Gravitational slingshots around black holes in a binary

The speed gain of a test mass from taking a gravitational slingshot around a celestial object (scattering centre) increases with the latter's speed and compactness (stronger deflection of the mass' trajectory becomes possible without it hitting the surface of the object). The black holes (BHs) in a tight binary (consisting of two black holes; we are not considering X-ray binaries), themselves moving at relativistic speeds, represent optimal scattering centres. Therefore, a sub-population of accreting matter particles, swept up into chaotic orbits around a BH binary, might repeatedly take slingshots and become accelerated to ultra-relativistic speeds (as seen by observers on Earth), ultimately escaping from the binary, as well as the fate of being devoured by a BH. The escaped particles can plausibly be observed on Earth as ultra-high-energy cosmic rays (UHECRs). Investigating such a possibility would require general relativistic slingshot formulae due to the high speeds involved and the close encounters with BHs. We derive them in this paper, and show that the percentage gain per slingshot in a particle's Lorentz factor remains undiminished even as the particle energizes up, thus demonstrating that the slingshot mechanism can in principal accelerate particles to extreme energies.