  We perform a linear stability analysis of dynamical Chern-Simons modified
gravity in the geometric optics approximation and find that it is linearly
stable on the backgrounds considered. Our analysis also reveals that
gravitational waves in the modified theory travel at the speed of light in
Minkowski spacetime. However, on a Schwarzschild background the characteristic
speed of propagation along a given direction splits into two modes, one
subluminal and one superluminal. The width of the splitting depends on the
azimuthal components of the propagation vector, is linearly proportional to the
mass of the black hole, and decreases with the third inverse power of the
distance from the black hole. Radial propagation is unaffected, implying that
as probed by gravitational waves the location of the event horizon of the
spacetime is unaltered. The analysis further reveals that when a high
frequency, pure gravitational wave is scattered from a black hole, a scalar
wave of comparable amplitude is excited, and vice-versa.
