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Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics
Abstract
Algebraically special solutions of Einstein's empty-space field ; equations that are characterized by the existence of a geodesic and shear-free ; ray congruence are considered. A class of solutions is presented for which the ; congruence is diverging and is not necessarily hypersurface orthogonal. (C.E.S.);
... Exact solutions are particularly important in General Relativity to understand the subtle local and global properties of the spacetime. Famous examples include the Schwarzschild [1] and Kerr [2] solutions for black holes, the Friedmann solutions [3] for cosmology, plane-wave solutions and Robinson-Trautman [4] solutions for the existence of gravitational radiation without linearization. However, finding exact solutions of the Einstein equation is very challenging because of its high non-linearity so that the perturbation schemes may not be valid. ...
... The uniqueness of the Schwarzschild solution can be extended to the static assumption [7]. A further extension is the uniqueness theorem of the Kerr metric [2] for the stationary axisymmetric spacetime with a smooth convex event horizon [8,9], and a further no-hair conjecture for relevant configurations [10]. Another line of physical assumptions is from the algebraic classification of the Weyl tensor [11], e.g. the original derivation of the Kerr solution [2] and the renowned Ricci-flat Plebanski-Demianski (PD) solution [12] that contains two parameters in addition to the usual mass and angular momentum of the Kerr metric, namely the NUT [13,14] and the acceleration parameters. ...
... A further extension is the uniqueness theorem of the Kerr metric [2] for the stationary axisymmetric spacetime with a smooth convex event horizon [8,9], and a further no-hair conjecture for relevant configurations [10]. Another line of physical assumptions is from the algebraic classification of the Weyl tensor [11], e.g. the original derivation of the Kerr solution [2] and the renowned Ricci-flat Plebanski-Demianski (PD) solution [12] that contains two parameters in addition to the usual mass and angular momentum of the Kerr metric, namely the NUT [13,14] and the acceleration parameters. It is well believed that the PD solution is the general algebraic Petrov Type-D solution. ...
We derive the Ricci-flat metrics in four dimensions that are stationary and algebraically special, together with the locally asymptotically flat conditions in the Bondi-Sachs framework. The solutions consist of a pair of arbitrary holomorphic and antiholomorphic functions analogous to the Virasoro modes, and also three constant parameters originated from the modes in spherical harmonic expansion. We show that the higher modes of the (anti-)holomorphic function contain an infinite tower of soft hairs from the perspectives of both the local gravitational degree of freedom and the asymptotic supertranslation charges. Within our general ansatz, we obtain from the zero modes the complete set of algebraic Petrov type-D solutions of four free parameters. We show that one parameter does not belong to the Plebanski-Demianski class and, therefore, yields a new type-D metric.
... Schwarzschild black holes are spherically symmetric, non-rotating, and uncharged black holes, while Kerr black holes are spherically symmetric, uncharged, and rotating. [2][3][4] In fact, black holes that strictly meet the conditions of ideal Schwarzschild and Kerr are dark black holes composed of dark matter. 3,4 The event horizon radius of a Schwarzschild black hole (the Schwarzschild radius) is: ...
... [2][3][4] In fact, black holes that strictly meet the conditions of ideal Schwarzschild and Kerr are dark black holes composed of dark matter. 3,4 The event horizon radius of a Schwarzschild black hole (the Schwarzschild radius) is: ...
This paper explores the theoretical existence and implications of dark quarks and their role in forming dark hadrons, dark neutron stars, and dark black holes. By simplifying the relationship between hadron charge and quark number, we propose a more transparent and theoretically elegant framework for understanding the charge structure of fundamental particles. We introduce a classification system that distinguishes between bright and dark particles based on their charge and color charge properties, leading to potential extensions in particle physics models. Furthermore, we investigate the nature of dark black holes, differentiating them from bright black holes, and discuss their potential observational signatures in astrophysics. This study provides a theoretical foundation for future explorations of dark matter and its interactions in the universe.
... Modeling spinning black holes, particularly those with significant classical spin, presents a formidable theoretical challenge deeply linked to the complex behavior of strongly coupled extended bodies. A prominent example of a spinning black hole is the Kerr black hole [1], which, in the thorough analysis of Israel [2], was shown to be isomorphic to a hypersurface exhibiting a disc topology. It motivates the exploration of the hypersurface covariant action principle for spinning black holes, which we will pursue here. ...
We examine a hypersurface model for the classical dynamics of spinning black holes. Under specific rigid geometric constraints, it reveals an intriguing solution resembling expectations for the Kerr Black three-point amplitude. We explore various generalizations of this formalism and outline potential avenues for employing it to analyze spinning black hole attraction.
... Nonetheless, the rotational metric ds 2 ω can be easily compared with these solutions in geometry. For instance, the solution behaves like that of Schwarzchild [4] at the poles where the inner SLS is tangent to the event horizon [3,20,22], but like that of Kerr [23] in the ergosphere where massive particles are necessarily dragged along with the hole's rotation, and finally like that of Schwarzschild-de Sitter (SdS) in the faraway region outside the outer SLS where it is not asymptotically flat [24][25][26]. Interestingly, there is still a normal space between the inner and outer SLSs, as shown in figure 1 by the white region. ...
The rotational metric offers an exact solution to Einstein's clock-rate problem in curved spacetime regarding whether time flows more slowly at the equator of a compact object like a neutron star than at its poles. Similar to what Einstein had done for a uniformly rotating disk, the metric can be obtained directly from the Schwarzschild metric through rotational transformations. In accordance with the equivalence of inertia and gravity, the rotational metric exhibits more complex geometric structures due to the effects of inertia-gravity arising from rotation. For instance, it has an extra ergosphere compared to the original Schwarzschild metric. The presence of the ergosphere could serve as a compelling example that illustrates how the rotational transformations can introduce new structures into a gravitational system. Exactly, there are additional physical degrees of freedom, carried by the rotational transformations, that are 'eaten' by the gravitational system to form an extra ergosphere inside the system itself, which is in analogy to the Higgs mechanism in particle physics. In this process, the rotational transformations play a role of 'symmetry breaking'. Besides, the rotational metric can describe the gravitational system that rigidly corotates with a spinning mass. It has potential applications in describing the rotationally-induced gravitational effects within various rotating magnetospheres and relevant astronomical phenomena. The understanding of the rotational metric is therefore expected to bring novel insights into general relativity and high-energy astrophysics.
... To extend our analysis, we now consider black hole spacetimes beyond the Schwarzschild black hole. Specifi-cally, we investigate the impact of scalar potentials on Hawking radiation in the context of the Kerr, Kerr-Newman, and Reissner-Nordström black holes, which describe rotating and charged black holes, as well as the Schwarzschild black hole [20,21]. The geometry of these spacetimes plays a crucial role in the propagation of scalar fields near the event horizon, affecting the particle creation process and the resulting radiation spectrum [22] [23][24][25]. ...
The study of Hawking radiation provides a critical avenue for exploring the interface between quantum mechanics, gravity, and black hole physics. Traditionally, the emission from black holes is expected to follow a thermal spectrum, as predicted by Hawking’s theory. However, recent advancements in quantum gravity and scalar field theories suggest that deviations from this ther- mal behavior may arise due to interactions between scalar fields and the black hole’s spacetime curvature. This paper investigates the potential modifications to Hawking radiation caused by scalar fields in different black hole spacetimes, including Schwarzschild, Kerr, Kerr-Newman, and Reissner-Nordstr¨om geometries. We examine the role of scalar field potentials—massive, massless, and self-interacting—and their impact on the radiation spectrum. Additionally, we explore the observational challenges of detecting these deviations, emphasizing the role of gravitational wave astronomy and multi-messenger astrophysics. Our results suggest that while traditional thermal spectra may serve as a baseline, new spectral features arising from scalar field interactions could offer unique observational signatures, advancing our understanding of black hole thermodynamics and quantum gravity.
... However, classical BH solutions -such as the Schwarzschild [4], the Reissner-Nordström [5,6], and the Kerr [7] metrics -possess singularities, posing fundamental challenges for gravitational physics. To overcome this issue, regular black hole (RBH) models have been proposed, introducing modifications to the geometry, often through the coupling with nonlinear electrodynamics (NED). ...
In this work, we explore general relativistic effects and geometric properties of the Fan-Wang spacetime, one of the simplest regular solutions that can be obtained in nonlinear electrodynamics. In particular, we investigate the motion of test particles, the capture cross-section of neutral massive and massless particles, such as neutrinos and photons, and the gravitational redshift. Additionally, using a perturbative approach, we derive analytical expressions for the perihelion shift and gravitational deflection of massless particles. By identifying the one-parameter corrections to the Schwarzschild spacetime, induced by the magnetic charge contained in the Fang-Wang metric, we show that this spacetime can be falsified, since it modifies classical general relativity predictions even at the local level. Moreover, we argue that these modifications could be experimentally tested with advanced observational instrumentation.
... The no-hair conjecture states that black holes cannot be described by any quantity apart from their mass, electric charge, and angular momentum [1]. This means that in an appropriately chosen reference frame, isolated black holes can be described by the well-known Kerr-Newman solution [2][3][4]. According to this conjecture, in the final stage of gravitational collapse, black holes are defined solely by these quantities, which adhere to a Gauss law and are measurable at infinity. ...
We study a class of solutions within the context of modified gravity theories, characterized by a non-trivial field that does not generate any back-reaction on the metric. These stealth configurations are effectively defined by the stealth conditions, which correspond to a vanishing stress-energy tensor. In this work, we introduce a novel approach to constructing this class of solutions. In contrast to the standard procedure, the starting point requires satisfying the stealth conditions for a given ansatz independently of the gravitational dynamics. This approach simultaneously determines the non-trivial field and the geometries capable of supporting it as a stealth configuration. Consequently, a gravity model can accommodate a stealth field only if its vacuum solution falls within the geometries permissible under stealth conditions. By applying this reverse procedure in the non-minimal Rϕ ² coupling, we recover all previously known stealth configurations and present new solutions. Although it seems intuitive to assume that this “gravitationally undetectable” scalar field leaves no physical traces, it remarkably reveals thermodynamic imprints, as its presence screens the black hole mass and modifies the entropy according to the first law.
... With the advent of a large amount of observational data [1][2][3][4][5][6], people's belief in the existence of black holes has strengthened. However, in General Relativity [7], black holes are always associated with singularities [8][9][10], where the curvature of spacetime becomes infinite, causing physical laws to break down. And the singularity theorems proved by Penrose and Hawking [11,12] indicate that if the strong energy condition and global hyperbolic spacetime assumption are satisfied, singularities are unavoidable. ...
In this paper, we construct a static spherical symmetric Bardeen-Proca star (BPS) model, which consists of the electromagnetic field and Proca field minimally coupled with gravity. The introduction of the Proca field disrupts the formation of event horizons, ensuring that these solutions are globally regular throughout the spacetime. We obtain families of BPS solutions under several magnetic charge conditions. Based on these results, we further investigate the ADM mass, Noether charge, and energy density distribution of them. We find that when the magnetic charge is sufficiently large, solutions with a critical horizon emerge as , and the time component of the metric approaches zero inside . To an observer at infinity, the collapse process of the matter near the critical horizon appears frozen. Consequently, we refer to the solution with as the frozen Bardeen-Proca star (FBPS). Additionally, we also investigate the circular geodesic orbits of BPS. For the light ring, we find that the light rings always appear in pairs, located on both sides of the critical horizon and moving further apart as the frequency decreases. For timelike circular orbits, we investigate their distribution in the spacetime of BPSs and highlight four representative families of BPS solutions.
This paper is devoted to studying the optical and thermal geometrical properties of Hot NUT–Kerr–Newman–Kasuya-anti-de Sitter (HNKNK-AdS) black hole. This black hole is characterized by the NUT charge and a parameter Q that comprises the electric [Formula: see text] and magnetic mono-pole parameter [Formula: see text] (Kasuya parameter). We compute the image of the BH shadow in two types: (1) at [Formula: see text], (2) at [Formula: see text] by analytical approach. We also investigate the effect of NUT, spin, inclination angle, and cosmological constant on the shape of shadow. We analyze that for type 1, the shadow in increasing for higher values of NUT charge, the cosmological constant, rotation parameters, and inclination angle, while for type 2, by increasing these parameters, the circular symmetry of the image of the BH shadow variate. Moreover, we discuss well-known thermal geometries such as Weinhold, Ruppeiner, HPEM and Quevedo case I and II spacetime. It is found that Ruppeiner, HPEM and Quevedo (II) formulations provide physical information about the microscopic structure as compared to Weinhold and Quevedo (I) geometries. Our findings provide distinctive characteristics in the shadow and thermal geometries of this BH as compared to other black holes types.
Einstein's equations for empty space are solved for the class of metrics which admit a family of hypersurface-orthogonal, non-shearing, diverging null curves. Some of these metrics may be considered as representing a simple kind of spherical, outgoing radiation. (Among them are solutions admitting no Killing field whatsoever.) Examples of solutions to the Maxwell-Einstein equations with a similar geometry are also given.