High energy collision of particles in the vicinity of extremal black holes in higher dimensions: Banados-Silk-West process as linear instability of extremal black holes

We study high energy particle collisions around higher dimensional black holes. It is shown that the center of mass energy can be arbitrarily large in the vicinity of the event horizon like the Banados, Silk and West (BSW) process in four dimensions if the following two conditions are satisfied: (i) the horizon is extremal and (ii) the parameters of either of the two colliding particles are fine-tuned, which is called a critical particle condition. We also show that a test particle which satisfies the critical particle condition can reach the event horizon from a distant region for a simple case. Finally, we discuss the relation between the BSW process and the linear instability of test fields around extremal black holes, the latter of which has been recently found by Aretakis [Commun. Math. Phys. 307, 17 (2011); Ann. Inst. Henri Poincare 12, 1491 (2011); J. Funct. Anal. 263, 2770 (2012); arXiv: 1206.6598; Classical Quantum Gravity 30, 095010 (2013); Phys. Rev. D 87, 084052 (2013)].