The case of the mass quadrupole is of particular importance because in compact objects it represents the highest multipole contribution. We propose the quadrupolar metric (q-metric), which is obtained by applying a Zipoy-Voorhees transformation, as the simplest generalization of the Schwarzschild metric which includes a quadrupole moment. As for the interior solutions, we follow the conceptual principle that the inner structure of compact objects other than black holes can be described by using the classical approaches of gravity with higher multipole moments and thermodynamics, whereas inner black hole configurations are of pure quantum nature.

We propose a new alternative method to match interior and exterior solutions. The C3 matching method is coordinate invariant because it is based upon the use of the eigenvalues of the Riemann curvature tensor and its derivatives (C3 conditions). It has been applied to spherically symmetric configurations, obtaining physically meaningful results.

We investigate the physical properties of interior and exterior quadrupolar solutions by analyzing the motion of test particles in the corresponding gravitational field. Moreover, the effects of repulsive gravity, as defined by using the C3 matching procedure, are investigated in the vicinity of astrophysical objects as well as in the context of cosmological models.