The phantom field at the heart of the study is not an exotic novelty pulled from thin air. Phantom energy is a hypothetical form of dark energy whose equation-of-state parameter w is less than minus one, meaning its energy density grows as the universe expands. First proposed by Robert Caldwell in the early 2000s, phantom matter would drive ever-accelerating expansion and, in the most dramatic scenarios, a cosmic big rip. When such a field is threaded through a black hole spacetime instead of the cosmos at large, it leaves a measurable fingerprint: the phantom scalar charge alpha. Regular phantom black hole solutions, developed by Bronnikov and Fabris and others, incorporate this charge directly into the metric, altering how spacetime curves around the object.

To understand how alpha changes the behavior of light, Halder turns to a powerful geometric idea: the optical manifold. Photons in a static spacetime do not trace arbitrary paths; their trajectories are geodesics, the straightest possible lines, on a specially constructed curved surface known as the optical geometry. By computing two fundamental quantities on this surface, the Gaussian curvature and the geodesic curvature, one can read off how light rays bend, converge and spread without solving the full ray equations every time. Gaussian curvature measures how the optical surface bulges or saddles at each point, while geodesic curvature describes how a light path deviates from the natural straight lines of that surface. Together they encode the entire bending environment a photon experiences.

The central finding is that the phantom parameter drives a smooth crossover between two qualitatively distinct regimes. For small values of alpha, the optical geometry is strongly curved, and the effective potential that governs photon motion is steep and unforgiving. This is essentially the Schwarzschild picture: light approaching the photon sphere, the critical radius where light can circle the black hole, sits on a razor’s edge, and the slightest perturbation sends it spiraling into the hole or flinging it back to infinity. As alpha grows, however, the Gaussian curvature of the optical manifold diminishes, the potential softens, and the system transitions continuously into a stable weak-field regime in which gravitational focusing is substantially weakened. The crossover is smooth rather than abrupt, meaning the black hole’s optical personality can be dialed from one extreme to the other.

To quantify the stability of circular light orbits, the study employs one of the standard tools of nonlinear dynamics: the Lyapunov exponent, lambda. This number measures how fast two initially neighboring trajectories separate from one another. A large positive exponent means chaos-adjacent behavior, where even infinitesimal deviations grow exponentially and circular photon orbits are hopelessly unstable. Halder shows that the Lyapunov exponent attains its maximum precisely at the photon sphere radius and decreases as that radius increases. Crucially, increasing the phantom charge systematically suppresses lambda, smoothing the effective potential and lowering the orbital instability. In the small-alpha limit the familiar Schwarzschild-like strong instability is recovered, while at large alpha the photon region becomes markedly more stable. This connects the work to a well-established result, due to Cardoso and collaborators, linking Lyapunov exponents of geodesic instability to the imaginary part of black hole quasinormal modes, the characteristic ringdown frequencies of perturbed black holes.

The observational stakes of this analysis are considerable, because the quantities Halder tracks are not abstract. The photon sphere defines the edge of the black hole shadow, the dark silhouette imaged by the Event Horizon Telescope collaboration in M87 and in Sagittarius A-star. If a phantom scalar charge were present, it would shift the photon-sphere radius, reshape the shadow’s apparent size, and modify both the weak and strong deflection angles of light passing near the hole. Strong gravitational lensing in particular is exquisitely sensitive to the structure of spacetime near the photon sphere, where photons may loop around the black hole one or more times before escaping. A softened effective potential at large alpha would change the pattern of relativistic images, potentially offering a way to constrain or detect phantom fields through precision shadow and lensing measurements.

Methodologically, the paper combines differential geometry with dynamical-systems visualization. Halder employs phase-portrait analysis, presented through logarithmic and semi-logarithmic plots, to display how photon trajectories evolve as the phantom charge is varied. Phase portraits are the standard graphical language of stability theory: they map out the full space of possible motions, showing at a glance which orbits spiral away, which settle into closed loops, and where the separatrices between qualitatively different behaviors lie. Reading these portraits alongside the curvature calculations gives a two-pronged picture: geometry tells you why the photon paths bend as they do, and phase-space analysis tells you what happens to a photon that starts slightly off the ideal circular orbit. That the two approaches agree, both pointing to alpha as the governing parameter, strengthens the physical interpretation.

The broader context of the study sits at the intersection of dark energy physics and strong-field gravity. Observational campaigns such as the Wilkinson Microwave Anisotropy Probe measurements analyzed by Dunkley, Komatsu and their colleagues established that the universe’s expansion is accelerating, but whether the responsible component violates the w greater than minus one bound remains an open question. Quintom cosmology, surveyed by Cai, Saridakis and coauthors, allows dark energy to evolve across that boundary, meaning a phantom phase is a live theoretical possibility. Vikman showed that dark energy can in principle evolve into the phantom regime, and scalar-tensor constructions by Elizalde, Nojiri and Odintsov provide concrete field-theoretic realizations. If the universe passed through or currently occupies a phantom-like state, the question of how phantom fields interact with compact objects becomes unavoidable, and black holes, with their maximally curved environments, are the most sensitive probes available.

Halder’s result reframes the phantom charge in a useful way: rather than treating alpha as an arbitrary deformation of an exact solution, the study demonstrates that it functions as a true control parameter for the photon dynamics. Everything an observer might measure about light near these holes, the effective potential, the photon-sphere radius, the shadow boundary, the deflection angles, and the instability exponent, depends monotonically and smoothly on alpha. That coherence across independent observables is what elevates the parameter from mathematical decoration to physical dial. It also suggests a practical program: by fitting shadow radii and strong-lensing image separations across a family of observed black holes, one could in principle bound the phantom charge and, by extension, place limits on phantom matter in the strong-field regime, a domain where cosmological surveys cannot directly look.

Of course, the analysis is theoretical, built on idealized static spacetimes and classical null geodesics, and no datasets were generated in the study. Real black holes rotate, accrete plasma, and live in environments threaded by ordinary matter as well as whatever dark components pervade the cosmos. Yet the qualitative message is robust and tantalizing: the universe’s most enigmatic energy component, if it couples to black holes, would not merely decorate their metrics but would actively tame them, softening the brutal light-bending environment of the photon sphere and stabilizing the orbits that define the shadows we photograph. As Event Horizon Telescope observations sharpen and next-generation facilities come online, parameters like alpha may move from the pages of theoretical journals into the error bars of real measurements, turning black hole shadows into laboratories for the darkest physics we know.

Subject of Research: Effect of a phantom scalar field charge on null geodesics, optical geometry and photon orbit stability around black holes

Article Title: Phantom scalar charge as a control parameter for optical geometry and photon orbit stability around black holes

Article References: Phantom scalar charge as a control parameter for optical geometry and photon orbit stability around black holes. (n.d.). https://doi.org/10.1007/s10509-026-04635-8

Image Credits: AI Generated

DOI: 10.1007/s10509-026-04635-8

Keywords: black holes, phantom scalar fields, dark energy, null geodesics, photon sphere, Gaussian curvature, Lyapunov exponent, black hole shadow, gravitational lensing, optical geometry, general relativity, geodesic stability

The research team, led by Y. Elaima, H. Lekbich, A. Daassou and F. Oubbad of Cadi Ayyad University and Moulay Ismail University in Morocco, begins from a static, spherically symmetric seed metric that carries two distinct signatures. The first is a regularization parameter, denoted r0, which replaces the point singularity with a de Sitter-like core. The second is a density parameter, epsilon, which characterizes a background cloud of strings following the framework introduced by P. S. Letelier in 1979. In such a model, the gravitational source is an anisotropic effective fluid whose radial pressure equals minus its energy density, a relation that mimics a dark-energy-like tension along the strings. The resulting seed metric function takes the elegant form f(r) = 1 − (2M/r + ε)Ψ(r), where the regularization function is Ψ(r) = 1 − exp(−r³/r0³), smoothly switching off the gravitational contribution of the mass and the string cloud at the origin.

Turning this static configuration into a rotating one is a delicate business. The authors employ the non-complexified Newman-Janis algorithm, a technique refined by M. Azreg-Aïnou in 2014 that avoids the mathematically questionable complexification step of the original 1965 procedure. By applying this method, the team generates a stationary, axisymmetric spacetime that rotates like Kerr but retains the regularity and the string-cloud content of the seed. The authors verify in detail, through an explicit evaluation of the Einstein tensor and the associated energy-momentum tensor, that the resulting metric is a genuine solution of Einstein’s field equations sourced by a well-defined anisotropic fluid. Far from the black hole, where the regularization function approaches unity, the energy density falls off as epsilon over r squared, precisely recovering the Letelier cloud of strings limit. The construction therefore interpolates seamlessly between known physics at large distances and a novel regular geometry at small radii.

The cure for the singularity is demonstrated with full mathematical rigor. Near the origin, the regularization function behaves like r³/r0³, so the metric function approaches 1 − 2Mr²/r0³, which is exactly the form of a de Sitter spacetime with a positive effective cosmological constant. The curvature invariants confirm this: the Ricci scalar tends to the finite value 24M/r0³ and the Kretschmann scalar to 96M²/r0⁶ as r goes to zero. There is no divergence anywhere in the spacetime. This de Sitter core, inherited from the tradition of Bardeen, Hayward and Ayón-Beato–García regular black holes, means that infalling matter and information would never encounter an infinite-curvature boundary, offering a concrete arena in which the quantum-gravity endgame of gravitational collapse might be modeled without the fatal flaw of classical relativity.

With the geometry in hand, the authors turn to thermodynamics, the field inaugurated by Hawking’s discovery that black holes radiate and Bekenstein’s identification of horizon area with entropy. Black hole temperature is tied to the surface gravity of the horizon, and its behavior as a function of mass encodes the stability of the object. The analysis reveals a rich structure. The heat capacity, whose sign determines whether a black hole responds to fluctuations by returning to or fleeing from equilibrium, develops divergences that signal a second-order phase transition in the Davies sense. On one side of the critical point the black hole is thermodynamically unstable and sheds energy through Hawking evaporation; on the other side it settles into a stable branch. Remarkably, the study shows that in a certain parameter regime a thermodynamically stable state exists in which Hawking evaporation simply ceases, leaving behind a long-lived remnant. Such remnants are of great theoretical interest because they could provide endpoints of evaporation that avoid information-loss puzzles, and the string cloud density epsilon and regularization scale r0 both shift the location and character of these transitions.

The second major line of investigation concerns the black hole shadow, the dark silhouette a black hole casts against the glow of background light. Since the Event Horizon Telescope’s landmark 2019 image of M87*, shadow calculations have become the standard phenomenological bridge between abstract metrics and actual observation. Following the established framework of Synge, Luminet and Bardeen’s geodesic analysis, and using the observables proposed by Hioki and Maeda, the authors compute the photon trajectories in their rotating regular spacetime and reconstruct the apparent shape seen by a distant observer. The result is a striking phenomenological decoupling of two physical effects that are usually entangled. The spin parameter governs the geometric distortion of the shadow: as in Kerr, faster rotation drags the silhouette sideways into the familiar D-shaped asymmetry. The string cloud density, by contrast, acts as a gravitational magnifying lens, systematically inflating the angular diameter of the shadow without substantially changing its distortion.

This decoupling has immediate observational significance. In realistic comparisons with horizon-scale imaging, degeneracies between black hole spin and environmental effects are a persistent obstacle, since different combinations of parameters can produce similar images. A scenario in which one parameter controls the size of the shadow while another independently controls its shape offers a cleaner diagnostic handle. If supermassive black holes are indeed threaded by a cloud of strings, or by some medium with an analogous anisotropic equation of state, then precision measurements of shadow diameter and distortion together could, in principle, disentangle the intrinsic rotation of the object from the properties of the exotic matter permeating its surroundings. The authors explicitly suggest that this phenomenological decoupling could be tested by future interferometric observations, including upgrades to the Event Horizon Telescope and proposed space-based very long baseline interferometry missions that would sharpen the image of Sagittarius A* and other targets.

The broader context makes the result timely. Regular black holes have been explored extensively in recent years, including rotating versions generated by Bambi and Modesto and models incorporating nonlinear electrodynamics, dark energy, quintessence and noncommutative geometry. Black holes have also been studied in the presence of perfect fluid dark matter and plasma environments, each of which modifies the shadow in characteristic ways. The string cloud channel, however, carries a distinctive theoretical pedigree: strings are the fundamental objects of quantum gravity’s leading candidate framework, and a universe threaded with cosmic strings or a stringy medium is a serious possibility in the early cosmos. Building a rotating, regular, string-embedded black hole therefore welds together three lines of thought — the removal of the singularity, the inclusion of string-inspired matter, and the phenomenology of shadows — that have mostly been pursued separately.

Caveats remain, as they do in any theoretical construction. The anisotropic fluid sourced by the metric is phenomenological, and identifying it with a concrete microscopic string model will require further work; the energy-momentum tensor derived by the authors is self-consistent but not derived from fundamental string theory. The parameters r0 and epsilon are not yet constrained by observation, and present-day shadow imaging is far from the precision needed to detect the magnifying effect of a weak string cloud. Nonetheless, the paper provides a complete, self-contained package: an exact rotating solution, a proof of its regularity, a full thermodynamic stability analysis with a well-defined phase transition and a stable remnant branch, and shadow observables that map directly onto measurable quantities. As horizon-scale experiments accumulate sharper and sharper images of the black holes at the centers of our galaxy and of M87, models of precisely this kind will define the vocabulary in which any deviation from classical Kerr expectations is expressed — and perhaps, one day, the language in which the first hints of quantum gravity are read.

Subject of Research: A new rotating regular black hole solution in a cloud of strings background and its thermodynamics and shadow properties.

Article Title: Rotating regular black hole in a string cloud background: thermodynamics and shadows

Article References: Rotating regular black hole in a string cloud background: thermodynamics and shadows. (n.d.). https://doi.org/10.1007/s10714-026-03598-z

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03598-z

Keywords: black holes, regular black hole, cloud of strings, string cloud, Newman-Janis algorithm, black hole thermodynamics, phase transition, black hole shadow, Event Horizon Telescope, Hawking radiation, general relativity, de Sitter core