Adiabatic radial perturbations of relativistic stars: analytic solutions to an old problem

We present a new system of equations that fully characterizes adiabatic, radial perturbations of perfect fluid stars within the theory of general relativity. The properties of the system are discussed, and, provided that the equilibrium spacetime verifies some general regularity conditions, analytical solutions for the perturbation variables are found. As illustrative examples, the results are applied to study perturbations of selected classical exact spacetimes, and the first oscillation eigenfrequencies are computed. Exploiting the new formalism, we derive an upper bound for the maximum compactness of stable, perfect fluid stars, which is equation-of-state-agnostic and significantly smaller than the Buchdahl bound.