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SJ Model Black hole

The SJ black hole model addresses the information paradox by proposing that information becomes progressively inaccessible to external observers as physical energy continues inward, without assuming the interior is a singularity, tunnel, wormhole, or another universe.

Original authors: Saroj Joshi

Published 2026-09-14

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Saroj Joshi

Original paper licensed under CC BY 4.0 (https://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

In the vast theater of modern physics, few mysteries are as profound as the fate of information inside a black hole. To understand the puzzle, one must first accept the basic rules of these cosmic traps. A black hole is a region of space where gravity is so intense that nothing, not even light, can escape once it crosses a specific boundary known as the event horizon. In the standard view provided by classical physics, this horizon acts as a one-way door; anything that passes through is lost to the outside universe forever. The central question that has kept scientists awake for decades is what happens to the "information" carried by that matter. In physics, information is not just a file on a computer; it is the fundamental description of a particle's state, its position, and its energy. The laws of nature generally suggest that this information cannot simply vanish. If a black hole swallows a star, does the information about that star disappear from the universe, or does it hide somewhere? This tension between the idea that information is lost and the idea that it must be preserved is known as the black hole information problem, a conflict that sits at the very edge of our understanding of reality.

A new proposal, called the SJ model, attempts to resolve this conflict by suggesting that we have been confusing two different things: the physical energy of an object and our ability to access the information it carries. The model, developed by researcher Saroj Joshi, does not claim to have solved the mystery with a final answer, but rather offers a fresh mathematical framework to investigate how information might fade from view without actually being destroyed. The core idea is simple yet radical: as an object falls past the event horizon, the physical energy it contains remains real and nonzero, but the information it holds becomes progressively inaccessible to an observer standing outside. The paper argues that the event horizon is not a surface that smashes or erases matter, but rather a causal boundary where the connection to the outside world is severed. The energy continues to exist and evolve inside, but the signal telling us what that energy is doing simply cannot get back out.

The researcher introduces a concept called physical progression to describe what happens inside the black hole. Imagine a journey where an object moves through a series of internal states, one after another, as it travels deeper. The model treats this movement not just as a change in time, but as a specific physical progression that drives the loss of external visibility. As this progression continues, the ability of the outside world to see or measure the object drops. The paper proposes a mathematical way to track this drop, suggesting that the information accessible to an external observer can approach zero while the physical energy remains positive. This distinction is crucial because it allows the model to respect the law that energy cannot disappear, while simultaneously explaining why we see nothing coming back out. The model does not assume that the information is destroyed; it assumes that the pathway for that information to reach us is effectively closed off by the physics of the black hole's interior.

To explore this, the paper defines several new parameters that describe how information moves or fails to move. It introduces an effective index that describes how signals propagate, similar to how light bends in glass, but applied here to the flow of information itself. It also defines a loss coefficient that measures how quickly the connection to the outside world fades as the object moves inward. The mathematics show that if this loss continues long enough, the information accessible to the outside world becomes vanishingly small, even though the physical object is still there. The model deliberately avoids making assumptions about what lies at the very center of the black hole. It does not claim that the interior is a tunnel to another universe, a wormhole, or a point of infinite density where physics breaks down. Instead, it leaves these questions open, suggesting that the true nature of the interior must be discovered by solving the equations, not by guessing the answer beforehand.

The paper is careful to state that this is a hypothesis, not a proven fact. The work is a proposal for a new way of looking at the problem, one that separates the concept of "being there" from the concept of "being seen." The author acknowledges that for this idea to become a true physical theory, it must eventually produce a unique prediction that can be tested against real-world observations. Potential tests could involve looking at the ripples in space-time caused by colliding black holes, the shape of the shadow a black hole casts against the background of stars, or the faint radiation that black holes emit. Until such a test is performed and the equations are derived from a deeper, more fundamental theory, the SJ model remains a suggestion. It offers a path forward for thinking about the black hole information problem, proposing that the disappearance of information is a matter of accessibility rather than destruction.

Ultimately, the SJ model suggests that the event horizon is not a place where things are unmade, but a place where they become unreachable. The energy and the physical states of anything that falls in continue to exist and evolve, driven by the internal progression of the black hole's own physics. The information is not gone; it is simply cut off from the rest of the universe. This perspective shifts the focus of the scientific inquiry from asking "where did the information go?" to asking "what physical mechanism makes it impossible for us to see it?" By treating the loss of information as a gradual fading of connection rather than a sudden erasure, the model provides a new lens through which to view one of the most stubborn puzzles in science. The work invites further investigation, urging researchers to derive the specific formulas that govern this progression and to look for the subtle signs that might confirm whether information truly becomes invisible without ever ceasing to exist.

Technical Summary: The SJ Model Black Hole

Problem Statement
The paper addresses the black hole information problem, a central unresolved issue in modern physics concerning the fate of information associated with physical states (light, matter, energy) that cross a black hole's event horizon. According to classical general relativity, the event horizon acts as a causal boundary where future-directed light paths cannot return information to an external observer. The fundamental question posed is whether this loss of external accessibility equates to the physical destruction of energy and information, or if the two phenomena can be decoupled.

Methodology and Framework
The author, Saroj Joshi, proposes the "SJ Model," a mathematical framework that distinguishes between physical energy ( $E$ ) and externally accessible information ( $I_{ext}$ ). The methodology relies on the following conceptual and mathematical constructs:

Separation of Energy and Accessibility: The core hypothesis posits that while $I_{ext}$ may approach zero as a system progresses inward, the physical energy $E$ remains non-zero ( $E > 0$ ). The event horizon is treated as a causal boundary rather than a destructive physical surface.
Physical Progression Parameter ( $\lambda$ ): The model introduces $\lambda$ to represent physical progression within the black hole interior. This parameter is distinct from ordinary coordinate time. The evolution of the system is modeled as a sequence of states ( $S_1 \to S_2 \to \dots \to S_n$ ).
Inward Radial Parameter ( $d$ ): Defined as $d = r_h - r$ , where $r_h$ is the Schwarzschild radius. At the horizon, $d=0$ ; as the system moves inward, $d > 0$ . This is treated as a mathematical parameter, not necessarily a physical depth of a tunnel.
Information Accessibility Dynamics:
- The model defines an information-accessibility loss coefficient, $\alpha(d)$ , governed by the differential equation:
  $\frac{dI_{ext}}{dd} = -\alpha(d)I_{ext}$
  Leading to the solution:
  $I_{ext}(d) = I_0 \exp\left[-\int_0^d \alpha(x)dx\right]$
- Alternatively, using the progression parameter $\lambda$ , the evolution is described by:
  $\frac{dI_{ext}}{d\lambda} = -\Gamma I_{ext}$
  Where $\Gamma = \Gamma(E, \mu, n, \mathcal{R})$ is a coefficient dependent on physical energy ( $E$ ), an effective information-propagation parameter ( $\mu$ ), an effective propagation index ( $n$ ), and spacetime curvature ( $\mathcal{R}$ ).
Discrete Progression: The model also considers a discrete representation where $I_{n+1} = P_n I_n$ (with $0 < P_n \leq 1$ ), allowing information accessibility to approach zero asymptotically as $n \to \infty$ .

Key Contributions
The paper's primary contribution is the theoretical proposal to separate the concept of "information destruction" from "information inaccessibility." Specifically:

Non-Destructive Horizon: The model suggests that the event horizon does not necessarily destroy physical energy; rather, it causes information to become progressively inaccessible to external observers.
New Parameters: It introduces specific parameters for investigation, including the effective information-propagation parameter ( $\mu$ ), the effective propagation index ( $n$ ), and the progression coefficient ( $\Gamma$ ).
Open-Ended Interior: Unlike models that presuppose the interior terminates in a singularity, connects to a wormhole, or leads to another universe, the SJ model does not assume a specific outcome. Instead, it asserts that the nature of the interior (finite, infinite, bouncing, or a new spacetime region) must be determined mathematically from the derived equations.
Consistency with Conservation Laws: The framework explicitly acknowledges that the stress-energy tensor must satisfy $\nabla_\mu T^{\mu\nu} = 0$ . Consequently, the model requires that energy must either remain in the interior, transform into another state, be encoded in gravitational/quantum degrees of freedom, or transfer to another region, rather than simply vanishing.

Results and Mathematical Direction
The paper does not present a finalized solution or a specific derived form for $\Gamma$ or $\mu$ . Instead, it establishes the mathematical direction for the hypothesis:

The central limiting condition is $\lim_{\lambda \to \lambda^*} I_{ext}(\lambda) = 0$ while $E > 0$ .
The model provides the differential equations governing the decay of external information accessibility as a function of physical progression.
It identifies the determination of the functional form of $\mu(E, \mathcal{R}, \lambda)$ and $\Gamma$ as the critical next step in the research.

Significance and Claims
The author positions the SJ Model as a hypothesis rather than a completed physical theory. Its significance lies in reframing the black hole information problem:

Reframing the Question: The paper shifts the focus from "where does the information go?" to "what physical mechanism drives externally recoverable information toward zero while physical energy persists?"
Testability Requirement: The author emphasizes that the model only becomes a physical theory if it yields a "unique prediction" that is "experimentally distinguishable."
Proposed Observational Tests: The paper lists potential areas for future investigation to validate the model, including gravitational-wave ringdown, black hole shadow structure, gravitational lensing, near-horizon radiation, Hawking radiation, and quantum correlations.
Modesty of Claim: The paper explicitly states it does not replace general relativity or the Schwarzschild solution. It proposes an additional mathematical investigation into information accessibility as a function of internal progression. The ultimate goal is to derive a testable prediction from the proposed equations, moving the hypothesis from a philosophical interpretation toward a testable physical theory.

Technical Summary: The SJ Model Black Hole

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