Shadow Images of a Rotating Dyonic Black Hole with a Global Monopole Surrounded by Perfect Fluid

In this paper we revisit and extend the prior work of Filho and Bezerra [Phys. Rev D, 64, 084009 (2001)] to rotating dyonic global monopoles in presence of a perfect fluid. We then show that the surface topology at the event horizon, related to the metric computed, is a 2- sphere using the Gauss-Bonnet theorem. By choosing $\omega=-1/3, 0, 1/3$ we investigate the effect of dark matter, dust, radiation on the silhouette of black hole. The presence of the global monopole parameter $\gamma$ and the perfect fluid parameters $\upsilon$, also deforms the shape of black hole's shadow, which has been depicted through graphical illustrations. In the end we analyse energy emission rate of rotating dyonic global monopole surrounded by perfect fluid with respect to parameters.


I. INTRODUCTION
Black holes are fascinating objects predicted to exist by Einstein's theory of general relativity.
Recent astrophysical observation shows that such objects may exist at the center of almost every galaxy. By studying the light-like geodesics around black holes it is shown that photons can be absorbed by the black hole or can escape from black holes [2]. That is to say a boundary is defined between these two categories of light-like geodesics, giving rise to a dark region known as the shadow. Very recently, a project known as the Event Horizon Telescope (EHT) [3], aimed of collecting signals from radio sources has been developed. In fact, EHT is expected to observe the first silhouette of a supermassive black hole. This data may also eventually provide us with means for testing the general theory of relativity in the strong-field regime. That being said, the black hole shadow has recently become a hot topic among the researchers for the simple fact to best evaluate the soon-expected observational data. Historically, Synge was the first to propose the apparent shape of a spherically symmetric black hole [4]. After that Luminet [5] discussed the appearance of a Schwarzschild black hole, the shadow of a Kerr black hole was studied by Bardeen [6], shadow of Kerr-Newman black holes [7], naked singularities with deformation parameters [8], Kerr-Nut spacetimes [9], while shadows of black holes in Chern-Simons modified gravity, Randall-Sundrum braneworlds, and Kaluza-Klein rotating black holes have been studied in [10][11][12], and many other interesting studies concerning the effect of dark matter and cosmological constant on the shadow images [13][14][15][16], Kerr-like wormholes as well as traversable wormholes and many others interesting studies [17][18][19][20][21][22][23][24][25]. Some authors have also tried to test theories of gravity by using the observations obtained from shadow of Sgr A* [27][28][29][30].
Global monopoles are topological defects which may have been produced during the phase transitions in the early universe. In fact, global monopoles are just one type of topological defects.
Other types of topological objects are expected to exist including domain walls and cosmic strings (e.g. [32]). More precisely, a global monopole is a heavy object characterized by spherically symmetry and divergent mass. Such objects which may have been formed during the phase transition of a system composed of a self-coupling triplet of scalar fields φ a which undergoes a spontaneous breaking of global O(3) gauge symmetry down to U(1). The gravitational field of a static global monopole for the first time was found by Barriola and Vilenkin and are expected to be stable against spherical as well as polar perturbations [33]. According to their model, global monopoles are configurations whose energy density decreases with the distance as r −2 and whose spacetimes exhibit a solid angle deficit given by ∆ = 8π 2 γ 2 , where γ is the scale of gauge-symmetry breaking.
Gravitational lensing by rotating global monopoles has been investigated in Ref. [34] and more recently in Ref. [35]. Among other things, global monopoles are expected to rotate and to carry magnetic charges.
In this paper we aim to study the impact of the rotating global monopole black hole surrounded by perfect fluid on the black hole shadow. In Section I, we consider the the gravitational field of a static dyonic black hole (SDBH) with a global monopole surrounded by perfect fluid. In Section II, by applying a complex coordinate transformation known as the Newman-Janis method [36] we find the spacetimes of a rotating dyonic black hole (RDBH) with a global monopole surrounded by perfect fluid. In Section IV, we consider the null geodesics using Hamilton-Jacobi equation. In Section V, we study the impact of dark matter, dust and radiation on the shape of global monopole shadow. In Section VII, we study the energy emission rate. Finally in Section VIII, we comment on our results.

II. A SDBH WITH A GLOBAL MONOPOLE IN PERFECT FLUID
Let us write the action of a four-dimensional Einstein-Maxwell theory minimally coupled to a scalar field with surrounding matter fields A global monopole is a heavy object formed in the phase transition of a system composed by a self-coupling scalar triplet field φ a . The simplest model which gives rise to a global monopole is described by the Lagrangian density [33] where φ 2 = ∑ a φ a φ a with a = 1, 2, 3, while λ is the self-interaction term and γ is the scale of a gauge-symmetry breaking. The field configuration describing a monopole is in which x a = {r sin θ cos ϕ, r sin θ sin ϕ, r cos θ } , such that ∑ a x a x a = r 2 , and h(r) is a function of radial coordinate r. The electromagnetic potential is given by [41] where r + is the horizon of the black hole. In our case the total stress-energy momentum reads T µν = T em µν + T gm µν + T µν (6) in which and T µν is the energy-momentum tensor of the surrounding matter. The corresponding Einstein field equations reads While the corresponding Maxwell equations reads With these equations in mind, and without loss of generality we can choose a spherically symmetric metric written as follows Such a metric imposes the only non-vanishing components of the electromagnetic tensor The field equations for the scalar field φ a reduces to a single equation for h(r) given as The surrounding matter which generally can be a dust, radiation, quintessence, cosmological constant, phantom field or even any combination of them. The energy momentum-tensor of the surrounding fluid has the following components [40] T t t = T r r = −ρ, and Outside the core h → 1 and the energy-momentum tensor has the following components Thus, the Einstein's field equations yield: Now by solving the set of differential equations (18) and (19) one obtains the following general solution for the metric with the energy density in the form Note that, υ is an integration constant related to the perfect fluid parameter. From the weak energy condition it follows the positivity of the energy density of the surrounding field, ρ ≥ 0, which should satisfy the following constraint ωυ ≤ 0.

III. A RDBH WITH A GLOBAL MONOPOLE IN PERFECT FLUID
We now extend the study of static global monopole solution and obtain its rotating counterpart. For this we apply Newman-Janis formalism to the metric (11) along with (20). As a first step to this formalism, we transform Boyer-Lindquist (BL) coordinates (t, r, θ, φ) to Eddington-Finkelstein (EF) coordinates (u, r, θ, φ). This can be achieved by using the cordinate transformation dt = du + dr which yields line element of the form where dΩ 2 = dθ 2 + sin 2 θdφ 2 . It's worth noting that compared to the previous work in [1], we shall use the metric form (11) along with f (r) given by Eq. (20) to obtain a simple metric to obtain a rotating black hole with a global monopole. This metric can be expressed in terms of null tetrads where the null tetrads are defined as These null tetrads are constructed in such a way that l µ and n µ while m µ andm µ are complex. It is obvious from the notation thatm µ is complex conjugate of m µ . These vectors further satisfy the conditions for normalization, orthogonality and isotropy as −l µ n µ = m µm µ = 1.
Following the Newman-Janis prescription we write, in which a stands for the rotation parameter. Next, let the null tetrad vectors Z a = (l a , n a , m a ,m a ) undergo a transformation given by Z µ = (∂x µ /∂x ν )Z ν , following where Σ = r 2 + a 2 cos 2 θ. With the help of the above equations the contravariant components of new metric are computed as Note that F is some function of r and θ. The metric is found as follows Finally, we revert the EF coordinates back to BL coordinates by using the following transformation where in order to simplify the notation we introduce the following quantities where f (r) is given by Eq. (20). In this work, we consider three different cases of ω = −1/3 dark matter dominant, 0 (dust dominant) and 1/3 (radiation dominant).
Hence the the rotating spacetime metric has the form

A. Surface Topology
It is interesting to determine the surface topology of the global monopole spacetime at the event horizon. At a fixed moment in time t, and a constant r = r + , the metric (41) reduces to The above metric has the following determinant det g (2) = 2Mr + + 8πr 2 Theorem: Let M be a compact orientable surface with metric g (2) , and let K be the Gaussian curvature with respect to g (2) on M. Then, the Gauss-Bonnet theorem states that Note that dA is the surface line element of the 2-dimensional surface and χ(M) is the Euler characteristic number. It is convenient to express sometimes the above theorem in terms of the Ricci scalar, in particular for the 2-dimensional surface there is a simple relation between the Gaussian curvature and Ricci scalar given by Yielding the following from A straightforward calculation using the metric (42) yields the following result for the Ricci scalar R = 2(r 2 + + a 2 )(r 2 + − 3a 2 cos 2 θ) r 2 + + a 2 cos 2 θ 3 (47) From the GBT we find 2π 0 π 0 2(r 2 + + a 2 )(r 2 + − 3a 2 cos 2 θ) r 2 + + a 2 cos 2 θ 3 det g (2) dθdϕ.
Finally, solving the integral we find Hence the surface topology of the rotating global monopole is a 2-sphere at the event horizon, since we know that χ(M) sphere = 2.

B. Shape of Ergoregion
Let us now proceed to study the shape of the ergoregion of a RDBH with a global monopole.
In particular we shall be interested to plot the shape of the ergoregion in the xz-plane. Recall that the horizons of the RDG can be found by solving ∆ = 0, on the other hand, the static limit or inner and outer ergosurface is given by g tt = 0, i.e., There is an interesting process which relies on the presence of an ergoregion, namely from such a rotating black hole energy can be extracted, and this process is known as the Penrose process.
In Figure 1 we plot the shape of ergoregion for different values of a, ω, γ, and υ. One can observe that the event horizon and static limit surface meet at the poles while the region between them is the ergoregion which supports negative energy orbits. Furthermore the shape of ergoregion, depends on the spin a, however due to the small values of υ we observe small changes related to the value of ω.

IV. NULL GEODESICS
Our main objective is to study the shadow casted by the black hole defined by metric 41. To do so, we first need to analyze the geodesics structure of photons moving around the compact gravitational source. This will enable us to detect the unstable photon orbits which in turn defines the boundary of the shadow.
To observe the null geodesics around the RDGM present in perfect fluid, we consider the Hamilton-Jacobi method. The Hamilton-Jacobi equation is given by In the above equation On Left Side: J is the Jacobi action, defined as the function of affine parameter τ and coordinates x µ i.e. J = J (τ, x µ ).

On Right Side:
H is the Hamiltonian of test particle's motion and is equivalent to g µν ∂ µ J ∂ ν J .
In the spacetime under consideration, along the photon geodesics the energy E and momentum L, defined respectively by Killing fields ξ t = ∂ t and ξ φ = ∂ φ , are conserved. The mass m = 0 of the photon is also constant. Using these constants of motion we can thus separate the Jacobi function as where the functions J r (r) and J θ (θ) respectively depends on coordinates r and θ. Combining Eq.
(52) and Eq. (53) yields a set of equations, which describes the dynamics of a test particle around the rotating black hole in perfect fluid matter, as: where R(r) and Θ(θ) read as with K the Carter constant.

V. CIRCULAR ORBITS
Now we consider a gravitational source placed between a light emitting source and an observer at infinity. The photons emitted from the light source will form two kinds of trajectories: the ones which eventually fall into the black hole and the ones which scatter away from it. The region separating these trajectories, contains unstable circular orbits. These unstable circular orbits form a dark region in sky thus forming the contour of the shadow. In this section we intend to discuss the presence of unstable circular orbits around the black hole under consideration. For this we consider photon as a test particle and hence take m = 0. We can express the radial geodesic equation in terms of effective potential V eff of photon's radial motion as For our convenience we introduce two independent parameters ξ and η [39] as The effective potential in terms of these two parameters is then expressed as where we have replaced V eff /E 2 by V eff . Figure where Combining Eqs. (63-64) results in It is worth mentioning here that impact parameters, ξ and η, will be affected not just by radial coordinate r, spin parameter a and mass of black hole M but also by electric charge Q E , magnetic charge Q M , monopole parameter γ and perfect fluid parameter υ. The unstable circular orbits are located at local maxima of the potential curves i.e. when V eff < 0 or In this section, we extend our calculations to observe shadow of RDGM surrounded by perfect fluid. To gain the optical image we specify the observer at position (r o , θ o ), where r o = r → ∞ and θ o is the angular coordinate at infinity, on observer's sky. The new coordinates, also widely known as celestial coordinates, α and β are then introduced. These coordinates are selected such that α and β correspond to the apparent perpendicular distance of the image from axis of symmetry and its projection on the equatorial plane, respectively. For an observer at infinity, these coordinates are thus given [39] by We can relate the above coordinates to parameters ξ and η, which then yield We expect that the parameters involve in RDGM in presence of a perfect fluid will effect the shape of its shadow. This can be clearly confirmed through Eq. 71 as it depends not only on spin parameter a and angular coordinate θ o but also on γ, ω and perfect fluid parameter υ. Later, we will justify our results also through graphical interpretations.
As our observer is placed in the equatorial plane (θ = π/2), α and β reduce to Figure (3) and (4) show deformation in shapes of the shadow with respect to monopole parameter γ and and perfect fluid parameter υ, respectively. It is a well known observation now that the rotational effect in a black hole distorts its shape. That being said, we notice in Figure (3) that for small spin parameter, a, the shadow of the black hole maintains a circular shape along with the increase in its size with the inclination of γ. As for larger spin value, the shadow is clearly distorted and matches with its Kerr counter part in perfect fluid [15] for γ = 0. Figure (4 shadow also increases. A distortion is noticed in shape of the shadow when the spin parameter a is increased. Also, in case of dark matter and dust, there is significant change in the size of the shadow with respect to υ. On the other hand, in case of radiation we do not observe any significant effect of perfect fluid parameter υ, in fact the effect is negligibly small. In [26], the authors introduces two observables, radius R s and distortion δ s , to analyze the size and form of the shadow. The first observable R s is the approximate radius of the shadow. It is defined by considering a reference circle passing through three points on the boundary of the shadow, such that (α tp , β tp ) is the top most point on the shadow, (α bm , β bm ) is the bottom most point on the shadow and (α r , 0) is the point corresponding to unstable circular orbit seen by an observer on reference frame. Thus The second observable δ s is the distortion parameter. Let D CS be the difference between the contour of shadow and reference circle. Then for the point (α p , 0) lying on the reference circle and the This tells us that with respect to circumference of reference circle, the shadow of the rotating black hole is significantly distorted for γ ∈ [0, 0.1] but for γ > 0 it may not show any distortion and thus we may obtain a perfect circle. As we have considered our observer to be at infinity so in this case the area of the black hole shadow will be approximately equal to high energy absorption cross section as discussed in [14]. we adopt the value of Π ilm as calculated by [14] Π ilm πR 2 s .
The energy emission rate of the black hole is thus defined by where σ is the frequency of the photon and T represents the temperature of the black hole at outer horizon i.e. r + , given by For all three cases, radiation, dust and dark matter, the energy emission rate is graphically presented in Figure (6) where we notice that the energy emission rate decreases with increase in parameter γ. A slight shift to the lower frequency is also observed while γ increases. The spin parameter a also effects the shape of the energy emission rate as an abrupt decrease in energy emisiion rate is noticed for higher spin value.

VII. CONCLUSION
In this paper we have used the complex transformations pointed out by Newman and Janis to obtain a RDGM solution in presence of a perfect fluid matter. Using the Gauss-Bonnet theorem we have shown that the surface topology of a RDGM is indeed a 2-sphere. Furthermore by choosing ω = −1/3, 0, 1/3 we have explored the impact of dark matter, dust, radiation, as well as the global monopole parameter γ, and perfect fluid parameters υ, on the silhouette of black hole. We have found that a rotating dyonic black hole with a global monopole retains a circular shape for small spin parameter. Whereas for high spin like a = 0.98M the shadow of RDGM is distorted. Also as monopole parameter γ increases, a slight shift towards the right is also noticed in shape of shadow of black hole under consideration. The two observables, R s and δ s , are also being discussed. In the end we analyze energy emission rate of rotating dyonic global monopole surrounded by perfect fluid with respect to parameters.