  In this short Note, I answer the titular question: Yes, a radiation gauge can
be horizon-locking. Radiation gauges are very common in black hole perturbation
theory. It's also very convenient if a gauge choice is horizon-locking, i.e.
the location of the horizon is not moved by a linear metric perturbation.
Therefore it is doubly convenient that a radiation gauge can be
horizon-locking, when some simple criteria are satisfied. Though the
calculation is straightforward, it seemed useful enough to warrant writing this
Note. Finally I show an example: the $\ell$ vector of the Hartle-Hawking tetrad
in Kerr satisfies all the conditions for ingoing radiation gauge to keep the
future horizon fixed.
