String-Filled Black Holes May Show Bigger Shadows and Endless Stability

Black holes are the most extreme objects predicted by Einstein’s general relativity, and yet the classical theory breaks down at their very centers. In the standard Kerr solution, which describes a rotating black hole, the mass is compressed into a singularity where curvature diverges and the known laws of physics cease to apply. Resolving this central pathology is one of the enduring puzzles of gravitational physics, and it has motivated theorists to build so-called regular black holes: spacetimes that behave like black holes on the outside but remain perfectly smooth at the core. A new study published in the journal General Relativity and Gravitation takes this program a significant step further by constructing a rotating regular black hole immersed in a cloud of strings, and then interrogating the resulting object with two of the sharpest available tools: the thermodynamics of horizons and the geometry of black hole shadows.

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