# Dynamics of particles near black hole with higher dimensions

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## Abstract

This paper explores the dynamics of particles in higher dimensions. For this purpose, we discuss some interesting features related to the motion of particles near a Myers–Perry black hole with arbitrary extra dimensions as well as a single non-zero spin parameter. Assuming it as a supermassive black hole at the center of the galaxy, we calculate red–blue shifts in the equatorial plane for the far away observer as well as the corresponding black hole parameters of the photons. Next, we study the Penrose process and find that the energy gain of the particle depends on the variation of the black hole dimensions. Finally, we discuss the center of mass energy for 11 dimensions, which indicates a similar behavior to that of four dimensions but it is higher in four dimensions than five or more dimensions. We conclude that higher dimensions have a great impact on the particle dynamics.

### Keywords

Black Hole Angular Momentum Event Horizon Blue Shift Equatorial Plane## 1 Introduction

Gravity in more than four dimensions has been the subject of interest in recent years for a variety of reasons. This leads to significant features of black holes (BHs) like uniqueness, dynamical stability, spherical topology, and the laws of BH mechanics. It has been found that the laws of BH mechanics are universal, while the properties of the BH are dimension dependent. The concept of higher dimensions became prominent in the 20th century with the Kaluza–Klein theory which unified gravitation and electromagnetism in five dimensions [1, 2]. Later on, development of string and M-theories led to further progress in higher dimensional gravity. String theory is the most promising candidate of quantum gravity—the fascinating theory of high energy physics. M-theory is the generalization of superstring theory that gave the concept of 11 dimensions. Charged BHs in string theory play an important role in understanding the BH entropy near the extremal limits [3]. Callan and Maldacena [4] calculated the Hawking temperature, the radiation rate, and the entropy for the extremal Reissner–Nordstr\(\ddot{\mathrm{o}}\)m BH in the context of string theory and proposed that quantum evolution of BH does not lead to information loss. Itzhaki et al. [5] studied D-brane solutions in string theory for the region where curvature is very small.

The study of BHs in higher dimensions has attracted many researchers. Tangherlini [6] was the first who generalized the Schwarzschild BH to arbitrary extra dimensions (\(D>4\)), while Myers and Perry generalized the Kerr BH [7]. There also exist black rings [8, 9] and multi BH solutions like black Saturns and multi black rings [10, 11, 12, 13]. Carter and Neupane [14] studied stability and thermodynamics of higher dimensional Kerr–anti de Sitter BH and found stability for equal rotation parameters. Dias et al. [15] investigated perturbations of the Myers–Perry (MP) BH and found stability in five and seven dimensions. Murata [16] found instabilities of *D*-dimensional MP BH and concluded that there is no evidence of instability in five dimensions, however, for \(D=7,9,11,13\), the spacetime became unstable due to large angular momenta.

Galactic rotation curves are based on the measurement of red–blue shifts of emitted light from distant stars. Due to rotation of the galaxy, one side of the galaxy will appear to be blue shifted as it rotates toward the observer and the other will be red-shifted as it rotates away from the observer [17, 18]. Nucamendi et al. [19] studied the rotation curves of galaxies by measuring the frequency shifts of spherically symmetric spacetime. Lake [20] showed that the galactic potential can be linked to red–blue shifts of the galactic rotation curves. Bharadwaj and Kar [21] proposed that the flat rotation curves of the spiral galaxies can be explained by dark matter halos having anisotropic pressure. Moreover, the deflection of light ray is sensitive to the pressure of the dark matter. Faber and Visser [22] argued that observations of galactic rotation curves together with gravitational lensing describe the deduction of galactic mass and this provides information as regards the pressure of the galactic fluid. Herrera-Aguilar et al. [23] presented a useful technique to study red–blue shifts for a spiral galaxy by generalizing the galactic rotation curves for spherically symmetric spacetime to an axisymmetric metric. This approach has been used to express the Kerr BH parameters in terms of red–blue shift functions [24].

The Penrose process is related to the energy extraction from a rotating BH which depends upon the conservation of angular momentum. Chandrasekhar [25] studied the Penrose process for the Kerr BH and discussed the nature of this process and examined the limits on the extracted energy. He found that in the equatorial plane, only retrograde particles possess negative energy and the particles should remain inside the static limit (ergosphere). Bhat et al. [26] investigated the Penrose process for the Kerr–Newman BH and concluded that the energy becomes highly negative in the presence of electromagnetic field, while for neutral particles, the gain energy decreases in the presence of charge of the BH. Recently, Lasota et al. [27] presented the generalized Penrose process and stated that “for any matter or field, tapping the BH rotation energy is possible if and only if negative energy and angular momentum are absorbed by BH and no torque at the BH horizon is necessary (or possible)”. There are some other important results [28, 29, 30] in the context of Penrose process.

The collision energy of particles in the frame of the center of mass results in the formation of new particles and the energy produced in this process is known as the center of mass energy. The center of mass energy of two colliding particles is infinitely high near the event horizon of a maximally spinning Kerr BH [31]. This approach is very useful as it describes the rotating BH as a particle accelerator at the Planck energy scale. Lake [32] examined particle collisions for a non-extremal Kerr BH at the inner horizon and found center of mass energy to be finite. The center of mass energy is also analyzed for the Kerr–Newman BH, which shows the dependence on the spin and charge of the BH [33]. The same mechanism was employed on the Kerr–Newman Tuab [34] and rotating Hayward BH [35]. Other important aspects related to the center of mass energy have been explored in [36, 37, 38, 39, 40, 41, 42, 43, 44].

In this paper, we study the dynamics of particles for a *D*-dimensional MP BH in the equatorial plane. The paper is organized as follows. In the next section, we review timelike geodesics in higher dimensions. In Sect. 3, we study red–blue shifts of MP BH and formulate BH parameters in terms of red–blue shift functions. Section 4 explores the Penrose process and Sect. 5 is devoted to the study of the center of mass energy for this BH. We conclude our results in the last section.

## 2 Review of geodesics in higher dimensions

*a*. The

*D*-dimensional MP BH in Boyer–Lindquist coordinates is given as [45, 46]

*D*may be even or odd. For \(D=4\) it reduces to the Kerr BH, while \(a=0\) leads to the Schwarzschild BH. The event horizon of (1) is the largest root of \(\Delta =0\),

*D*-velocity and \(\tau \) is the affine parameter. In the equatorial plane (\(\theta =\frac{\pi }{2},~\dot{\theta }=0\)), Eq. (2) takes the form

*t*and \(\phi \), therefore \(k_{t}\) and \(k_{\phi }\) are conserved and hence this describes stationary and axisymmetric characteristics of MP BH.

*M*is a parameter related to the BH mass). Here, we do not consider this substitution, as we are interested in finding our results with the ADM mass.

## 3 Red–blue shifts of Myers–Perry BH

This section is devoted to the study of red–blue shifts in higher dimensions for MP BH. Herrera-Aguilar et al. [23] discussed red–blue shifts for an axially symmetric spacetime and presented a convenient approach to study the galactic rotational curves of spiral galaxies. Since spiral galaxies possess axial symmetry, this method provides information as regards the interior of the gravitational field of such galaxies. Herrera-Aguilar [24] extended this technique for the Kerr BH and found parameters in terms of red–blue shifts in the equatorial plane. Following [23, 24], we generalize these results for the MP BH. We consider two observers \(O_{d}\) and \(O_{e}\), corresponding to detector and light emitter (star) placed at points \(P_{d}\) and \(P_{e}\), respectively. The detector and emitter possess *D*-velocities \(u^{\nu }_{d}\) and \(u^{\nu }_{e}\). We assume that the stars are moving in the galactic plane such that the polar angle is fixed (\(\theta =\frac{\pi }{2}\)). In this case, we have \(u^{\nu }_{e}=(u^{t},~u^{r},0,\ldots ,0,~u^{\phi })_{e}\), where \(u^{\nu }=\mathrm{d}x^{\nu }/\mathrm{d}\tau \) and \(\tau \) is the proper time of the particle. The *D*-velocity of the detector, \(u_{d}=(u^{t},~u^{r},0,\ldots ,0,~u^{\phi })_{d}\), is located far away from the source. The component \(u^{\phi }\) is related to the observer’s dragging at point \(P_{d}\) due to galactic rotation and its effect is present when measuring red–blue shifts in our galaxy (Milky Way) or nearby galaxies [23].

*D*-momentum in the equatorial plane and the index

*c*corresponds to emitter (

*e*) or detector (

*d*) for the spacetime at point \(P_{c}\). The light frequencies detected by an observer at \(P_{d}\) and measured by an observer moving with the emitted particle at point \(P_{e}\) are

*b*to calculate the red–blue shifts. The impact parameter can be calculated from the radial null geodesic \(k^{\nu }k_{\nu }=0\) [23],

*D*-velocity components corresponding to (9) and (10) for the circular orbits in the equatorial plane

*D*dimensions can be obtained from Eq. (18),

### 3.1 The Penrose process

*E*and

*L*, we obtain

*E*can be negative (seen by an observer at infinity).

### 3.2 The original Penrose process

## 4 Center of mass energy in higher dimensions

*D*dimensions becomes

## 5 Final remarks

When a light wave (from an approaching galaxy which is moving toward an observer) gets scrunched to the shorter wavelength, this is known as the galactic blue shift. On the other hand, if a light wave from a galaxy moving away from the observer gets stretched to the longer wavelength then this is called galactic red shift. Slipher discovered that the Andromeda galaxy possesses a large blue shift which indicates that this galaxy moves toward the Earth. He further investigated other spiral galaxies and found that most of them have a large red shift, indicating that they are moving away from us. Hubble observations indicate that relative to the Earth and all the observed galactic objects, galaxies are receding in every direction. The velocities calculated from their observed red shifts are directly proportional to their distance from each other as well as from the Earth. Hubble was the pioneer in explaining the expanding universe and red shift phenomena [48, 49].

It is well known that there is a supermassive BH (SgrA*) in the center of Milky Way, as is the case in many other spiral galaxies. In this paper, we assume a higher dimensional MP BH as a supermassive BH at the galactic center. Motivated by [23, 24], we generalize the mass and angular momentum parameter in arbitrary extra dimensions in terms of red–blue shifts of the photons emitted from circular timelike geodesic and traveling along the null geodesic. For this purpose, we have first calculated red–blue shifts of the photons in higher dimensions for an observer located far away. We have taken circular as well as equatorial orbits to find these shifts. We have expressed the corresponding mass, rotation parameter and radius of the detector in terms of red–blue shifts. In this way, we have generalized the results for the Kerr BH. We have only discussed the analytical model, however, these results may be useful if they can be calculated using the observational data. The generalized results can provide information as regards the behavior of BH parameters in dimensions \(D\ge 4\).

It is believed that the supermassive BHs (powering the active galaxies and quasars) are the rapidly rotating BHs. Such BHs produce powerful jets of gas (whose direction is sometimes stable over a million of years) whose source of energy may be the rotation of BH. It may be possible that this rotational energy is extracted due to the Penrose process [47]. Following [25], we have studied the Penrose process for the MP BH. We have found that a particle will have negative energy for a retrograde motion (\(L<0\)) in higher dimensions. We have also seen that the energy gain of the particle is dimension dependent. The energy gains for \(D=4\) and \(D=6\) have the same values, while for \(D=5\), it has the highest value.

We have also examined the influence of higher dimensions on the center of mass energy of two colliding particles. We have plotted \(E_\mathrm{cm}\) by considering two different values of the rotating parameters. The center of mass energy decreases with increasing radius. For \(a=0.25\), the center of mass energy for \(D=6~\text {and}~ 8\), it lies inside the event horizon, while for other dimensions, it crosses the event horizon. For \(a=0.6\), the center of mass energy crosses the event horizon in all dimensions. In both cases, the center of mass energy for \(D=4\) is greater than that of \(D\ge 5\). Finally, we conclude that the motion of particles in higher dimensions experiences a very different behavior than the four dimensions.

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