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

This paper investigates the existence and viability of three-dimensional black bounce solutions in $f(R)$ gravity, analyzing their generalization from general relativity, constructing new zero-curvature solutions supported by scalar-nonlinear electrodynamics couplings, and evaluating the associated energy conditions and scalaron mass behavior.

The Gist

Imagine the universe as a giant, stretchy trampoline. In our standard understanding of gravity (Einstein's General Relativity), if you put a heavy bowling ball on this trampoline, it creates a deep dip. If you put something really heavy, like a black hole, the trampoline stretches so much that it eventually tears, creating a bottomless pit called a "singularity." At the bottom of this pit, the laws of physics break down, and we don't know what happens.

For decades, physicists have been trying to fix this tear. They want to know: Is there a way to make a black hole that doesn't have a bottomless pit? One idea is the "Black Bounce."

Think of a Black Bounce not as a pit, but as a tunnel. If you fall into a Black Bounce, instead of hitting a singularity and disappearing, you hit a soft, rubbery floor at the bottom and bounce right back out the other side. It's like a cosmic trampoline that never tears.

This paper is about exploring these "Black Bounces" in a specific, simplified version of the universe (2 dimensions of space + 1 of time, like a flat sheet instead of a 3D room) and asking a big question: Can we build these tunnels using a new, upgraded version of gravity called f(R) gravity?

Here is a breakdown of their journey, using simple metaphors:

1. The Problem with the Old Blueprint

In the old blueprint (General Relativity), to build a Black Bounce, you need "exotic" ingredients. Imagine trying to build a house, but the bricks you need are made of "anti-gravity" or "ghostly" material that pushes things apart instead of pulling them. In physics terms, this is called a phantom scalar field. It's weird stuff that violates the usual rules of energy, but it's necessary to keep the tunnel open.

2. The New Gravity Engine (f(R))

The authors decided to swap the old gravity engine for a new one called f(R) gravity. Think of f(R) as a gravity engine with an extra "turbo button" or a "smart sensor" that reacts to how curved space is. It's more complex than Einstein's engine, but maybe it can handle the job better.

They asked: If we use this new engine, do we still need the weird "ghostly" bricks? Or can we build the tunnel with normal, everyday bricks?

3. The Four Experiments

The team ran four different simulations to see what happens when they mix the Black Bounce shape with the new gravity engine.

• Experiment A & B (The Custom Engines): They tried tweaking the new gravity engine in two specific ways.

  • The Result: They found that to keep the tunnel open, they still needed some of that weird "ghostly" material (phantom fields). However, the new engine made the "ghosts" behave a bit differently. Sometimes, the ghosts were only "half-ghosts" (partially phantom), which is a slight improvement, but you can't get rid of them entirely if you want the math to work.
  • The Catch: The new engine has strict safety rules (viability conditions). To keep the engine from exploding (instability), the "ghostly" material must be there. If you try to use normal bricks, the engine breaks.

• Experiment C (The Starobinsky Engine): They tried a very famous, popular version of the new engine (the Starobinsky model), which is often used to explain how the universe began.

  • The Result: This engine is powerful but finicky. To build the tunnel, the "ghostly" material had to change its personality depending on where you were. Near the center of the tunnel, it was a ghost; far away, it acted normal.
  • The Catch: The "ghosts" were still needed, and the energy required to hold the tunnel together was even more "exotic" (violating energy rules more severely) than in the old universe.

• Experiment D (The Flat Engine): They tried a scenario where the curvature of space was zero (flat).

  • The Result: They found a solution, but it was a strange one. It was an "Inverted Black Hole." Imagine a black hole where the "outside" is actually the inside, and the "inside" is the outside. The tunnel exists, but the rules of time and space flip over.
  • The Catch: This solution required the "ghostly" material to be 100% ghostly everywhere. It was the most extreme violation of energy rules of all.

4. The Big Takeaway

The authors discovered a fascinating trade-off:

1. You can build the tunnel: Yes, you can create these "Black Bounce" shapes in this new, upgraded gravity theory.
2. But the cost is high: You cannot escape the need for "exotic" matter. In fact, the new gravity engine often demands that you use even weirder, more "ghostly" matter than the old engine did to keep the math consistent.
3. The "Ghost" is a Feature, not a Bug: In this new theory, the "phantom" nature of the matter isn't just a mistake; it's a requirement. The engine is designed in such a way that it needs this weird energy to function without breaking.

The Final Analogy

Imagine you are trying to build a bridge across a canyon.

• Old Gravity (Einstein): You need a very special, expensive, and slightly magical steel to build the bridge.
• *New Gravity (f(R)):* You have a new, super-strong steel. But, because this new steel is so sensitive to the wind (curvature), it actually requires you to glue it together with even more magical, unstable glue to keep it from snapping.

Conclusion: The paper shows that while we can mathematically construct these "Black Bounce" tunnels in the new gravity theories, nature (or at least the math) insists that we still need "exotic," rule-breaking ingredients to make them work. The new gravity doesn't magically fix the need for weird stuff; it just changes how that weird stuff behaves.

Technical Summary

1. Problem Statement

The paper addresses the existence and physical viability of Black Bounce (BB) solutions in $(2+1)$-dimensional spacetime within the framework of $f(R)$ gravity.

• Context: General Relativity (GR) predicts singularities in black holes (BHs). Regular black holes and "black bounces" (geometries that transition between BHs and traversable wormholes) have been proposed to resolve these singularities. While BB solutions are well-studied in GR and $(3+1)$ dimensions, their generalization to modified gravity theories like $f(R)$ in lower dimensions remains an open problem.
• Challenge: Constructing exact solutions in $f(R)$ gravity is difficult due to the fourth-order nature of the field equations. Furthermore, regular geometries often require "exotic" matter (violating energy conditions) or specific couplings between scalar fields and nonlinear electrodynamics (NED). The authors aim to determine if the regularized BTZ (Bañados-Teitelboim-Zanelli) geometry can be sustained by consistent matter sources in $f(R)$ gravity and to analyze the viability of such models.

2. Methodology

The authors employ a reverse-engineering approach (ansatz method) rather than solving the differential equations for the metric directly.

• Metric Ansatz: They adopt the regularized BTZ metric (Simpson-Visser regularization) in $(2+1)$ dimensions:
  $$ds^2 = A(r)dt^2 - \frac{1}{A(r)}dr^2 - (r^2 + a^2)d\phi^2$$
  where $a$ is the regularization parameter.
• Matter Sector: The source is modeled as a scalar field ($\phi$) coupled to Nonlinear Electrodynamics (NED). The action includes an $f(R)$ term, a scalar kinetic term with a coupling function $h(\phi)$, a potential $V(\phi)$, and an NED Lagrangian $L(F)$.
• Dual Formalism: To handle the complexity of inverting the electromagnetic invariant $F(r)$ to find the Lagrangian $L(F)$, the authors utilize the H(P) formalism (Legendre transformation). This avoids the multi-valued branch issues inherent in $L(F)$ when $F(r)$ has extrema, allowing for a single-valued, analytic description of the Lagrangian in terms of the invariant $P$.
• Investigation Strategy: The paper analyzes four distinct scenarios:
  1. Model A: $f_R = 1 + a_2 r^2$ (where $f_R \equiv df/dR$).
  2. Model B: $f_R = 1 + a_\Sigma \Sigma$ (where $\Sigma = \sqrt{r^2+a^2}$).
  3. Model C: The Starobinsky model, $f(R) = R + a_R R^2$.
  4. Model D: A solution with vanishing curvature scalar ($R=0$) within the Starobinsky framework.

3. Key Contributions

• Generalization to Modified Gravity: The work successfully demonstrates that regularized BTZ black bounce geometries can be consistently generated in $f(R)$ gravity, not just in GR.
• Matter Source Construction: The authors explicitly derive the required matter sources (scalar field potential $V(\phi)$, kinetic coupling $h(\phi)$, and NED Lagrangian $L(P)$) for each $f(R)$ model. They show that $f(R)$ corrections necessitate significantly higher nonlinearity in the matter sector compared to GR.
• H(P) Formalism Application: The paper provides a robust method for constructing analytic Lagrangians for charged black bounces in modified gravity by bypassing the inversion problems of $L(F)$.
• Inverted Black Hole Solution: In the $R=0$ case, the authors discover a new class of "inverted" black holes where the static region lies inside the event horizon ($r < r_h$), and the asymptotic region ($r \to \infty$) corresponds to a non-static interior.

4. Key Results

A. Viability Conditions and Scalaron Mass

For an $f(R)$ model to be physically viable, it must satisfy $f_R > 0$ (positive effective gravitational coupling) and $f_{RR} > 0$ (absence of Dolgov-Kawasaki instability).

• Models A & B: Viability ($f_R, f_{RR} > 0$) requires the coupling constants ($a_2, a_\Sigma$) to be positive.
• Model C (Starobinsky): Viability requires $a_R > 0$ and constraints on the curvature $R$.
• Scalar Field Nature: A critical finding is that viability conditions enforce the scalar field to be phantom ($h(\phi) < 0$) in most cases.
  • In Models A and B, satisfying $f_R > 0$ and $f_{RR} > 0$ forces the scalar field to be phantom.
  • In the Starobinsky model, a positive $a_R$ leads to a scalar field that is phantom in central regions and canonical at infinity (partially phantom).
  • Conclusion: It is generally impossible to have an everywhere-canonical scalar field while satisfying the stability conditions of these $f(R)$ models.

B. Energy Conditions

The authors analyze the Null (NEC), Weak (WEC), Strong (SEC), and Dominant (DEC) energy conditions.

• General Violation: Consistent with regular black hole and wormhole physics, energy conditions are violated in these spacetimes.
• Impact of $f(R)$: The introduction of $f(R)$ corrections generally increases the degree of exoticity.
  • In Model A, no single sign of $a_2$ satisfies all energy conditions simultaneously.
  • In Model B, if $a_\Sigma > 0$ (required for viability), the Strong Energy Condition is violated everywhere, though other conditions can be satisfied inside the horizon.
  • In the Starobinsky model, violations are often more severe than in GR, though some inequalities that are always violated in GR can be satisfied in restricted regions.
  • In the $R=0$ model, all standard energy conditions are violated throughout the spacetime.

C. Specific Model Findings

• Model A ($f_R = 1 + a_2 r^2$): The source functions are symmetric under $r \to -r$. The NED Lagrangian $L(P)$ is derived analytically.
• Model B ($f_R = 1 + a_\Sigma \Sigma$): Yields simpler source functions. The Lagrangian $L(P)$ is quadratic in $P$ for $n=2$, deviating from Maxwellian behavior.
• Model C (Starobinsky): The electromagnetic invariant $F(r)$ exhibits maxima and minima, creating three distinct branches. The $L(P)$ Lagrangian is complex but analytic.
• Model D ($R=0$): This solution represents an asymmetric, inverted black bounce. The metric function $A(r)$ changes sign such that the "universe" for a stationary observer is the region $r < r_h$. The scalar field is always phantom.

5. Significance and Conclusion

• Theoretical Robustness: The study confirms that the geometric features of black bounces (regularity, horizon-wormhole transitions) are robust enough to be embedded in $f(R)$ gravity, provided specific, highly nonlinear matter sources are present.
• Trade-off between Viability and Exoticity: The paper highlights a tension: the conditions required to make the $f(R)$ theory stable (viability) often force the matter sector to be more exotic (phantom fields) and violate energy conditions more severely than in standard GR.
• New Geometries: The identification of "inverted" black holes with vanishing curvature scalars expands the catalog of exact solutions in modified gravity.
• Future Directions: The authors suggest that these results pave the way for studying perturbative stability, quasinormal modes, and observational signatures (like gravitational lensing and shadows) of 2+1 dimensional black bounces in modified gravity.

In summary, this work provides a systematic construction of field sources for 2+1 dimensional black bounces in $f(R)$ gravity, demonstrating that while such solutions are mathematically consistent, they rely heavily on phantom scalar fields and violate classical energy conditions, with the degree of violation depending on the specific $f(R)$ model chosen.