The technical foundation of the work is a carefully chosen seed metric. The authors begin with a static, spherically symmetric Schwarzschild-type line element whose metric function f(r) gathers every energy contribution into a single expression: the black hole’s mass, its electric charge squared, its magnetic charge squared, a term proportional to the cosmological constant Λ, and a quintessence term of the form α times r to the power minus one minus three omega. Here α measures the strength of the dark energy field and the parameter ω, constrained to lie between −1 and −1/3, encodes how the pressure of that field relates to its energy density. This functional form descends from the classic 2003 solution of V. V. Kiselev, which first showed how the Einstein equations can be solved exactly for a black hole surrounded by quintessence. Crucially, in the new construction the cosmological constant and the quintessence are treated as genuine external matter sources distributed throughout spacetime, not merely as geometric corrections appended to a vacuum solution.

To set this static configuration spinning, de Oliveira and Schmidt employed the Newman–Janis algorithm, a mathematical recipe invented in 1965 that transforms non-rotating black hole metrics into their rotating counterparts. The algorithm is powerful but notorious: different choices in its intermediate steps can lead to ambiguous results. The authors sidestepped this pitfall by adopting a unique and general complexification rule for the radial coordinate, one that systematically converts every power of r in the metric function into its rotating equivalent. Under this rule, powers of r are rewritten in terms of the quantity Σ, which equals the radial coordinate squared plus the square of the spin parameter a times the square of the cosine of the polar angle. The result is a fully rotating line element in Boyer–Lindquist coordinates, whose structure function Δr combines the mass, both charges, the quintessence contribution, and a fourth-power term in r carrying the cosmological constant. Notably, Δr contains no coupling terms between Λ and the spin parameter a—a departure from the standard Kerr–Newman–AdS geometry, where such couplings are introduced precisely to keep the vacuum Einstein equations satisfied.

The payoff of that choice is conceptual clarity about what is holding the spacetime together. Because the cosmological constant and quintessence enter as matter from the very beginning, the Einstein tensor of the new solution is nonzero in ways the vacuum AdS case is not. The authors computed the full stress–energy tensor by evaluating the Ricci tensor and the Einstein tensor for the rotating metric, and they found components that scale with each energy ingredient: pieces proportional to α times ω that encode the quintessence fluid, pieces proportional to the combined charge squared, and pieces proportional to Λ. They also derived the electromagnetic field tensor of the dyonic configuration. Starting from the static solution, where Maxwell’s equations yield a radial electric field of Q_e over r squared and a radial magnetic field of Q_m over r squared, they used the Newman–Penrose formalism with null tetrad vectors to carry the field into the rotating geometry. The resulting Maxwell scalar takes the compact form involving the complex combination Q_e plus i times Q_m divided by two Σ, and the electromagnetic tensor, by symmetry, turns out to be entirely independent of Λ and α.

With the geometry in hand, the researchers mapped its most dramatic features: the horizons and the ergosphere. Setting Δr equal to zero yields the event horizon condition, a fourth-degree equation in the radial coordinate whose roots include an inner horizon, the outer Schwarzschild-like horizon, a cosmological horizon, and a quintessence horizon. Numerical analysis across representative parameter values showed that only three of the four roots are ever real, the fourth remaining complex. The trends are physically intuitive: increasing either the cosmological constant or the quintessence parameter shrinks the real horizon radii, meaning the dark energy content of the universe effectively compresses the region of no return. The ergosphere—the region outside the horizon where nothing can remain static because the spinning black hole drags spacetime itself around with it—was found by setting the time-time metric component to zero. In the equatorial plane this reduces to static circumferences that depend on Λ, α, ω, and the charges but are, curiously, independent of the spin parameter a. Larger quintessence strength again produces smaller static circumferences.

Singularities, the points where the mathematical description of spacetime breaks down entirely, were probed using the Kretschmann scalar, a coordinate-invariant measure of curvature built from the Riemann tensor. For this solution the Kretschmann scalar takes the form of a complicated parameter-dependent function divided by Σ raised to the twelfth power. It diverges only when Σ vanishes, which happens at r equals zero on the equatorial plane—exactly the ring-like singular structure familiar from the Kerr and Kerr–Newman solutions. The dark energy ingredients, however dramatic their effect on the horizon structure, leave this singularity region unchanged. The Ricci scalar tells a subtler story. In the limits where both spin and quintessence vanish, it reduces to minus four Λ, the textbook vacuum value for a spacetime containing only a cosmological constant. But whenever α is nonzero, even in the limit where the cosmological constant is absent, the scalar curvature departs from the vacuum result—a direct fingerprint of treating dark energy as matter rather than geometry.

To see how matter would actually move in this environment, the authors studied a unit-mass test particle on a circular orbit in the equatorial plane, observed from a locally non-rotating frame, a reference system in which the particle carries no angular momentum. From the geodesic equations they extracted expressions for the particle’s angular momentum, mechanical energy, angular velocity, and rotational velocity. Every energy term in the model leaves its mark on these quantities, and in the limit where both α and Λ go to zero the standard Kerr–Newman results are recovered. Intriguingly, even when the black hole itself has no spin, the locally non-rotating frame analysis yields nonzero angular and rotational velocities, a reminder that this reference frame is itself defined by the geometry rather than by the black hole’s rotation alone.

The final piece of the analysis concerned the energy conditions, the classical sanity checks that general relativity imposes on any legitimate matter source. Because the rotating metric contains off-diagonal terms, the usual shortcut expressions cannot be applied; instead, the authors constructed an orthonormal tetrad basis in the locally non-rotating frame and expressed the stress–energy tensor in terms of an energy density ε and three principal pressures. Their findings delineate when this exotic spacetime can be supported by ordinary matter and when it cannot. When the null energy condition is violated, every other energy condition falls with it. Divergences in the energy density and pressures occur at the event horizon and at a second locus defined by the vanishing of ξ—but that second locus always lies inside the horizon, so it does not affect the exterior region of physical interest. For concrete parameter choices, such as ω equal to −2/3 with modest charge, mass, quintessence strength, and cosmological constant, there exists an exterior band of radii where the weak energy condition holds, meaning any observer measures positive energy density and the geometry is supported by non-exotic, classical matter. Outside those bands, the spacetime demands exotic ingredients—precisely the quintessence field of the model, or potentially quantum effects.

The new solution is a rich family rather than a single object. Switching off parameters recovers a parade of familiar geometries: without spin, the dyonic cosmological black hole surrounded by quintessence; without quintessence, the rotating charged cosmological black hole; without a cosmological constant, the dyonic Kerr–Newman black hole in quintessence; with both dark energy terms removed, the Kasuya–Kerr–Newman solution; and with both charges zeroed, the rotating cosmological black hole surrounded by quintessence. The work thus gives theorists a versatile laboratory for exploring how dark energy reshapes black hole shadows, accretion disks, quasinormal ringing, and gravitational lensing—observables that upcoming surveys and the next generation of gravitational wave detectors may one day constrain. For now, the result stands as an exact, matter-inclusive answer to a deceptively simple question: what does a spinning, charged black hole look like in a universe that refuses to sit still?

Subject of Research: An exact general relativistic solution for a rotating dyonic black hole surrounded by quintessence dark energy

Article Title: Dyonic rotating cosmological black hole surrounded by quintessence

Article References: Dyonic rotating cosmological black hole surrounded by quintessence. (n.d.). https://doi.org/10.1140/epjc/s10052-026-16309-4

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16309-4

Keywords: black holes, general relativity, quintessence, dark energy, Kerr–Newman solution, Newman–Janis algorithm, event horizon, ergosphere, energy conditions, cosmological constant, dyonic black hole, theoretical physics

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