# Cylindrical symmetric, non-rotating and non-static or static black hole solutions and the naked singularities

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## Abstract

In this work, a four-dimensional cylindrical symmetric and non-static or static space-times in the backgrounds of anti-de Sitter (AdS) spaces with perfect stiff fluid, anisotropic fluid and electromagnetic field as the stress–energy tensor, is presented. For suitable parameter conditions in the metric function, the solution represents non-static or static non-rotating black hole solution. In addition, we show for various parameter conditions, the solution represents static and/or non-static models with a naked singularity without an event horizon.

## 1 Introduction

In General Relativity, it is a subject of long-standing interest to look for the exact solutions of Einstein’s field equations. Among these exact solutions, black hole solutions take an important position because thermodynamics, gravitational theory, and quantum theory are connected in quantum black hole physics. In addition, black holes might play an important role in developing a satisfactory full quantum theory of gravity which does not exist till today. Black holes are regions of space-time from which nothing, not even light, can escape. A typical black hole is the result of the gravitational force becoming so strong that one would have to travel faster than light to escape its pull. Such black holes generically contain a space-time singularity at their center; thus we cannot fully understand a black hole without also understanding the nature of singularities. Black holes, however, raise several additional conceptual problems and questions on their own. When quantum effects are taken into account, black holes, although they are nothing more than regions of space-time, appear to become thermodynamic entities, with a temperature and an entropy. The fundamental features of a black hole is the topology. In four-dimensional asymptotically flat stationary space-time, Hawking showed that a black hole has necessarily a \(S^{2}\) topology, provided that the dominant energy condition holds [1]. This result extends to outer apparent horizons in black hole space-times that are not necessarily stationary [2]. Such restrictive uniqueness theorems do not hold in higher dimensions, the most famous counter-example being the black ring of Emparan and Reall [3], with horizon topology \(S^2\times S^1\). Nevertheless, Galloway and Schoen [4] have shown that in arbitrary dimension, cross sections of the event horizon (in the stationary case) and outer apparent horizons (in the general case) admit metrics of positive scalar curvature. Hawking’s theorem was later generalized by Gannon [5] who had proved, replacing stationary by some weaker assumptions, that the horizon must be either spherical or toroidal. The topological censorship theorem [6] also plays an important role in black hole physics which states that in asymptotically flat space-times, only spherical horizons can give rise to well defined causal structure for a black hole. This is circumvented by the presence of a negative cosmological constant, in which case well defined black holes with locally flat or hyperbolic horizons have been shown to exist [7, 8, 9]. This kind of black holes with topologically non-trivial anti-de Sitter (AdS) asymptotics are relevant in testing the AdS/CFT correspondence.

In the framework of four-dimensions, it is well known that generic black hole solutions to Einstein–Maxwell equations are Kerr–Newman solutions [10, 11], which are characterized by only three parameters: the mass (M), charge (q), and angular momentum (J). It is often referred to as the non-hair conjecture of black holes and the space-time is asymptotically flat. When a non-zero cosmological constant is introduced, the space-time will become asymptotically de Sitter (dS) or anti-de Sitter (AdS) spaces depending on the sign of the cosmological constant. In a recent paper [12], Lemos constructed a cylindrical symmetric rotating and/or non-rotating and static black hole solutions (black strings) in four-dimensional Einstein gravity with a negative cosmological constant. The black string solutions of Lemos is asymptotically AdS not only in the transverse directions, but also in the string direction. Huang and Liang [13] further constructed the so-called torus-like black holes with the topology \(R^2\times S^1\times S^1\). The topological black holes in four-dimensional non-static space-time with non-zero cosmological constant, were investigated in [14]. Wu et al. [15] constructed a topologically charged static black hole solution in four-dimensions. Once the energy condition is relaxed, a black hole can have quite different topology. Such examples can occur even in four-dimensional space-times where the cosmological constant is negative [8, 16, 17, 18, 19, 20, 21].

The curvature singularities of matter-filled as well as vacuum space-times are recognized from the divergence of the energy-density and/or the scalar curvature invariant, such as the zeroth order scalar curvature \(R_{\mu \nu \rho \sigma }\,R^{\mu \nu \rho \sigma }\) (so called the Kretschmann scalar), and \(R_{\mu \nu \rho \sigma }\,R^{\rho \sigma \lambda \tau }\,R^{\mu \nu }_{\lambda \tau }\). In addition, by analysing the outgoing radial null geodesics of a space-time containing space-time singularity, one can determine whether the curvature singularity is naked or covered by an event horizon (see Refs. [22, 23] for detail discussion). For the strength of curvature singularities, two conditions are generally used: first one being the *strong curvature condition* (SCC) given by Tipler [24], and second one is the *limiting focusing condition* (LFC) given by Krolak [25]. Meanwhile, the curvature singularities are basically of three types: first one being a space-like singularity (e.g. Schwarzschild singularity), second one is the time-like singularity, and third one is the null singularity. In time-like singularity, two possibilities arise: (i) there is an event horizon around a time-like singularity (e.g. RN black holes). Here an observer cannot see a time-like singularity from an outside region of the space-time, that is, the singularity is covered by an event horizon; (ii) there is no event horizon around a time-like singularity which we called naked singularities (NS), and it would observable for far away observers. Therefore existence of a naked singularity would present opportunities to observe the physical effects near the very dense regions that formed in the very final stages of a gravitational collapse. But in a black hole scenario, such regions are necessarily hidden within the event horizon. Attempts have been made to provide the theoretical framework to devise a technique to distinguish between black holes and naked singularities from astrophysical data mainly through gravitational lensing (GL) method. Some significant works in this direction are the study of strong gravitational lensing in the Janis–Newman–Winicour space-time [26, 27], and its rotating generalization [28], and notable work in [29, 30, 31, 32]. Other workers have shown that naked singularities and black holes can be differentiated by the properties of the accretion disks that accumulate around them. Consequently, the study of naked singularities and space-time with such objects are of considerable current interest. In [33], the authors have enumerated three possible end states of gravitational collapse. There are examples of gravitational collapse model which formed a naked curvature singularity known. The earliest model is the Lemaitre–Tolman–Bondi (LTB) [34, 35, 36] solutions, a spherically symmetric in-homogeneous collapse of dust fluid that admits both naked and covered singularity. Papapetrou [37] pointed out the formation of naked singularities in Vaidya [38] radiating solution, a null dust fluid space-time generated from Schwarzschild vacuum solution. The examples of spherically symmetric gravitational collapse space-times with naked singularities would be [39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49] and many more. To counter the occurrence of naked singularity in a solution of the Einstein’s field equations, Penrose proposed a Cosmic Censorship Conjecture (CCC) [50, 51, 52]. However, the general and/or detail proofs of this Conjecture has not yet been given. On the contrary, there is no mathematical details yet known which forbid the appearance of naked singularity in a solution of the field equations.

The Gravitational collapsing of cylindrically symmetric models that is formed a naked singularity has been discussed in [53, 54]. Apostolatos and Thorne [55] investigated the collapse of counter-rotating dust shell cylinder. Echeverria [56] has studied the evolution of cylindrical dust shell analytically at late times and numerically for all times. Guttia et al. [57] have studied the collapse of non-rotating, infinite dust cylinders. Nakao and Morisawa [58] have studied the high-speed collapse of cylindrically symmetric thick shell composed of dust, and perfect fluid with non-vanishing pressure [59]. Recent work describes the cylindrically symmetric collapse of counter-rotating dust shell [60, 61, 62], self-similar scalar field [63], axially symmetric vacuum space-time [64], cylindrically symmetric anisotropic fluid space-time [65], axially symmetric null dust space-time [66], cylindrically symmetric type N anisotropic and null fluid space-time [67], cylindrically symmetric vacuum space-time [68], and cylindrically symmetric and static anisotropic fluid space-time [69]. Some other examples of non-spherical gravitational collapse with naked singularity would be discussed [70, 71, 72, 73, 74, 75, 76, 77, 78, 79].

*p*is the isotropic pressure.

In the present work, taking into account the energy conditions we attempt to generalize a four-dimensional cylindrically symmetric black hole into non-static case which may represent model of black holes and/or naked singularities under various parameters condition.

## 2 A four-dimensional non-static space-times

*t*is time-like and

*r*is space-like defined by

*f*(

*r*) and the quantity

*k*(

*r*) under this case are

*k*(

*r*) for \(f(r_0)=0\) is \(k(r=r_0)=\frac{3}{2}\,b^2\,r_0>0\) which is space-like where, \(r=r_0=(\frac{2\,\beta }{b^2})^{\frac{1}{3}}\). Here

*v*is time-like as long as \(r > r_0\) and

*k*(

*r*) is space-like for all \(r>0\). One can easily check that the scalar curvature invariant constructed from the Riemann tensor is singular at \(r=0\) which is covered by an event horizon \(r=r_0\) and vanish rapidly at \(r\rightarrow \infty \).

*f*(

*r*) and the surface gravity

*k*(

*r*) are given by

*t*is time-like and

*r*is space-like for \(r>r_0\) and vice-versa in the region \(r <r_0\).

*S*, of constant

*v*and

*r*, from the metric (21) we have two null vector fields

Once we have the expansions, following Penrose definition [82] one can define that the two-surface, *S*, of constant *v* and *r* is *trapped*, *marginally trapped*, or *un-trapped*, according to whether \(\Theta _{l}\,\Theta _{k}>0\), \(\Theta _{l}\,\Theta _{k}=0\), or \(\Theta _{l}\,\Theta _{k}<0\). An *apparent horizon*, or *trapping horizon* in Hayward’s terminology [54, 83] is defined as a hypersurface foliated by *marginally trapped* surfaces. However, this need not be the case in regions of strong curvature.

*A*is

*A*is a null hypersurface.

*S*with \(\Theta _{k}<0\) [80, 81, 82, 83, 84]

*future outer trapping horizon*(FOTH) is the closure of a surface foliated by

*marginal surfaces*, such that \(k^{\mu }\,\nabla _{\mu }\,\Theta _{l}|_{r=r_0} < 0\), \(\Theta _{l}=0\) and \(\Theta _{k}<0\). In our case, we have

*z*-

*planes*in this case is given by

*k*, is connected with the temperature \(T_h\) known as Hawking temperature, given by,

*f*(

*r*) is given by

*M*,

*g*) is cylindrically symmetric if and only if it admits a \(G_2\) on \(S_2\) group of isometries containing an axial symmetry (see also, Refs. [91, 92, 93]). For the space-time (44), the metric tensor is independent of the coordinates \((\phi , z)\) so that the Killing vectors are \((\xi _{(\phi )}, \xi _{(z)})=(\partial _{\phi }, \partial _{z})\). The existence of axial symmetry about

*z*-axis (since \(|g_{\phi \phi }|\rightarrow 0\), as \(r\rightarrow 0\)) implies that the orbits of one of these Killing vectors, namely, \(\partial _{\phi }\) are closed but those of the other (\(\partial _{z}\)) is open. In addition, these spacelike Killing vectors generates a two-dimensional isometry group \(G_2\) and they commute. Here the assumption on the existence of two-surfaces (\(S_2\)) orthogonal to the group orbits is not necessary for the definition of cylindrical symmetry. So in the astrophysical context, the orthogonal transitivity in the definition of cylindrical symmetry is removed and we assume the space-time is cylindrically symmetric.

**Sub-case**(i): If \(\beta =0=\delta \).

*t*, \(\phi \) and

*z*. These are

*z*-

*plane*, we have from (61)

*t*and

*r*are bounded for finite value of the affine parameter \(\lambda \), and thus the presented space-time is radially null geodesically complete but incomplete for non-radial null geodesics.

*strong curvature condition*developed by Tipler [24] (see also, [95]) nor the

*limiting focusing condition*developed by Krolak [25]. These two criteria are as follows:

**Sub-case** (ii): If \(\beta =0=\alpha \).

**Sub-case**(iii): For \(\delta =0=\alpha \).

*C*(which is identical to the Riemann tensor in empty space) via

*strong curvature condition*nor the

*limiting focusing condition*. Recently, the author with other [68] constructed a cylindrical symmetric type D vacuum solution of the field equations which is free-from causality possesses a naked singularity.

Sub-case (i): For \(b^2<0\), that is, \(\Lambda >0\), *t* is space-like and *r* is a time-like coordinate. The matter-energy sources the stiff perfect fluid satisfies the energy condition. The physical parameter \(\rho (=p)\) is singular at \(r=0\) and tends to zero at spatial infinity along the radial distances i.e., \(r\rightarrow \infty \). Thus the non-static type D solution corresponds to a stiff perfect fluid in the backgrounds of de-Sitter (dS) spaces with naked singularities.

Sub-case (ii): For \(b^2>0\), that is, \(\Lambda <0\), *t* is time-like and *r* is a space-like coordinate. In that case, the stiff perfect fluid violates the weak energy condition. Therefore the non-static type D solution corresponds to a stiff fluid in the background of anti-de Sitter (AdS) spaces with naked singularities violating the weak energy condition.

*f*(

*r*) and the surface gravity

*k*(

*r*) is given by

## 3 Conclusions

The General Theory Relativity (GTR) is the successful theory not only because of the predictions of black holes but also its capability of predicting its shortcomings. One of the apparent shortcomings of General Relativity is that it predicts the existence of singularities, space-time geometrical points where curvature becomes infinite and laws of physics are no longer working. Interestingly the Penrose–Hawking singularity theorems do not say much about geometrical locations of singularities [100]. Since the introduction of cosmic censorship conjecture by Penrose, many endeavours attempted to argue in favour (please see, [101]) or against the weak and strong versions of cosmic censorship conjecture. None came with conclusive definitive proof whether naked singularities could or could not physically exist. On the contrary, there are a number of spherical and non-spherical gravitational collapse solution with naked singularities exist (For a review, see, [23]). For regular or non-singular black hole solution, Bardeen proposed the first static and spherically symmetric regular black hole solution [102]. After that several authors investigated this type black hole solution. For a comprehensive review of this type black hole solution, see [103]. In the present work, we are mainly interested on singular black hole solution where the central singularity is covered by an event horizon.

A four dimensional non-static or static space-time with the matter-energy sources the stiff fluid, anisotropic fluid and an electromagnetic field, is analyzed here. The space-time represents naked singularity models and/or singular black hole solutions depending on the various parameters conditions on the metric functions in the background of de-Sitter (dS) or anti de-Sitter (AdS) spaces. In *Case 1*, we have presented a generalization of cylindrical symmetric and static Lemos black hole solution into non-static one with a perfect fluid satisfying the weak energy condition (WEC) within the black hole region. In *Case 2*, a cylindrical symmetric and static black hole solution with a negative cosmological constant and anisotropic fluid as the stress–energy tensor, is discussed in details. In *Case 3*, type D cylindrical symmetric and non-static solutions of non-null electromagnetic field and anisotropic fluid as the stress–energy tensor with a naked curvature singularity, is presented. There we have discussed three sub-cases by (i)–(iii) of the space-time. In sub-case (i), we have constructed a type D cylindrical symmetric and static space-time of non-null electromagnetic field only which possesses a naked curvature singularity. We have shown that the space-time is radially null geodesically complete. Also we have shown there that the singularity which is formed due to scalar curvature (or the Kretschmann scalar) is naked and satisfies neither the strong curvature condition nor the limiting focusing condition. In sub-case (ii), a cylindrical symmetric and conformally flat type *O* solution of anisotropic fluid only with a naked curvature singularity, is presented. It is worth mentioning the conformally flat condition for static and cylindrically symmetric case with anisotropic fluids was obtained by Herrera et al. [96]. Furthermore, all static, cylindrical symmetric solutions (conformally flat or not) for anisotropic fluids have been found in [97]. Thus cylindrical symmetric and conformally flat static space-time discussed in sub-case (ii) is a special case of these known solutions. In sub-case (iii), we constructed a cylindrical symmetric and static type D vacuum space-time with a naked curvature singularity. In *Case 4*, type D cylindrical symmetric and non-static solution of stiff perfect fluid in the backgrounds of de-Sitter (dS) and anti-de Sitter (AdS) spaces with a naked curvature singularity, is presented. In *case 5*, cylindrical symmetric and static solution of anisotropic fluid coupled with a non-null electromagnetic field in the backgrounds of de-Sitter (dS) and anti-de Sitter (AdS) spaces with a naked singularity, is presented. We have plotted graphs (Figs. 1 and 2) showing variation of the function *f*(*r*) against *r* in the backgrounds of de-Sitter and anti-de Sitter spaces by choosing suitable parameters \(\alpha \) and \(\delta \).

## Notes

### Acknowledgements

I would like to thank the anonymous kind referee(s) for his/her valuable comments and suggestions. I am also very thankful to Prof. Luis Herrera, Universidad de Salamanca, Spain, for fruitful discussion and valuable suggestions.

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