Because there is large gauge freedom in choice of coordinates and random choice of gauge will likely produce a coordinate singularity somewhere and you will not see what’s beyond. So you don’t reconstruct whole spacetime history, but you can reconstruct at least something, and perhaps use different coordinates to move further. Of course there are problems with precision whenever the equations are enough non-linear, but that’s nothing specific to 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.
Because there is large gauge freedom in choice of coordinates and random choice of gauge will likely produce a coordinate singularity somewhere and you will not see what’s beyond. So you don’t reconstruct whole spacetime history, but you can reconstruct at least something, and perhaps use different coordinates to move further. Of course there are problems with precision whenever the equations are enough non-linear, but that’s nothing specific to 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.