Stellar model in a fourth order theory of gravity

Abstract

Within a fourth order theory of gravity, we obtain the approximation equations in first order of the coupling constant $\ensuremath{\alpha}$ of the quadratic term in the curvature and apply these equations to discuss a spherically symmetric perfect fluid stellar model in a weak field limit up to second order in the mass density $\ensuremath{\rho}$. We find, unlike general relativity (GR), that the continuity of the metric does not allow for a discontinuous mass density; i.e., for any bounded distribution of matter the pressure and the mass density have to be zero at the boundary. We show that the active mass of the fourth order theory is different than the active mass in GR. Furthermore, for a hard core star model, we find the explicit solution for the pressure and investigate the upper bound on the active mass of the star by assuming that matter couples minimally in the Jordan conformal frame and by applying the dominant energy condition to the perfect fluid at the center of the star. We show that there exist values of $\ensuremath{\alpha}$ and of the radius $R$ for which this mass of the system does not have an upper bound.