  We study the scenario in which a massive particle is thrown into a rapidly
rotating Kerr black hole in an attempt to spin it up beyond its extremal limit,
challenging weak cosmic censorship. We work in black-hole perturbation theory,
and focus on non-spinning, uncharged particles sent in on equatorial orbits. We
first identify the complete parameter-space region in which overspinning occurs
when back-reaction effects from the particle's self-gravity are ignored. We
find, in particular, that overspinning can be achieved only with particles sent
in from infinity. Gravitational self-force effects may prevent overspinning by
radiating away a sufficient amount of the particle's angular momentum
("dissipative effect"), and/or by increasing the effective centrifugal
repulsion, so that particles with suitable parameters never get captured
("conservative effect"). We analyze the full effect of the self-force, thereby
completing previous studies by Jacobson and Sotiriou (who neglected the
self-force) and by Barausse, Cardoso and Khanna (who considered the dissipative
effect on a subset of orbits). Our main result is an inequality, involving
certain self-force quantities, which describes a necessary and sufficient
condition for the overspinning scenario to be overruled. This "censorship"
condition is formulated on a certain one-parameter family of geodesics in an
extremal Kerr geometry. We find that the censorship condition is insensitive to
the dissipative effect (within the first-order self-force approximation used
here), except for a subset of perfectly fine-tuned orbits, for which a separate
censorship condition is derived. We do not obtain here the self-force input
needed to evaluate either of our two conditions, but discuss the prospects for
producing the necessary data using state-of-the-art numerical codes.
