  We study how the nature of a hybrid system (perfect fluid, solid or a mixture
of them) could be related to the induction of general relativistic surface
degrees of freedom on phase-splitting surfaces upon perturbation of its phases.
We work in the scope of phase conversions in the vicinity of sharp phase
transition surfaces whose timescales are either much smaller (rapid
conversions) or larger (slow conversions) than the ones of the perturbations
($\omega^{-1}$, where $\omega$ is a characteristic frequency of oscillation of
the star). In this first approach, perturbations are assumed to be purely
radial. We show that surface degrees of freedom could emerge when either the
core or the crust of a hybrid star is solid and phase conversions close to a
phase-splitting surface are rapid. We also show how this would change the usual
stability rule for solid hybrid stars, namely $\partial M_0/\partial \rho_c\geq
0$, where $M_0$ is the total mass to the background hybrid star and $\rho_c$
its central density. Further consequences of our analysis for asteroseismology
are also briefly discussed.
