Charged Regular Black Holes From Quasi-topological Gravities in $D\ge 5$

This paper constructs a unique spherically symmetric charged black hole solution in $D \ge 5$ dimensions within a quasi-topological gravity framework featuring an infinite tower of higher-curvature corrections, demonstrating that an appropriate choice of coupling coefficients progressively mitigates and ultimately resolves the central singularity into a globally regular spacetime.

The Gist

The Big Problem: The "Singular" Glitch

Imagine General Relativity (Einstein's theory of gravity) as a very powerful video game engine. It works perfectly for 99% of the universe. But when you try to simulate a black hole, the engine crashes. It hits a point called a singularity—a place where the math says gravity becomes infinite, space-time tears apart, and the laws of physics stop making sense. It's like a video game character falling through the floor into an endless void of "NaN" (Not a Number).

For decades, physicists have tried to fix this glitch. Some tried patching the code with "exotic matter" (imaginary stuff that doesn't exist), while others just edited the map to hide the hole. This paper proposes a different solution: upgrade the engine itself.

The New Engine: "Quasi-Topological Gravity"

The authors are working in a universe with 5 or more dimensions (imagine our 3D space plus time, plus a few extra hidden dimensions, like in Interstellar or Stranger Things).

They use a theory called Quasi-topological Gravity. Think of this theory as a "smart" version of gravity that doesn't just stop at the basic rules (Einstein's equations). Instead, it adds an infinite tower of correction terms.

• The Analogy: Imagine you are trying to describe the shape of a bumpy road.
  • Standard Gravity is like drawing a straight line. It works okay for flat roads but fails miserably at the bumps.
  • This New Theory is like adding more and more detailed curves to your drawing. First, you add a curve for the first bump, then a curve for the next, and so on.
  • The Magic: The authors show that if you add enough of these correction curves (an infinite number), the "bump" (the singularity) doesn't just get smaller; it disappears entirely. The road becomes smooth again.

The Challenge: Adding "Charge"

In the real world, black holes can have electric charge (like a giant static shock). In standard physics, adding charge to a black hole makes the singularity problem worse, not better. It's like trying to fix a glitchy video game while someone is constantly throwing more bugs into the code.

Previous attempts to fix charged black holes required inventing new, weird types of electricity (Non-Linear Electrodynamics). This paper is special because it keeps the electricity normal (standard Maxwell equations) but fixes the gravity side instead.

The Solution: The "Gravity Sponge"

The authors found that in this 5+ dimensional universe, the infinite tower of gravity corrections acts like a super-sponge.

1. The Divergence: As you get closer to the center of the black hole, the electric field tries to become infinitely strong (it wants to blow up).
2. The Absorption: The infinite gravity corrections kick in. They don't fight the electricity; they absorb the energy.
3. The Result: Instead of a point of infinite density (a singularity), the center of the black hole becomes a smooth, finite ball of space. It's not a hole anymore; it's a regular core.

Think of it like a pressure cooker. In standard gravity, the pressure builds up until the lid blows off (the singularity). In this new theory, the "safety valve" (the infinite corrections) opens up just enough to release the pressure, keeping the pot intact and safe.

The "Zoo" of Black Holes

The paper also explores what happens when you change the mass and charge of these new black holes. They found a "zoo" of different states:

• The Normal Black Hole: Heavy enough to have an event horizon (the point of no return) and a Cauchy horizon (a second inner boundary).
• The Extremal Black Hole: The "Goldilocks" state where the inner and outer horizons merge into one. It's the smallest possible black hole that still has a horizon.
• The Naked Soliton: If the black hole isn't heavy enough, it has no event horizon at all. The smooth, regular core is visible to the outside world. It's a "naked" star-like object that looks like a black hole but doesn't trap light.

Why Does This Matter?

1. No Magic Required: We don't need to invent new, fake particles to fix the universe. We just need to understand that gravity is more complex than Einstein thought, especially at tiny scales.
2. Quantum Gravity Clue: Since these corrections look like what we expect from a theory of Quantum Gravity (the theory that unifies gravity and quantum mechanics), this suggests that nature might actually be "regular" at the center of black holes. The "singularity" might just be a sign that our current math is too simple.
3. Higher Dimensions: It gives us a playground to test how gravity behaves in 5, 6, or more dimensions, which is crucial for String Theory.

The Bottom Line

This paper says: "If you look at black holes through the lens of a more advanced gravity theory with infinite corrections, the scary, infinite singularity at the center vanishes. It gets replaced by a smooth, finite core. The universe is safe, the math works, and the black hole is just a very dense, very smooth ball of space."

It's like realizing the "crash" in the video game wasn't a bug in the universe, but just a bug in our old, simple version of the game engine.

Technical Summary

1. Problem Statement

General Relativity (GR) predicts that gravitational collapse leads to black holes containing physical singularities where curvature invariants diverge, signaling a breakdown of causality and the theory itself. While quantum gravity is expected to resolve this, a complete theory remains elusive.

• The Gap: Existing approaches to regular black holes often rely on ad hoc metric modifications or the introduction of exotic matter (e.g., Non-Linear Electrodynamics), which lack fundamental physical motivation or suffer from instabilities.
• The Specific Challenge: Extending regular black hole solutions from vacuum (Schwarzschild-like) to charged (Reissner-Nordström-like) scenarios in higher dimensions ($D \ge 5$) is non-trivial. In standard GR, the electric charge term in the metric function leads to a $1/r^{2D-6}$ divergence at the origin, creating a singularity. The authors aim to resolve this singularity using a modified gravity framework that retains the standard linear Maxwell field.

2. Methodology

The authors employ Quasi-topological Gravity (QTG), a class of higher-curvature theories in $D \ge 5$ dimensions, extended to include an infinite tower of curvature corrections.

• Action Formulation: The model uses an effective action containing the Einstein-Hilbert term, an infinite series of higher-curvature invariants ($Z_n$) with coupling constants $\alpha_n$, and the standard Maxwell Lagrangian ($L_M = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$).
  $$S = \int d^Dx \sqrt{|g|} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{max}} \frac{\alpha_n Z_n}{16\pi G} + L_M \right]$$
• Ansatz: A static, spherically symmetric metric is assumed:
  $$ds^2 = -N(r)^2 f(r) dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega_{D-2}^2$$
  coupled with a Maxwell potential $A_\mu dx^\mu = \Phi(r) dt$.
• Reduction to Algebraic Equations: Using reduced Lagrangian methods, the authors derive the field equations. A key feature of QTG is that the complex fourth-order differential equations reduce to a single algebraic master equation for the metric function $f(r)$ (or the auxiliary variable $\psi = (1-f)/r^2$):
  $$h(\psi) = \frac{m}{r^{D-1}} - \frac{q}{r^{2D-4}}$$
  where $h(\psi) = \psi + \sum \alpha_n \psi^n$ is the characteristic polynomial of the theory.
• Constraints on Couplings: To ensure a unique, globally regular solution, specific constraints are imposed on the coupling coefficients $\alpha_n$:
  1. Odd Function: Even-order couplings vanish ($\alpha_{2k}=0$) to allow $h(\psi)$ to cover the full range of the source term (which can be negative).
  2. Monotonicity: Odd-order couplings are non-negative ($\alpha_{2k+1} \ge 0$) to prevent branch singularities.
  3. Finite Convergence Radius: The series must have a finite radius of convergence such that as the source term diverges ($r \to 0$), $\psi$ saturates to a finite constant, preventing the metric function from diverging.

3. Key Contributions

1. Algebraic Derivation of Charged Solutions: The paper successfully generalizes previous vacuum QTG results to charged black holes. It demonstrates that even with a minimally coupled linear Maxwell field, the field equations remain algebraic, yielding exact analytic solutions for specific correction orders ($n=3$) and the infinite-order limit.
2. Mechanism of Regularization: The authors identify a novel regularization mechanism where the gravitational sector (via infinite-order curvature corrections) absorbs the divergence of the linear Maxwell field. Unlike Bardeen or Hayward models that require Non-Linear Electrodynamics (NLED), this model retains standard Maxwell electrodynamics; the singularity is resolved purely by the non-perturbative response of the geometry.
3. Phase Structure Analysis: The paper provides a comprehensive analysis of the horizon structure and phase space for $D=5$, defining the conditions for non-extremal black holes, extremal black holes, and horizonless regular solitons.

4. Key Results

• Asymptotic Behavior at the Origin ($r \to 0$):
  • Finite Order ($n < \infty$): The solution exhibits a divergence, though the severity decreases as the order $n$ increases.
  • Infinite Order ($n \to \infty$): The solution becomes globally regular. The metric function behaves as $f(r) \approx 1 + C r^2$ near the origin, indicating an Anti-de Sitter (AdS) core.
  • Independence from Charge: In the infinite-order limit, the core geometry becomes independent of the electric charge $q$. The effective cosmological constant of the core is determined solely by the asymptotic behavior of the gravitational couplings $\alpha_n$.
• Curvature Invariants: The Kretschmann scalar ($K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$) remains finite everywhere. However, the paper notes that $K$ exhibits non-monotonic behavior and sharp peaks (a "curvature wall") during the transition from the classical exterior to the saturated core, which are physical features of the regularization mechanism, not numerical artifacts.
• Extremality Conditions: For $D=5$, the authors derive parametric equations for the critical mass ($m_{ext}$) and charge ($q_{ext}$) that define the boundary between black holes and regular solitons.
  • Case I ($m > m_{ext}$): Regular non-extremal black hole (two horizons).
  • Case II ($m = m_{ext}$): Extremal regular black hole (degenerate horizon).
  • Case III ($m < m_{ext}$): Regular soliton (no horizon, naked regular core).
• Recovery of GR: In the limit where higher-curvature corrections vanish ($\alpha \to 0$), the results reduce to the standard 5D Tangherlini-Reissner-Nordström black hole.

5. Significance

• Theoretical Robustness: The work provides a compelling argument that spacetime singularities can be resolved within a classical framework extended by effective field theory (EFT) arguments (infinite curvature series) without invoking exotic matter or modifying Maxwell's equations.
• Universality: It suggests that the regularization of singularities is a generic feature of quasi-topological gravities with infinite-order corrections, applicable to both vacuum and charged scenarios.
• Astrophysical and String Theory Relevance: While astrophysical black holes may discharge rapidly, these solutions serve as crucial analogues for the internal structure of rotating black holes and offer insights into the extremal limits relevant to String theory and M-theory.
• Stability: The algebraic nature of the field equations in the static sector avoids Ostrogradsky ghosts, suggesting the model is theoretically healthy, though full dynamical stability analysis is left for future work.

In summary, the paper establishes that Quasi-topological Gravity with infinite-order corrections provides a mathematically consistent and physically motivated framework for constructing charged, regular black holes in higher dimensions, where the gravitational self-interaction naturally cures the singularity associated with electric charge.