Acceleration of particles by black holes as a result of deceleration: ultimate manifestation of kinematic nature of BSW effect

The recently discovered so-called BSW effect consists in the unbound growth of the energy E_{c.m.} in the centre of mass frame of two colliding particles near the black hole horizon. We consider a new type of the corresponding scenario when one of two particles (“critical”) remains at rest near the horizon of the charged near-extremal black hole due to balance between the attractive and repulsion forces. The other one hits it with a speed close to that of light. This scenario shows in a most pronounced way the kinematic nature of the BSW effect. In the extremal limit, one would gain formally infinite E_{c.m.} but this does not happen since it would have require the critical massive particle to remain at rest on the null horizon surface that is impossible. We also discuss the BSW effect in the metric of the extremal Reissner-Nordstr"om black hole when the critical particle remains at rest near the horizon.

💡 Research Summary

The paper revisits the Banãdos‑Silk‑West (BSW) effect, which predicts that the centre‑of‑mass energy (E_cm) of two particles colliding near a black‑hole horizon can become arbitrarily large. While the original formulation considered two freely falling particles (or particles on fine‑tuned geodesics) around a rotating or charged black hole, the authors introduce a qualitatively different configuration that they call a “deceleration‑acceleration” scenario. In this setup one particle, termed the “critical” particle, is held almost at rest just outside the horizon of a near‑extremal Reissner‑Nordström (RN) black hole. The particle’s static equilibrium is achieved by a precise balance between the attractive gravitational force and the repulsive electrostatic force, which is possible only for a specific charge‑to‑mass ratio q/m. The second particle is a usual massive test particle that falls freely from infinity and, as it approaches the horizon, attains a velocity arbitrarily close to the speed of light with respect to a locally static observer.

The key point is that the relative Lorentz factor γ between the two particles is given by γ = –u₁·u₂, where u₁ is the four‑velocity of the critical particle and u₂ that of the infalling particle. Because u₁ is almost null (its spatial components are negligible while its temporal component is tuned to the horizon’s red‑shift), γ grows like 1/√(1–v²) where v is the infalling particle’s local speed. Consequently the standard expression for the centre‑of‑mass energy,

E_cm² = m₁² + m₂² + 2γ m₁ m₂,

predicts an unbounded increase of E_cm as the collision point approaches the horizon. This reproduces the familiar BSW divergence, but the authors emphasize that the divergence is purely kinematic: it stems from the extreme relative velocity rather than any exotic property of the spacetime itself.

However, the authors also point out a crucial physical limitation. In the exact extremal limit (Q → M) the horizon becomes a null surface. A massive particle cannot remain at rest on a null surface; doing so would require an infinite charge‑to‑mass ratio, which is not realizable for any known particle. Therefore, although the mathematical expression suggests E_cm → ∞, the physical configuration that would be needed – a massive critical particle exactly on the horizon – cannot exist. The same conclusion holds for the extremal RN metric when the critical particle is placed at the “static equilibrium radius” that coincides with the horizon.

The paper proceeds to analyse the extremal RN case explicitly. The authors demonstrate that the balance point where gravitational attraction equals electrostatic repulsion lies infinitesimally outside the horizon for a near‑extremal black hole, and merges with the horizon in the extremal limit. They compute the four‑velocity of the critical particle and confirm that its norm approaches zero as the horizon is approached, reinforcing the null‑like character of the equilibrium. Yet, because the particle’s rest mass remains finite, the equilibrium cannot be maintained exactly on the horizon, and the collision energy remains large but finite.

By constructing this “deceleration‑acceleration” picture, the authors make the kinematic nature of the BSW effect especially transparent: the essential ingredient is a huge relative boost between the colliding particles, which can be engineered either by having both particles move at relativistic speeds (the original BSW set‑up) or by holding one particle essentially still while the other approaches the horizon at near‑light speed. The latter configuration highlights that the black‑hole’s rotation or charge is not the source of the energy amplification; rather, it is the spacetime geometry that permits a static equilibrium point arbitrarily close to a null surface, thereby enabling an arbitrarily large Lorentz boost.

In the concluding discussion the authors stress that, despite the theoretical possibility of arbitrarily high E_cm, realistic astrophysical constraints (finite charge‑to‑mass ratios, back‑reaction, radiation losses, and the impossibility of a massive particle residing on a null surface) impose an upper bound on the achievable centre‑of‑mass energy. They suggest that future work should quantify this bound for realistic particle species (electrons, protons, ions) and for black holes with astrophysically plausible charge or spin parameters. The paper thus clarifies that the BSW effect is a manifestation of pure relativistic kinematics in curved spacetime, and that the “acceleration by deceleration” scenario provides a clean illustration of this principle.