Regular black hole with sub-Planckian curvature and suppressed exponential mass inflation

This paper constructs a static, spherically symmetric regular black hole featuring a Minkowski core and a degenerate inner horizon, which ensures sub-Planckian curvature in the large-mass regime and suppresses exponential mass inflation to a power-law growth, resulting in a finite internal mass at late times.

The Gist

The Problem: The "Black Hole Singularity" and the "Inner Trap"

Imagine a black hole as a cosmic vacuum cleaner. According to old-school physics (General Relativity), if you fall into one, you eventually hit a point of infinite density called a singularity. Think of this like a mathematical "glitch" where the rules of the universe break down completely. It's like trying to divide by zero; the computer crashes.

Physicists have tried to fix this by building "Regular Black Holes" (RBHs). These are like upgraded models where the center isn't a broken glitch, but a smooth, safe zone (like a calm, flat room).

However, there is a second problem.
Most of these "safe" black holes have a hidden trap inside called an inner horizon. In standard physics, this inner horizon is unstable. It acts like a feedback loop in a microphone: a tiny whisper gets amplified into a deafening scream. In a black hole, this "scream" is a massive buildup of energy called mass inflation. Even if the center is safe, this inner trap might still explode with infinite energy, destroying the black hole's stability.

The Solution: A New "Regular" Black Hole

The authors of this paper have designed a new type of black hole that solves both problems at once. Here is how they did it, using three main features:

1. The "Soft Landing" Center

Instead of a singularity, the center of their black hole is a Minkowski core.

• Analogy: Imagine falling into a deep hole. In an old black hole, you hit a jagged, infinite spike at the bottom. In this new model, the bottom is a soft, flat trampoline. As you get closer to the center, the space becomes perfectly flat and calm, just like empty space far away from any stars.

2. The "Silent" Inner Horizon

The biggest breakthrough is how they handled the inner horizon.

• The Old Way: Usually, the inner horizon acts like a super-magnifying glass. Light and energy bounce back and forth, getting stronger and stronger exponentially (like a snowball rolling down a hill, getting huge very fast). This causes the "mass inflation" explosion.
• The New Way: The authors built a degenerate inner horizon.
• Analogy: Imagine the inner horizon is a door. In the old model, the door slams shut with infinite force, creating a shockwave. In this new model, the door is "stuck" in a way that it opens and closes with zero force. Because the "surface gravity" (the force pulling things in) is zero, the feedback loop is broken.
• The Result: Instead of an exponential explosion (a runaway train), the energy buildup slows down to a power-law behavior (like a gentle slope). It grows, but it grows slowly and stays finite. It never explodes.

3. The "Curvature Cap"

One of the biggest worries with these models is: "Is the gravity inside so strong that it breaks the laws of physics again?"

• The Finding: The authors found that for very large black holes, the maximum amount of "bending" in space (curvature) doesn't depend on how heavy the black hole is. Instead, it depends entirely on the size of that inner "safe zone."
• Analogy: Think of a rubber sheet. If you put a heavy bowling ball on it, the sheet bends a lot. Usually, the heavier the ball, the deeper the bend. But in this new model, the authors added a "stiffener" inside. No matter how heavy the bowling ball gets, the deepest bend in the sheet is limited by the size of the stiffener, not the weight of the ball.
• The Guarantee: By choosing the right size for this inner zone, they proved the bending of space never gets so extreme that it reaches "Planck scale" (the point where quantum gravity takes over). The universe stays "sub-Planckian" everywhere, meaning the math stays valid.

How They Tested It

To make sure their idea works, they ran two different "stress tests" using mathematical models:

1. The Double Shell Test: They imagined two shells of energy crashing into each other inside the black hole. In old models, this crash would cause infinite energy buildup. In their model, the crash happened, but the energy stayed finite and settled down to a specific, safe number.
2. The Ori Model: They simulated a continuous stream of rain (radiation) falling into the black hole while a shockwave moved out. Again, instead of the energy blowing up to infinity, it stabilized and settled at a value determined by the size of the inner horizon.

The Bottom Line

This paper presents a blueprint for a black hole that:

1. Has a smooth, safe center (no singularity).
2. Has an inner horizon that doesn't explode with infinite energy (no mass inflation).
3. Keeps the bending of space gentle enough that the laws of physics don't break, even for massive black holes.

It's like upgrading a car that used to crash into a wall (singularity) and had a steering wheel that spun out of control (inner horizon instability). The new model has a soft bumper and a steering system that locks gently, ensuring a safe ride even at high speeds.

Technical Summary

Technical Summary: Regular Black Hole with Sub-Planckian Curvature and Suppressed Exponential Mass Inflation

Problem Statement
Classical general relativity predicts spacetime singularities within black holes, indicating a breakdown of the theory in strong-field regimes. Regular Black Holes (RBHs) have been proposed as effective models to resolve these singularities, typically by modifying the deep interior to approach a regular core (e.g., Minkowski) while recovering Schwarzschild behavior at large radii. However, removing the central singularity is insufficient for physical viability. Most RBHs possess an inner (Cauchy) horizon, which is notoriously unstable due to the "mass inflation" mechanism. In standard scenarios, ingoing and outgoing radiation undergo infinite blue-shifting near the inner horizon, causing the effective internal mass to grow exponentially, potentially destabilizing the geometry and challenging semiclassical descriptions. While previous works suggested that degenerate inner horizons (with vanishing surface gravity, $\kappa_- = 0$) could soften this instability, a comprehensive model that simultaneously ensures sub-Planckian curvature for large masses and suppresses exponential mass inflation has remained a central consistency test.

Methodology
The authors construct a new static, spherically symmetric RBH metric defined by the line element $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2d\Omega^2$. The construction is guided by four specific requirements:

1. Asymptotic Schwarzschild behavior at large $r$.
2. Regularity at the center ($r \to 0$), approaching a Minkowski core.
3. A non-extremal outer horizon ($r_+$) and a degenerate inner horizon ($r_-$) with vanishing surface gravity ($\kappa_- = 0$).
4. A curvature scale that can remain below the Planck scale everywhere, particularly in the large-mass regime.

To achieve this, the metric function is formulated as $f(r) = N(r)/D(r)$, where the numerator $N(r)$ is designed to possess an effective triple root at $r_-$ (ensuring $\kappa_- = 0$ and the correct sign change for the metric inside the core) and a simple root at $r_+$. The authors introduce an auxiliary function $g(r) = \exp[(1 - r_-/r)^2] - 1$, which provides a double zero at $r_-$, making the effective order of the zero at the inner horizon cubic. The denominator $D(r)$ is constructed to preserve the root structure while ensuring the correct asymptotic and central limits.

The physical viability of the spacetime is analyzed through:

• Curvature Analysis: Calculation of the Kretschmann scalar ($K = R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$) to determine global curvature bounds and dependence on the ADM mass $M$ and inner horizon radius $r_-$.
• Energy Conditions: Examination of the Null Energy Condition (NEC) to identify potential pathologies in the effective stress-energy tensor.
• Mass Inflation Tests: Application of two effective models to test the stability of the inner horizon against perturbations:
  1. The Double Null Shell (DTR) model, analyzing the intersection of ingoing and outgoing null shells.
  2. The Ori model, modeling ingoing radiation as a continuous stream and outgoing perturbations as a null shell.

Key Contributions and Results

1. Sub-Planckian Curvature in the Large-Mass Regime:
   The analysis reveals that in the limit of large black hole mass ($r_+ = 2M$ with $r_- \ll r_+$), the Kretschmann scalar becomes nearly independent of the ADM mass $M$. Instead, the global curvature scale is governed primarily by the inner horizon radius $r_-$.

   • Numerical and analytical results show that the maximum curvature $K_{max}$ scales as $r_-^{-4}$.
   • By choosing an appropriate $r_-$, the spacetime can be made sub-Planckian ($K < 1$ in Planck units) everywhere. Specifically, for the subclass considered, $r_- \gtrsim 6.10174$ is sufficient to ensure $K_{max} < 1$ for arbitrarily large $M$.
   • The global maximum of the curvature is found to occur inside the inner horizon ($r_{max} < r_-$).

2. Suppression of Exponential Mass Inflation:
   The degenerate nature of the inner horizon ($\kappa_- = 0$) fundamentally alters the dynamics of perturbations.

   • DTR Model: The standard exponential amplification of the Misner-Sharp mass is replaced by power-law behavior. The shell-induced mass perturbations scale as $(r - r_-)^3$, and the shell-crossing point approaches the horizon as a power law in advanced time $v$. Consequently, the nonlinear term responsible for mass inflation is suppressed, and the future Misner-Sharp mass approaches a finite limit: $\lim_{r \to r_-} m_f = r_-/2$.
   • Ori Model: Using a Price-law decay for ingoing radiation, the numerical evolution confirms that the Misner-Sharp mass $m_2$ remains bounded. The near-horizon amplification factor grows only as a power law, which is insufficient to trigger exponential mass inflation given the rapid decay of the flux. The mass again stabilizes at $r_-/2$.

3. Energy Conditions:
   The radial Null Energy Condition (NEC) is identically saturated ($\rho + p_r = 0$). The angular NEC is strictly satisfied at the center, the inner horizon, and asymptotically. Numerical analysis indicates that any violation of the angular NEC is confined to a finite interval strictly below the inner horizon.

Significance
The paper claims to provide an explicit example of a regular black hole spacetime that simultaneously achieves three critical goals often in tension with one another:

1. Singularity Resolution: The central singularity is removed, replaced by a regular Minkowski core.
2. Controlled Curvature: The spacetime curvature remains sub-Planckian everywhere, even for macroscopic black holes, by decoupling the curvature scale from the ADM mass and linking it to the inner horizon scale.
3. Stabilized Inner Horizon: The standard exponential mass inflation mechanism is suppressed in favor of power-law behavior, resulting in a finite, stable internal mass scale determined by the inner horizon radius.

The authors note that while this model demonstrates the simultaneous achievement of these features within effective shell descriptions, a full semiclassical stability analysis and a derivation from a complete dynamical matter model remain beyond the scope of the current work. The construction serves as a proof of concept that the interplay between a degenerate inner horizon and specific metric modifications can yield a physically consistent regular black hole geometry.