Suppose in Cartesian coordinate system a Minkowski metric for flat spacetime can be written as : $$ ds^2 = a^2 (t)[-(1 + 2ψ(t,x,y,z))dt^2 - 2B_idx^idt + (1 - 2ψ(t,x,y,z))dx^{(i)2}] $$ This is a symmetric metric where x^i is comoving Cartesian coordinates. How can I produce pseudo orthonormal frame for this metric? For that case, what will be the components of this metric in the orthonormal frame?
Again, how can I convert the orthonormal frame to the general frame of spacetime?