On Black Hole Echoes
  We consider a very simple model for gravitational wave echoes from black hole
merger ringdowns which may arise from local Lorentz symmetry violations that
modify graviton dispersion relations. If the corrections are sufficiently soft
so they do not remove the horizon, the reflection of the infalling waves which
trigger the echoes is very weak. As an example, we look at the dispersion
relation of a test scalar field corrected by roton-like operators depending
only on spatial momenta, in Gullstrand-Painlev\'e coordinates. The near-horizon
regions of a black hole do become reflective, but only very weakly. The
resulting ``bounces" of infalling waves can yield repetitive gravity wave
emissions but their power is very small. This implies that to see any echoes
from black holes we really need an egregious departure from either standard GR
or effective field theory, or both. One possibility to realize such strong
echoes is the recently proposed classical firewalls which replace black hole
horizons with material shells surrounding timelike singularities.
