannot produce the BSW effect in Kerr or Kerr-Newman black holes.

Finite forces impose additional constraints on particle trajectories for high-energy collisions.

Abstract

According to the Banados-SIlk-West (BSW) effect, two particles moving towards a black hole, can collide near the horizon with an unbounded energy in the center of mass frame. This requires one of particles to have fine-tuned parameters in such a way that the time component of generalized momentum is zero $X=0$. Thus the existence of such trjectories is a necessary condition for the BSW effect. However, it is insufficient since the forward-in-time condition requires $X>0$ outside the horizon. We examine this condition for different types of partricles and horizons and find configurations for which the BSW effect is possible. In doing so, we take into account a finite force of unspesified nature exerted on particles. It includes relationships between numbers characterizing the rate with which four-velocity, acceleration and metric functions change near the horizon. For some aforementioned relations, parameters of a system control the sign of $X$, in other cases they are required for $X$ to be real quantity. In the simplest case of free particles the BSW effect for the Kerr or Kerr-Newman black hole is impossible if a fine-tuned particle has a negative energy, so in this sense combination of the Penrose process and the BSW effect is forbidden.