⚛️ high-energy theory

Constrained Scenarios of Non-Commutative Schwarzschild Black Holes with Global Monopoles from Van der Waals Behaviors

This paper investigates the Van der Waals behaviors of non-commutative Schwarzschild black holes with global monopoles by combining thermodynamic constraints and optical shadow analysis, ultimately using machine learning to demonstrate that the proposed model is compatible with observational data from the M87* black hole.

Original authors: Ismail Benyaich, Maryem Jemri

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ismail Benyaich, Maryem Jemri

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

Deep in the fabric of space and time, gravity acts as a sculptor, carving out regions so dense that not even light can escape. These are black holes, the most extreme objects in the universe. For decades, scientists have studied them as perfect spheres of darkness, but recent theories suggest that space itself might be slightly "fuzzy" at the tiniest scales, much like a photograph that loses sharpness when zoomed in too far. This idea, known as non-commutative geometry, proposes that the coordinates of space do not line up perfectly, introducing a fundamental graininess to the cosmos. When this fuzzy space is combined with other theoretical ingredients, such as global monopoles—topological defects that act like cosmic scars left over from the early universe—black holes behave in ways that are surprisingly similar to everyday fluids. Specifically, they begin to mimic the behavior of gases and liquids that change phase, like water boiling into steam or condensing into rain. This connection between the physics of the very large and the very small offers a unique way to test our theories against the real universe.

A team of researchers in Morocco has taken this theoretical framework and put it to the test against real-world observations. They focused on a specific type of black hole that exists in a universe with a negative cosmological constant, a setting that allows for a rich variety of thermodynamic behaviors. Their goal was to see if these exotic black holes could reproduce the familiar patterns of a Van der Waals fluid, a model used to describe how real gases deviate from ideal behavior under pressure and temperature changes. By carefully adjusting the parameters of their model, particularly the strength of the global monopole, they discovered a precise setting where the black hole's thermodynamics align perfectly with the Van der Waals laws. At this specific point, the black hole exhibits a critical ratio—a mathematical fingerprint of its phase transition—that matches the universal value found in simple fluids. This was not just a lucky guess; the researchers confirmed this alignment using two different thermodynamic methods, ensuring that the black hole behaves exactly like a fluid in this specific state.

Once they established this thermodynamic match, the team turned their attention to what these black holes would actually look like to an observer. They calculated the "shadow" of the black hole, which is the dark silhouette cast against the bright background of surrounding light, a feature famously captured by the Event Horizon Telescope. Using the specific parameters that created the fluid-like behavior, they simulated how the shadow would appear for different values of the non-commutative parameter, which controls the fuzziness of space. They found that changing this parameter altered the size of the shadow while keeping its shape circular, as expected for a non-rotating black hole. To see if their theoretical shadows matched reality, they compared their results with the actual data collected from two famous black holes: M87*, the giant at the center of a distant galaxy, and Sgr A*, the one at the center of our own Milky Way.

The comparison revealed a striking result. When the researchers adjusted their model to fit the observational data from the Event Horizon Telescope, they found that their proposed black hole model matched the observations of M87* with high precision. The model's predicted shadow size fell comfortably within the range of uncertainty provided by the telescope's measurements. However, the same model did not fit the data for Sgr A* as well, suggesting that while this specific type of fuzzy, monopole-embedded black hole could explain the giant in M87, it might not be the correct description for the one in our own galaxy. To ensure this conclusion was robust, the team employed machine learning, training a computer algorithm to recognize the difference between parameter combinations that fit the data and those that did not. The algorithm, which analyzed thousands of simulated scenarios, confirmed that the model is highly consistent with the M87* data, achieving an accuracy of over 98 percent in distinguishing valid configurations from invalid ones.

This work bridges the gap between abstract mathematical theories and the tangible images captured by our most powerful telescopes. It demonstrates that by tweaking the fundamental properties of space and adding specific cosmic defects, we can create a model of a black hole that behaves like a fluid and casts a shadow that matches what we see in the sky. While the model successfully explains the observations of M87*, it also highlights that not all black holes are the same; the one in our own galaxy may require a different explanation. The study leaves the door open for future research, suggesting that exploring rotating or charged versions of these black holes could reveal even more about the nature of gravity and the structure of the universe. For now, the findings offer a compelling glimpse into how the quantum graininess of space might shape the giants that rule the cosmos.

Technical Summary: Constrained Scenarios of Non-Commutative Schwarzschild Black Holes with Global Monopoles from Van der Waals Behaviors

Problem Statement
Recent investigations into black hole thermodynamics, particularly within Anti-de Sitter (AdS) spacetimes, have highlighted similarities between charged black holes and Van der Waals (VdW) fluid systems. While the Reissner-Nordström-AdS solution exhibits a critical ratio $\chi = P_c v_c / T_c = 3/8$ , characteristic of VdW fluids, non-commutative (NC) Schwarzschild black holes have been found to yield a different universal value ( $\chi \approx 0.3667$ ). This paper addresses the gap in recovering standard VdW behaviors in NC Schwarzschild geometries. Furthermore, it seeks to constrain the parameters of such deformed black holes by linking thermodynamic criticality with optical signatures (shadows) observed by the Event Horizon Telescope (EHT) for the black holes M87*, SgrA* (VLTI), and SgrA* (Keck).

Methodology
The study employs a multi-stage approach combining analytical thermodynamics, numerical relativity, and machine learning:

Theoretical Model Construction: The authors consider a static, spherically symmetric NC Schwarzschild black hole in the presence of a Global Monopole (GM). The metric incorporates a smeared mass distribution derived from NC geometry (characterized by parameter $\Theta$ ) and a GM term (characterized by energy scale $\eta$ ). The line element includes a cosmological constant $\Lambda$ and a NC correction parameter $a$ .
Thermodynamic Analysis:
- The authors derive the equation of state $P(T, v)$ within the extended phase-space formalism, treating the cosmological constant as thermodynamic pressure.
- They analyze the $P-v$ criticality to determine critical points ( $P_c, T_c, v_c$ ) and the critical ratio $\chi$ .
- To recover the standard VdW ratio ( $\chi = 3/8$ ), they impose a constraint on the GM parameter $\eta$ .
- The validity of this constraint is cross-verified using the Joule-Thomson effect, specifically analyzing the inversion temperature ratio $\zeta = T_i^{min}/T_c$ .
Optical Properties and Shadow Calculation:
- Using the Hamilton-Jacobi formalism, the authors calculate the photon sphere radius ( $r_p$ ) and the resulting black hole shadow radius ( $R_s$ ) for the constrained model.
- They compute the deviation $d = R_s/r_{sh} - 1$ relative to the standard Schwarzschild shadow to compare with EHT observational bounds (1- $\sigma$ and 2- $\sigma$ intervals).
Machine Learning Classification:
- A Fully Connected Neural Network (FCNN) is trained to classify parameter space configurations ( $\Lambda, a$ ) as compatible or incompatible with EHT data for M87*, SgrA* (VLTI), and SgrA* (Keck).
- The dataset is generated via numerical simulations of shadow radii under VdW limits.
- Model robustness is tested using a Monte Carlo voting strategy with Gaussian perturbations.

Key Contributions and Results

Recovery of Van der Waals Behavior: The study demonstrates that the NC Schwarzschild black hole with GMs can reproduce standard VdW thermodynamic behaviors. By solving for the critical ratio $\chi = 3/8$ , the authors identify a specific constraint on the GM parameter: $\eta^2 = 0.039788736$ . This value is independently confirmed by the Joule-Thomson inversion analysis, which yields a ratio $\zeta \approx 3/4$ under the same constraint.
Shadow Constraints: With $\eta^2$ $η^{2}$ fixed to the VdW-recovering value, the authors constrain the NC parameter $a$ $a$ and the cosmological constant $\Lambda$ $Λ$ against EHT data.
- For M87*, the model is consistent with observations within the 1- $\sigma$ and 2- $\sigma$ bounds for specific ranges of $\Lambda$ (e.g., $-0.14 \leq \Lambda \leq -0.09$ at 1- $\sigma$ ).
- For SgrA* (both VLTI and Keck datasets), the model also yields allowed parameter regions, though the constraints differ slightly between datasets.
Machine Learning Performance: The FCNN successfully distinguishes between compatible and incompatible parameter configurations.
- The model achieves high test accuracy, ranging from 97.36% to 98.44% across the different black hole datasets and confidence levels.
- The classifier performs slightly better under 1- $\sigma$ constraints due to the more restrictive nature of the data, which creates a clearer separation between classes.
- The M87* dataset consistently yields the highest classification accuracy.

Significance and Claims
The paper claims to provide a unified framework where thermodynamic criticality (VdW behavior) and optical observations (EHT shadows) converge to constrain non-commutative gravity models. The primary significance lies in the finding that the proposed NC Schwarzschild black hole with Global Monopoles matches the observational data of M87*. The authors assert that this model successfully recovers the universal VdW ratio through specific geometric constraints, offering a viable scenario for describing black hole thermodynamics and shadows in non-commutative spacetime. The work also demonstrates the efficacy of machine learning in efficiently mapping complex parameter spaces against empirical astronomical data without requiring repeated, computationally expensive shadow calculations.

The authors conclude modestly, noting that the current study is limited to neutral, non-rotating solutions and suggests that future work should extend these findings to include rotation and additional internal or external parameters.

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