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HWF Analysis of TRN Black Holes

This paper extends the horizon wave-function formalism to charged, spinless quantum sources in $\mathcal{D}+1$ dimensions to calculate and analyze the formation probabilities of both inner and outer horizons of Tangherlini–Reissner–Nordström black holes, revealing that the ratio of these probabilities deviates significantly from unity as the source moves away from the extremal limit and that formation probabilities are strongly suppressed in higher spatial dimensions.

Original authors: Ashfaque Ahmad, Sudhir Kumar Gupta, Abbas Ali

Published 2026-10-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashfaque Ahmad, Sudhir Kumar Gupta, Abbas Ali

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

In the classical view of the universe, gravity is a force that pulls matter together until it collapses into a point of infinite density, hidden behind a boundary known as an event horizon. This is the black hole, a region where space and time are so warped that nothing, not even light, can escape. For decades, physicists have treated these objects as purely mechanical, governed by the rigid laws of general relativity. However, when we zoom in to the scale of the very small, where quantum mechanics rules, the picture becomes fuzzy. At this level, particles do not sit in fixed positions; they exist as clouds of probability, defined by their energy and the uncertainty of their location. The question arises: if the matter trying to form a black hole is a fuzzy quantum cloud rather than a solid ball, can we still say a black hole has formed? The answer depends on whether the cloud is dense enough to be contained within its own gravitational boundary, a concept that requires a new way of thinking about how horizons appear and disappear in the quantum realm.

Researchers at Aligarh Muslim University have taken a significant step toward answering this question by applying a specific mathematical tool called the horizon wave-function to a complex type of black hole. They focused on a theoretical object known as the Tangherlini–Reissner–Nordström black hole, which exists in a universe with more than the usual three dimensions of space and carries an electric charge. In our everyday experience, we only see three dimensions, but many theories of physics suggest that extra, hidden dimensions might exist. The team wanted to understand how the presence of these extra dimensions and the electric charge of the source material would change the likelihood of a black hole forming. Instead of assuming a black hole forms automatically when mass is compressed, they treated the formation as a probability, calculating the chances that a quantum source would be found inside its own horizon.

To do this, the scientists modeled the source of the black hole as a spherical cloud of energy, similar to a wave spreading out in space. They then calculated the probability that this cloud would be small enough to fit inside the outer boundary of the black hole, as well as the probability of it fitting inside the inner boundary, which exists only in charged black holes. Their calculations revealed a striking sensitivity to the number of dimensions in the universe. When the researchers increased the number of spatial dimensions from five to ten, the probability of a black hole forming dropped dramatically, especially for sources that were not extremely massive. For instance, in a universe with ten dimensions, a source with a mass roughly half the scale of the fundamental gravity limit had a chance of forming a black hole that was thousands of times smaller than it would be in a five-dimensional universe. This suggests that in higher-dimensional worlds, it is much harder for gravity to trap matter into a black hole unless the matter is incredibly heavy and tightly packed.

The study also uncovered a fascinating difference between the outer and inner horizons of these charged black holes. When the electric charge of the source is low, the chance of forming the outer horizon is high, while the chance of forming the inner horizon is nearly zero. However, as the charge increases toward its maximum possible limit, the probability of forming the inner horizon rises sharply, eventually catching up to the probability of the outer horizon. This indicates that the two boundaries, which are distinct in most cases, begin to merge and behave similarly as the system approaches its most extreme state. The researchers found that the difference between the likelihood of forming the outer versus the inner horizon is most pronounced when the charge is low and the mass is moderate, but this gap closes as the charge grows stronger.

These findings offer a clearer picture of how quantum mechanics and extra dimensions interact to shape the birth of black holes. The work confirms that the dimensionality of space is not just a background setting but an active player that suppresses the formation of these cosmic objects. It also highlights that the electric charge of the source plays a critical role in determining which type of horizon is likely to appear. By mapping out these probabilities, the researchers have provided a more nuanced understanding of black hole formation, moving beyond simple yes-or-no answers to a detailed landscape of possibilities. Their results suggest that if extra dimensions do exist, they would make the creation of microscopic black holes in high-energy collisions far less probable than previously thought, unless the energy involved is significantly higher than current expectations. This adds a layer of caution to predictions about what we might observe in future experiments, grounding the search for these elusive objects in a more rigorous quantum framework.

Technical Summary: HWF Analysis of TRN Black Holes

Problem Statement
In classical general relativity, black hole formation is deterministic, governed by criteria such as the hoop conjecture. However, this description breaks down for quantum sources where the spatial localization width is comparable to the Compton wavelength. In this regime, the horizon radius cannot be treated as a fixed classical quantity but must be described probabilistically. While the Horizon Wave-Function (HWF) formalism has been applied separately to charged sources in $(3+1)$ dimensions and to uncharged sources in higher-dimensional $(D+1)$ spacetimes, the combined effect of electric charge and spatial dimensionality on black hole formation probabilities had not been explored. This paper addresses the gap in understanding how charge and higher dimensions jointly influence the quantum probability of black hole formation for a Tangherlini–Reissner–Nordström (TRN) geometry.

Methodology
The authors extend the HWF formalism to a spherically symmetric, charged, spinless quantum source in $(D+1)$ -dimensional spacetime.

Quantum Source: The source is modeled as a Gaussian wave function in position space with width $\ell$ , which is related to the $D$ -dimensional Compton wavelength. The analysis utilizes the momentum-space representation of this wave function to derive the energy distribution.
Geometry: The background is the TRN metric, characterized by mass $M$ , charge $Q$ , and $D$ spatial dimensions. The horizon locations ( $R_+$ for the outer horizon and $R_-$ for the inner Cauchy horizon) are determined by the roots of the metric function $f_D(r) = 0$ . A dimensionless charge parameter $\alpha = 2Q/\Sigma_D$ is introduced, where $|\alpha| \leq 1$ ensures the existence of real horizons.
HWF Construction: The ADM mass is promoted to a Hamiltonian operator. By inverting the mass-shell relation, the horizon radius is treated as a quantum operator dependent on the source's energy. The HWFs ( $\psi^\pm_H$ ) for the outer and inner horizons are constructed by mapping the momentum-space wave function onto the respective horizon branches, subject to a constraint that the horizon radius must exceed a minimum value determined by the source's rest mass.
Probability Calculation: The joint probability density $P^\pm_J$ is defined as the product of the horizon probability density and the probability of the source being localized within that radius. The total black hole formation probabilities ( $P^\pm_{BH}$ ) for the outer and inner horizons are obtained by integrating these joint densities over the allowed horizon radii.

Key Contributions and Results
The paper provides a quantitative analysis of $P^\pm_{BH}$ as a function of the source mass ( $m$ ), localization width ( $\ell$ ), charge parameter ( $\alpha$ ), and spatial dimension ( $D$ ).

Dependence on Charge ( $\alpha$ ):
- For the outer horizon ( $P^+_{BH}$ ), the formation probability is relatively insensitive to $\alpha$ at low values but remains high for massive, localized sources.
- For the inner horizon ( $P^-_{BH}$ ), the probability is extremely sensitive to $\alpha$ . At small $\alpha$ , $P^-_{BH}$ is highly suppressed (nearly vanishing) because the inner horizon radius approaches zero, falling below the resolution of the finite-width source. As $\alpha$ approaches the extremal limit ( $\alpha \to 1$ ), $P^-_{BH}$ increases drastically, converging with $P^+_{BH}$ as the two horizons coincide.
- The ratio $P^-_{BH}/P^+_{BH}$ changes from being negligible to unity as the system moves from non-extremal to extremal limits.
Dependence on Dimensionality ( $D$ ):
- Black hole formation probabilities are significantly suppressed as the number of spatial dimensions $D$ increases.
- For a fixed mass and charge, higher dimensions require a larger mass-to-gravity-scale ratio ( $m/\Lambda_D$ ) to achieve comparable formation probabilities.
- The paper cites specific numerical examples: for $\alpha = 0.5$ , the ratio of outer-horizon probabilities $P^+_{BH}(D=5)/P^+_{BH}(D=10)$ is approximately $1.9 \times 10^4$ , $1.1 \times 10^4$ , and $8.4 \times 10^2$ for $m/\Lambda_D = 0.5$ , $1$, and $2$, respectively. Even at higher masses ( $m/\Lambda_D = 2$ ), the suppression remains significant (ratio $\approx 840$ ).
- The joint probability densities peak sharply just above the minimum allowed radius, with the width of this peak narrowing as $D$ increases.
Asymmetry ( $\Delta P_{BH}$ ):
- The difference $\Delta P_{BH} = P^+_{BH} - P^-_{BH}$ is predominantly positive across the parameter space, indicating that outer-horizon formation is generally more probable than inner-horizon formation.
- This asymmetry is largest at low charge ( $\alpha$ ) and intermediate masses. As $\alpha \to 1$ , $\Delta P_{BH} \to 0$ .
- Increasing $D$ suppresses the maximum value of $\Delta P_{BH}$ and shifts the peak of the asymmetry toward higher masses.

Significance
The authors claim that their analysis reveals the critical role of spatial dimensionality and charge in the quantum formation of black holes. The results indicate that in higher-dimensional models (such as ADD or RS scenarios), the probability of forming a black hole is highly suppressed unless the source mass is significantly larger than the reduced gravity scale $\Lambda_D$ . Furthermore, the study highlights that while the outer horizon is the primary candidate for formation in non-extremal cases, the inner horizon becomes increasingly relevant as the source approaches the extremal limit. These findings suggest that experimental searches for microscopic black holes in high-energy collisions or cosmic-ray interactions must account for the strong suppression effects introduced by higher dimensions and the specific charge-to-mass ratios of the colliding particles. The paper concludes that the distinction between inner and outer horizon formation probabilities is a strong function of both the charge parameter and the dimensionality of space.

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