Spherically symmetric black holes in Gravity from Entropy and spontaneous emission

This paper investigates static and dynamical spherically symmetric black holes within the Gravity from Entropy framework, demonstrating that the theory yields $r^{-4}$ corrections to the Schwarzschild metric, aligns with current astrophysical observations, and predicts both a standard Hawking-like mass loss at intermediate scales and a constant background evaporation rate for large black holes due to inherent entropic leakage.

The Gist

Imagine the universe as a giant, complex fabric. For over a century, we've understood gravity as the way this fabric bends and stretches when heavy objects (like stars or black holes) sit on it. This is Einstein's General Relativity. But this paper asks a different question: What if gravity isn't just about the shape of the fabric, but about the information hidden inside it?

The authors are exploring a theory called "Gravity from Entropy" (GfE). Think of "entropy" as a measure of disorder or, in this case, the amount of hidden information a system holds. The core idea is that gravity emerges because the universe is constantly trying to manage this information, much like how a messy room naturally tends to get messier unless you actively clean it.

Here is a simple breakdown of what they found, using everyday analogies:

1. The Black Hole Gets a "Makeover"

In standard physics, a black hole is like a perfect, smooth hole in a trampoline. The math describing it (the Schwarzschild solution) is very clean.

The authors found that when you apply the "Gravity from Entropy" rules, this smooth hole gets a tiny, subtle wrinkle.

• The Analogy: Imagine a perfectly round balloon. If you look at it from far away, it looks like a perfect circle. But if you zoom in very close, you see tiny bumps and textures on the rubber that weren't there before.
• The Result: The black hole's event horizon (the point of no return) isn't exactly where Einstein said it would be. It shifts slightly. The paper calculates exactly how much it shifts based on a "coupling parameter" (let's call it β), which measures how strong this new "information-based" gravity is.

2. Checking the Theory Against Real Life

The authors didn't just do math on a whiteboard; they checked if their "wrinkled" black holes match what we see in the sky. They looked at two things:

• The S2 Star: This is a star orbiting the supermassive black hole at the center of our galaxy. It moves in a weird, elongated loop. The authors calculated how the "wrinkles" in gravity would change the star's path. They found that as long as the "wrinkle" strength (β) is within a certain reasonable range, the star's path still matches what telescopes see.
• The Black Hole Shadow: The Event Horizon Telescope took a picture of a black hole's "shadow" (the dark circle surrounded by a ring of light). The authors calculated how the "wrinkles" would change the size of this shadow. They found that their theory predicts a shadow size that fits perfectly with the actual photo, provided the "wrinkle" strength isn't too wild.

The Takeaway: Their new theory is consistent with what we currently observe. It doesn't break the universe; it just adds a tiny, subtle layer of complexity that we haven't been able to see clearly until now.

3. The Black Hole That "Leaks"

This is the most surprising part. In standard physics, black holes are supposed to be eternal unless they are hit by something. However, the authors found that in their "Gravity from Entropy" framework, black holes naturally lose mass over time, even without anything falling into them.

• The Analogy: Imagine a bucket of water with a tiny, invisible hole in the bottom. Even if you don't tip the bucket, water slowly drips out.
• The Mechanism: The authors call this "entropic leakage." Because the black hole is made of this "information fabric," the fabric itself is slightly unstable. It naturally wants to shed energy to reach a more "disordered" state.
• The Result: They derived a formula showing that the black hole loses mass at a rate that looks very similar to the famous Hawking Radiation (a quantum effect predicted by Stephen Hawking).
  • The Twist: In standard Hawking radiation, the temperature of the black hole depends heavily on its size (smaller = hotter). In this new theory, the black hole stays warmer for longer as it shrinks. It's like a campfire that doesn't cool down as quickly as you'd expect when the wood gets small.

4. Why This Matters (According to the Paper)

The paper suggests that this "leaking" isn't a quantum trick happening on top of gravity; it is a classical consequence of the gravity theory itself.

• The "Remnant" Idea: The authors hint that this mass loss might stop at a certain point, leaving behind a tiny, stable "remnant" of the black hole.
• The Information Puzzle: If black holes don't disappear completely but leave behind these stable remnants, it might solve a huge mystery in physics called the Information Paradox. It suggests that the information swallowed by a black hole isn't destroyed; it's just stored in these tiny, leftover pieces of the "entropic fabric."

Summary

This paper proposes that gravity is driven by information (entropy). When they applied this to black holes, they found:

1. Black holes are slightly "wrinkled" compared to Einstein's predictions, but these wrinkles fit our current telescope data.
2. Black holes naturally "leak" energy and lose mass, similar to Hawking radiation, but driven by the geometry of space itself.
3. This process might leave behind tiny, stable leftovers, potentially solving the mystery of where the information inside black holes goes.

It's a new way of looking at the universe where the "shape" of space and the "information" inside it are two sides of the same coin.

Technical Summary

Technical Summary: Spherically Symmetric Black Holes in Gravity from Entropy and Spontaneous Emission

Problem Statement
The paper addresses the need to understand the deviations from classical Schwarzschild geometry within the "Gravity from Entropy" (GfE) framework. While previous work established that GfE naturally yields inflationary solutions and that black holes obey the area law for entropy, a rigorous derivation of static, spherically symmetric solutions using the full modified vacuum equations was lacking. The authors aim to determine how entropic modifications, which treat the spacetime metric as a quantum operator and utilize quantum relative entropy as the fundamental action, alter the geometry of black holes and their dynamical evolution, specifically regarding mass loss and evaporation.

Methodology
The authors employ the GfE framework, where the gravitational action is identified with the Araki quantum relative entropy between the spacetime metric ($\tilde{g}$) and an induced metric ($\tilde{G}$) modified by topological Ricci-Riemann tensors. The study proceeds in two main phases:

1. Static Analysis: The authors derive the modified vacuum Einstein equations for a static, spherically symmetric spacetime. They adopt a diagonal metric ansatz and restrict the analysis to a specific class of diagonal metrics to ensure the Riemann curvature matrix remains diagonal, rendering the trace of the logarithmic action computationally tractable. They solve these highly nonlinear equations using an asymptotic series expansion at spatial infinity ($r \to \infty$) to determine boundary conditions, followed by numerical integration to find global solutions.
2. Dynamical Analysis: To study black hole evolution, the authors transition to a dynamical regime using comoving Lemaître coordinates. They solve the generalized field equations ($G_{\mu\nu} = -\beta H_{\mu\nu}$), where $H_{\mu\nu}$ encapsulates higher-order geometric stresses. They introduce a time-dependent mass parameter $M(t)$ and a lapse function $g(t,r)$ to resolve the system, treating the mass evolution as a perturbative response to the modified background.

Key Contributions and Results

• Perturbative Corrections to Schwarzschild Geometry: The analysis reveals that the classical Schwarzschild geometry receives perturbative corrections scaling as $r^{-4}$. Specifically, the metric functions $A(r)$ and $B(r)$ acquire terms proportional to the coupling parameter $\beta$ and powers of the mass $M$. The leading corrections are $a_4 = -\beta M^2/12$ and $b_4 = \beta M^2/3$. These corrections deform the event horizon, shifting its location from the classical Schwarzschild radius $r_S$ to $r_h \approx r_S + \beta/(48r_S)$.
• Observational Constraints: The authors test the theory against current astrophysical data:
  • S2 Star Orbit: Using data from the GRAVITY Collaboration regarding the perihelion shift of the S2 star orbiting Sagittarius A*, they find that the theory is consistent with observations for a broad range of $\beta$ (specifically $-1 < \beta < 1$ in dimensionless units relative to $r_S^2$).
  • Event Horizon Telescope (EHT): The shadow diameter of Sagittarius A* provides a more sensitive probe of the near-horizon geometry. By comparing theoretical predictions with EHT measurements, the authors derive a tight constraint: $-9.04r_S^2 \leq \beta \leq 18.63r_S^2$. This confirms the GfE framework is consistent with strong-field observations.
• Spontaneous Mass Loss and Evaporation: In the dynamical analysis, the higher-order geometric stresses inherent to the GfE vacuum drive a consistent mass-evolution profile. The derived mass-loss law is:
  $$\dot{M} \approx -\frac{\beta}{24} - \frac{0.17\beta}{M^2}$$
  • Large Mass Limit: For black holes with mass $M \gg 1$ (Planck mass), the theory predicts a constant background evaporation rate of $-\beta/24$. The authors interpret this as an inherent "entropic leakage" of the vacuum, suggesting the vacuum itself is not strictly stationary but possesses a dissipative flow.
  • Intermediate Scales: The $M^{-2}$ term replicates the standard Hawking radiation mass-loss law ($\dot{M} \propto M^{-2}$) through a purely classical response of the modified background, without requiring a separate treatment of quantum fields on a fixed background.
• Thermodynamic Implications: The effective temperature of the black hole in this framework scales as $T \propto M^{-1/2}$, differing from the standard Hawking scaling of $T \propto M^{-1}$. This implies that macroscopic black holes in GfE remain warmer than their classical counterparts. The heat capacity remains negative, indicating thermodynamic instability, though the rate of instability is modified at small scales.

Significance and Claims
The paper claims that the GfE framework offers a unified description where Hawking-like radiation emerges as an intrinsic property of the modified vacuum geometry rather than a quantum field effect on a fixed background. The discovery of a constant background mass-loss term suggests a mechanism for "geometric evaporation" that persists even for large black holes, potentially leading to stable black hole remnants if the mass-loss rate vanishes at a finite non-zero mass. This outcome is presented as a potential resolution to the black hole information paradox, where remnants could serve as repositories for quantum information. Furthermore, the consistency of the theory with EHT and S2 orbital data demonstrates that GfE is a viable candidate for a UV-complete theory of gravity that avoids naked singularities while remaining compatible with current strong-field astrophysical observations.