Shadow cast and deflection angle of Kerr-Newman-Kasuya spacetime
We investigate the shadow cast or silhouette by rotating dyon black hole, namely Kerr-Newman-Kasuya (KNK) spacetime. It was shown that in addition to the angular momentum of the black hole, the dyon charge also affects the shadow image of the KNK black hole. Moreover, we analyze the weak gravitational lensing by the KNK black hole by using the Gauss-Bonnet theorem. Finally, we find that extra dyon charge decreases the deflection angle and also shadow of the KNK black hole.
pacs:04.40.-b, 95.30.Sf, 98.62.Sb
The center of galaxies is a playground of a gigantic black holes. Because of the gravitational lens effect, the background would have cast a shade larger than its horizon size synge . The shape and size of this shadow can be calculated. Black holes are the most interesting objects in the universe. Black hole has a event horizon, as a boundary where nothing can escape include the light so that they are called as black luminet . But beyond that line, particles can escape. Black hole always try to pull the surrounding matter which is known as accretion cunha ; Cunha:2018acu . During the accretion of black holes, they release large amounts of energy into their large-scale environments, so that black holes may play a prominent role in the processes that control galaxy formation Falcke:1999pj ; Tremblay:2016ijg . Moreover, this accreting matter heats up through viscous dissipation and radiate light in various frequencies. For instance, the radio waves are one of them which can be detected through the radio telescopes Shen:2005cw ; Huang:2007us ; Johannsen:2015mdd . When the accretion happens onto black hole, shining material pass through the event horizon, which results in a dark area over a bright background: black hole shadow (BHS) Cunha:2018gql . We can say that this shadow is actually an image of the event horizon. In 1970s, Bardeen, Press and Teukolsky bardeen and then Chandrasekhar Chandra never escape the vicinity of the black hole. As seen by an outside observer, a BHS has a radius of over the background light source. As reported by many numerical calculations, rotating black holes cast shadows of approximately the same size as well Hioki:2009na ; Johannsen:2010ru ; Nedkova:2013msa ; Amarilla:2013sj ; Abdujabbarov:2012bn ; Grenzebach:2014fha ; Johannsen:2015hib ; Giddings:2014ova ; Atamurotov:2013sca ; Wei:2013kza ; Sakai:2014pga ; Perlick:2015vta ; Abdujabbarov:2015xqa ; Tinchev:2013nba ; Wang:2018eui ; Tsukamoto:2017fxq ; Amarilla:2010zq ; Yumoto:2012kz ; Takahashi:2005hy ; Papnoi:2014aaa ; Dexter:2012fh ; Moffat:2015kva ; Younsi:2016azx ; Johannsen:2015qca ; Zakharov:2014lqa ; Cunha:2016bjh ; Freivogel:2014lja ; Cunha:2016bpi ; Ohgami:2015nra ; Zakharov:2011zz ; Hennigar:2018hza ; Pu:2016qak ; Sharif:2016znp ; Abdujabbarov:2016hnw ; Xu:2018mkl ; Gyulchev:2018fmd ; Vetsov:2018mld ; Hou:2018bar ; Dokuchaev:2018kzk ; Mizuno:2018lxz ; Perlick:2018iye ; Stuchlik:2018qyz ; Shaikh:2018lcc ; Eiroa:2017uuq ; Mars:2017jkk ; Wang:2017hjl ; Singh:2017vfr ; Mureika:2016efo ; Huang:2016qnl ; Ghasemi-Nodehi:2016wao ; Tsukamoto:2014tja ; Vincent:2016sjq . On the other hand, null geodesics method and gravitational lensing are also useful tools to gain information about the black holes Bartelmann:1999yn ; Bozza:2009yw . Strong and weak gravitational lensing by black holes, wormholes or other exotic objects have been investigated by several authors Virbhadra:2007kw ; Zschocke:2011mm ; Stefanov:2010xz ; Manna:2018jxb ; Ono:2018ybw .
The main purpose of studying the gravitational lensing is to detect the black holes in the universe, which are believed that they are mostly located at the center of the galaxies. The strong gravitational lensing helps us to find the position, magnification, and time delays of the images by black holes. Moreover, in ”weak lensing” the effect is much weaker but can still be detected statistically Hu:2000ee . To do so, Gibbons and Werner generated a new technique to calculate the deflection angle of light rays within the asymptotic source and receiver Gibbons:2008rj . Their method is based on the famous Gauss-Bonnet theorem (GBT), which solves the integral in an infinite domain bounded by the light ray. After locating the source and receiver to the asymptotic Minkowski regions, the deflection angle of optical metric of a static black hole was computed Gibbons:2008hb . Then Werner extended to the stationary black holes by employing the Finsler-Randers type optical geometry with Nazım’s osculating Riemannian manifold Werner:2012rc . In the sequel, this topic has been thoroughly studied by several authors for different types of spacetimes Gibbons:2015qja ; Gibbons:2011rh ; Gibbons:2008zi ; Sakalli:2017ewb ; Jusufi:2017lsl ; Crisnejo:2018uyn ; Jusufi:2017hed ; Ovgun:2018ran ; Jusufi:2017vta ; Ovgun:2018prw ; Jusufi:2017mav ; Ovgun:2019prw ; Jusufi:2017xnr ; Cvetic:2016bxi ; Jusufi:2017uhh ; Ovgun:2018fnk ; Jusufi:2018jof . Recently, A. Ishihara et al. showed for the spherical symmetric static objects that it is possible to find deflection angle with considering finite-distance corrections instead of using the asymptotic receiver and source Ishihara:2016vdc ; Ishihara:2016sfv . Then T. Ono et al. extended for the axisymmetric spacetimes Ishihara:2016vdc ; Ishihara:2016sfv ; Arakida:2017hrm . The expectation of the detection of the Milky Way’s central supermassive black hole (Sagittarius A* is the site of that black hole) by the Event Horizon Telescope (EHT), which is simulated in Fig. 1 increases the impact on the studies about the BHS. EHT tries to observe the extremely hot gas around the event horizon of the black hole.
In this paper, we consider the Kerr-Newman-Kasuya (KNK) black hole kasuya , which is nothing but the Kerr-Newman spacetime involved with extra magnetic monopole charge. It is believed that magnetic monopoles exist on symmetric backgrounds governed by the field equations of electromagnetism. Monopole hypothesis was first introduced by Dirac, long time ago. But, the existence of the magnetic monopoles was neglected due to the failing attempts to detect them. On the other hand, recent studies of gauge theories have given promising results about the existence of the monopole Chanyal:2017xqf ; Mavromatos:2016ykh . Besides, there is a renewed interest for the cosmological constant since it can be one of the theoretical models to explain the inflationary scenario of the early universe. In this scenario the universe undergoes a stage which is geometrically described by hot de-Sitter(dS) spacetime which is related with KNK spacetime PLBds . In addition to this, it was shown that primordial universe can be described by KNK spacetime Comtet:1979kq .
This paper is organized as follows. Section II briefly describes the KNK black hole and its physical properties. We calculate the deflection angle by folloing the GBT of Gibbons:2008rj in Sec. III. In Sec. IV, we reveal the shadows of the KNK black holes by manipulating the rotation parameter and the dyon (a hypothetical particle in 4-dimensional theories with both electric and magnetic charges) charge. Section V is devoted to summary and conclusions.
Ii Kerr-Newman-Kasuya spacetime
The metric of the KNK black hole in the Boyer-Lindquist coordinates is given by kasuya
Note that is the mass, represents the rotation parameter, which is the angular momentum per unit mass, and denote electric and magnetic charges, respectively. The spacetime of the KNK black hole reduces to the Kerr-Newman black hole when , the Kerr black hole if , the Reissner-Nordström black hole for and the Schwarzschild black hole if .
Event horizon of the KNK black hole is obtained by using the following equation
whose solutions are
The surface gravity Wald of the KNK black hole can be obtained as follows:
Thus, the Hawking temperature Wald of the KNK black hole becomes
The surface area of the horizon is given by
where . So, the entropy Wald of the KNK black hole at the event horizon reads
The angular velocity of KNK black hole is given by
Furthermore, the electric potential with magnetic and electric charges is as follows
Note that stands for the two gauges. At the event horizon of KNK black hole, the first law of the thermodynamics is satisfied via the following expression:
Iii Deflection angle of light by Kerr-Newman-Kasuya spacetime
In this section, by using the GBT, we shall study the deflection angle for the KNK black hole. We use the null condition and solve the KNK spacetime for as follows:
Note that goes as (). Then, we obtain the components of the optical metric of KNK spacetime in terms of and as follows:
The spatial metric stands for an arc-length () along the light ray. At the equatorial plane (), we have
where . In practice, we define the deflection angle () with the angles , and which correspond to receiver, source, and coordinate, respectively. Namely, we have
Locating the receiver’s and the source’s positions to suitable locations where the endpoints of light rays lie in the Euclidean space, the GBT Ono:2017pie admit the following expression for the deflection angle:
We should note that is embedded quadrilateral. Moreover, is for the geodesic curvature and is for an arc length. For evaluating the first integral of Eq. (20), one should first calculate the Gaussian optical curvature (related with 2-dimensional Riemann tensor) in the weak field approximation:
and a straightforward calculation yields the following geodesic curvature Ono:2017pie :
For the KNK black hole, the geodesic curvature is computed as
We now examine the net contribution of the geodesic part:
To visualize the boundary of the integration domain, we define the angle by using the outgoing radial direction of light rays:
and the solution for the photon orbit is found as follow:
Afterwards, we calculate the integral of the Gaussian curvature of the optical metric of KNK black hole Gibbons:2008hb
In sequel, we combine the solutions of the Gaussian optical curvature integral and geodesic curvature integral to obtain the total deflection angle of light on the equatorial plane of the KNK black hole:
We note that the positive sign stands for the retrograde and negative sign is for the prograde case of the photon orbit. The deflection angle of the KNK black hole is agreed with the Kerr case with the limit of kerr and the non-rotating dyon black hole if Clem1 . Moreover, the deflection angle of the charged black hole is recovered if and the deflection angle of the Schwarzschild black hole is obtained for . The deflection angle is linearly decreased with the extra magnetic charge compared to Kerr-Newman black hole.
Iv Shadows of Kerr-Newman-Kasuya spacetime
The KNK spacetime in Boyer-Lindquist coordinates is described by the following metric tensors:
The motion of the particle on the KNK is obtained by the following Lagrangian: Chandra
where , stands for four velocity of the particle with the affine parameter . Due to the symmetry of black hole, conjugate momenta and are conserved because of metric-independent variables and . Hence the energy and angular momentum are obtained as:
and we derive
where . To calculate other geodesics equations, we use the Hamilton-Jacobi equation :
According to the ansatz:
For the KNK spacetime, the Hamilton-Jacobi equation yields:
Then we solve it for and as Chandra :
and is the Carter constant defined by . and should be non-negative for the photon motion. Two impact parameters and are introduced in terms of energy , angular momentum and carter constant as Chandra
For photon case, Eq. (31) can be rewritten in terms of dimensionless quantities and :
Note that and are as a effective potential for moving particle in and . Equation can be represented as
where is the effective potential in terms of impact parameters given by
We can achieve the most critical and unstable circular orbit by maximizing the effective potential, which satisfies the following conditions
where is the radius of the unstable circular null orbit. We assume that the photons and the observer are located at the infinity () and photons come near the equatorial plane . We solve the Eq. (43) and obtain the following celestial coordinates of the image:
Note that observer at spatial infinity can observe the celestial coordinates of the image and determine the contour of the BHS. We briefly review the expressions of the images of photon rings around black holes. There are three independent constants of motion in KNK metric. The inertial reference frame of an observer with the basis vectors and coordinate basis of the metric Johannsen:2010ru :
which are defined, at a large distance from the KNK black hole, with the relation between local observer’s basis vector and coordinate basis of metric: with for the Minkowski metric . From the orthonormal property of and , one obtains the following equations bardeen ; Johannsen:2015qca :
Then, the energy and angular momentum are obtained as:
On the other hand, the quantities and are known as the impact parameters. They are the axes of the Cartesian coordinate of the image plane of the observer located at a distance and inclination angle between the rotation axis of KNK black hole and observer’s line of sight:
All the metric elements seen in the above expressions are evaluated at and . In the limit , the shape of the BHS for a far away observer can be determined by the following celestial coordinates bardeen ; Chandra :
The celestial coordinates and show the apparent perpendicular distances of the image around the black hole. For the KNK black hole, at , the celestial coordinates become
In the Figs. 2 and 3, we plot the shadows of the KNK black hole with different values of spin and magnetic charge . Then, we plot the shadows of KNK black hole with the different inclination angles and illustrate them in Fig. 4.
Alongside the recent successes of the LIGO detector and gravitational wave astronomy LIGO , the EHT has a potential outcome to answer many questions at strong gravitational field regime in general relativity by using the millimeter wavelength radio astronomy EHT and observe the BHS which is embedded on the image of hot gas.
First, we have studied the weak gravitational lensing by using the GBT for the KNK optical spacetime, where the optical geometry gives a more geometrical view on weak gravitational lensing than the quasi-Newtonian lensing method. Remarkably, the method of GBT is easier than the null geodesics method since we only use the optical geometry dealing with spatial light rays. As a consequence of the GBT, the result obtained has a global effect. When the location of receiver and source is at null infinity, the deflection angle in the weak field approximation has been found as follows Eq. (28):
which has the Kerr limit at and the non-rotating dyon black hole when . Furthermore, the deflection angle of the charged black hole is recovered if and the deflection angle of the Schwarzshild black hole is found for . It is worth noting that the deflection angle is linearly decreased with the extra magnetic charge compared to Kerr-Newman black hole.
Second, we have investigated the shadow cast of the KNK black hole. With the choice of this black hole, we have analyzed how and parameters affect the image of the shadow. From the numerical plots, we have shown that the magnetic charge or magnetic monopole dramatically decreases the size of the BHS cast. On the other hand, as seen in Fig. 3, the shadow cast of KNK black hole decreases with decreasing inclination angle .
In the near future, we are very hopeful that the EHT will provide the event horizon visualization within black hole shadow. If this observation occurs, we expect to see some of the experimental results of this theoretical work.
Acknowledgements.This work is supported by Comisión Nacional de Ciencias y Tecnología of Chile (CONICYT) through FONDECYT Grant N 3170035 (A. Ö.). A. Ö. is grateful to Institute for Advanced Study, Princeton for hospitality.
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