Self-gravitating black hole scalar wigs
  It has long been known that no static, spherically symmetric, asymptotically
flat Klein-Gordon scalar field configuration surrounding a nonrotating black
hole can exist in general relativity. In a series of previous papers we proved
that, at the effective level, this no-hair theorem can be circumvented by
relaxing the staticity assumption: for appropriate model parameters there are
quasi-bound scalar field configurations living on a fixed Schwarzschild
background which, although not being strictly static, have a larger lifetime
than the age of the universe. This situation arises when the mass of the scalar
field distribution is much smaller than the black hole mass, and following the
analogies with the hair in the literature we dubbed these long-lived field
configurations wigs. Here we extend our previous work to include the
gravitational backreaction produced by the scalar wigs. We derive new
approximate solutions of the spherically symmetric Einstein-Klein-Gordon system
which represent self-gravitating scalar wigs surrounding black holes. These
configurations interpolate between boson star configurations and Schwarzschild
black holes dressed with the long-lived scalar test field distributions
discussed in previous papers. Nonlinear numerical evolutions of initial data
sets extracted from our approximate solutions support the validity of our
approach. Arbitrarily large lifetimes are still possible, although for the
parameter space that we analyze in this paper they seem to decay faster than
the quasi-bound states. Finally, we speculate about the possibility that these
configurations could describe the innermost regions of dark matter halos.
