# Rotating black hole and quintessence

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## Abstract

We discuss spherically symmetric exact solutions of the Einstein equations for quintessential matter surrounding a black hole, which has an additional parameter (\(\omega \)) due to the quintessential matter, apart from the mass (*M*). In turn, we employ the Newman–Janis complex transformation to this spherical quintessence black hole solution and present a rotating counterpart that is identified, for \(\alpha =-e^2 \ne 0\) and \(\omega =1/3\), exactly as the Kerr–Newman black hole, and as the Kerr black hole when \(\alpha =0\). Interestingly, for a given value of parameter \(\omega \), there exists a critical rotation parameter (\(a=a_{E}\)), which corresponds to an extremal black hole with degenerate horizons, while for \(a<a_{E}\), it describes a non-extremal black hole with Cauchy and event horizons, and no black hole for \(a>a_{E}\). We find that the extremal value \(a_E\) is also influenced by the parameter \(\omega \) and so is the *ergoregion*.

The purpose of this letter is to seek the generalization of the solution (5) to the axially symmetric case or to the Kerr-like metric. The Kerr metric [9] is beyond question the most extraordinary exact solution in the Einstein theory of general relativity, which represents the prototypic black hole that can arise from gravitational collapse, which contains an event horizon [10]. It is well known that Kerr black holes enjoy many interesting properties distinct from its non-spinning counterpart, i.e., from the Schwarzschild black hole. However, there is a surprising connection between the two black holes of Einstein theory, and it was analyzed by Newman and Janis [11, 12, 13, 14] that the Kerr metric [9] could be obtained from the Schwarzschild metric using a complex transformation within the framework of the Newman–Penrose formalism [15]. A similar procedure was applied to the Reissner–Nordstrm metric to generate the previously unknown Kerr–Newman metric [12]. It is an ingenious algorithm to construct a metric for rotating black hole from static spherically symmetric solutions in Einstein gravity. The Newman–Janis method has proved to be prosperous in generating new stationary solutions of the Einstein field equations and the method has also been studied outside the domain of general relativity [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], although it may lead to additional stresses [20, 21, 26, 27]. For possible physical interpretations of the algorithm, see [28, 29], and for discussions on more general complex transformations, see [16, 28, 29]. For a review of the Newman–Janis algorithm see, e.g., [30].

*r*factor in the null vectors to take on complex values. We rewrite the null vectors in the form [25, 26]

*r*. Following the Newman–Janis prescription [11], we now write

*a*is the rotation parameter. Simultaneously, let the null tetrad vectors \(Z^a\) undergo a transformation \(Z^a = Z'^a{\partial x'^a}/{\partial x^b} \) in the usual way; we obtain

The effect of quintessence parameter \(\omega \) on the extremal rotation parameter (\(a_{E}\)) and extremal horizon (\(r^{E}\))

\(\omega \) | \(\theta =\pi /4\) | \(\theta =\pi /2\) | ||
---|---|---|---|---|

\(a=a_{E}\) | \(r^{E}\) | \(a_{E}\) | \(r^{E}\) | |

\(-\)0.50 | 0.9270821091720760 | 0.884066 | 0.9556124089886700 | 0.894288 |

\(-\)0.66 | 0.9243197545134466 | 0.860520 | 0.9578436527014000 | 0.883037 |

\(-\)0.77 | 0.9228228129512000 | 0.845995 | 0.9593257246846765 | 0.877225 |

\(-\)0.88 | 0.9215809590176600 | 0.832531 | 0.9607794407624000 | 0.872548 |

The Cauchy and event horizons of the black hole for different values of \(\omega \) and *a* (\(\theta =\pi /4\))

\(a < a_{E}\) | \(\omega =-0.50\) | \(\omega =-0.66\) | \(\omega =-0.77\) | \(\omega =-0.88\) | ||||
---|---|---|---|---|---|---|---|---|

\(r^{-}\) | \(r^{+}\) | \(r^{-}\) | \(r^{+}\) | \(r^{-}\) | \(r^{+}\) | \(r^{-}\) | \(r^{+}\) | |

0.7 | 0.304099 | 1.46241 | 0.299547 | 1.41561 | 0.297165 | 1.38497 | 0.295200 | 1.35543 |

0.8 | 0.437164 | 1.32997 | 0.430741 | 1.28681 | 0.427068 | 1.25913 | 0.423826 | 1.23282 |

0.9 | 0.671981 | 1.09592 | 0.665427 | 1.05488 | 0.660976 | 1.02987 | 0.656497 | 1.00698 |

The Cauchy and event horizons of the black hole for different values of \(\omega \) and *a* (\(\theta =\pi /2\))

\(a < a_{E}\) | \(\omega =-0.50\) | \(\omega =-0.66\) | \(\omega =-0.77\) | \(\omega =-0.88\) | ||||
---|---|---|---|---|---|---|---|---|

\(r^{-}\) | \(r^{+}\) | \(r^{-}\) | \(r^{+}\) | \(r^{-}\) | \(r^{+}\) | \(r^{-}\) | \(r^{+}\) | |

0.7 | 0.289008 | 1.48923 | 0.287523 | 1.45172 | 0.286949 | 1.42732 | 0.286574 | 1.40387 |

0.8 | 0.408981 | 1.37298 | 0.405586 | 1.34310 | 0.404079 | 1.32413 | 0.402983 | 1.30616 |

0.9 | 0.596924 | 1.18918 | 0.588102 | 1.17125 | 0.583599 | 1.16050 | 0.579960 | 1.15067 |

*r*in the function \(\Delta \) as well as in \(\Sigma \). Then this metric could be cast in the more familiar Boyer–Lindquist coordinates to read

In general, as envisaged the black hole horizon is spherical and it is given by \(\Delta = 0\), which has two positive roots giving the usual outer and inner horizon and no negative roots. The numerical analysis of the algebraic equation \(\Delta =0\) reveals that it is possible to find non-vanishing values of the parameters \(a,\; \omega \) and \(\alpha \) for which \(\Delta \) has a minimum, and that \(\Delta =0\) admits two positive roots \(r^{\pm }\) (cf. Fig. 1).

*ergoregion*admitting negative energy orbits, i.e., the region between \(r_+^{EH}\, < r\, < r_+^{SLS}\) is called the

*ergoregion*, where the asymptotic time translation Killing field \(\xi ^a=(\frac{\partial }{\partial t})^a\) becomes spacelike and an observer follows the orbit of \(\xi ^a\). It turns out that the shape of the

*ergoregion*, therefore, depends on the spin

*a*and the parameter \(\omega \). Interestingly, the quintessence matter does influence the shape of the

*ergoregion*as described in Fig. 3 when compared with the analogous situation of the Kerr black hole. Indeed, we have demonstrated that the

*ergoregion*is vulnerable to the parameter \(\omega \) and the

*ergoregion*becomes more prolate,; the ergoregion area increases as the value of the parameter \(\omega \) increases. Further, we find that for a given value of \(\omega \), one can find a critical parameter \(a^C\) such that the horizons are disconnected for \(a > a^C\) (cf. Fig. 3).

Penrose [31] surprised everyone when he suggested that energy can be extracted from a rotating black hole. The Penrose process [31] relies on the presence of an *ergoregion*, which for the solution (19) grows with the increase of the parameter \(\omega \) as well with spin *a* (cf. Fig. 3). Thus the parameter \(\omega \) is likely to have an impact on the energy extraction. It will also be useful to further study the geometrical properties, causal structures, and thermodynamics of this black hole solution. All these and related issues are being investigated.

In this letter, we have used the complex transformations pointed out by Newman and Janis [11], to obtain rotating solutions from the static counterparts for the quintessential matter surrounding a black hole. Interestingly, the limit as \(a \rightarrow 0\) is still correct from the point of view of the obtained solution, but it is easy to see that the metric obtained by the complex transformation is likely to generate additional stress [20, 26, 27]. It may be pointed out that in the general relativity case the source, if it exists, is the same for both a black hole and its rotating counterpart (obtained by Newman–Janis complex transformations), e.g., the vacuum for both Schwarzschild and Kerr black holes, and the charge for the Reissner–Nordstrom and Kerr–Newman black holes. The source for the solution (5) is just quintessence matter, whereas its rotating counterpart (19), in addition to quintessence matter, has some additional stresses.

## Notes

### Acknowledgments

We would like to thank SERB-DST Research Project Grant No. SB/S2/HEP-008/2014, and M. Amir for help in the plots. We also thank IUCAA for hospitality while this work was being done and ICTP for Grant No. OEA-NET-76.

### Added in proof

After this work was completed, we learned of a similar work by Toshmatov et al. [32], which appeared in the arXiv a couple of days before.

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