Corrected Hawking Temperature and Final State of Black Hole Evaporation Under GEVAG Framework

This paper demonstrates that within the GEVAG framework, promoting the horizon-area-dependent effective gravitational constant to the horizon neighborhood yields a corrected Hawking temperature with a second term that drives the temperature to zero at a minimum mass, thereby resolving the nonzero remnant temperature inconsistency found in Generalized Uncertainty Principle (GUP) approaches while providing general formulas for thermodynamic energy and the Bekenstein bound.

The Gist

The Big Picture: The Black Hole's Last Breath

Imagine a black hole not as a cosmic vacuum cleaner, but as a glowing ember in the dark. For decades, physicists have known that these embers slowly lose heat and shrink, a process called "Hawking evaporation."

The big mystery has always been: What happens when the ember gets really, really small?

Does it vanish completely? Does it stop shrinking and leave behind a tiny, hot "dust bunny" (a remnant) that never goes away? Or does it cool down to absolute zero and fade into nothingness?

This paper argues that the ember doesn't just stop; it cools down to zero right as it reaches its smallest size. The author, Yen Chin Ong, uses a new way of thinking about gravity to fix a problem in previous theories.

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The Old Problem: The "Stuck" Ember

For a long time, physicists used a tool called the Generalized Uncertainty Principle (GUP) to predict the end of a black hole. Think of GUP as a rulebook that says, "You can't squeeze space and time too tightly; there's a minimum size limit."

Using this rulebook, the math suggested that as a black hole shrinks, it gets hotter and hotter until it hits a minimum size. At that point, the math said the temperature would stop changing. It would become a tiny, super-hot rock that stays hot forever.

The Analogy: Imagine you are blowing out a candle. As the flame gets smaller, it flickers and gets hotter. Suddenly, the flame stops shrinking but stays at a scorching 1,000 degrees. It never goes out. This feels wrong. If something is losing energy, it should eventually run out of steam and go cold, not stay hot forever.

This "stuck hot remnant" was a major headache for physicists because it didn't make sense physically.

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The New Solution: The "Variable Gravity" Engine

The author introduces a framework called GEVAG (Generalized Entropy Varying-G).

To understand this, imagine gravity isn't a fixed setting on a thermostat (like "always 70 degrees"). Instead, imagine gravity is a smart thermostat that changes its setting depending on how much "stuff" (entropy) is in the room.

In standard physics, the "gravitational constant" ($G$) is a fixed number. In GEVAG, this number becomes effective gravity ($G_{eff}$), which changes as the black hole's surface area changes.

The Two-Part Temperature Formula

The author realized that previous calculations treated this "smart gravity" as if it were frozen in place once the black hole reached the horizon (the edge). But if the black hole is shrinking, the gravity at the edge is actually changing right alongside it.

When you account for this change, the temperature formula splits into two parts:

1. The Standard Part: This matches the old GUP prediction (the part that gets hot).
2. The Correction Part: This is a new term that appears because gravity is "running" or changing.

The Analogy: Think of driving a car down a hill.

• Old Theory: You press the gas, and the car speeds up until you hit a speed limit sign, then you just cruise at that speed forever.
• New Theory (GEVAG): You press the gas, but as you get closer to the bottom, the road gets steeper and the brakes automatically engage harder. The car speeds up, reaches a peak, and then the brakes take over, slowing the car down until it comes to a gentle, perfect stop.

The Result: A Gentle Fade to Zero

When the author applied this new math to the black hole:

1. As the black hole shrinks, the temperature rises (just like the old theory).
2. But as it gets very close to the minimum size, the "correction term" kicks in.
3. This term acts like a brake, pulling the temperature down.
4. The Magic Moment: At the exact moment the black hole hits its minimum size, the temperature hits absolute zero.

The black hole doesn't leave behind a hot, stubborn remnant. Instead, it evaporates completely, fading away into a cold, empty state. This solves the "stuck hot ember" problem perfectly.

Why This Matters: The "Bekenstein Bound"

The paper also touches on a concept called the Bekenstein Bound. Think of this as a "storage limit" for information. It says a black hole can only hold a certain amount of data (entropy) relative to its size and energy.

The author shows that with this new "smart gravity" view, the universe's storage limit remains stable and sensible. It suggests that the laws of physics are "natural"—meaning they don't require weird, fine-tuned tricks to work. The math flows smoothly, like a river finding its way to the sea, rather than hitting a sudden, impossible wall.

Summary in a Nutshell

• The Problem: Old theories said black holes would shrink until they became tiny, permanently hot rocks. This felt wrong.
• The Fix: The author realized that as a black hole shrinks, the strength of gravity at its edge changes dynamically.
• The Outcome: This change in gravity acts like a brake. It slows the black hole's temperature down just as it reaches its smallest size.
• The Conclusion: Black holes don't get stuck; they cool down to zero and vanish completely. The universe is tidier and more logical than we thought.

It's like realizing that a melting ice cube doesn't turn into a hot, solid pebble at the end; it just melts away into a puddle of water and disappears.

Technical Summary

1. Problem Statement

The paper addresses a critical inconsistency in the standard Generalized Uncertainty Principle (GUP) approach to black hole evaporation.

• The GUP Paradox: In standard GUP models, the Hawking temperature ($T$) approaches a non-zero finite value as the black hole mass ($M$) approaches a minimum remnant mass ($M_{min}$). This implies the black hole stops evaporating at a finite temperature, leaving a "hot" stable remnant. This is physically puzzling because a stable remnant with non-zero temperature suggests a continuous emission of energy without mass loss, or a violation of thermodynamic equilibrium.
• The Limitation of Previous GEVAG Models: Previous work using the Generalized Entropy Varying-G (GEVAG) framework reproduced the GUP temperature result but failed to resolve this "non-zero temperature" pathology. This was because previous models treated the effective gravitational constant ($G_{eff}$) as a static parameter defined strictly on the horizon, ignoring its variation in the immediate vicinity of the horizon.

2. Methodology

The author employs the GEVAG (Generalized Entropy Varying-G) framework, which posits that generalizing the Bekenstein-Hawking entropy ($S = f(A)/4G$) necessitates a field equation where the gravitational constant becomes an area-dependent effective constant ($G_{eff}$).

Key Methodological Steps:

1. Off-Shell Extension: The core innovation is promoting $G_{eff}$ from a quantity defined solely on the horizon ($r = r_h$) to a function valid in the neighborhood of the horizon, $G_{eff}(r)$. This is motivated by the highly dynamical nature of spacetime during the final stages of evaporation.
2. Modified Surface Gravity Calculation: The Hawking temperature is derived from the surface gravity $\kappa = g'(r_h)/2$.
   • In standard treatments, $G_{eff}$ is treated as a constant during differentiation.
   • In this work, because $G_{eff}$ is a function of $r$ (via the area $A=4\pi r^2$), the derivative of the metric function $g(r)$ yields two terms:
     $$T = \frac{1}{4\pi} \left( \frac{1}{r_h} - \frac{G'_{eff}}{G_{eff}} \bigg|_{r=r_h} \right)$$
3. Entropy Correction: The analysis specifically applies a logarithmic correction to the entropy, a common feature in various quantum gravity approaches:
   $$S = \frac{A}{4G} + c_1 \ln\left(\frac{A}{G}\right)$$
   This leads to a specific form for $G_{eff}$ dependent on the horizon area.

3. Key Contributions

• Derivation of the Full Temperature Formula: The paper derives a complete expression for Hawking temperature under GEVAG, showing it consists of two distinct parts:
  $$T = T_{GUP} + T_{correction}$$
  • $T_{GUP}$: The first term matches the standard GUP result exactly.
  • $T_{correction}$: A new term arising from the derivative of the varying gravitational constant ($G'_{eff}$).
• Resolution of the Remnant Problem: The paper demonstrates that the $T_{correction}$ term is negative and exactly cancels the positive $T_{GUP}$ term as the black hole approaches the minimum mass ($M_{min}$).
• General Thermodynamic Formulas: The author derives general formulas for thermodynamic energy ($E$) and the Bekenstein bound ($C_B$) for any arbitrary entropy correction $f(A)$, not just the logarithmic case.
• RG-Scaling Interpretation: The paper interprets the generalized Bekenstein bound in terms of the Renormalization Group (RG) scaling dimension of the entropy functional, linking thermodynamic bounds to the "naturalness" of the theory.

4. Key Results

• Zero-Temperature Remnant: For the case of positive logarithmic correction ($c_1 > 0$), the total temperature $T$ goes to zero as $M \to M_{min}$.
  • This resolves the GUP inconsistency. Instead of a "hot" remnant, the black hole evolves into a "cold" stable remnant, asymptotically approaching zero temperature.
  • This behavior aligns with various other quantum gravity models (e.g., non-commutative geometry, asymptotic safety, loop quantum gravity).
• Thermodynamic Energy: The thermodynamic energy $E$ is derived as:
  $$E(A) = \frac{\sqrt{A}}{4G\sqrt{\pi}} \left( 1 + \frac{c_1}{A} \right)$$
  This recovers the ADM mass ($E=M$) in the limit where corrections vanish ($c_1 \to 0$).
• Bekenstein Bound Behavior:
  • The Bekenstein constant $C_B$ is found to be finite and bounded.
  • In the large area limit, $C_B \to 2\pi$ (recovering General Relativity).
  • At the minimum area, $C_B$ deviates but remains finite (e.g., $C_B = \pi$ for $c_1=1$).
  • The paper argues that the condition $C_B \sim O(1)$ implies the entropy functional has a natural RG scaling dimension ($\gamma \approx 1$).
• Instability of Negative Corrections: The paper notes that if the correction parameter is negative ($c_1 < 0$, corresponding to a negative GUP parameter $\alpha$), the temperature diverges and the Bekenstein bound is violated. This suggests that negative GUP parameters may be unphysical or require higher-order corrections to stabilize.

5. Significance

• Theoretical Consistency: The work provides a self-consistent classical effective theory (GEVAG) that reproduces GUP results without explicitly invoking the Generalized Uncertainty Principle, while simultaneously curing the GUP's thermodynamic pathology (the non-zero temperature remnant).
• Physical Mechanism: It identifies the "running" of the gravitational constant near the horizon as the physical mechanism responsible for cooling the black hole to absolute zero, offering a more robust picture of the final state of evaporation.
• Unification of Concepts: By linking the varying-$G$ framework, entropy corrections, and RG scaling dimensions, the paper bridges gaps between thermodynamic gravity, quantum gravity phenomenology, and renormalization group theory.
• Validation of Off-Shell Assumptions: The fact that the off-shell extension of $G_{eff}$ leads to an exact cancellation resulting in $T=0$ serves as a strong theoretical validation for assuming $G_{eff}$ varies in the horizon neighborhood.

In conclusion, the paper argues that the "pathology" of the GUP black hole remnant is an artifact of treating the gravitational constant as static. By consistently applying the GEVAG framework where $G$ varies with the horizon area, the model naturally predicts a zero-temperature final state, aligning with the expectations of a stable quantum gravitational remnant.