  We present the whole class of Gaussian coordinate systems for the Kerr
metric. This is achieved through the uses of the relationship between Gaussian
observers and the relativistic Hamilton-Jacobi equation. We analyze the
completeness of this coordinate system. In the appendix we present the
equivalent JEK formulation of General Relativity -- the so-called
quasi-Maxwellian equations -- which acquires a simpler form in the Gaussian
coordinate system. We show how this set of equations can be used to obtain the
internal metric of the Schwazschild solution, as a simple example. We suggest
that this path can be followed to the search of the internal Kerr metric.
