# Gravitational Collapse in General Relativity and in $R^2$-gravity: A Comparative Study

## TL;DR

This study compares gravitational collapse dynamics in General Relativity and $R^2$ gravity across different matter types, revealing how modifications in gravity and energy conditions influence collapse rates and potential expansion scenarios.

## Contribution

It provides a comparative analysis of collapse behavior in GR and $R^2$ gravity, including effects of vacuum and phantom energy, highlighting new dependencies on initial conditions and energy parameters.

## Key findings

- Collapse rate varies with equation of state parameter w.
- Vacuum energy can slow or reverse collapse, leading to expansion.
- $R^2$ gravity introduces additional initial condition degrees of freedom.

## Abstract

We compare the gravitational collapse of homogeneous perfect fluid with various equations of state in the framework of General Relativity and in $R^2$ gravity. We make our calculations using dimensionless time with characteristic timescale $t_{g}\sim (G\rho)^{-1/2}$ where $\rho$ is a density of collapsing matter. The cases of matter, radiation and stiff matter are considered. We also account the possible existence of vacuum energy and its influence on gravitational collapse. In a case of $R^2$ gravity we have additional degree of freedom for initial conditions of collapse. For barotropic equation of state $p=w\rho$ the result depends from the value of parameter $w$: for $w>1/3$ the collapse occurs slowly in comparison with General Relativity while for $w<1/3$ we have opposite situation. Vacuum energy as expected slows down the rate of collapse and for some critical density gravitational contraction may change to expansion. It is interesting to note that for General Relativity such expansion is impossible. We also consider the collapse in the presence of so-called phantom energy. For description of phantom energy we use Lagrangian in the form $-X-V$ (where $X$ and $V$ are the kinetic and potential energy of the field respectively) and consider the corresponding Klein-Gordon equation for phantom scalar field.