Numerical simulations of the accretion of test scalar fields with non-standard kinetic terms (of the k-essence type) onto a Schwarzschild black hole are performed. We find a full dynamical solution for the spherical accretion of a Dirac-Born-Infeld type scalar field. The simulations show that the accretion eventually settles down to a well known stationary solution. This particular analytical steady state solution maintains two separate horizons. The standard horizon is for the usual particles propagating with the limiting speed of light, while the other sonic horizon is for the k-essence perturbations propagating with the speed of sound around this accreting background. For the case where the k-essence perturbations propagate superluminally, we show that one can send signals from within a black hole during the approach to the stationary solution. We also find that a ghost condensate model settles down to a stationary solution during the accretion process.
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