Circular orbits and acceleration of particles by near-extremal dirty rotating black holes: general approach

We study the effect of ultra-high energy particles collisions near the black hole horizon (BSW effect) for two scenarios: when one of particle either (i) moves on a circular orbit or (ii) plunges from it towards the horizon. It is shown that such circular near-horizon orbits can exist for near-extremal black holes only. This includes the innermost stable orbit (ISCO), marginally bound orbit (MBO) and photon one (PhO). We consider generic “dirty” rotating black holes not specifying the metric and show that the energy in the centre of mass frame has the universal scaling dependence on the surface gravity $\kappa $. Namely, $E_{c.m.}\sim \kappa ^{-n}$ where for the ISCO $n=1/3$ in case (i) or $n=1/2$ in case (ii). For the MBO and PhCO $n=1/2$ in both scenarios that agrees with recent calculations of Harada and Kimura for the Kerr metric. We also generalize the Grib and Pavlov’s observations made for the Kerr metric. The magnitude of the BSW effect on the location of collision has a somewhat paradoxical character: it is decreasing when approaching the horizon.

💡 Research Summary

The paper investigates the Banãdos‑Silk‑West (BSW) effect – the possibility of arbitrarily high centre‑of‑mass (CM) energy in particle collisions near a rotating black‑hole horizon – in two concrete scenarios. In the first scenario one of the colliding particles is confined to a circular orbit that lies extremely close to the horizon; in the second scenario the same particle starts from that circular orbit and plunges inward toward the horizon. The authors deliberately keep the spacetime as general as possible: they consider a generic stationary, axisymmetric “dirty” rotating black hole, i.e. a black hole whose exterior may be deformed by surrounding matter, electromagnetic fields, or any other non‑vacuum contribution. No explicit metric (such as Kerr) is assumed, only the standard near‑horizon expansion of the lapse function (N) and the definition of the surface gravity (\kappa = \frac12 \partial_r N^2|_{r_H}).

The analysis begins with the conserved energy (E) and angular momentum (L) of a test particle moving in a metric of the form
( ds^{2}= -N^{2}dt^{2}+g_{\phi\phi}(d\phi-\omega dt)^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}).
From the normalization of the four‑momentum one obtains the effective potential
(V_{\text{eff}} = E-\omega L - N\sqrt{m^{2}+L^{2}/g_{\phi\phi}}).
Circular orbits satisfy (V_{\text{eff}}=0) and (\partial_{r}V_{\text{eff}}=0). Solving these equations near the horizon yields a relation between the orbital radius (r_{0}) and the surface gravity, \