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Charged black holes in Weyl conformal gravity

This paper presents a parametric study of charged black holes in Weyl conformal gravity, deriving analytic expressions for horizons and photon spheres to reveal exotic spacetime structures—such as nested black holes with trapped Cauchy horizons and triple-horizon collisions—that arise due to the absence of the inverse quadratic term and differ fundamentally from standard general relativity.

Original authors: Reinosuke Kusano, Miguel Yulo Asuncion, Keith Horne

Published 2026-02-05

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CC BY 4.0

Original authors: Reinosuke Kusano, Miguel Yulo Asuncion, Keith Horne

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, stretchy trampoline. In the standard story of gravity (Einstein's General Relativity), heavy objects like stars and black holes make deep dents in this trampoline. If you put a marble on the trampoline, it rolls toward the dent. This theory works great for our solar system, but when we look at huge galaxies, the math gets messy. We have to invent invisible "dark matter" to explain why galaxies spin the way they do, and we have to invent "dark energy" to explain why the universe is speeding up.

This paper explores a different theory called Weyl Conformal Gravity. Think of this theory not just as a trampoline, but as a trampoline that can also be stretched or shrunk locally without changing the physics. It's a "fourth-order" theory, which is a fancy way of saying the rules for how it bends are more complex and flexible than Einstein's.

The authors of this paper decided to ask a specific question: What happens if we put an electric charge on a black hole in this stretchy, flexible universe?

In standard Einstein gravity, a charged black hole (called Reissner-Nordström) has a very predictable structure. It has an outer shell (the event horizon) and an inner shell (the Cauchy horizon). The electric charge acts like a repulsive force, pushing the inner shell out.

However, the authors found that in Weyl Conformal Gravity, the rules are completely different. Here is what they discovered, using simple analogies:

1. The Missing "Push"

In Einstein's gravity, the electric charge creates a specific kind of repulsive push that gets stronger the closer you get to the center (like a $1/r^2$ force). In Weyl gravity, this specific push is missing. Instead, the charge changes the "weight" of the black hole itself.

Because this repulsive push is gone, the inner structure of the black hole behaves strangely. In Einstein's world, the inner horizon is always a "Cauchy horizon" (a boundary where predictability breaks down). In this new theory, that isn't guaranteed.

2. The "Russian Nesting Doll" Black Hole

The most surprising discovery is a structure the authors call a "nested black hole."

Imagine a set of Russian nesting dolls.

In a normal black hole, you have an outer shell (Event Horizon) and an inner core.
In this new Weyl black hole, for certain settings, you can have an Event Horizon, then a Cauchy Horizon inside it, and then another Event Horizon inside that!

It's like a black hole trapped inside another black hole, with a weird "safe zone" (the Cauchy horizon) sandwiched between them. This is a structure that simply cannot exist in standard Einstein gravity. It's a "three-layer cake" of spacetime that the authors found is possible only in this specific theory.

3. The "Photon Spheres" (The Light Traps)

Around black holes, there are rings where light can orbit the hole like a planet orbits a star. These are called "photon spheres."

Usually, there is one unstable ring (if you nudge the light, it flies away or falls in).
Sometimes, there is a stable ring (light can orbit safely).

The authors found a "critical value" for the electric and magnetic charges. When the charge hits this specific number, the stable ring and the unstable ring crash into each other and merge into a single, "saddle-point" ring. It's like two dancers spinning in opposite directions suddenly locking arms and spinning together in a precarious balance. If you add even a tiny bit more charge, these rings disappear entirely.

4. The "Triple Collision"

In standard physics, you can sometimes have three horizons (inner, outer, and a cosmological one) merge into one. The authors found that in Weyl gravity, you can have a "Triple Limit" where three horizons collide.

Sometimes this is the standard "inner, outer, and cosmological" trio.
But they also found a new type: Two Event Horizons and one Cauchy Horizon merging together. This creates a very strange, ultra-cold point in spacetime that we don't see in normal gravity.

5. The "Switch" Effect

The authors noticed that the electric charge acts like a switch depending on a parameter they call $\gamma$ (gamma).

If $\gamma$ is positive, the charge acts like it does in normal gravity: it pushes the outer horizon in and the inner horizon out.
If $\gamma$ is negative, the charge does the exact opposite. It pushes the outer horizon out and pulls the inner horizon in.

It's as if the electric charge has a "reverse gear" in this theory, flipping its behavior based on the background of the universe.

Summary

The paper is a detailed map of all the possible shapes a charged black hole can take in Weyl Conformal Gravity. They found that:

You can get exotic "nested" black holes with layers of horizons that don't exist in Einstein's theory.
The electric charge can flip its behavior, acting repulsive or attractive in ways that surprise us.
There are critical limits where light rings merge and horizons collide, creating unique "extremal" states.

The authors conclude that while this theory offers these fascinating, exotic possibilities, it also has its own challenges (like explaining how binary stars lose energy). But for now, they have successfully mapped out this strange new landscape of "charged black holes" where the rules of gravity are written in a different language.

Technical Summary: Charged Black Holes in Weyl Conformal Gravity

Problem Statement
General Relativity (GR) faces significant tensions on galactic and cosmological scales, necessitating the introduction of dark matter and dark energy, alongside the vacuum energy problem. Weyl Conformal Gravity (CG) offers an alternative, fourth-order theory of gravitation that is conformally invariant and potentially renormalizable. While previous studies have analyzed the CG analogues of the Schwarzschild and Kerr metrics, the spacetime structure of the CG analogue to the charged Reissner-Nordström (RN) metric had not been comprehensively explored. A key theoretical distinction is that the CG field equations, being fourth-order, do not naturally generate the inverse quadratic ( $1/r^2$ ) term characteristic of the charged RN metric in GR. This absence fundamentally alters the behavior of the central singularity and the horizon structure, potentially allowing for exotic spacetime configurations not permitted in standard GR.

Methodology
The authors present a parametric study of the static, spherically symmetric, non-rotating dyonic (electric and magnetic charge) Reissner-Nordström solution in Weyl Conformal Gravity. The analysis proceeds through the following steps:

Metric Derivation: The study utilizes the Weyl-Maxwell system to derive the "blackening factor" $B(r)$ $B (r)$ for the CGRN metric. Two distinct cases are treated separately based on the linear parameter $\gamma$ $γ$ :
- Case $\gamma \neq 0$ : The metric includes constant, $1/r$ , linear ( $\gamma r$ ), and quadratic ( $\kappa r^2$ ) terms. The charge modifies the $1/r$ coefficient rather than introducing a $1/r^2$ term.
- Case $\gamma = 0$ : The metric reduces to a form resembling a GR Schwarzschild-(A)dS metric but with a modified constant term $w_0$ dependent on the charge.
Curvature and Singularity Analysis: The authors calculate the Ricci scalar ( $R$ ) and the Kretschmann scalar ( $K$ ) to identify physical singularities and determine their nature (timelike vs. spacelike) based on the parameters $\beta$ (mass), $\gamma$ , $\kappa$ , and the dimensionless dyonic parameter $D_g$ .
Geodesic and Photon Sphere Analysis: Equations of motion for null particles are derived to determine the effective potential $V_{eff}(r)$ . The locations of stable and unstable photon spheres are found by solving $dV_{eff}/dr = 0$ .
Horizon and Extremal Limit Analysis: Horizon radii are determined by solving $B(r)=0$ , resulting in cubic equations. The authors derive analytic formulae for extremal limits where horizons coincide (e.g., Event Horizon $H_E$ and Cauchy Horizon $H_C$ merging). Hawking temperatures are calculated via surface gravity.
Parameter Mapping: The spacetime structures are classified and mapped across the dimensionless parameter space of $\beta\gamma$ , $\beta^2\kappa$ , and $D_g^2$ .

Key Contributions and Results

Absence of $1/r^2$ Term: The study confirms that the CGRN metric lacks the $1/r^2$ term found in GR's RN metric. Consequently, the central singularity is not guaranteed to be timelike; its nature (spacelike or timelike) depends on the interplay between the charge parameter $D_g$ and the linear parameter $\gamma$ .
Exotic "Nested Black Hole" Structures: A primary finding is the existence of a "nested black hole" configuration (denoted as the $IUOI$ domain). In this spacetime, a Cauchy horizon is trapped between two event horizons ( $H_E^{inner} < H_C < H_E^{outer}$ ). This structure, where a Cauchy horizon does not surround the central singularity but is instead enclosed by event horizons, is not obtainable in standard GR.
Critical Charge Threshold ( $D_g^2 = 1$ ): The authors identify a critical value for the dyonic parameter.
- At $D_g^2 = 1$ , the stable and unstable photon spheres merge to form a marginally stable saddle-point photon sphere.
- This limit corresponds to an "extremal triple limit" ( $H_{TL}$ ) where three horizons coincide. Depending on the parameters, this can involve the merger of two event horizons and a Cauchy horizon, a configuration distinct from the standard "ultracold" limit in GR.
- For $D_g^2 > 1$ (in the $\gamma \neq 0$ case), photon spheres vanish entirely, and the spacetime structure simplifies to empty space, naked singularities, or metrics resembling Schwarzschild-Anti-de Sitter without photon spheres.
Charge Behavior Reversal: In the $\gamma \neq 0$ case, the effect of charge on horizon positions depends on the sign of $\beta\gamma$ . Unlike GR, where charge always pulls horizons together, in CG, positive charge can push event horizons outward and pull Cauchy horizons inward if $\beta\gamma < 0$ .
Extremal Limits: Analytic expressions are derived for four types of extremal limits:
1. Extremal Charge ( $H_{CE}$ ): $H_C$ and $H_E$ merge.
2. Nested Limit ( $H_{EC}$ ): $H_E$ is interior to $H_C$ and they merge.
3. Nariai Limit ( $H_{E\Lambda}$ ): $H_E$ and the cosmological horizon $H_\Lambda$ merge.
4. Triple Limit ( $H_{TL}$ ): Three horizons coincide.

Significance
The paper claims that the CGRN metric admits a significantly richer variety of spacetime configurations (20 distinct varieties) compared to the uncharged CG Schwarzschild metric (13 varieties). The most significant contribution is the demonstration that Weyl Conformal Gravity allows for "nested black hole" spacetimes and specific extremal limits involving the merger of three horizons in ways that are fundamentally different from General Relativity. The authors emphasize that the lack of the $1/r^2$ term in the fourth-order metric is the direct cause of these exotic structures, particularly the non-standard arrangement of Cauchy and event horizons. The work provides a comprehensive classification of these spacetimes, offering analytic formulae for horizons, photon spheres, and Hawking temperatures, which serves as a foundation for further investigation into the thermodynamics and near-horizon geometries of charged black holes in conformal gravity.