Not just that, you are free to choose a gauge that only “kicks in” the future. In fact there is no unique well-defined future history, just a future defined up,to gauge even if you fix a choice of gauge for the present.
Gauge fixing has to be done for all history, else there is fewer equations than dynamical variables, of course.
The point is that this is very hard to do for general relativity.
Not just that, you are free to choose a gauge that only “kicks in” the future. In fact there is no unique well-defined future history, just a future defined up,to gauge even if you fix a choice of gauge for the present.
Gauge fixing has to be done for all history, else there is fewer equations than dynamical variables, of course.
The point is that this is very hard to do for general relativity.