# Spinning higher dimensional Einstein–Yang–Mills black holes

## Abstract

We construct a Kerr–Newman-like spacetime starting from higher dimensional (HD) Einstein–Yang–Mills black holes via complex transformations suggested by Newman–Janis. The new metrics are a HD generalization of Kerr–Newman spacetimes which has a geometry that is precisely that of Kerr–Newman in 4D corresponding to a Yang–Mills (YM) gauge charge, but the sign of the charge term gets flipped in the HD spacetimes. It is interesting to note that the gravitational contribution of the YM gauge charge, in HD, is indeed opposite (attractive rather than repulsive) to that of the Maxwell charge. The effect of the YM gauge charge on the structure and location of static limit surface and apparent horizon is discussed. We find that static limit surfaces become less prolate with increase in dimensions and are also sensitive to the YM gauge charge, thereby affecting the shape of the ergosphere. We also analyze some thermodynamical properties of these BHs.

## Keywords

Black Hole Event Horizon Black Hole Solution High Dimensional Spacetime Dimension## 1 Introduction

The Kerr metric [1] is an explicit exact solution of the Einstein field equations describing a spinning black hole (BH) in four dimensional (4D) spacetime. It is well known that a BH with non-zero spinning parameter, i.e., a Kerr BH, enjoys many interesting properties distinct from its non-spinning counterpart, i.e., from a Schwarzschild BH [2]. However, there is a surprising connection between the two BHs of Einstein theory, and this is analyzed by Newman and Janis [3]. They demonstrated that applying a complex coordinate transformation, it was possible to construct both the Kerr and Kerr–Newman solutions starting from the Schwarzschild metric and Reissner–Nordstr\(\ddot{o}\)m metric, respectively, [3]. The Kerr–Newman describes the exterior of a spinning massive charged BH [4]. The Newman–Janis Algorithm (NJA) is successful in analyzing several spinning BH metrics starting from their non-spinning counterparts [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. For a review on the NJA (see, e.g., [18]). However, the NJA has often be considered that there is arbitrariness and physicists considered this as an ad-hoc procedure [19]. But Schiffer et al. [20] gave a very elegant mathematical proof as to why the Kerr metric can be considered as a complex transformation of the Schwarzschild metric.

It is rather well established that higher dimensions provide a natural playground for string theory and they are also required for its consistency [21, 22]. Even from the classical standpoint, it is interesting to study the higher dimensional (HD) extension of Einstein’s theory, and in particular its BH solutions [23]. There seems to be a general belief that endowing general relativity with a tunable parameter, namely the spacetime dimension, should also lead to valuable insights into the nature of the theory, in particular into its most basic objects: BHs. For instance, 4D BHs are known to have a number of remarkable features, such as uniqueness, spherical topology, dynamical stability, and the laws of BH mechanics. One would like to know which of these are peculiar to 4D, and which are true more generally. At the very least, such probings into HD will lead to a deeper understanding of classical BHs and of what space-time can do at its most extreme. There is a growing realization that the physics of HD BHs can be markedly different, and much richer than their counterparts in 4D [2, 24, 25, 26], e.g., the event horizon may not be spherical in HD and also may have no BH uniqueness [22]. It is of interest to consider models based on different interacting fields including the Yang–Mills (YM). In general, it is difficult to tackle Einstein–Yang–Mills (EYM) equations because of the non-linearity both in the gauge fields and in the gravitational field. The solutions of the classical YM fields depend upon the particular *ansatz* one chooses. Wu and Yang [27] found static spherically symmetric solutions of the YM equations in flat space for the gauge group SO(3). A curved spacetime generalization of these models has been investigated by several authors (see, e.g., [28]). Indeed Yasskin [28] has presented an explicit procedure based on the Wu–Yang *ansatz* [27] which gives the solution of EYM rather trivially. Using this procedure, Mazharimousavi and Halilsoy [29, 30, 31] have found a sequence of static spherically symmetric HD EYM BH solutions. The remarkable feature of this *ansatz* is that the field has no contribution from the gradient; instead, it has a pure YM non-Abelian component. It, therefore, has only the magnetic part.

The strategy of obtaining the familiar Kerr–Newman solution, both in 4D and HD, in general relativity is based on either using the metric ansatz in the Kerr–Schild form or applying the method of complex coordinate transformation to a non-rotating charged black hole. Surprisingly, it has been demonstrated that when employing HD dimensional spacetime the two approaches lead to the same result [32, 33] The main purpose of this work is to apply NJA to the HD EYM BH metric previously discovered in [29, 30, 31] and the spinning HD EYM BH metric is obtained. This result shows that NJA works well also in HD spacetime. We further discuss the properties of the spinning HD EYM BH such as horizons and ergosphere. Spinning BH solutions in higher dimensions are known as Myers–Perry BHs [26]. The thermodynamical quantities associated with the spinning HD EYM BH are also calculated. Further we demonstrate that the thermodynamical quantities of this BH go over to the corresponding quantities of Myers–Perry BH and Kerr BH.

## 2 Static BH in HD EYM theory

*ansatz*in \((N+3)\) dimensions [29, 30, 31] as

*ansatz*[27] facilitates obtaining the solution.

*ansatz*[27] is given by

## 3 Spinning HD EYM BH via NJA

## 4 Horizon properties

It is natural to discuss not only spinning BH solutions but their various properties. It is known that the structure of a spinning BH is very different from that of a stationary BH. The EH of a spinning BH is smaller than the EH of an otherwise identical but non-spinning one. Similar to Kerr solutions in asymptotically flat spacetimes the above metric has two types of horizon-like hypersurface: a stationary limit surface (SLS) and an EH. Within the stationary limit, no particles can remain at rest, even though they are outside the EH. We shall explore the two horizons SLS and EH of spinning HD EYM BH, and also we shall discuss the effects which come from the YM gauge charge and also the effects due to the spacetime dimension.

### 4.1 SLS and EH

#### 4.1.1 4D case

#### 4.1.2 6D case

#### 4.1.3 7D Case

### 4.2 Ergosphere

*time-axis*are not time-like but space-like. An ergosphere, thus, is a region of spacetime where no observer can remain still with respect to the coordinate system in question. Thus the ergosphere of a spinning HD EYM BH, as in Kerr/Kerr–Newman BH, is bounded by the EH on the inside and an oblate spheroid SLS, which coincides with the EH at the poles and is noticeably wider around the equator. It is the region of spacetime where time-like geodesics remain stationary and time-like particles can have negative energy relative to infinity. It is theoretically possible to extract energy and matter from the BH from the ergosphere [37]. In Fig. 4 we show the dependence of the shape of the ergosphere on the YM gauge charge. It is noticed that for \(D=4\) the shape of the ergosphere is increasing with the increase in the YM gauge charge while decreasing for \(D\ge 6\). This shows that the shape of the ergosphere is sensitive to the YM gauge charge. In Fig. 5 the variation of the shape of ergosphere with \(a\) is shown. We note that the relative shape of the ergosphere becomes more prolate, thereby increasing the area of the ergosphere with rotation parameter \(a\), i.e., the faster the BH rotates, the more the ergosphere grows. This may have direct consequences on the Penrose energy extraction process [37].

## 5 Conclusion

The physical properties of the solutions have not yet been fully investigated, this being a very severe job. However, we are currently working on this project. We have also shown that the presence of a YM gauge charge decreases the temperature with increase in gauge charge parameter Q. Such a change could have a significant effect in the thermodynamics of a BH. Hence, it will be of interest to see how the YM gauge charge affects the thermodynamics by deriving a Smarr-like relation and the first law, and also analyzing stability. Further analysis of these solutions and the role of the YM gauge charge and spacetime dimension in an energy extraction process remains an interesting issue to explore in the future.

## Notes

### Acknowledgments

The authors would like to thank the University Grant Commission (UGC) for major research project Grant No. F-39-459/2010(SR) and to IUCAA, Pune, for kind hospitality while part of this work was being done.

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