Three dimensional black bounces in $f(R)$ gravity

TL;DR

This work investigates the existence and properties of black bounce geometries in $2+1$ dimensions within $f(R)$ gravity, focusing on whether GR BBs can be consistently generalized to modified gravity and which matter sources can support them. The authors employ a regularized BTZ-like baseline metric and explore several $f(R)$ models, including $f_R=1+a_2 r^2$, $f_R=1+a_Sigma$, the Starobinsky form $f(R)=R+a_R R^2$, and a zero-curvature case $R=0$, with a scalar field nonminimally coupled to nonlinear electrodynamics. They derive the necessary field-content and potentials, analyze viability through the scalaron mass $m_bpsi^2$ and $f_R,f_{RR}$ positivity, and assess energy conditions reinterpreted as an effective anisotropic fluid. The results show that BBs can be realized in $2+1$D $f(R)$ gravity when the matter sector includes NED and a (partially) phantom scalar, but global viability and energy conditions typically impose strong constraints, with the $R=0$ case yielding an inverted BH and a curvature-vanishing solution. These findings illuminate how modified gravity alters BB regularity, stability considerations, and exotic-matter requirements in lower dimensions, laying groundwork for stability analyses and potential observational signatures such as lensing or shadows in related settings.

Abstract

We investigate the existence of black bounce solutions in $2+1$ dimensions within the framework of $f(R)$ gravity. We analyze whether black bounce geometries originally obtained in general relativity can be consistently generalized to $f(R)$ theories and identify the matter sources capable of supporting such solutions. We also construct a new class of solutions by imposing a vanishing curvature scalar. In the matter sector, we consider models involving a coupling between a scalar field and nonlinear electrodynamics, while in the gravitational sector we analyze both the Starobinsky model and more general forms of $f(R)$. We further examine the viability conditions of the $f(R)$ models that give rise to these spacetimes, including the behavior of the scalaron mass. Finally, we study the associated energy conditions, in order to assess the degree of exoticity of the matter content required to sustain these black bounce solutions and how the $f(R)$ theory modifies the energy conditions.

Figures (8)

• Figure 1: Behavior of the function $F(r)$ (left panel) and the electromagnetic Lagrangian $L(F)$ (right panel) for the parameters $M=1$, $q=a=0.2$, $l=10$, and different values of $a_{2}$ considering the model $f_R=1+a_2r^2$.
• Figure 2: Behavior of the function $F(r)$ (left panel) and the electromagnetic Lagrangian $L(F)$ (right panel) for the parameters $M=1$, $q=a=0.2$, $l=10$, and different values of $a_{\Sigma}$, considering the model $f_R=1+a_\Sigma \Sigma$.
• Figure 3: Behavior of the function $F(r)$ (left panel) and the electromagnetic Lagrangian $L(F)$ (right panel) for the parameters $M=1$, $q=a=0.2$, $l=10$, and different values of $a_{R}$, considering the Starobinsky gravity.
• Figure 4: Behavior of the function $A(r)$, Eq. \ref{['Amodel4']}, in terms of the radial coordinate with $M=Q=1$.
• Figure 5: Behavior of the function $F(r)$ and the electromagnetic Lagrangian $L(F)$ for the parameters $M=1=q=a=1$, considering the Starobinsky gravity and $R=0$.