  The cold dark matter at the center of a galaxy will be redistributed by the
presence of a massive black hole. The redistribution may be determined using an
approach pioneered by Gondolo and Silk: begin with a model distribution
function for the dark matter, and ``grow'' the black hole adiabatically,
holding the adiabatic invariants of the motion constant. Unlike the approach of
Gondolo and Silk, which adopted Newtonian theory together with ad hoc
correction factors to mimic general relativistic effects, we carry out the
calculation fully relativistically, using the exact Schwarzschild geometry of
the black hole. We find that the density of dark matter generically vanishes at
r=2R_S, not 4R_S as found by Gondolo and Silk, where R_S is the Schwarzschild
radius, and that the spike very close to the black hole reaches significantly
higher densities. We apply the relativistic adiabatic growth framework to
obtain the final dark matter density for both cored and cusped initial
distributions. Besides the implications of these results for indirect detection
estimates, we show that the gravitational effects of such a dark matter spike
are significantly smaller than the relativistic effects of the black hole,
including frame dragging and quadrupolar effects, for stars orbiting close to
the black hole that might be candidates for testing the black hole no-hair
theorems.
