Collapse in $f(R)$ gravity and the method of $R$ matching

TL;DR

This paper explores spherically symmetric gravitational collapse in $f(R)$ gravity with an $ ext{R}^2$ term, using $R$-matching to find new solutions that can lead to black holes or naked singularities while satisfying energy conditions.

Contribution

It introduces novel collapse solutions in $f(R)$ gravity employing $R$-matching, including shear-free and shear-viscous models, highlighting the impact of modified gravity on collapse outcomes.

Findings

Existence of solutions leading to black holes or naked singularities.

Analytic expressions for energy-momentum tensor and heat flux evolution.

Modified gravity influences the collapse dynamics and end states.

Abstract

Collapsing solutions in $f(R)$ gravity are restricted due to junction conditions that demand continuity of the Ricci scalar and its normal derivative across the time-like collapsing hypersurface. These are obtained via the method of $R$-matching, which is ubiquitous in $f(R)$ collapse scenarios. In this paper, we study spherically symmetric collapse with the modification term $\alpha R^2$, and use $R$-matching to exemplify a class of new solutions. After discussing some mathematical preliminaries by which we obtain an algebraic relation between the shear and the anisotropy in these theories, we consider two metric ansatzes. In the first, the collapsing metric is considered to be a separable function of the co-moving radius and time, and the collapse is shear-free, and in the second, a non-separable interior solution is considered, that represents gravitational collapse with non-zero shear viscosity. We arrive at novel solutions that indicate the formation of black holes or locally naked singularities, while obeying all the necessary energy conditions. The separable case allows for a simple analytic expression of the energy-momentum tensor, that indicates the positivity of the pressures throughout collapse, and is further used to study the heat flux evolution of the collapsing matter, whose analytic solutions are presented under certain approximations. These clearly highlight the role of modified gravity in the examples that we consider.