Varying Newton constant, entropy and the black hole evaporation law

This paper investigates how a time-dependent Newton constant and non-conserved energy-momentum tensor, constrained by the Bianchi identity and thermodynamic principles, lead to specific power-law relations between $G$ and black hole mass $M$ that determine the black hole's evaporation behavior and temperature evolution based on the chosen entropy definition.

The Gist

Imagine the universe as a giant, cosmic balloon. For decades, physicists have believed that the "rules" written on this balloon are fixed. One of the most important rules is Newton's Constant ($G$), which dictates how strongly gravity pulls things together. Another rule is the Cosmological Constant ($\Lambda$), which acts like a mysterious pressure pushing the balloon apart.

In this paper, two physicists from Poland, Julia and Zbigniew Haba, ask a bold question: What if these rules aren't fixed, but change as the universe ages?

Here is the story of their discovery, broken down into simple concepts and analogies.

1. The Cosmic Ledger (The Bianchi Identity)

In physics, there is a fundamental rule called the Bianchi identity. Think of this as the universe's "accounting law." It says that energy and momentum cannot just appear or disappear; they must balance the books.

Usually, we assume the "currency" (gravity, represented by $G$) stays the same. But the authors suggest that if $G$ changes over time, the universe's accounting law forces a trade-off. If the strength of gravity changes, it must be balanced by a change in the energy of the fluid filling space or the cosmological constant. It's like a seesaw: if one side (gravity) goes up, the other side (energy or expansion) must go down to keep the balance.

2. The Black Hole as a Leaky Bucket

Now, let's talk about Black Holes. Imagine a black hole as a super-hot bucket of water sitting in a cold room. According to Stephen Hawking, this bucket slowly leaks water (energy) in the form of radiation. Eventually, the bucket empties, and the black hole disappears. This process is called evaporation.

The speed at which the bucket leaks depends on two things:

1. Temperature: How hot the bucket is.
2. Surface Area: How big the bucket is.

In standard physics, as the bucket gets smaller (loses mass), it gets hotter and leaks faster, leading to a massive explosion at the very end.

3. The Twist: Gravity Changes the Rules

The Haba brothers introduce a new variable: What if gravity ($G$) gets stronger or weaker as the black hole loses mass?

They use a mathematical "recipe" to connect the changing gravity to the changing mass. They propose a relationship where Gravity ($G$) is related to Mass ($M$) by a power law:
$$G \approx \frac{1}{M^\gamma}$$
(The symbol $\gamma$ is just a number that changes the outcome).

Depending on what this mysterious number $\gamma$ is, the black hole behaves in three very different ways:

Scenario A: The Steady Stream ($\gamma = 1$)

If gravity changes in a specific way (where $\gamma = 1$), the black hole behaves like a steady faucet.

• What happens: As the black hole loses mass, the changing gravity perfectly cancels out the heating effect.
• The Result: The black hole stays at a constant temperature and leaks energy at a constant rate until it vanishes. No explosion. It just slowly fades away.

Scenario B: The Slow Fade ($\gamma > 1$)

If the relationship is even stronger (where $\gamma$ is greater than 1), the black hole acts like a cooling cup of coffee.

• What happens: As it loses mass, the changing gravity makes it get colder and leak slower.
• The Result: The black hole becomes dimmer and colder over time. It never explodes; it just slowly evaporates into nothingness over an infinite amount of time. This is interesting for scientists looking for Dark Matter, as these "cold" black holes could be hiding in plain sight.

Scenario C: The Classic Explosion ($\gamma = 2/3$)

This is the "standard" Bekenstein-Hawking view, but applied to a changing universe.

• What happens: The black hole gets hotter and hotter as it shrinks.
• The Result: It ends in a violent explosion, releasing a massive burst of energy at the very end of its life.

4. Why Does This Matter?

Why should we care if gravity changes?

1. Solving Mysteries: Our current model of the universe (Lambda-CDM) has some holes, like the need for "Dark Energy" and "Dark Matter." Maybe these aren't invisible particles, but just the result of gravity changing over time.
2. Primordial Black Holes: Scientists are currently searching for tiny black holes formed at the beginning of the universe. Some recent observations of giant neutrino bursts (ghostly particles) might be the "burps" of these black holes exploding.
   • If the Haba brothers are right and $\gamma > 1$, these black holes wouldn't explode violently. They would just slowly fade away, which changes how we should look for them.
3. The Thermodynamics Connection: The paper beautifully links gravity (the force that holds stars together) with thermodynamics (the science of heat and entropy). It suggests that the "heat" of a black hole is directly tied to how the laws of gravity are evolving.

The Bottom Line

The paper suggests that the universe might be more dynamic than we thought. Gravity might not be a fixed constant, but a variable that shifts as the universe evolves.

If this is true, the dramatic "explosions" of dying black holes might actually be gentle, slow fades. It's a bit like realizing that a firework that we thought would go BOOM is actually just a candle that slowly burns out, changing the color of the sky as it goes.

This theory offers a new lens through which to view the cosmos, potentially explaining dark matter and the mysterious expansion of the universe without needing to invent new, invisible particles.

Technical Summary

Here is a detailed technical summary of the paper "Varying Newton constant, entropy and the black hole evaporation law" by Julia Haba and Zbigniew Haba.

1. Problem Statement

The paper addresses the theoretical inconsistencies and potential modifications arising when the Newtonian gravitational constant ($G$) and the cosmological term ($\Lambda$) are treated as time-dependent variables within the framework of Einstein's field equations.

• The Conflict: In standard General Relativity, the Bianchi identity ($\nabla_\mu G^{\mu\nu} = 0$) enforces the strict conservation of the energy-momentum tensor ($\nabla_\mu T^{\mu\nu} = 0$). However, if $G$ and $\Lambda$ vary with time, the standard form of the Einstein equations implies a non-conservation of the fluid's energy-momentum tensor ($T^{\mu\nu}$).
• The Gap: There is a lack of a unified thermodynamic description linking the variation of fundamental constants ($G, \Lambda$) with the non-conservation of energy-momentum, specifically in the context of compact systems like black holes undergoing evaporation.
• The Goal: To derive the relationship between a time-varying $G$ and the mass ($M$) of a compact system by reconciling the Bianchi identity with the First Law of Thermodynamics, and to determine how this affects the black hole evaporation law (Stefan-Boltzmann law).

2. Methodology

The authors employ a combination of classical General Relativity, thermodynamics, and phenomenological modeling:

• Modified Einstein Equations: They start with the Einstein equations where the energy-momentum tensor includes a fluid component and a cosmological term: $G_{\mu\nu} = 8\pi G T_{\mu\nu} - g_{\mu\nu}\Lambda$.
• Bianchi Identity Analysis: By applying the covariant divergence ($\nabla_\mu$) to the modified equations, they derive a constraint equation linking the time derivatives of $G$ and $\Lambda$ to the non-conservation of $T^{\mu\nu}$:
  $$8\pi \partial_\mu G T^\mu_\nu + 8\pi G \nabla_\mu T^\mu_\nu - \partial_\nu \Lambda = 0$$
• Thermodynamic Mapping: The authors identify the covariant divergence term ($\nabla_\mu T^\mu_\nu$) with heat flow ($dQ = TdS$). By integrating this over a volume in a comoving frame, they derive a generalized First Law of Thermodynamics for the system:
  $$M \partial_t G + G T \partial_t S = -\frac{1}{8\pi} \int dx \sqrt{\gamma} \partial_t \Lambda$$
  (Where $M$ is the mass, $T$ is temperature, and $S$ is entropy).
• Power-Law Ansatz: They propose a power-law relationship between the Newton constant and the mass of a compact system: $G \propto M^{-\gamma}$.
• Evaporation Dynamics: They apply the Stefan-Boltzmann law for mass loss ($dM/dt = -\lambda T^4 A$) to a black hole with an event horizon, substituting the derived $G(M)$ relations and standard Bekenstein-Hawking formulas for Temperature ($T_{BH}$) and Area ($A$).

3. Key Contributions

• Thermodynamic Derivation of $G(M)$: The paper establishes that the variation of $G$ is not arbitrary but is constrained by the thermodynamic state (entropy and pressure) of the system via the Bianchi identity.
• Identification of the Exponent $\gamma$: The authors derive specific values for the exponent $\gamma$ in the relation $G \propto M^{-\gamma}$ based on different entropy definitions:
  • $\gamma = 1$: Occurs if the system has zero pressure or if entropy is defined via the thermodynamic relation $dS = T^{-1}dM$ (without assuming the Bekenstein-Hawking area law for time-dependent $G$).
  • $\gamma = 2/3$: Occurs if the standard Bekenstein-Hawking entropy formula ($S \propto Area/G$) is assumed to hold true even when $G$ is time-dependent.
  • $\gamma > 1$: Possible in other effective gravity models (e.g., loop quantum gravity corrections, scalar-tensor theories).
• Unified Evaporation Framework: The paper provides a closed-form solution for the time evolution of black hole mass and the Newton constant for any given $\gamma$, replacing the standard Hawking evaporation model with a parameterized family of solutions.

4. Results

The behavior of the black hole during evaporation depends critically on the value of $\gamma$:

Value of $\gamma$	Behavior of Temperature ($T$)	Behavior of Luminosity ($L$)	Mass Evolution ($M$)	Physical Interpretation
$\gamma = 1$	Constant	Constant	Linear decrease to zero	No "explosion"; constant evaporation rate. Consistent with $dS = T^{-1}dM$.
$\gamma = 2/3$	Diverges to $\infty$	Diverges to $\infty$	Finite time explosion	Standard Bekenstein-Hawking entropy applied to varying $G$ leads to a "black hole explosion."
$1 < \gamma < 3/2$	Decreases to 0	Decreases to 0	Finite time evaporation	Temperature cools as mass decreases.
$\gamma \geq 3/2$	Decreases to 0	Decreases to 0	Asymptotic to 0 (infinite time)	The black hole never fully evaporates in finite time; density trends to zero (if $\gamma > 2/3$).

• Neutron Star Connection: The authors note that the result $\gamma = 2/3$ (derived from Bekenstein-Hawking entropy) coincides with the mass-radius relation derived from the Chandrasekhar limit for neutron stars, suggesting a deep link between compact object stability and the variation of $G$.
• Cosmological Term: If $\Lambda$ varies, the evaporation rate is further modified by the sign of $\partial_t \Lambda$.

5. Significance

• Primordial Black Holes (PBHs) as Dark Matter: The finding that $\gamma > 1$ leads to a decreasing temperature and luminosity is significant for cosmology. Standard Hawking evaporation predicts a final burst of high-energy radiation (explosion). If PBHs have $\gamma > 1$, they would cool down and fade away, making them viable candidates for Cold Dark Matter without contradicting current non-detection of gamma-ray bursts from PBH explosions.
• Observational Tests: The paper suggests that the specific evaporation law (constant vs. decreasing temperature) could be tested against observations of gamma-ray bursts and neutrino emissions (e.g., from KM3NeT). A lack of high-energy explosions could support models with varying $G$ and $\gamma > 1$.
• Theoretical Consistency: The work provides a rigorous thermodynamic justification for allowing $G$ to vary in classical gravity, framing it not as a violation of conservation laws but as a redistribution of energy between the gravitational field and matter fields, mediated by the cosmological term.

In conclusion, the paper demonstrates that the "law" of black hole evaporation is not universal but depends on the underlying thermodynamic definition of entropy and the behavior of the Newton constant. This offers a new perspective on the final stages of black hole life and the nature of dark matter.