  We study adiabatic, radial perturbations of static, self-gravitating perfect
fluids within the theory of general relativity employing a new perturbative
formalism. We show that by considering a radially static observer, the
description of the perturbations can be greatly simplified with respect to the
standard comoving treatment. The new perturbation equations can be solved to
derive analytic solutions to the problem for a general class of equilibrium
solutions. We discuss the thermodynamic description of the fluid under
isotropic frame transformations, showing how, in the radially static,
non-inertial frame, the stress-energy tensor of the fluid must contain momentum
transfer terms. As illustrative examples of the new approach, we study
perturbations of equilibrium spacetimes characterized by the Buchdahl I,
Heintzmann IIa, Patwardhan-Vaidya IIa, and Tolman VII solutions, computing the
first oscillation eigenfrequencies and the associated eigenfunctions. We also
analyze the properties of the perturbations of cold neutron stars composed of a
perfect fluid verifying the Bethe-Johnson model I equation of state, computing
the oscillation eigenfrequencies and the $e$-folding time.
