  We present a novel formulation of the Kerr spacetime solution, based on the
Lema\^itre coordinates. Such an approach allows one to avoid the coordinate
singularities of the Boyer-Lindquist metric, thus offering the possibility to
explore in some detail the Kerr physics near and beyond the event horizon,
adopting the point of view of a freely falling observer, whose adapted tetrad
is moreover proven to be Fermi-Walker transported. We use the Lema\^itre form
of the Kerr metric discussing the motion of radial photons close to the Cauchy
horizon, also showing that their behaviour fairly agrees with other similar
results presented in the literature. Being synchronous, the Lema\^itre form of
the Kerr metric allows a Hamiltonian formulation of a quantum field theory
nearby and beyond the Kerr event horizon. This could provide a further tool in
the investigation of the intriguing issue concerning the Cauchy horizon
(in)stability.
