  We show that the locally constant force necessary to get a stable hyperbolic
motion regime for classical charged point particles, actually, is a combination
of an applied external force and of the electromagnetic radiation reaction
force. It implies, as the strong Equivalence Principle is valid, that the
passive gravitational mass of a charged point particle should be slight greater
than its inertial mass. An interesting new feature that emerges from the
unexpected behavior of the gravitational and inertial mass relation, for
classical charged particles, at very strong gravitational field, is the
existence of a critical, particle dependent, gravitational field value that
signs the validity domain of the strong Equivalence Principle. For electron and
proton, these critical field values are $g_{c}\simeq 4.8\times 10^{31}m/s^{2}$
and $g_{c}\simeq 8.8\times 10^{34}m/s^{2}$, respectively.
