Local and approximate classification of spacetimes in the transverse frames

We revisit the definition of transverse frames and tetrad choices with regards to its application numerically generated spacetimes, in particular those from merger binary black holes. introduce concept local approximate algebraic Petrov types strong field regime. define an index $\mathcal{D}=\sqrt{12/I}\left(\Psi_2 - \Psi_3^2/\Psi_4\right)$ able discriminate between II D regions spacetime when used conjunction speciality invariant $S=27J^2/I^3$. provide explicit example applying this method Brill-Lindquist initial data corresponding two nonspinning holes rest at a given separation. find doughnut-like region that is approximately type surrounded by region. complete study proposing totally symmetric fixing frame can be simply implemented spacetimes through computation spin coefficients ratios. explicitly deriving Kerr-perturbative equations tetrad.