Oppenheimer–Snyder Collapse in Metric f(R) Gravity: Constraints from Generalized Vaidya Exteriors.

We study the Oppenheimer--Snyder (OS) collapse problem in metric $f(R)$ gravity by matching a homogeneous dust Friedmann--Lema\^itre--Robertson--Walker (FLRW) interior to a generalized Vaidya exterior across a timelike hypersurface. In metric $f(R)$ gravity, regular matching requires the continuity not only of the induced metric and extrinsic curvature, but also of the Ricci scalar and its normal derivative. For the homogeneous dust interiors considered here, these additional conditions generically exclude Ricci-flat exteriors, such as the Schwarzschild solution. We show that, for an unrestricted generalized Vaidya exterior, the matching conditions fix the boundary data but do not uniquely determine the bulk extension, leaving open the possibility of a physical resolution of the collapse problem. However, once the exterior matter content is restricted to the generalized Vaidya form, the field equations impose a strong constraint, forcing f,R to be linear in the areal radius, f,R=A(v)\,r+B(v). For locally invertible f,R with f,RR≠ 0, this sharply reduces the admissible class of exteriors, so that the matching data uniquely determine the exterior solution on each interval where the boundary map is locally invertible. For branches with f,R>0, f,RR≠0, and f,R regular and finite at finite curvature, a globally extended exterior with finite asymptotic curvature requires $A(v)=0$. The matching conditions then force constant curvature. Imposing the null energy condition on both sides leaves only the vacuum-energy component and a locally maximally symmetric geometry. Under these assumptions, nontrivial collapse of spatially flat FLRW perfect-fluid interiors is therefore excluded within the restricted exterior class.