  Warp drives are space-times allowing for superluminal travel. However, they
are quantum mechanically unstable because they produce a Hawking-like radiation
which is blue shifted at their front wall without any bound. We reexamine this
instability when local Lorentz invariance is violated at ultrahigh energy by
dispersion, as in some theories of quantum gravity. Interestingly, even though
the ultraviolet divergence is now regulated, warp drives are still unstable.
Moreover the type of instability is different whether one uses a subluminal or
a superluminal dispersion relation. In the first case, a black-hole laser
yields an exponential amplification of the emitted flux whereas, in the second,
infrared effects produce a linear growth of that flux. These results suggest
that chronology could still be protected when violating Lorentz invariance.
