⚛️ general relativity

Complementarity of Gravitational Collapse (I) Origin of the Bekenstein-Hawking Entropy

This paper proposes that the inner structure of black holes formed via gravitational collapse is described by two complementary frameworks—collapsars in Schwarzschild time and oscillatory solid-balls in Lemaître time—and uses this duality to derive the Bekenstein-Hawking area law by directly counting the degeneracy of the collapsing material's wave functional.

Original authors: Ding-fang Zeng

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ding-fang Zeng

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

The Big Question: What's Inside a Black Hole?

For decades, physicists have been stuck on a fundamental question: What actually exists inside a black hole?

The traditional answer, based on Einstein's General Relativity, is that everything collapses into a singularity—a tiny, infinitely dense point where the laws of physics break down. However, this creates a problem. We also know black holes have entropy (a measure of disorder or "information"). If a black hole is just a single, boring point, it shouldn't have any complexity or "information" inside it. It's like saying a hard drive with a trillion files is actually just a single empty dot.

This paper proposes a new solution. The author, Ding-fang Zeng, suggests that the "singularity" view and the "entropy" view are actually two sides of the same coin. They are complementary descriptions of the same object, depending on how you look at it.

The Two Ways to Watch the Movie

Imagine you are watching a movie of a star collapsing to form a black hole. The paper argues that the story looks completely different depending on which "camera" (or time definition) you use.

1. The Schwarzschild Camera (The Outside Observer)

The View: If you are an observer far away watching the collapse, time seems to slow down for the falling star.
The Analogy: Imagine a runner trying to reach the finish line, but every time they take a step, the finish line moves slightly further away. To the outside observer, the star never actually crosses the "event horizon" (the point of no return). It just gets closer and closer, freezing in time.
The Result: The star becomes a "collapsar"—a dense ball of matter that is constantly shrinking but never quite finishes the job. Because it never fully forms a perfect, impenetrable horizon, the star retains a "fingerprint" of its original shape (its mass distribution). It's like a squishy ball that is being squeezed but never becomes a perfect, smooth marble.

2. The Lemaître Camera (The Inside Observer)

The View: If you were riding along with the falling star (moving at the same speed), time feels normal.
The Analogy: Imagine a ball bouncing inside a box. In this view, the star doesn't stop at a singularity. Instead, it hits the center, bounces back, expands, and then collapses again. It oscillates (wiggles) back and forth through the center forever.
The Result: The star is like a bouncing, oscillating solid ball. It crosses the "singularity" point repeatedly. Because the physics at that crossing point is unpredictable, the ball explores every possible way it can wiggle and vibrate.

The "Complementarity" (The Magic Link)

The paper's main claim is that these two views are actually the same thing.

View A (Outside): A star that is slowly shrinking, frozen just outside the horizon, with a messy, complex internal structure.
View B (Inside): A star that is bouncing wildly through the center, exploring every possible vibration mode.

The author calls this Complementarity of Gravitational Collapse (CGC). Just as a coin has a head and a tail, the black hole has these two descriptions. They are mathematically equivalent. If you translate the "bouncing ball" math into the "frozen star" math, you get the exact same result.

Solving the Entropy Mystery

Why does this matter? It solves the mystery of Black Hole Entropy (the Bekenstein-Hawking entropy).

The Old Problem: If a black hole is just a singularity, where does all the "information" (entropy) come from?
The New Solution: The entropy comes from the frozen star's internal structure.
- In the "Outside View," the star is a collection of many layers (shells) of matter. Each layer can be arranged in many different ways.
- In the "Inside View," the star is bouncing in many different vibration patterns.
- The author counts all the possible ways these layers can be arranged or how the ball can vibrate. When you do the math, the number of possibilities matches the famous formula for black hole entropy perfectly.

Think of it like a library. The "Outside View" sees a library where the books are stacked in a specific, messy order that never changes. The "Inside View" sees the books flying around the room in every possible pattern. Both views describe the same library, and the "messiness" (entropy) is real in both cases.

What Does This Mean for the Real World?

The paper suggests that black holes aren't the "hairless," featureless monsters of old theory. Instead, they are more like giant, super-dense atoms or squishy, contracting stars that never quite finish collapsing.

Predictions for Observations:
Because these black holes have an "inner structure" and don't have a perfect, impenetrable horizon, the paper predicts we might see:

Gravitational Wave Echoes: When two black holes merge, the "ringing" sound (gravitational waves) might have extra echoes or a different "damping" pattern because the objects are squishy and exchanging mass, rather than just two perfect spheres merging.
Radio Bursts: These "squishy" black holes might vibrate in a way that creates fast radio bursts (FRBs), acting like a giant, vibrating magnet.

The Bottom Line

This paper argues that we don't need to invent new, unknown laws of physics to explain black holes. We just need to accept that General Relativity allows for two different, equally valid ways of describing the same object.

To the outside world, a black hole is a frozen, shrinking star with a complex internal map.
To the inside, it's a bouncing, oscillating ball exploring all its possibilities.

This "Complementarity" explains why black holes have entropy without needing to break the rules of Einstein's gravity. It turns the black hole from a mysterious, information-destroying void into a complex, understandable object that we might one day be able to "see" through its gravitational ripples.

Technical Summary: Complementarity of Gravitational Collapse

Problem Statement
The paper addresses a fundamental contradiction in modern theoretical physics regarding the internal structure of physical black holes (pBHs) formed through gravitational collapse. On one hand, singularity theorems (Penrose, Hawking, Geroch) suggest that gravitational collapse inevitably leads to spacetime singularities where matter density and curvature become infinite. On the other hand, black hole thermodynamics implies that black holes possess entropy, suggesting a microscopic diversity of states that contradicts the deterministic, singular nature of the classical collapse scenario. While theories like string theory and loop quantum gravity attempt to resolve this by abandoning the singularity concept, the paper argues that a reconciliation can be achieved within General Relativity (GR) itself by utilizing the principle of general coordinate invariance.

Methodology
The authors employ a dual-description framework based on two distinct time definitions within GR:

Schwarzschild Time Definition: The authors analyze the collapse of spherical dust balls, deriving exact solution families to the Einstein equations. They establish a gauge choice where the collapsing rate of a shell matches the free-fall velocity of test particles outside a Schwarzschild black hole of equivalent mass. This results in a description where the horizon is "close-to-implementing" but never successfully formed in finite time for outside observers.
Lemaître Time Definition: Using the proper time of co-moving probes, the authors describe the collapse as a process where matter crosses the central singularity. In this frame, the singularity is not a terminal point; rather, the system undergoes periodic, ergodic oscillations where the horizon and singularity form and resolve cyclically.

The core methodology involves proving a one-to-one correspondence between the free parameters of these two solution families. The authors then quantize the microscopic states of the collapsing material in the Schwarzschild frame. They model the collapsar as a system of concentric shells, applying canonical quantization to the geodesic equations of these shells to derive a wave functional. The degeneracy of this wave functional is calculated to determine the entropy.

Key Contributions and Results

Complementarity of Gravitational Collapse (CGC): The paper introduces the concept of CGC, asserting that the ensemble of collapsars in Schwarzschild time (EnS) and the ergodic, over-cross-oscillatory modes in Lemaître time (ErL) constitute two equally valid and complete descriptions of the same microscopic state ($EnS = ErL$). This equivalence arises naturally from general covariance, not as an ad-hoc principle for information puzzles.
Derivation of Bekenstein-Hawking Entropy: By quantizing the radial mass profile parameters ( $m(0, r)$ ) of the collapsing shells, the authors calculate the degeneracy of the resulting wave functional. They demonstrate that the number of distinguishable microscopic states scales exponentially with the area of the horizon. Specifically, they derive the Bekenstein-Hawking entropy formula ( $S \propto A/4G\hbar$ ) with logarithmic corrections, showing that the entropy arises from the "abandoned information" regarding the radial mass profile deviations in the Schwarzschild frame or the oscillatory modes in the Lemaître frame.
Microscopic Origin: The fundamental degrees of freedom are identified not as constituent particles, but as the collective motion modes of the collapsing material. The typical mass of these modes is linked to Hawking particles, providing a mechanism for unitary evolution without information loss.
Exact Solutions: The paper provides explicit metric forms for both time definitions, showing that the Lemaître metric (containing oscillatory functions $a[\tau, \varrho]$ ) reduces to the Schwarzschild metric via coordinate transformation, confirming the mathematical consistency of the dual description.

Significance and Claims
The paper claims to resolve the tension between the deterministic singularity theorems and the statistical nature of black hole thermodynamics without invoking physics beyond General Relativity.

Reconciliation: It posits that pBHs are not singular objects wrapped by an event horizon but are "horizonless" compact objects with extended radial mass profiles (in Schwarzschild time) or ergodic oscillating solid balls (in Lemaître time).
Observational Predictability: The authors claim their picture is "easily dis-verifiable." They predict that gravitational wave signals from binary mergers will exhibit late-time damping features distinct from standard numerical relativity predictions, as the merger involves mass redistribution rather than just apparent horizon deformation. Additionally, they propose a mechanism for Fast Radio Bursts (FRBs) arising from perturbations in the dynamically contracting crust of these objects, a feature not predicted by standard black hole or neutron star models.
Unitary Evolution: The framework suggests that black hole radiation is a unitary process analogous to spontaneous radiation in atoms, maintaining a Hermitian Hamiltonian description and avoiding information loss paradoxes.

The authors emphasize that this work offers a first-principle, intuitive microscopic interpretation of black hole entropy derived directly from Einstein's theory, challenging interpretations based on string theory or loop quantum gravity while remaining fully compatible with GR.