⚛️ general relativity

Globally regular charged black holes in non-polynomial quasi-topological gravity with Born-Infeld electrodynamics

This paper constructs exact static, spherically symmetric charged solutions in four-dimensional non-polynomial quasi-topological gravity coupled to Born-Infeld electrodynamics, demonstrating that nonlinear electrodynamics can resolve the finite-radius curvature singularity of the vacuum branch to yield globally regular black holes with finite-curvature AdS cores and nontrivial inner-horizon structures.

Original authors: Hong-Lin Liu, Zhong-Wen Feng, Qing-Quan Jiang, Xia Zhou, Xue-Ling Mu

Published 2026-09-09

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CC BY 4.0

Original authors: Hong-Lin Liu, Zhong-Wen Feng, Qing-Quan Jiang, Xia Zhou, Xue-Ling Mu

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

Deep within the heart of a black hole, the laws of physics as we know them seem to collapse. According to the standard theory of gravity, the center of these cosmic traps is a point of infinite density and zero volume, a place where the fabric of space and time tears apart. This "singularity" is not just a mathematical quirk; it signals that our current understanding of the universe breaks down completely in these extreme environments. For decades, physicists have wondered if nature actually allows for such a tear, or if a more complete theory of gravity would smooth out this rough edge, replacing the infinite point with a dense, finite core. The search for these "regular" black holes—objects that look like black holes from the outside but possess a smooth, singularity-free interior—has become a major frontier in theoretical physics.

The challenge lies in finding a theory that naturally produces such objects without requiring artificial adjustments. While some models have successfully described regular black holes in empty space, a persistent problem has been that adding electric charge to these models often destroys their smoothness, reintroducing the very singularity scientists hoped to avoid. It seemed that the delicate balance required to keep the center of a black hole smooth could not survive the introduction of real-world matter like electricity. However, a new study by researchers at China West Normal University and Chengdu University suggests that this balance can be restored, but only if we change how we view the interaction between gravity and the electromagnetic field at the most extreme scales.

The researchers set out to test a specific, complex theory of gravity known as non-polynomial quasi-topological gravity. This framework modifies Einstein's classic equations by adding terms that become important only when gravity is incredibly strong, such as near the center of a black hole. In a vacuum, or empty space, this specific version of the theory actually predicts a black hole with a fatal flaw: a singularity that appears not at the very center, but at a specific, finite distance away from it. It is as if the space-time fabric snaps at a certain radius, preventing the geometry from being extended all the way to the middle. The team wanted to see if coupling this theory with a specific type of electricity, known as Born-Infeld electrodynamics, could fix this broken geometry. Unlike standard electricity, which grows infinitely strong as you get closer to a charge, Born-Infeld electrodynamics imposes a natural limit on how strong the electric field can become, preventing it from blowing up to infinity.

By combining these two ideas, the team constructed exact mathematical models of charged black holes and found that the nonlinear behavior of the electric field acts as a stabilizing force. The electric field, which is capped at a maximum strength, interacts with the modified gravity in such a way that it prevents the gravitational equations from reaching the critical point where they would otherwise break. Instead of snapping at a finite distance, the geometry of the black hole remains smooth all the way to the center. The result is a globally regular black hole: an object with an event horizon that traps light, an exterior that looks like a standard black hole, and an interior that transitions into a finite, smooth core with a specific type of curvature, rather than an infinite tear.

The study reveals that this solution is not a fragile accident that requires precise tuning of numbers to work. Instead, the researchers identified a broad region of parameters where these smooth, charged black holes exist naturally. Within this region, they found two distinct types of objects: regular black holes that possess an event horizon, and "horizonless" configurations that look like dense stars but lack a point of no return. These two types are separated by a sharp boundary where the horizon becomes degenerate, or "frozen." Perhaps most surprisingly, the team discovered a continuous family of these regular black holes that possess a unique internal structure. These objects feature an outer event horizon and an inner horizon that is "triple-degenerate," meaning it is a highly stable, flattened surface where the usual forces that would destabilize a black hole's interior are effectively suppressed.

This finding offers a converse to a long-standing problem in the field. Previously, it was thought that introducing charge to a regular vacuum black hole would inevitably spoil its smoothness. Here, the researchers show the opposite can happen: a theory that produces a singular black hole in a vacuum can be "cured" by adding charge, provided the electric field follows the Born-Infeld rules. The electric field does not just add weight to the system; it fundamentally alters the radial dependence of the gravitational field, steering the solution away from the singularity and into a stable, regular state. The work demonstrates that the path to a singularity-free universe might not require discarding our current theories of gravity, but rather understanding how they interact with matter in the most extreme conditions imaginable. The universe, it seems, may have a way of keeping its own secrets smooth, even at the very edge of a black hole.

Technical Summary: Globally Regular Charged Black Holes in Non-Polynomial Quasi-Topological Gravity with Born-Infeld Electrodynamics

Problem Statement
Classical black hole solutions in general relativity inevitably contain spacetime singularities, signaling the breakdown of the classical description in high-curvature regimes. While Regular Black Holes (RBHs) have been proposed as nonsingular alternatives, a fundamental challenge remains: whether such geometries arise naturally from well-defined gravitational theories or require phenomenological metric modifications. Furthermore, recent studies indicate that introducing charge or matter fields into theories with regular vacuum solutions can often induce curvature singularities that were absent in the uncharged case. Conversely, this paper investigates the complementary scenario: can nonlinear matter fields (specifically Born-Infeld electrodynamics) remove a finite-radius curvature singularity present in the vacuum solution of a higher-curvature gravity theory, thereby yielding globally regular charged geometries?

Methodology
The authors construct exact static, spherically symmetric charged solutions within the framework of four-dimensional non-polynomial quasi-topological gravity (NPQTG) coupled to Born-Infeld electrodynamics.

Gravitational Sector: The study utilizes the single-function static, spherically symmetric sector of NPQTG. The dynamics are governed by a characteristic function $h(p)$ , where $p = [1-f(r)]/r^2$ is a curvature variable. The authors specifically select the model $h(p) = p/(1 + \ell^2 p)$ . Unlike the Hayward-type model ( $h(p) = p/(1 - \ell^2 p)$ ), this choice leads to a vacuum solution that develops a curvature singularity at a finite radius $r_s$ where the characteristic relation diverges.
Matter Sector: The theory is coupled to Born-Infeld (BI) electrodynamics, characterized by a Lagrangian that recovers Maxwell theory in the weak-field limit but modifies electromagnetic dynamics in the strong-field regime via a parameter $b$ .
Solution Construction: The authors derive the reduced field equations for the spherical sector. By solving the electromagnetic equation, they obtain the electric field $E(r)$ . Substituting this into the gravitational equation yields an algebraic relation $h(p) = H_{BI}(r)$ , where $H_{BI}(r)$ incorporates both the mass term and the BI electromagnetic contribution. The metric function $f(r)$ is then determined exactly by inverting this relation.
Regularity Analysis: The global regularity of the solution is analyzed by examining the behavior of $H_{BI}(r)$ . The solution is singular if $H_{BI}(r)$ reaches the critical value $1/\ell^2$ at any finite radius. The authors analyze the dimensionless profile of $H_{BI}(r)$ to determine the parameter space where the maximum value of $H_{BI}(r)$ remains strictly below this critical threshold.

Key Contributions and Results

Removal of Vacuum Singularities: The primary result is that Born-Infeld nonlinearities can indeed remove the finite-radius curvature singularity inherent in the vacuum branch of the chosen NPQTG model. While the vacuum solution ( $Q=0$ ) inevitably hits the singularity at $r_s = (\mu \ell^2)^{1/3}$ , the inclusion of charge modifies the radial profile of the gravitational source term. Within a specific finite region of the parameter space (defined by the mass $M$ , charge $Q$ , BI scale $b$ , and curvature scale $\ell$ ), the function $H_{BI}(r)$ never reaches the critical value $1/\ell^2$ .
Global Geometry: The resulting charged solutions are globally regular. The spacetime extends from an asymptotically flat exterior to a finite-curvature Anti-de Sitter (AdS)-type core. Specifically, near the center ( $r \to 0$ ), the metric function behaves as $f(r) \approx 1 + r^2/\ell^2$ , indicating a regular core with curvature scale $\ell$ .
Curvature Invariants: The authors verify that both the Ricci scalar $R$ and the Kretschmann scalar $K$ remain finite at the center, with values $R(0) = -12/\ell^2$ and $K(0) = 24/\ell^4$ . Notably, this regularity is achieved despite the fact that the Born-Infeld energy density and radial pressure diverge as $r^{-2}$ near the center; the regularity arises from the combined effect of the divergent matter stress tensor and the non-polynomial gravitational dynamics.
Horizon Structure and Phase Space: The globally regular solution space contains both horizonless configurations and RBHs. These two classes are separated by a degenerate-horizon boundary. The authors identify a continuous branch of regular black holes characterized by a triple-degenerate inner horizon and a simple outer event horizon. For this branch, the surface gravities satisfy $\kappa_- = 0$ (vanishing inner horizon surface gravity) and $\kappa_+ \neq 0$ .
Contrast with Previous Models: The paper highlights a converse mechanism to previous findings. While some theories possess regular vacuum solutions that become singular upon adding charge, this model features a singular vacuum branch that becomes globally regular upon the introduction of charge.

Significance
The paper demonstrates that nonlinear electrodynamics can act as a mechanism to "cure" singularities in higher-curvature gravity theories that are otherwise singular in vacuum. This provides a concrete example where the introduction of matter fields alters the global causal structure of a gravitational theory, transforming a singular vacuum branch into a sector of globally regular charged black holes. The identification of a specific parameter region supporting RBHs with nontrivial inner-horizon structures (specifically the triple-degenerate inner horizon with $\kappa_- = 0$ ) offers a potential candidate for studying the stability of black hole interiors, as configurations with vanishing inner surface gravity are known to potentially suppress the mass-inflation instability. The work underscores that the regularity of a gravitational theory coupled to matter cannot be inferred solely from its vacuum solutions.

Technical Summary: Globally Regular Charged Black Holes in Non-Polynomial Quasi-Topological Gravity with Born-Infeld Electrodynamics

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