  A test fluid composed of relativistic collisionless neutral particles in the
background of Kerr metric is expected to generate non-isotropic equilibrium
configurations in which the corresponding stress-energy tensor exhibits
pressure and temperature anisotropies. This arises as a consequence of the
constraints placed on single-particle dynamics by Killing tensor symmetries,
leading to a peculiar non-Maxwellian functional form of the kinetic
distribution function describing the continuum system. Based on this outcome,
in this paper the generation of Kerr-like metric by collisionless N-body
systems of neutral matter orbiting in the field of a rotating black hole is
reported. The result is obtained in the framework of covariant kinetic theory
by solving the Einstein equations in terms of an analytical perturbative
treatment whereby the gravitational field is decomposed as a prescribed
background metric tensor described by the Kerr solution plus a self-field
correction. The latter one is generated by the uncharged fluid at equilibrium
and satisfies the linearized Einstein equations having the non-isotropic
stress-energy tensor as source term. It is shown that the resulting self-metric
is again of Kerr type, providing a mechanism of magnification of the background
metric tensor and its qualitative features.
