Generalized black-bounces solutions in f(R) gravity and their field sources

TL;DR

This work shows that Simpson–Visser black-bounce geometries can be realized as solutions in metric $f(R)$ gravity when coupled to nonlinear electrodynamics and (partially) phantom or canonical scalar fields. The authors develop a general reconstruction method for $f(R)=R+H(R)$ to determine the corresponding matter sources from a known GR bounce source, and they apply it to four representative $H(R)$ models, deriving explicit expressions for $L^{(H)}(F)$, $L_F^{(H)}(F)$, $V^{(H)}(\,\\phi)$, and $h^{(H)}(\,\\phi)$; they also analyze viability via the scalaron mass $m_ abla^2$ and energy conditions. Notably, in several models, the scalar field can be canonical in some regions and the energy conditions can be satisfied locally, revealing a relaxation of GR’s ubiquitous violations for BBs. The results connect modified gravity, nonlinear electrodynamics, and black-bounce phenomenology, with potential observational implications through photon propagation in the effective metric and by guiding tests with strong-gravity data. The framework is general and extendable to other BB solutions, offering a pathway to test modified gravity in the strong-field regime.

Abstract

In this work, following our recent findings in [1], we extend our analysis to explore the generalization of spherically symmetric and static black-bounce solutions, known from General Relativity, within the framework of the $f(R)$ theory in the metric formalism. We develop a general approach to determine the sources for any model where $f(R) = R + H(R)$, provided that the corresponding source for the bounce metric in General Relativity is known. As a result, we demonstrate that black-bounce solutions can emerge from this theory when considering the coupling of $f(R)$ gravity with nonlinear electrodynamics and a partially phantom scalar field. We also analyzed the energy conditions of these solutions and found that, unlike in General Relativity, it is possible to satisfy all energy conditions in certain regions of space-time.

Figures (10)

• Figure 1: Behavior of $f(R)$ as a function of the radial coordinate (left panel) and as a function of the curvature scalar (right panel), fixing $a = m = 1$ and varying the values of the parameter $a_1$. As we can see, $f(R(r))$ is not symmetric under the transformation $r \to -r$ when $a_1 \neq 0$. This happens because $f_R(r)$ is linear in the radial coordinate, $f_R = 1 + a_1 r$. Nevertheless, the curvature scalar is symmetric under $r \to -r$. This means that under this transformation we obtain the same value of the curvature scalar but a different value of $f(R(r))$, leading to the multivalued behavior shown in the right-hand figure: the same value of the curvature scalar can correspond to two different values of $f(R)$.
• Figure 2: Scalaron mass $m_\psi^2$ as a function of the radial coordinate $r$ for $a_1 = 1$ (left) and $a_1 = -1$ (right), with $a = m = 1$. Again, for both $a_1 > 0$ and $a_1 < 0$, there are regions where the scalaron mass is positive.
• Figure 3: Behavior of $f(R)$ as a function of the radial coordinate (left panel) and as a function of the curvature scalar (right panel), fixing $a = m = 1$ and varying the values of the parameter $a_2$. In this case, the function $f(R(r))$ is symmetric under the transformation $r \to -r$, even when $a_2 \neq 0$. This ensures that we obtain a single value of $f(R)$ for each value of the curvature scalar, thereby avoiding any multivalued behavior.
• Figure 4: Scalaron mass $m_\psi^2$ as a function of the radial coordinate $r$ for $a_2 = 1$ (left) and $a_2 = -1$ (right), with $a = m = 1$. In this case, $m_\psi^2 <0$ in more central regions for $a_2>0$, and positive in some regions for $a_2<0$, indicating that stability regions exist for this case.
• Figure 5: Behavior of the function $f(R)$ as a function of the radial coordinate (left panel) and as a function of the curvature scalar (right panel), fixing $a = m = 1$ and varying the values of the parameter $a_1$. In this case, the function $f(R(r))$ is symmetric under the transformation $r \to -r$, even when $a_\Sigma \neq 0$. This ensures that we obtain a single value of $f(R)$ for each value of the curvature scalar, thereby avoiding any multivalued behavior.