In ADM formalism there are evolution equations only for spatial metric and extrinsic curvature, but nothing about lapse and shift. I understand that the latter two are essentially just choice of coordinates. However, if I already choose coordinates, shouldn't be their evolution also be determined? Unfortunately, all I found was that there are "gauge conditions" for this (https://arxiv.org/pdf/gr-qc/9412071). How can i determine the gauge from my choice of coordinates? Also, are ADM equations depend on gauge?

The ADM formalism by itself does not fix the gauge. It is just the Einstein equations, rewritten in terms of ##N## and ##N^i## (and the spatial metric), rather than ##g_{00}## and ##g_{0i}## (and the spatial metric). The gauge has to be fixed for the same reason as it does for solving the Einstein equations.

Fixing ##N## and ##N^i## does not fix the gauge completely, some gauge freedom still exists. This is analogous to electrodynamics where fixing ##A^0## does not fix the gauge completely.

Demystifier said:

The ADM formalism by itself does not fix the gauge. It is just the Einstein equations, rewritten in terms of ##N## and ##N^i## (and the spatial metric), rather than ##g_{00}## and ##g_{0i}## (and the spatial metric). The gauge has to be fixed for the same reason as it does for solving the Einstein equations.

Fixing ##N## and ##N^i## does not fix the gauge completely, some gauge freedom still exists. This is analogous to electrodynamics where fixing ##A^0## does not fix the gauge completely.

Yes, I understand there is also freedom related to gravitational waves. However, I have solution for some short time and want to simulate the corresponding evolution. So, there should be no freedom watsoever, the evolution is determined. But in ADM there are no equations for evolution of lapse and shift, which are clearly parts of Einstein equations. Are they not always with explicit first only time derivative? If so, I should explicitly adjust my solution to specific gauge.