Shadow cast by a rotating charged black hole in quintessential dark energy

https://doi.org/10.1016/j.dark.2020.100644Get rights and content

Abstract

The existence of quintessential dark energy around a black hole has considerable consequences on its spacetime geometry. Hence, in this article, we explore its effect on horizons and the silhouette generated by a Kerr–Newman black hole in quintessential dark energy. Moreover, to analyse the deflection angle of light, we utilize the Gauss–Bonnet theorem. The obtained result demonstrates that, due to the dragging effect, the black hole spin elongates its shadow in the direction of the rotational axis, while increases the deflection angle. On the other hand, the black hole charge diminishing its shadow, as well as the angle of light’s deflection. Besides, both spin and charge significantly increase the distortion effect in the black hole’s shadow. The quintessence parameter γ, increases the shadow radius, while decreases the distortion effect at higher values of charge and spin parameters.

Introduction

In recent years, inspecting the invisible aspects of our Cosmos got a considerable interest of many researchers. Various cosmic data-sets specified that dark energy (DE) is a crucial source for the accelerated expansions of our universe [1]. About 70% of our observable universe is consists of DE [2], which could be explained with the help of a repellent cosmological constant Λ>0 or by a quintessential field [3], [4], [5], [6], [7], [8]. The parameter Λ can be assumed as homogeneous, while everywhere in space, its value remains the same (Λ1.3×1056cm2) [9]. Apart from cosmological constant, another essential model of the DE is known as the quintessence [10]. Spacetime geometry has greatly influenced by the contents of DE, i.e., a quintessence field and/or a cosmological constant of the black hole (BH). In the presence of a cosmological constant, the asymptotic structure of a BH changes to the asymptotic de Sitter spacetime [11], [12].

The idea of BHs shadow was initiated by Bardeen [13] with the conclusion that a BH has a radius of rshadow=5.2M, over a background light source (visible to an external observer). As described by many researchers, the size of shadows cast by spinning BHs is almost the same as well [14], [15], [16], [17], [18], [19]. Theoretical investigation of the shadow cast by the BH horizons can be studied as the existence of a photon sphere and null geodesics. To an external observer, BH appears to be a dark disc termed as the shadow of a BH. In the case of non-rotating BHs, the shadow appeared to be a circular disc, whereas, in rotating BHs the shadow appeared to be flattened on one side, rather than a circular disc [20]. In recent years, many researchers have been motivated by the astrophysical advances to investigate shadows cast by BHs [21], [22], [23], [24]. Researchers accept that very soon, direct examinations of the BHs could be possible [25], [26], [27], [28]. Henceforth, studying a BH’s shadow will be a fruitful way for a better understanding of astrophysical BHs, as well as for the comparison of general relativity to those of modern theories [29]. To have information on BHs, the approaches of null geodesics and gravitational lensing are of compelling interest [30], [31]. Recently, scientists have finally succeeded in obtaining the first-ever image of a supermassive BH, at the centre of the M87 galaxy [32], [33], [34]. Among other evidence, this is the strongest ever assurance to the existence of supermassive BHs, which opens up new windows onto the investigation of BHs.

Gravitational lensing (GL), can be defined as the bending of light due to gravity. They can be used to examine the distribution of dark matter in our Cosmos, as well as to explain the far-away galaxies. Using the strong GL, we can detect the location, magnification and time delays of shadows by BHs, whereas, in weak GL the consequence is much weaker yet could still be analysed statistically [35]. Moreover, the weak, as well as strong GL by wormholes and BHs, can be found in [36], [37], [38]. The assumption that massive particle bends light rays while passing through it is remarkable in general relativity. The applications of GL consist of exploring the deflection angle, as well as detecting BHs in our Cosmos. Gibbons and Werner [39] by making use of the Gauss–Bonnet theorem (GBT), introduced a new approach to acquired the light’s deflection angle. In this approach, bending of light can be viewed as a global topological effect, rather than corresponding to a region with a radius compared to the impact parameter. Besides BHs, the Gibbon–Warner approach for the deflection angle of light ray has also studied for both of the asymptotically flat and non-flat spacetimes and wormholes [40], [41], [42].

In current work, our primary focus is on a non-singular domain exterior to a light ray. In the case of an asymptotically flat spacetime geometry, the deflection angle Θˆ could be calculated as [39] Θˆ=RSKdS,where K and dS, respectively denote the Gaussian optical curvature and the surface element of optical geometry. It should be noted that the above expression of Θˆ, can only be satisfied in asymptotically flat spacetimes, where one can only consider a finite distance correction for the non-asymptotically flat spacetime geometries.

In this article, besides charge and the rotational parameter a, our main goal is to understand the effects of quintessential DE on BH’s shadow. The following section will provide a brief review of the Kerr–Newman (KN) BH in quintessential DE and a cosmological constant (KNdSQ BH) and the angle of light’s deflection. In Section 3, we will investigate the null geodesics in detail. The principal purpose of Section 4 is to explore the images generated by a KNdSQ BH. Finally, the last section will provide a discussion and a thorough summary of our obtained results.

Section snippets

Spacetime metric of the KNdSQ BH

Motivated by previous work, this section aimed to investigate the KNdSQ BH. The KNdSQ BH is the solution of Einstein–Maxwell equation and in Boyer–Lindquist coordinates takes the form [12] ds2=Δθsin2θρ2adtΣa2+r2dϕΣ2+ρ2Δrdr2+ρ2Δθdθ2Δrρ2dtΣasin2θdϕΣ2, with Δr=r22Mr+a2+Q2γr13ωΛ3(r2+a2)r2,ρ2=r2+a2cos2θ,Δθ=1+a23Λcos2θ,Σ=1+a23Λ. In the above model, M, a and Λ, respectively represent mass, spin and cosmological constant of the BH. The parameter Q denotes BH charge, while γ is the intensity of

Null geodesics

Generally, the shadow of a BH can be defined as the boundary of photon capture and scattering orbits [48]. In our case, we consider photon orbits around a KNdSQ BH, with the condition of pμpμ=m2. Henceforth, by making use of the Lagrangian equation L=12gμνẋμẋν,here dot means derivative with respect τ (τ is the proper time) and ẋμ=dxμdτ where xμ represent the corresponding four velocities of the particle. BH symmetry allows us to define the conserved energy and angular momentum as, Lṫ=pt

Shadow of KNdSQ BH

This section is devoted to the study of BH shadow. Since an observer at a special infinity can distinguish the shadow of a BH over a bright background, as a dark region. In principle, the shadow of stationary BHs occurs just like a circular disc, whereas spin parameter causes distortion to the shadow of spinning BHs. The apparent shape of a BH image can be preferably visualized with the help of celestial coordinates α and β, which could be defined as [15], [48] α=limrr2sinϑdϕdr,β=limrr2

Conclusion

This article aimed to establish the theoretical investigation of BH shadow for a rotating charged BH in quintessential DE. We hope that our investigation would be helpful for the upcoming Event Horizon Telescope observations. We have started our investigation by reviewing the spacetime geometry of a KNdSQ BH and determined the necessary expressions for the exploration of BH shadow. Firstly, we have investigated BH horizons and the deflection angle of light using the GBT, which is found to be Θˆ

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work is supported by the National Natural Science Foundation of China (11771407).

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