⚛️ general relativity

Complete evaporation of black holes and Page curves

This paper proposes that by considering specific mass-charge relationships in Reissner-Nordström black holes (and extending the analysis to Kerr and de Sitter variants), complete evaporation can occur without a temperature blow-up, resulting in radiation entropy that follows the Page curve and vanishes at the end of the process.

Original authors: Irina Aref'eva, Igor Volovich

Published 2026-10-20

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Irina Aref'eva, Igor Volovich

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

In the vast, quiet theater of the cosmos, black holes are often imagined as eternal, unchanging pits of gravity. However, decades of theoretical work have revealed a different story: these objects are not static. They slowly leak energy in the form of radiation, a process known as evaporation. As a black hole loses this energy, it shrinks, and paradoxically, it gets hotter. This relationship creates a terrifying scenario for the smallest black holes. If a black hole were to shrink all the way to nothing, its temperature would theoretically shoot up to infinity, releasing an infinite amount of energy in a final, catastrophic explosion. This "temperature explosion" is not just a mathematical curiosity; it suggests a fundamental breakdown in our understanding of physics, implying that information about what fell into the black hole is lost forever, violating a core principle of quantum mechanics that states information must be preserved.

Two physicists from the Steklov Mathematical Institute in Moscow, Irina Aref'eva and Igor Volovich, have proposed a way to resolve this paradox without requiring a violent end to the black hole's life. They suggest that if a black hole possesses certain extra characteristics, such as an electric charge or a spin, it can evaporate completely without ever reaching that infinite temperature. Instead of a fiery explosion, the black hole could fade away gently, with its temperature dropping to zero as its mass vanishes. This scenario allows the black hole to disappear entirely while keeping the laws of physics intact, ensuring that the radiation it emits carries away all the information it once held.

The researchers focused their study on specific types of black holes that are more complex than the simplest, neutral kind. They looked at charged black holes and spinning black holes, examining how their properties change as they lose mass. In the standard model of a simple black hole, the temperature rises uncontrollably as the mass drops. However, the authors found that if the charge or spin of the black hole is carefully linked to its mass in a specific way, the temperature behaves differently. They described these specific relationships as paths or curves on a landscape of possible states. If a black hole follows one of these special paths during its evaporation, it avoids the temperature spike.

To visualize this, imagine a black hole not as a single point, but as an object that can be described by several numbers at once: its mass, its charge, and its temperature. These numbers form a surface, like a map of a mountain range. The standard evaporation path leads to a cliff where the temperature goes to infinity. The authors identified new paths on this map that curve away from the cliff. As the black hole travels along these new paths, losing mass, its charge or spin adjusts in a precise manner. This adjustment acts as a brake, preventing the temperature from rising. Instead, the temperature peaks at a certain point and then begins to fall, eventually reaching zero just as the black hole's mass disappears.

The study explored several mathematical shapes for these paths. In one scenario, the relationship between the black hole's charge and its mass creates a bell-shaped curve. As the black hole shrinks, its charge changes in a way that keeps the temperature under control. In another scenario, the temperature itself follows a semi-circular pattern as the mass decreases. In both cases, the result is the same: the black hole evaporates completely, and the radiation it emits follows a specific pattern of entropy, or disorder, that rises and then falls back to zero. This pattern, known as the Page curve, is exactly what physicists need to see to confirm that information is not lost.

The researchers also applied this logic to black holes in universes with different cosmic forces, such as those with a cosmological constant that pushes space apart or pulls it together. They found that similar paths exist for these more complex black holes as well. By adjusting the cosmological constant in relation to the black hole's mass, they could again find a route where the temperature remains finite and eventually vanishes. This suggests that the mechanism for a gentle, complete evaporation is not limited to just one type of black hole but could be a general feature of how these objects behave in classical gravity.

The implications of this work are significant for the long-standing puzzle of what happens to information inside a black hole. If the temperature does not explode, the black hole does not vanish in a flash of infinite energy. Instead, it fades away, and the radiation it emits carries the story of everything that fell into it. The entropy of this radiation increases as the black hole shrinks, reaches a maximum, and then decreases as the black hole nears its end. This rise and fall mirrors the behavior of a pure quantum state evolving into a mixed state and then back again, preserving the unity of the physical laws.

The authors note that for this to happen in reality, the black hole must be in a state very close to what is called "extremal," where its charge or spin is nearly as large as its mass allows. While this might seem like a rare condition, the paper suggests that if nature follows these specific curves, the explosion is avoided. The study does not claim to have observed this in the real world, as black holes are too distant and their evaporation too slow to measure directly. Instead, it provides a theoretical framework showing that complete evaporation without a temperature explosion is mathematically possible within the known laws of gravity.

Ultimately, this work offers a way to reconcile the violent image of a black hole explosion with the gentle requirements of quantum mechanics. By finding the specific paths a black hole can take as it dies, the authors show that the universe might have a built-in mechanism to prevent the loss of information. The black hole does not need to be a monster that destroys everything; it can be a system that, under the right conditions, dissolves completely, leaving behind a trail of radiation that tells the whole story of its existence.

Technical Summary: Complete Evaporation of Black Holes and Page Curves

Problem Statement
The paper addresses the problem of the complete evaporation of black holes within the framework of classical gravity, specifically focusing on the unphysical "temperature explosion" predicted by Hawking for Schwarzschild black holes. According to the standard Hawking formula, $T = 1/8\pi M$ , as the black hole mass $M$ approaches zero, the temperature diverges to infinity. This implies an infinite release of energy and a divergence in radiation entropy ( $S_R \sim M^{-3}$ ), which contradicts the expectation of unitary evolution where the final state should be a pure state with vanishing entropy. This divergence is closely linked to the black hole information paradox. The authors investigate whether complete evaporation can occur without this temperature blow-up, ensuring that the radiation entropy follows a "Page curve" (increasing initially, then decreasing to zero as the black hole vanishes).

Methodology
The authors propose that complete evaporation without temperature divergence is possible if the black hole possesses additional parameters—specifically electric charge $Q$ (Reissner-Nordström), angular momentum $a$ (Kerr), or a non-zero cosmological constant (de Sitter/Anti-de Sitter)—and if these parameters satisfy specific constraints relative to the mass $M$ during the evaporation process.

Instead of treating $Q$ or $a$ as constants, the paper models the evaporation as a trajectory (a curve) on the surface of state equations defined by the Hawking temperature $T(M, Q)$ or $T(M, a)$ . The core methodology involves:

Parametrization: Introducing a function $\lambda(M)$ (where $\lambda = \sqrt{M^2 - Q^2}$ for RN or $\lambda = \sqrt{M^2 - a^2}$ for Kerr) to describe the deviation from the extremal limit.
Constraint Analysis: Imposing bounds on $\lambda(M)$ such that as $M \to 0$ , the temperature $T \to 0$ . Specifically, the authors analyze cases where $\lambda(M) \sim M^\gamma$ with $\gamma > 2$ , or where the temperature is explicitly defined as a function of mass (e.g., a semi-circle dependence).
Dynamical Evolution: Solving coupled differential equations for mass and charge loss ($dM/dt$ and $dQ/dt$) to determine the time evolution of the system.
Entropy Calculation: Calculating the thermodynamic entropy of the radiation ( $S_R \propto T^3$ ) and the black hole entropy ( $S_{BH}$ ) to verify if they exhibit the Page curve behavior (vanishing at $M=0$ ).

Key Contributions and Results

Reissner-Nordström (RN) Black Holes:
The authors demonstrate that if the charge $Q$ and mass $M$ satisfy the bound $Q > M - CM^3$ (or more generally $\lambda(M) = o(M^2)$ ) for small $M$ , the Hawking temperature tends to zero as $M \to 0$ .
- Bell-shaped Curves: For $\lambda(M) = CM^\gamma$ with $\gamma > 2$ , the temperature and radiation entropy first increase as $M$ decreases, reach a maximum, and then decrease to zero. This yields a "deformed bell-shaped" dependence on mass. In the specific case where $\gamma = 2$ (i.e., $\lambda(M) = CM^2$ ), the limit of the temperature as $M \to 0$ is not zero but a non-zero constant ( $C/2\pi$ ).
- Semi-circle Curves: By prescribing a semi-circle temperature dependence $T(M) = C\sqrt{M(M_0 - M)}$ , the authors derive specific charge-mass trajectories where $T \to 0$ and $S_R \to 0$ at the end of evaporation.
- Time Evolution: Solving the evolution equations, they find that for $\gamma > 2$ , the mass decays in finite time, while for $\gamma = 2$ , it decays exponentially over infinite time. In both cases, the time-dependent radiation entropy follows the Page curve.
Kerr Black Holes:
The analysis is generalized to rotating black holes. By imposing a similar bound on the angular momentum $a$ ( $a > M - CM^3$ ), the temperature $T_{Kerr}$ vanishes as $M \to 0$ . The resulting entropy and temperature profiles mirror the RN case, exhibiting the necessary behavior to avoid singularities and support unitary evolution.
Kerr-Newman Black Holes:
The results are extended to charged, rotating black holes. The authors show that constraints on the combined parameters $a^2 + q^2$ allow for complete evaporation with vanishing temperature, maintaining the Page curve behavior for radiation entropy.
Schwarzschild-de Sitter (SdS) and RNdS/AdS Black Holes:
The study includes black holes with cosmological constants.
- For SdS, the cosmological constant $\Lambda$ is treated as a function of mass. If $\Lambda$ scales appropriately (via a function $\lambda(M)$ ), the temperature can vanish as $M \to 0$ .
- For RNdS and RNAdS, the authors analyze the state equation surfaces in the presence of a cosmological constant. They identify specific curves (parametrized by $\mu(x)$ or $\lambda(x)$ ) where the temperature vanishes at the end of evaporation. Numerical results indicate that for certain ranges of scaling exponents (e.g., $3/2 < \alpha < 2$ for RNdS), the temperature approaches zero rather than diverging.

Significance and Claims
The paper claims to provide a mechanism within classical gravity for the complete evaporation of black holes that resolves the temperature explosion problem. By restricting the black hole to follow specific trajectories in the parameter space (mass, charge, angular momentum, cosmological constant), the authors show that:

The Hawking temperature can tend to zero as the mass vanishes.
The radiation entropy vanishes at the end of the process, which is consistent with unitary evolution (the evolution of an initially pure quantum state to a final pure state).
The time evolution of the radiation entropy naturally reproduces the Page curve, where entropy increases initially and then decreases to zero.

The authors suggest that if realistic black holes follow such evaporation curves, the scenario is consistent with unitary evolution, avoiding the information paradox's implication of a mixed final state, without invoking quantum gravity corrections provided the mass remains above the Planck scale ( $M > M_{Planck}$ ). They note that the choice of such curves could be justified by an anthropic principle or a form of "black hole censorship" to prevent unphysical explosions. The work generalizes these findings across various black hole types, including those with charge, rotation, and cosmological constants.

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