Linear growth of massive vector superradiant instabilities of Kerr black holes

A recent breakthrough in the separability of the Proca equation in Kerr spacetime opened up new opportunities for a thorough study of fast-growing massive vector superradiant instabilities. Of particular interest is a time-domain analysis of the long-time evolution of test-field configurations. A possible approach to tackle the problem is to write the separated Proca equation as a coupled system of one-dimensional equations and solve it using the two-step, second-order Lax-Wendroff finite-difference scheme. In this short talk, I will sketch this method and (hopefully) present some preliminary results.