# Shadows of rotating five-dimensional charged EMCS black holes

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## Abstract

Higher-dimensional theories admit astrophysical objects like supermassive black holes, which are rather different from standard ones, and their gravitational lensing features deviate from general relativity. It is well known that a black hole shadow is a dark region due to the falling geodesics of photons into the black hole and, if detected, a black hole shadow could be used to determine which theory of gravity is consistent with observations. Measurements of the shadow sizes around the black holes can help to evaluate various parameters of the black hole metric. We study the shapes of the shadow cast by the rotating five-dimensional charged Einstein–Maxwell–Chern–Simons (EMCS) black holes, which is characterized by four parameters, i.e., mass, two spins, and charge, in which the spin parameters are set equal. We integrate the null geodesic equations and derive an analytical formula for the shadow of the five-dimensional EMCS black hole, in turn, to show that size of black hole shadow is affected due to charge as well as spin. The shadow is a dark zone covered by a deformed circle, and the size of the shadow decreases with an increase in the charge *q* when compared with the five-dimensional Myers–Perry black hole. Interestingly, the distortion increases with charge *q*. The effect of these parameters on the shape and size of the naked singularity shadow of the five-dimensional EMCS black hole is also discussed.

## 1 Introduction

Black holes are intriguing astrophysical objects and perhaps the most fascinating objects in the Universe and it is hard to find any other object or topic that attracts more attention. However, it is still not clear whether black holes can be observed. Observation of the shadow of black hole candidate Sagittarius \(\hbox {A}^*\) or Sgr \(\hbox {A}^*\) is one of the most important goals of the Very Large Baseline Interferometry (VLBI) technique. It should able to image the black hole with resolution at the level of the event horizon. At galactic center the black hole candidate Sgr \(\hbox {A}^*\), due to the gravitational lensing effect, casts a shadow; the shape and size of this shadow can be calculated. There is a widespread belief that evidence for the existence of black holes will come from the direct observation of its shadow. The shape of a shadow could be used to study extreme gravity near the event horizon and also to know whether the general relativity is consistent with the observations. Observation of a black hole shadow may allow us to determine the mass and spin of a rotating black hole [1, 2, 3, 4, 5, 6]. As black holes are basically non-emitting objects, it is of interest the study null geodesics around them where photons coming from other sources move to lead to a shadow. To a distant observer, the event horizons cast shadows due to the bending of light by a black hole [7]. A first step towards the study of a black hole shadow was done by Bardeen [8], who calculated the shape of a dark area of a Kerr black hole, i.e., its shadow over a bright background. Although the shadow of Schwarzschild black hole is a perfect circle [9, 10], the Kerr black hole does not have a circular shadow image; it has an elongated shape in the direction of rotation [11]. The pictures of individual spherical light-like geodesics in the Kerr spacetime can be found in [12], and one can find a discussion and the picture of the photon region in the background of Kerr spacetime in Ref. [13].

Thus, the shadow deviation from the circle can determine the spin parameter of black holes. The study of black holes has been extended for other black holes, such as Kerr black holes [8], Kerr-Newman black holes [14], regular black holes [15, 16, 17] black holes, multi-black holes [18], black holes in extended Chern–Simons modified gravity [19] and for the Randall–Sundrum braneworld case [20]. The shadows of black holes with nontrivial NUT charge were obtained in [21], while the Kerr–Taub–NUT black holes were discussed in [22]. The apparent shape of the Kerr–Sen black holes is studied in [23], and rotating braneworld black holes were investigated in [20, 24]. Further, the effect of the spin parameter on the shape of the shadow was extended to the Kaluza–Klein rotating dilaton black hole [25], the rotating Horava–Lifshitz black hole [26], the rotating non-Kerr black hole [27] and the Einstein–Maxwell–dilaton-axion black hole [28]. There are different approaches to calculating the shadow of the black holes, e.g., a coordinate-independent characterization [29] and by general relativistic ray-tracing [30].

Recent years witnessed black hole solutions in more than four spacetime dimensions, especially in five dimensions as the subject of intensive research motivated by ideas in the braneworld, string theory and gauge/gravity duality. Several interesting and surprising results have been found in [31]. The models with large, extra dimensions have been proposed to deal with several issues arising in modern particle phenomenology [31, 32, 33, 34]. The rotating black holes have many applications and display interesting structures, but they are also very difficult to find in higher dimensions and the bestiary for solutions is much wider and less understood [35, 36]. The uniqueness theorems do not hold in higher dimensions due to the fact that there are more degrees of freedom. The black-ring solution in five dimensions shows that higher-dimensional spacetime can admit nontrivial topologies [37]. The Myers–Perry black hole solution [38] is a higher-dimensional generalization of the Kerr black hole solution. However, the Kerr–Newman black hole solution in higher dimensions has not yet been discovered. Nevertheless, there is a related solution of the Einstein–Maxwell–Chern–Simons (EMCS) theory in five-dimensional minimal gauged supergravity [39, 40]. Remarkably, the exact five-dimensional solutions for rotating charged black holes are known in the Einstein–Maxwell theory with a Chern–Simons term. The extremal limits of the five-dimensional rotating charged black hole solutions are of special interest, since they encompass a two parameter family of stationary supersymmetric black holes. The shadow of the five-dimensional rotating black hole [41] suggests that the shadow is slightly smaller and less deformed than for its four-dimensional Kerr black hole counterpart. Recently, the shadow of a higher-dimensional Schwarschild–Tangherlini black hole was discussed in [42] and the results show that the size of the shadows decreases in higher dimensions.

The aim of this paper is to investigate the shadow of a five-dimensional EMCS minimal gauged supergravity black hole (henceforth five-dimensional EMCS black hole) and compare the results with the Kerr black hole/five-dimensional Myers–Perry black hole. We have discussed in detail the structure of the horizons and the shadows of EMCS black holes in five dimensions, focusing on solutions with equal magnitude angular momenta, to enhance the symmetry of the solutions, making the analytical analysis much more tractable, while at the same time revealing already numerous intriguing features of the black hole shadows.

The paper is organized as follows. In Sect. 2, we review the five-dimensional Myers–Perry black hole solution and also visualize the ergosphere for various values of the charge *q*. In Sect. 3, we present the particle motion around the five-dimensional EMCS black hole to discuss the black hole shadow. Two observables are introduced to discuss the apparent shape of the black hole shadow in Sect. 4. The naked singularity shadow of the five-dimensional EMCS spacetime is the subject of Sect. 5. We discuss the energy emission rate of the five-dimensional EMCS black hole in Sect. 6 and finally we conclude with the main results in Sect. 7.

We have used units that fix the speed of light and the gravitational constant via \(8\pi G=c=1\).

## 2 Rotating five-dimensional EMCS black holes

*R*is the curvature scalar, \( F_{\mu \nu } = \partial _\mu A_\nu -\partial _\nu A_\mu \) with \(A_{\mu }\) is the gauge potential, and \(\epsilon ^{\mu \nu \lambda \rho \sigma }\) is the five-dimensional Levi-Civita tensor. The Lagrangian (1) has an additional Chern–Simons term different from the usual Einstein–Maxwell term. The corresponding equations of motion reads [44, 45]

*q*is the charge and

*a*,

*b*are the two different angular momenta of the black hole. Furthermore, the non-zero components of the metric (3) can be expressed as

*x*via \(x=r^2\). One can check that, for \(q=0\), the five-dimensional EMCS black hole reduces to the five-dimensional Myers–Perry black hole, which is analyzed in [38, 41], and also in addition if \(a=0\), it reduces to the five-dimensional Schwarschild–Tangherlini black hole [46]. It may be noted that the metric (3) is independent of the coordinates (\(t, \phi , \psi \)), and hence it admits three Killing vectors, given by

### 2.1 Horizons and ergosphere

*q*on the horizons and also on the ergosphere. It can be seen that the metric (3) is singular at \(\rho ^2=0\) and \(\Delta =0\). Note that \(\rho ^2=0\) is a physical singularity and \(\Delta = 0\) gives a coordinate singularity which defines the horizons of the metric (3). It turns out that \(\Delta =0\) admits two roots [44],

*q*, both surfaces of the metric (3) coincide. Figure 1 shows the contour plots of \(\Delta =0\), for the cases when \(a\ne b\) and \(a=b\). The contours depicted in Fig. 1 indicate the boundary lines between a black hole region and the naked singularity. The coinciding roots arise on the colored line and there exist two roots inside the black hole region (cf. Fig. 1). It can also be seen from Fig. 1 that if there is no real root, then the region belongs to the naked singularity. An ergoregion is a region outside the event horizon where the time-like Killing vectors behave like space-like. A particle can enter into the ergoregion and leave again, and it moves in the direction of spin of the black hole and has relevance for the energy extraction process [47]. We have plotted the ergoregion of the five-dimensional EMCS black hole in Fig. 2 for different values of

*a*and

*q*, and we observe that there is an increase in the area of the ergoregion when we increase the values of the parameter

*q*and

*a*(cf. Fig. 2).

In Kerr black hole case, we have the region between the event horizon and static limit surface, known as the ergoregion, which can be seen from Fig. 2. In Fig. 2, we explicitly show the effect of the parameter *q* on the ergoregion, it shows that the area of ergoregion grows with the rotation parameter *a* as well as with the charge *q*. Thus, for the faster rotating black holes the ergoregion is enlarged. Figure 3 suggests that the ergoregion area is maximum for an extremal five-dimensional EMCS black hole.

## 3 Particle motion

*S*is the Jacobian action with the following separable ansatz for the Jacobian action [41]:

*E*, and \(L_{\phi }\), \(L_{\psi }\) correspond to the energy and angular momentum of the particle, respectively. However, \(S_{\text {r}}(r)\) and \(S_{\theta }(\theta )\) are, respectively, functions of

*r*and \(\theta \) only. It can be seen, using Eqs. (27) and (26), that one can find the complete geodesic equations for the five-dimensional EMCS black hole [44]:

*r*= constant, define the apparent shape of the black hole. The photons come from infinity and approach a turning point with zero radial velocity, which corresponds to an unstable circular orbit determined by

## 4 Five-dimensional EMCS Black hole shadow

Higher dimensions admit astrophysical objects such as supermassive black holes, which are rather different from standard ones. The gravitational lensing features for alternative gravity theories with extra dimensions are different from general relativity [52]. Several tests were proposed to discover the signatures of extra dimensions in supermassive black holes since the gravitational field may be different from the standard one in general relativity. In particular, it was shown how measurements of the shadow can have constraints on parameters of the higher-dimensional black hole [52]. When a black hole is situated between an observer and a bright object, the light reaches the observer after being deflected by the black hole’s gravitational field; but some part of the photons emitted by the bright object ends up with falling into the black hole, and this means that photons never reach the observer.

*a*) and the charge (

*q*) at different inclination angles. The celestial coordinates \(\alpha \) and \(\beta \) in Eqs. (52) and (53) satisfy the relation \(\alpha ^2 + \beta ^2 = \eta + a^2\), where \(\eta \) is given by Eq. (40). It is clear that \(\alpha \) and \(\beta \) depend on the charge

*q*and spin

*a*.

*q*. The effect of the charge

*q*can be seen by the radius of the circle with an increase in

*q*(cf. Fig. 4). When \(q=0\), the radius of the circle of the shadow is \(R_{\text {s}}=2\), which is similar to the five-dimensional Schwarzschild black hole [42]. Thus, the effect of the charge

*q*is to decrease the size of the shadow (cf. Fig. 4).

*a*, charge

*q*, and extra dimension. With increasing

*a*, the shadow gets more and more distorted and shifts to the rightmost on the vertical axis, as in the Kerr case [51]. In the absence of charge \(q=0\), one gets

*q*and the spin

*a*. The shape of the black hole shadow is a deformed circle instead of a perfect circle. We see that the shape of the shadow is largely affected due the parameters

*a*,

*q*and extra dimension. The size of the shadow decreases continuously (cf. Fig. 5) with the increase in

*q*. This can be understood as a dragging effect due to the rotation of the black hole. The extra dimension has the same effect on the size of the shadow [41].

*a*is depicted in Fig. 7, which suggests that the radius \(R_{\text {s}}\) monotonically decreases as

*q*increases, and the distortion \(\delta _{\text {s}}\) of the shadow increases with

*q*and the shadow gets more distortion for larger values of

*a*. When compared with the five-dimensional Myers–Perry black hole [41], the effective size of the shadow decreases for higher values of the charge

*q*. Also, a comparison with the shadow with Kerr–Newman black holes [14] indicates that the effective size of the shadow decreases due to the extra dimensions. Instead of using \(R_{\text {s}}\) and \(\delta _{\text {s}}\), one can also use the observables defined by Schee and Stuchlík [24] to arrive at the same conclusions.

## 5 Naked singularity

*q*, the condition for the existence of a naked singularity satisfies \(\mu <(a+b)^2\) (\(a \ne b\)) and \(\mu <4a^2\) (\(a=b\)). The most general condition for a naked singularity in the five-dimensional Myers–Perry spacetime is \(a>1/{\sqrt{2}}\) [41]. The shadow of the naked singularity for the five-dimensional Myers–Perry spacetime is studied in [41]. Here we plot the contour of the naked singularity shadows for different values of

*q*and

*a*in Fig. 8, and the shapes change dramatically as compared to black hole shadows. It is found that the shape of a naked singularity shadow forms an arc instead of a circle (cf. Fig. 8). The behavior is similar to the naked singularity in the Myers–Perry geometries [41] or in the braneworld case [20]. The unstable spherical photon orbits with a positive radius give way to an open arc instead of a closed curve due to the absence of the event horizon and photons can reach the observer. The shapes of the naked singularity shadows are affected by the charge

*q*as well as by the spin

*a*; its size decreases in both cases when we increase either

*q*or

*a*(cf. Fig. 8). Our results show that the naked singularity shadow is different from the five-dimensional Myers–Perry black hole [41] (cf. Fig. 8). We observe that the arc of the shadow tends to open with the increasing values of the spin

*a*as well as charge

*q*. The two observables, viz. \(R_{\text {s}}\) and \(\delta _{\text {s}}\) are no longer valid for a naked singularity and new observables are necessary.

## 6 Energy emission rate

*T*is the Hawking temperature of the black hole. The Hawking temperature of the five-dimensional EMCS black hole reads [60]

*q*as well as the spin

*a*. When \(q \rightarrow 0\), the temperature reduces to

*q*and the spin

*a*. It can be seen that an increase in the values of

*q*and

*a*decreases the peak of energy emission rate (cf. Fig. 9). We analyze how the energy emission rate of a five-dimensional EMCS black hole decreases in comparison with a five-dimensional Myers–Perry black hole [41].

## 7 Conclusion

The formation of a shadow due to the strong gravitational field near a black hole has received significant attention due to a possibility of observing the images of the supermassive black hole Sgr \(A^*\) situated at the center of our galaxy [62]. The gravitational theories with extra dimensions admit black hole solutions which have different properties from the standard ones. Several tests were proposed to discover the signatures of extra dimensions in black holes as the gravitational field likely to be distinct from the one in general relativity, e.g., gravitational lensing for gravities with extra dimensions has different characteristics from general relativity. It turns out that the measurements of shadow sizes for higher-dimensional black holes can put constraints on the parameters of black holes [52], e.g., evaluating the size of a shadow it was shown that the probability to have a tidal charge (\(-q\)) for the black hole at the galactic center is ruled out by the observations [63]. Thus, it suggests that black holes with positive charge (\(+q\)) are consistent with observations, but a significant negative charge (\(-q\)) black holes is ruled out [63], i.e., the Reissner–Nordström black holes which have positive charge (\(+q\)) are consistent with the observation, but this is not true for the braneworld black holes, which have a negative tidal charge (\(-q\)). A five-dimensional EMCS black hole solution has an additional charge parameter *q* when compared with the five-dimensional Myers–Perry black hole, and it provides a deviation from the Myers–Perry black hole. The five-dimensional EMCS black hole has a richer configuration for the horizons and ergosphere. It is interesting to note that the ergosphere size is sensitive to the charge *q* as well as the rotation parameter *a*.

We have extended the previous studies of the black hole shadow and derived analytical formulas for the photon regions for a five-dimensional EMCS black hole. We also make quantitative analyses of the shape and size of the black hole shadow cast by a rotating five-dimensional EMCS black hole. We have analyzed how the shadow of black hole is changed due to the presence of the charge *q*, and we explicitly show that the charge *q* apparently affects the shape and the size of the shadow. In particular, it is observed that the shadow of a rotating five-dimensional EMCS black is a dark zone covered by a more deformed circle as compared to a Myers–Perry black hole shadow. For a fixed value of spin *a*, compared to the rotating five-dimensional Myers–Perry black hole, the size of the shadow decreases with the charge *q*, whereas shadows become more distorted with an increase in charge *q*, and the distortion is maximal for an extremal black hole. In comparison with the four-dimensional Kerr–Newman black hole, we found that the shadow of the five-dimensional EMCS black hole decreases. The study of the shadow of a naked singularity of the five-dimensional EMCS metric suggests that the shape of the shadow decreases for higher values of the charge *q* when compared with the naked singularity shadow with the five-dimensional Myers–Perry metric. Obviously, our results in the limit \(q\rightarrow 0\) reduce exactly to the five-dimensional Myers–Perry black hole.

We think that the results obtained here are of interest in the sense that they do offer the opportunity to explore properties associated with a shadow in higher dimensions, which may be crucial in our understanding of whether the size of the shadow might suggest if there are signatures of a five-dimensional charged black hole. The possibility of a further generalization by adding a negative cosmological constant is an interesting problem for future research under active consideration.

## Notes

### Acknowledgements

S.G.G. would like to thank SERB-DST Research Project Grant No. SB/S2/HEP-008/2014 and DST INDO-SA bilateral project DST/INT/South Africa/P-06/2016. M.A. acknowledges the University Grant Commission, India, for financial support through the Maulana Azad National Fellowship For Minority Students scheme (Grant No. F1-17.1/2012-13/MANF-2012-13-MUS-RAJ-8679).

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