At the heart of this investigation lies the principle of Lorentz invariance, a cornerstone of Einstein’s theory of relativity. This principle asserts that the laws of physics remain the same for all observers moving at constant velocities, regardless of their motion. In simpler terms, whether you’re standing still or cruising in a spaceship at a steady speed, the fundamental rules governing how things interact should not change. However, theories that attempt to unify gravity with quantum mechanics, particularly those involving extra spatial dimensions or exotic particle physics at extremely high energies, sometimes predict subtle violations of this cherished symmetry. These potential violations, if they exist, could manifest as tiny, directional preferences in the universe, like a faint cosmic current that nudges particles in a particular way.

The researchers are focusing their attention on a specific, energetic type of particle: ultra-high-energy cosmic rays. These are not your everyday electrons or protons; these are particles that have been accelerated to absurdly high speeds, carrying energies billions of times greater than what we can achieve in terrestrial particle accelerators like the Large Hadron Collider. Their immense energies mean they are incredibly sensitive probes of the vacuum they traverse. As these cosmic travelers journey across vast cosmic distances, they interact with the very fabric of spacetime, and it is in these interactions that the subtle fingerprints of Lorentz violation might be imprinted.

The proposed observational signature of this nonminimal dimension-5 Lorentz violation is rooted in the concept of Vacuum Cherenkov Radiation. Normally, Cherenkov radiation is observed when a charged particle travels through a medium, like water or glass, faster than the speed of light in that medium. This speed limit is slower than the speed of light in a vacuum, c, due to interactions with the medium’s atoms. The result is a characteristic blue glow, famously seen in nuclear reactors. However, the scenario being investigated here is far more exotic: it posits that even in the seemingly empty vacuum of space, if Lorentz symmetry is broken in a specific way, charged particles could lose energy by emitting radiation. This “vacuum” Cherenkov radiation would be a direct consequence of the particle’s interaction with the anisotropic background.

The implications of detecting such vacuum Cherenkov radiation would be nothing short of revolutionary. It would provide the first direct experimental evidence that spacetime is not a perfectly isotropic arena but rather possesses a preferred direction or a subtle structural anisotropy. This discovery would fundamentally alter our understanding of the universe at its most basic level, potentially opening up entirely new avenues for theoretical physics and cosmology. Imagine the scientific frenzy, the countless new experiments designed to map this anisotropy and understand its origins. It would be akin to the discovery of electromagnetism or the confirmation of general relativity – a paradigm shift of monumental proportions.

The theoretical framework underpinning this idea involves extending the Standard Model of particle physics with higher-dimensional operators, specifically dimension-5 operators. These operators are mathematical terms that can be added to the fundamental equations of physics that become relevant at extremely high energy scales, beyond what we have direct access to. The “nonminimal” aspect suggests that these violations are not simple, but rather involve a more complex interplay of fields and symmetries, leading to a richer, more intricate set of potential observable effects. The dimension-5 classification refers to the power of energy or momentum involved in these hypothetical interactions, placing them at a significant, yet potentially accessible, energy scale for cosmic ray observations.

Petrov, Schreck, and Vieira’s paper meticulously lays out the theoretical underpinnings of this phenomenon. They’ve calculated how such Lorentz-violating effects would manifest in the energy spectra of ultra-high-energy cosmic rays. Specifically, they predict that charged particles traveling through this anisotropic vacuum would exhibit an energy-dependent damping effect due to the emission of this vacuum Cherenkov radiation. This damping would translate into a distortion of the observed cosmic ray spectrum, a deviation from what would be expected in a perfectly symmetric universe.

The challenge, of course, lies in identifying this subtle signature amidst the cosmic noise. Ultra-high-energy cosmic rays are incredibly rare events, and accurately measuring their energies and arrival directions is a formidable experimental task. Observatories like the Pierre Auger Observatory in Argentina and the Telescope Array in Utah are designed to detect these particles by observing the extensive air showers they produce when they collide with the Earth’s atmosphere. Analyzing the data from these experiments with the theoretical predictions of Petrov and his colleagues could be the key to unlocking this cosmic secret.

The beauty of this research lies in its predictive power and the potential for falsifiability. The theory doesn’t just speculate; it provides concrete, testable predictions. If ultra-high-energy cosmic rays exhibit the predicted spectral distortions, it would lend strong support to the idea of Lorentz violation. Conversely, if current and future observations show no such distortions, it would place stringent limits on the existence and strength of these hypothetical nonminimal dimension-5 Lorentz-violating effects, further refining our understanding of fundamental physics.

The source of such a Lorentz-violating anisotropy is still a subject of theoretical debate. Some speculative models suggest that it could arise from the fundamental structure of spacetime itself, perhaps related to quantum gravity effects or the presence of a background field that breaks perfect symmetry. Others might point to the distribution of matter or energy in the very early universe, leaving a lasting imprint on the cosmic fabric that influences particle propagation today. The discovery of such an anisotropy would undoubtedly spur intense efforts to understand its origin, potentially leading to breakthroughs in our understanding of the Big Bang and the evolution of the universe.

The researchers highlight that the detection of vacuum Cherenkov radiation would be particularly sensitive to dimension-5 operators because of how they modify the dispersion relations of charged particles. The dispersion relation describes the relationship between a particle’s energy and its momentum. In a Lorentz-invariant theory, this relationship has a well-defined form. However, Lorentz violation can alter this, leading to phenomena like modified speed limits or, in this case, the possibility of energy loss through radiation even in a vacuum. The dimension-5 operators contribute in a specific way to this modification, making them a prime target for observational searches.

The image accompanying this research, though abstract, visually hints at the complex symmetries and potential breaks being explored. It might suggest intersecting planes or warped geometries, alluding to the intricate mathematical structures that describe spacetime at its most fundamental level. Such visualizations, even if not direct depictions of the phenomenon, serve to engage the imagination and convey the profound nature of the questions being asked by theoretical physicists. They bridge the gap between abstract equations and the tangible universe we inhabit, prompting us to consider possibilities beyond our everyday intuition.

Furthermore, the implications extend beyond fundamental physics. If indeed spacetime has directional properties at very high energies, it could have subtle but measurable effects on the propagation of light from distant astronomical objects, potentially influencing everything from our measurements of cosmic distances to our understanding of the expansion of the universe. While the primary focus is on charged particles, the underlying theoretical framework might have broader consequences for our understanding of all fundamental forces and particles interacting with this potentially anisotropic spacetime.

The quest to understand the fundamental nature of the universe is an ongoing journey, marked by bold theoretical proposals and ingenious experimental endeavors. The work by Petrov, Schreck, and Vieira represents a significant step in this journey, offering a compelling new avenue to explore the very foundations of reality. By connecting the abstract realm of theoretical physics with the observable universe through the lens of ultra-high-energy cosmic rays and vacuum Cherenkov radiation, they are pushing the boundaries of our knowledge, inviting us to reconsider what we thought we knew about the ultimate nature of space and time. The universe, it seems, might be a far more interesting and complex place than we ever imagined.

Subject of Research: Probing for nonminimal dimension-5 Lorentz violation through Vacuum Cherenkov radiation in ultra-high-energy cosmic rays.

Article Title: Vacuum Cherenkov radiation for nonminimal dimension-5 Lorentz violation

Article References:
Petrov, A.Y., Schreck, M. & Vieira, A.R. Vacuum Cherenkov radiation for nonminimal dimension-5 Lorentz violation.
Eur. Phys. J. C 86, 30 (2026). https://doi.org/10.1140/epjc/s10052-025-15220-8

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-15220-8

Keywords: Lorentz violation, Vacuum Cherenkov radiation, ultra-high-energy cosmic rays, spacetime anisotropy, dimension-5 operators, theoretical physics, particle physics, cosmology.

The universe, as we perceive it, is a grand and stable stage for the unfolding of cosmic drama. Stars are born and die, galaxies collide, and the tapestry of spacetime hums with predictable regularity. However, lurking at the very foundations of reality might be a hidden instability, a cosmic secret that, if revealed, could fundamentally alter every atom, every force, and every fundamental particle we know. New research, published in the esteemed European Physical Journal C, delves into a theoretical concept so profound and potentially terrifying that it borders on the realm of science fiction, yet it is grounded in the rigorous mathematics of theoretical physics. This work, spearheaded by M.Y. Sassi and G. Moortgat-Pick, explores the unsettling possibility that our universe might be residing in a “false vacuum,” a state of energy that appears stable but is, in fact, a precarious plateau awaiting a collapse into a lower, more stable energy state, a true vacuum.

The concept of vacuum instability might seem counterintuitive at first. We often associate the vacuum with emptiness, with nothingness. In classical physics, the vacuum is simply the absence of matter and energy. However, in the quantum realm, the vacuum is a far more dynamic and energetic place. It is a sea of fluctuating quantum fields, constantly bubbling with virtual particles popping into and out of existence. This quantum vacuum is not necessarily the lowest possible energy state. Imagine a ball resting in a dip on a hillside. This dip represents our current vacuum state, a local minimum in energy. It appears stable, and the ball will likely stay there for a long time. But beyond this dip, perhaps over a small hill, lies an even deeper valley, a global minimum in energy. This deeper valley represents the true vacuum, a state of even lower energy and, theoretically, greater stability.

The paper by Sassi and Moortgat-Pick specifically investigates a theoretical framework known as the “N2HDM,” or the “Next-to-Minimal Two-Higgs-Doublet Model.” This extension of the Standard Model of particle physics proposes the existence of additional Higgs bosons beyond the single one discovered at the Large Hadron Collider. These extra Higgs particles, and the additional scalar fields they represent, introduce a complexity that can lead to the possibility of multiple vacuum states. In such a scenario, our universe, governed by these complex interactions, might indeed be trapped in a false vacuum, a metastable state that persists due to an energy barrier. This barrier, akin to the hill in our analogy, prevents the universe from spontaneously transitioning to the lower energy true vacuum state.

The mechanism by which such a catastrophic transition might occur is through a process called “vacuum decay.” Think of it as a bubble nucleating within our current vacuum. This bubble, representing the true vacuum, would expand outwards at the speed of light, obliterating everything in its path. The energy landscape of our universe would be fundamentally reshaped within this expanding bubble. The fundamental constants of nature, the masses of particles, the strengths of forces – everything could be drastically different, rendering our universe, and all life within it, utterly unrecognizable and impossible.

A particularly intriguing aspect of Sassi and Moortgat-Pick’s work is the role they assign to “domain walls” in inducing vacuum decay. In many cosmological models involving phase transitions or symmetry breaking, the universe can fragment into different regions or “domains,” each in a distinct vacuum state. The boundaries between these domains are called domain walls. These walls are not empty spaces; they are regions where the fundamental fields are undergoing significant changes, and they can possess substantial energy densities. The researchers propose that these domain walls, acting as catalysts or ignition points, could provide the necessary “kick” or quantum tunneling probability to overcome the energy barrier separating the false vacuum from the true vacuum.

Essentially, these domain walls, if they exist and possess certain characteristics, could lower the energy threshold required for vacuum decay. They act as critical seeds, lowering the energy cost of forming the initial bubble of true vacuum. Instead of a random quantum fluctuation needing to spontaneously create the bubble, the pre-existing structure of a domain wall could facilitate this process much more readily. This is a crucial distinction that brings a theoretical possibility much closer to a potential observable phenomenon, even if the observation itself would be the end of our observable universe.

The N2HDM, which serves as the theoretical playground for this research, is a sophisticated extension of the Standard Model designed to address various unanswered questions in particle physics. It introduces a richer spectrum of scalar particles, including the possibility of charged Higgs bosons and additional neutral Higgs bosons. This extended Higgs sector can lead to a more complex potential energy landscape for the vacuum, potentially presenting multiple minima. The precise configuration and interactions of these additional scalar fields in the N2HDM are what open the door to the possibility of our universe residing in a metastable false vacuum state.

The mathematics involved in determining the stability of vacuum states is exceptionally complex. It requires calculating the behavior of quantum fields at extremely high energies and considering the potential energy functions associated with these fields. Even small changes in the parameters of a theoretical model can dramatically alter the vacuum structure, leading from a stable vacuum to a metastable one, or vice versa. Sassi and Moortgat-Pick’s simulations and calculations, performed within the framework of the N2HDM, suggest that certain parameter choices within this model favor the existence of a false vacuum state that is susceptible to decay, with domain walls playing a pivotal role.

The implications of this research are, to put it mildly, staggering. If our universe is indeed in a false vacuum, then the existence of domain walls could mean that we are not passively waiting for a random, low-probability event to trigger our cosmic demise. Instead, we might be living in proximity to a domain wall, or perhaps the universe is already populated by these interfaces, gradually increasing the likelihood of a transition. The search for direct evidence of domain walls or other signatures of vacuum instability is an active, albeit challenging, area of research in cosmology and particle physics.

One of the key challenges in studying vacuum instability is the lack of direct observational evidence. We are, by definition, living within our current vacuum. Detecting the subtle signs of its instability or the presence of domain walls would require highly sensitive instruments and sophisticated analysis of cosmological data. However, theoretical predictions arising from models like the N2HDM can guide experimentalists in their search. For instance, if domain walls have specific gravitational effects or produce particular patterns in the cosmic microwave background radiation, these could be potential observational avenues.

The N2HDM, while a theoretical construct, draws inspiration from observed phenomena and well-established theories. The existence of the Standard Model Higgs boson strongly suggests that scalar fields play a crucial role in the universe. Extending this idea to include multiple Higgs bosons and their associated fields is a natural progression for theorists seeking to explain phenomena not fully accounted for by the Standard Model, such as the masses of neutrinos or the nature of dark matter and dark energy. The N2HDM offers a rich framework where such possibilities, including false vacuum scenarios, can be explored.

Gravitational waves could also potentially offer a window into vacuum decay events. A violent, universe-altering phase transition would likely generate a distinctive gravitational wave signature. Detecting such signals would be a monumental achievement and could provide the first concrete evidence that our universe is not as stable as we once believed. The precise characteristics of these gravitational waves would also shed light on the dynamics of the vacuum transition itself, including the role of any mediating structures like domain walls.

The concept of a “false vacuum” is not entirely new in theoretical physics. It has been a subject of discussion for decades in various cosmological contexts, including inflationary cosmology and theories of grand unification. However, the specific focus on domain walls as facilitators of decay within an extended Higgs model, such as the N2HDM, represents a nuanced and potentially more immediate pathway to exploring this existential threat. It moves the discussion from a purely abstract possibility to one that might be influenced by observable structures within the universe itself.

The implications for humanity are, of course, profound. If such a catastrophic event were imminent, our current understanding of physics would be incomplete, and our future would be drastically altered, or more likely, extinguished. However, it is crucial to emphasize that this remains a theoretical exploration. The universe might be perfectly stable in its true vacuum, or the energy barrier to decay might be so immense that it will take longer than the current age of the universe for even a single decay event to occur. Nevertheless, the pursuit of such questions is fundamental to our understanding of reality.

The research by Sassi and Moortgat-Pick serves as a stark reminder of the vast unknowns that still permeate our understanding of the cosmos. While we celebrate the triumphs of scientific discovery, such as the observation of dark energy or the precise measurement of the Higgs boson’s properties, we must also confront the humbling possibility that the very fabric of our existence might be resting on precognitive instability. The intricate dance of quantum fields, governed by laws we are still striving to fully comprehend, could hold the key to these ultimate cosmic secrets.

This line of inquiry, while perhaps unsettling, is a testament to the power of theoretical physics to push the boundaries of our imagination and knowledge. By exploring extreme scenarios, scientists can uncover fundamental truths about the universe that might otherwise remain hidden. The N2HDM, with its elegant mathematical structure, provides a fertile ground for such explorations, offering insights into potential instabilities that could redefine our cosmic destiny. The allure of understanding the universe at its most fundamental level, even if that understanding reveals uncomfortable truths, continues to drive scientific endeavor. The search for the true nature of the vacuum, whether stable or perilously close to decay, is a quest for the ultimate meaning of existence itself within the grand cosmic narrative.

Subject of Research: Vacuum instability and false vacuum decay induced by domain walls in the N2HDM.

Article Title: Vacuum instability and false vacuum decay induced by domain walls in the N2HDM

Article References: Sassi, M.Y., Moortgat-Pick, G. Vacuum instability and false vacuum decay induced by domain walls in the N2HDM.
Eur. Phys. J. C 85, 1230 (2025). https://doi.org/10.1140/epjc/s10052-025-14875-7

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-14875-7

Keywords: vacuum instability, false vacuum decay, domain walls, N2HDM, Higgs bosons, quantum fields, cosmology, particle physics

At the heart of this investigation lies the ModMax theory, a fascinating extension of Einstein’s general relativity designed to address certain shortcomings of the standard model of gravity. By introducing modifications to the gravitational action, ModMax aims to provide a more comprehensive description of gravitational phenomena, particularly in regimes where gravity behaves in unusual ways. Within this theoretical landscape, the researchers are examining a specific class of black hole solutions that exhibit unique characteristics. These ModMax black holes are not your everyday Schwarzschild or Kerr black holes; they possess properties that allow for a deeper exploration of the fundamental principles of gravity and spacetime. Understanding these exotic black hole solutions is crucial because they serve as theoretical laboratories where extreme conditions can be simulated and fundamental physical laws can be tested under immense gravitational stress, offering insights into how gravity might behave in the very early universe or near singularities.

One of the key concepts being investigated is the “weak gravity conjecture.” This conjecture, a cornerstone of modern theoretical physics, posits that a fundamental theory of gravity must be ‘weak’ enough to allow for the existence of certain exotic particles and phenomena that would otherwise be forbidden by strong gravitational interactions. In simpler terms, it suggests that gravity is not universally so overwhelmingly dominant that it prevents all possibility of exotic physics. The researchers are applying this conjecture to their ModMax black hole solutions to see if these solutions are consistent with the fundamental constraints imposed by this conjecture, thereby strengthening our confidence in the predictive power of ModMax gravity and its ability to describe the universe accurately. This connection to the weak gravity conjecture is significant because it links the behavior of astrophysical objects like black holes to overarching principles that are thought to govern all fundamental forces and particles in the universe.

Furthermore, the study delves into the critical concept of the “weak cosmic censorship conjecture.” This conjecture, proposed by the renowned physicist Roger Penrose, suggests that singularities, the points of infinite density and curvature predicted by general relativity, are always hidden behind event horizons, the one-way boundaries of black holes. In essence, it asserts that the universe is “well-behaved” and that naked singularities, which would violate causality and lead to unpredictable physical outcomes, do not exist in reality. The researchers are probing whether their ModMax black holes uphold this crucial conjecture, examining if any of these exotic spacetime geometries could potentially harbor naked singularities. The violation of cosmic censorship would have profound implications, suggesting that our universe might be far more chaotic and unpredictable than currently believed, and that our understanding of causality itself might need revision.

The “photon sphere” analysis also plays a pivotal role in this research. A photon sphere is a spherical region around a black hole where gravity is so strong that photons, particles of light, can be trapped in unstable orbits. This region is crucial for understanding how light behaves near black holes and provides a distinct observational signature. By studying the properties of the photon sphere in ModMax black holes, the researchers can gain valuable insights into the structure of spacetime around these exotic objects. The size and stability of the photon sphere are directly influenced by the underlying gravitational theory, making this analysis a powerful tool for discriminating between different models of gravity and for testing the validity of ModMax theory against observational data, should it become possible to observe such phenomena directly.

The meticulous calculations and theoretical explorations undertaken by Gashti and their team delve into the mathematical intricacies of Einstein-Hilbert action and its modifications within the ModMax framework. They are not just qualitatively discussing these concepts but are performing rigorous derivations to understand the precise conditions under which these conjectures hold or might be violated. This quantitative approach is essential for turning abstract theoretical ideas into testable predictions. The energy conditions, fundamental assumptions about the distribution of matter and energy in spacetime, are critically examined within the context of their black hole solutions. The behavior of quantum fields propagating in these modified spacetimes is also a significant area of interest, as it can reveal subtle deviations from standard general relativity and offer clues about quantum gravity.

The research paper likely involves complex mathematical tools, including differential geometry, tensor calculus, and potentially advanced techniques from quantum field theory in curved spacetime. The team is likely employing sophisticated numerical methods to solve the Einstein field equations, or their ModMax equivalents, for specific configurations of matter and energy. The stability of these black hole solutions under various perturbations is also a key aspect of the analysis, as unstable solutions would not be expected to persist in the real universe. This detailed mathematical framework allows them to make precise predictions about observable quantities, even if those observations are currently beyond our technological capabilities, thereby guiding future observational efforts in a more informed direction.

The implications for our understanding of the universe are far-reaching. If ModMax theory, with its unique black hole solutions, proves to be a more accurate description of gravity than standard general relativity, it could revolutionize our understanding of cosmological evolution, from the Big Bang to the formation of large-scale structures. It might also shed light on fundamental mysteries such as dark matter and dark energy, which currently lack satisfactory explanations within the standard model. The exploration of weak gravity and cosmic censorship in these exotic black holes could also provide crucial insights into the nature of quantum gravity, the elusive theory that aims to unify gravity with the other fundamental forces of nature.

The study of ModMax black holes and their adherence to the weak gravity and cosmic censorship conjectures can potentially lead to profound philosophical implications about the nature of reality. If naked singularities were to exist, it would imply a breakdown of predictability and causality, suggesting that the universe might not be as deterministic as we once assumed. This could fundamentally alter our understanding of free will, the arrow of time, and our place within the cosmic order. The very fabric of our comprehension of cause and effect could be challenged, forcing us to re-evaluate our most deeply held assumptions about the universe and our ability to understand it.

The researchers are likely also examining the thermodynamics of these ModMax black holes. Black holes, despite their seemingly simple exterior, possess a rich thermodynamic character, with properties such as temperature and entropy. Studying these thermodynamic properties in exotic black hole solutions can reveal deep connections between gravity, quantum mechanics, and thermodynamics, offering further insights into the fundamental nature of spacetime and the universe. The entropy associated with these black holes, for instance, could provide a crucial link to microscopic degrees of freedom that underly gravitational phenomena, furthering our quest for a quantum theory of gravity.

The precision with which these theoretical predictions are made is crucial. The researchers are not presenting vague notions but are formulating specific, mathematically derived consequences of their theoretical framework. This allows for the possibility of future experimental verification, even if that verification requires advancements in observational astronomy or particle physics. The ability to connect theoretical constructs with potentially measurable quantities is the hallmark of strong scientific inquiry and is what drives progress in our understanding of the cosmos. This iterative process of theory, prediction, and verification is what allows science to refine its models and approach a more accurate description of reality.

The potential for ModMax black holes to exhibit properties that challenge current understanding underscores the dynamic and ever-evolving nature of physics. The universe, it seems, is far more complex and surprising than we can readily imagine. Each new theoretical development, each novel mathematical exploration, opens up new avenues of inquiry and pushes the boundaries of our knowledge. The ModMax theory and its black hole solutions represent just one such frontier, but it is a frontier that promises to yield significant insights into the fundamental workings of the cosmos and the deep connection between gravity and the very essence of existence.

The quest to understand black holes is not merely about deciphering the behavior of these celestial objects; it is about unraveling the fundamental laws of physics that govern all of reality. The work of Gashti and their collaborators, by exploring the theoretical landscape of ModMax gravity and its implications for cosmic censorship and the weak gravity conjecture, is contributing to this grand endeavor. Their research serves as a beacon, illuminating the path towards a deeper, more unified understanding of the universe, from the smallest quantum scales to the largest cosmic expanse, and challenging us to think beyond the limits of our current, albeit highly successful, physical models.

In conclusion, the provided image and accompanying publication represent a significant step forward in our theoretical understanding of gravity and black holes. The research into ModMax black holes, the weak gravity conjecture, and cosmic censorship is not only intellectually stimulating but also has the potential to redefine our cosmic perspective. As our observational capabilities continue to advance, the theoretical frameworks laid out in works like this will become increasingly vital for interpreting the universe’s deepest secrets and for charting the future course of fundamental physics. The pursuit of knowledge in these extreme theoretical domains highlights humanity’s insatiable curiosity and its relentless drive to comprehend the profound mysteries of existence.

Subject of Research: Theoretical exploration of modified gravity theories, specifically the ModMax theory, and its implications for black hole physics, cosmic censorship, and fundamental conjectures in physics.

Article Title: Weak gravity conjecture in ModMax black holes: weak cosmic censorship and photon sphere analysis.

Article References: Gashti, S.N., Afshar, M.A.S., Alipour, M.R. et al. Weak gravity conjecture in ModMax black holes: weak cosmic censorship and photon sphere analysis. Eur. Phys. J. C 85, 1144 (2025). https://doi.org/10.1140/epjc/s10052-025-14890-8

Image Credits: AI Generated

DOI: https://doi.org/10.1140/epjc/s10052-025-14890-8

Keywords: ModMax black holes, weak gravity conjecture, weak cosmic censorship, photon sphere, modified gravity