Near-horizon properties of trajectories with finite force relevant for Bañados-Silk-West effect

According to the Bañados-Silk-West (BSW) effect, two particles moving toward a black hole can collide near the horizon with an unbounded energy in the center-of-mass frame. This requires one of the particles to have fine-tuned parameters in such a way that the time component of generalized momentum is zero <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>X</a:mi><a:mo>=</a:mo><a:mn>0</a:mn></a:math>. Thus the existence of such trajectories is a necessary condition for the BSW effect. However, it is insufficient since the forward-in-time condition requires <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>X</c:mi><c:mo>></c:mo><c:mn>0</c:mn></c:math> outside the horizon. We examine this condition for different types of particles and horizons and find configurations for which the BSW effect is possible. In doing so, we take into account a finite force of unspecified 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>X</e:mi></e:math>; in other cases they are required for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>X</g:mi></g:math> to be a 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 a combination of the Penrose process and the BSW effect is forbidden.