For Einstein equation R_{uv}=0, there are 10 independent
equations. Due to 4 Bianchi equations, there are only 6
independent equations left to solve 10 variationals in
g_{uv}. Four constrain equations must be chosen to fix
four freedom of Einstein equation.
I have known two sets to constrains equations. One is
diff[(g^{1/2}*g^{uv}),x_v]=0 (1)
The other base on the spacetime metric in the form