Thermodynamic Phase Transitions and Quantum Entropy Corrections in the Simpson-Visser Regular Black Hole

This paper investigates the thermodynamic phase transitions and quantum entropy corrections of the Simpson-Visser regular black hole, demonstrating that singularity resolution induces critical instabilities and alters the evaporation end-state through a discontinuous heat capacity and leading-order quantum effects.

The Gist

Imagine a black hole not as a cosmic vacuum cleaner that ends in a terrifying, infinitely small point of destruction (a singularity), but as a cosmic "trampoline" that bounces back. This is the core idea of the Simpson–Visser Regular Black Hole, a model explored in this paper by Vinayak and Ashok Joshi.

Here is a breakdown of their findings using simple analogies:

1. The "Trampoline" Instead of the "Pit"

In standard black hole theory, if you fall in, you eventually hit a point where the laws of physics break down—a singularity. It's like falling into a bottomless pit.

The Simpson–Visser model suggests that instead of a pit, there is a smooth bounce.

• The Analogy: Imagine a trampoline. If you jump on it, you don't fall through to the center of the Earth; you hit the fabric and bounce back up (or through to the other side).
• The Result: The "center" of this black hole is a smooth, finite surface. It doesn't tear space-time; it just curves it back around. The paper calls this a "black-bounce."

2. The "Thermostat" Switch (Phase Transitions)

The authors discovered that this "trampoline" black hole behaves very differently from a normal one when it comes to heat and stability. They found a specific "switch" that changes the black hole's personality.

• The Unstable Phase (The Wild Fire): When the "bounce" is small (close to a normal black hole), the black hole is unstable. It's like a campfire that gets hotter the more fuel you take away. As it loses mass, it gets hotter and evaporates faster, eventually running away with itself.
• The Stable Phase (The Calm Lake): When the "bounce" is large enough, the black hole becomes stable. It's like a lake that can sit calmly in the sun without boiling away. It can reach a comfortable balance with its surroundings.
• The Switch: There is a precise point (a "critical value") where the black hole flips from being a wild, unstable fire to a calm, stable lake. The paper calls this a Phase Transition, similar to water turning into ice, but for black holes.

3. The "Quantum Microscope" (Entropy Corrections)

The paper also looked at what happens when you zoom in with a "quantum microscope" to see the tiny, fuzzy details of the black hole's heat and disorder (entropy).

• The Old View: Previously, scientists thought the "bounce" in the middle didn't matter much for the black hole's total heat until you got to the very end.
• The New Discovery: The authors found that the "bounce" actually changes the black hole's heat signature immediately, right from the start. It's like realizing that the material of a trampoline changes how a person bounces, not just at the very bottom, but the moment they jump.
• The Safety Net: They also found that the "bounce" acts as a safety net for the math itself. If the bounce is too small (getting close to the old, dangerous singularity), the quantum math starts to break down and go crazy. The "bounce" parameter keeps the math from falling apart.

4. The Final Fate: The "Cosmic Seed"

What happens when the black hole runs out of fuel?

• Old Theory: It might vanish completely, taking all its secrets with it (the Information Paradox).
• This Paper's Theory: Because of the "bounce" and the stability switch, the black hole doesn't vanish. Instead, it shrinks down until it hits a "floor" (the extremal state) and stops.
• The Result: It leaves behind a tiny, stable, zero-temperature remnant—a "cosmic seed." This seed has a specific amount of "quantum information" stored in it, determined by the size of the bounce.

Summary

The paper argues that fixing the "broken" center of a black hole (the singularity) isn't just a geometric tweak; it's a thermodynamic revolution.

1. It turns a runaway, unstable object into a stable one.
2. It changes the black hole's heat signature from the very beginning.
3. It ensures the black hole doesn't disappear but settles into a stable, tiny remnant, potentially solving the mystery of where the black hole's information goes.

In short: Regularizing the center of a black hole turns a chaotic, disappearing act into a stable, permanent cosmic object.

Technical Summary

Technical Summary: Thermodynamic Phase Transitions and Quantum Entropy Corrections in the Simpson–Visser Regular Black Hole

Problem Statement
The central challenge in theoretical physics remains the reconciliation of general relativity with quantum mechanics, particularly regarding the breakdown of classical theory at spacetime singularities. While the semiclassical approximation has established black hole thermodynamics (Bekenstein-Hawking entropy and Hawking temperature), standard models predict complete evaporation, inevitably leading to a singular region and exacerbating the information loss paradox. Existing frameworks for "regular" black holes (which resolve the central singularity) often rely on specific matter sources or cosmological constants (e.g., AdS spacetime) to induce phase transitions. A critical gap exists in understanding whether the asymptotically flat Simpson–Visser geometry, which resolves the singularity purely through a geometric "black-bounce" mechanism, exhibits intrinsic thermodynamic phase transitions and how singularity resolution modifies the leading-order quantum entropy.

Methodology
The authors employ the Simpson–Visser metric, a static, spherically symmetric solution characterized by a single regularization parameter $a$ (a fundamental length scale). This geometry interpolates between a Schwarzschild black hole ($a \to 0$), a regular "black-bounce" ($0 < a < 2m$), and a traversable wormhole ($a > 2m$). The investigation proceeds in two complementary phases:

1. Classical Thermodynamic Stability: The authors derive the Hawking temperature ($T_H$) and Bekenstein-Hawking entropy ($S_{BH}$) for the regular black hole regime ($0 < a \leq 2m$). They then calculate the heat capacity ($C$) as a function of mass $m$ and the regularization parameter $a$ to probe for critical behavior and thermodynamic stability. Free energy ($F$) is also computed to determine the preferred thermodynamic state.
2. Quantum Entropy Corrections: Moving beyond the semiclassical limit, the authors utilize the Hamilton-Jacobi tunneling formalism (extended by Banerjee and Majhi) to model Hawking radiation as a quantum tunneling process. They expand the particle's action in powers of $\hbar$ to account for back-reaction effects and self-gravitational interactions. This allows for the derivation of a quantum-corrected temperature ($T_{corr}$) and, subsequently, the integration of the first law of thermodynamics ($dS = dm/T_{corr}$) to obtain the leading-order quantum corrections to the entropy.

Key Contributions and Results

• Intrinsic Davies-Type Phase Transition:
  The analysis of the heat capacity reveals a critical instability driven solely by the regularization parameter $a$, without the need for a cosmological constant. The heat capacity diverges at a critical value $a_{crit} = \sqrt{2}m$, signaling a Davies-type phase transition.

  • Unstable Phase ($0 < a < \sqrt{2}m$): The heat capacity is negative ($C < 0$), analogous to the Schwarzschild black hole, indicating thermodynamic instability and runaway evaporation.
  • Stable Phase ($\sqrt{2}m < a < 2m$): The heat capacity becomes positive ($C > 0$), indicating a thermodynamically stable phase where the black hole can achieve equilibrium with a thermal reservoir.
  • Free Energy Analysis: While the system exhibits a stable phase, the free energy is minimized at $a=0$ (the singular Schwarzschild limit). However, the authors argue that if $a$ represents a fundamental length scale, the singular state is inaccessible. Instead, the thermodynamic evolution is driven by mass loss; as $m$ decreases for a fixed $a$, the system naturally transitions from the unstable to the stable regime.

• Quantum-Corrected Entropy and Regularization Dependence:
  The derivation of the quantum-corrected entropy via the tunneling formalism yields a result that fundamentally differs from the standard area law application.

  • Leading-Order Dependence on $a$: Contrary to the standard Bekenstein-Hawking result ($S_{BH} = 4\pi m^2$), which is independent of $a$ when evaluated strictly at the horizon, the thermodynamically integrated leading-order entropy explicitly depends on the regularization parameter $a$. This demonstrates that the regular core structure influences the entropy at the leading order.
  • Logarithmic and Inverse-Mass Corrections: The entropy includes a leading logarithmic correction term proportional to $\beta_1 \ln(2m + \sqrt{4m^2 - a^2})$ and a higher-order correction scaling as $1/a^2$.
  • Regulator Role of $a$: The $1/a^2$ scaling of the higher-order term implies that as $a \to 0$ (approaching the singular limit), the perturbative quantum expansion diverges. Thus, $a$ acts not only as a geometric regulator but also as a regulator for the validity of the effective field theory description of quantum entropy.

• Black Hole End-State:
  The combined analysis suggests a self-consistent end-state for black hole evaporation. As a black hole evaporates and its mass decreases, it eventually crosses the critical point into the stable phase. The evaporation slows and ceases as the system approaches the extremal limit ($m = a/2$), leaving a stable, zero-temperature remnant. The entropy of this remnant is non-zero and logarithmic, determined by the quantum gravity scale $a$.

Significance
The paper claims that resolving the singularity is not merely a geometric modification but a profound thermodynamic event with direct implications for black hole stability and fate. The primary significance lies in demonstrating that:

1. Singularity resolution drives phase transitions: The regularization parameter acts as a control parameter that can drive a black hole from an unstable to a stable thermodynamic phase purely through geometric modification.
2. Thermodynamic sensitivity to the core: The entropy of a regular black hole is sensitive to the internal regular structure even at the leading order, challenging the notion that such effects are confined to subleading quantum corrections.
3. Quantum gravity scale as a regulator: The regularization parameter $a$ serves a dual role, regulating both the geometric singularity and the convergence of the quantum thermodynamic expansion.
4. Remnant stability: The results provide a statistical-mechanical basis for the existence of stable black hole remnants, offering a potential pathway to resolve the information loss paradox by preventing the system from reaching a singular state.

The authors conclude that the Simpson–Visser model offers a self-consistent framework for describing the black hole life cycle without singularities, where the interplay between classical thermodynamic stability and quantum corrections dictates the ultimate fate of evaporating black holes.