Eur. Phys. J. C (2019) 79:493 https://doi.org/10.1140/epjc/s10052-019-7017-8

Regular Article - Theoretical Physics

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

Faizuddin Ahmeda Ajmal College of Arts and Science, Dhubri, Assam 783324, India

Received: 3 April 2019 / Accepted: 2 June 2019 / Published online: 12 June 2019 © The Author(s) 2019

Abstract In this work, a four-dimensional cylindrical sym- necessarily a S2 topology, provided that the dominant energy metric and non-static or static space-times in the back- condition holds [1]. This result extends to outer apparent grounds of anti-de Sitter (AdS) spaces with perfect stiff fluid, horizons in black hole space-times that are not necessarily anisotropic fluid and electromagnetic field as the stressÐ stationary [2]. Such restrictive uniqueness theorems do not energy tensor, is presented. For suitable parameter condi- hold in higher dimensions, the most famous counter-example tions in the metric function, the solution represents non-static being the black ring of Emparan and Reall [3], with horizon or static non-rotating black hole solution. In addition, we topology S2 × S1. Nevertheless, Galloway and Schoen [4] show for various parameter conditions, the solution repre- have shown that in arbitrary dimension, cross sections of sents static and/or non-static models with a naked singularity the event horizon (in the stationary case) and outer appar- without an event horizon. ent horizons (in the general case) admit metrics of positive . Hawking’s theorem was later generalized by Gannon [5] who had proved, replacing stationary by some 1 Introduction weaker assumptions, that the horizon must be either spheri- cal or toroidal. The topological censorship theorem [6]also In , it is a subject of long-standing interest plays an important role in black hole physics which states to look for the exact solutions of Einstein’s field equations. that in asymptotically flat space-times, only spherical hori- Among these exact solutions, black hole solutions take an zons can give rise to well defined causal structure for a black important position because thermodynamics, gravitational hole. This is circumvented by the presence of a negative cos- theory, and quantum theory are connected in quantum black mological constant, in which case well defined black holes hole physics. In addition, black holes might play an important with locally flat or hyperbolic horizons have been shown to role in developing a satisfactory full quantum theory of grav- exist [7Ð9]. This kind of black holes with topologically non- ity which does not exist till today. Black holes are regions of trivial anti-de Sitter (AdS) asymptotics are relevant in testing space-time from which nothing, not even light, can escape. the AdS/CFT correspondence. A typical black hole is the result of the gravitational force In the framework of four-dimensions, it is well known becoming so strong that one would have to travel faster than that generic black hole solutions to EinsteinÐMaxwell equa- light to escape its pull. Such black holes generically con- tions are KerrÐNewman solutions [10,11], which are char- tain a space-time singularity at their center; thus we cannot acterized by only three parameters: the mass (M), charge fully understand a black hole without also understanding the (q), and angular momentum (J). It is often referred to as nature of singularities. Black holes, however, raise several the non-hair conjecture of black holes and the space-time additional conceptual problems and questions on their own. is asymptotically flat. When a non-zero cosmological con- When quantum effects are taken into account, black holes, stant is introduced, the space-time will become asymptoti- although they are nothing more than regions of space-time, cally de Sitter (dS) or anti-de Sitter (AdS) spaces depend- appear to become thermodynamic entities, with a tempera- ing on the sign of the cosmological constant. In a recent ture and an entropy. The fundamental features of a black hole paper [12], Lemos constructed a cylindrical symmetric rotat- is the topology. In four-dimensional asymptotically flat sta- ing and/or non-rotating and static black hole solutions (black tionary space-time, Hawking showed that a black hole has strings) in four-dimensional Einstein gravity with a negative cosmological constant. The black string solutions of Lemos a e-mails: [email protected]; [email protected] 123 493 Page 2 of 11 Eur. Phys. J. C (2019) 79 :493 is asymptotically AdS not only in the transverse directions, examples of gravitational collapse model which formed a but also in the string direction. Huang and Liang [13]fur- naked curvature singularity known. The earliest model is the ther constructed the so-called torus-like black holes with the LemaitreÐTolmanÐBondi (LTB) [34Ð36] solutions, a spher- topology R2 × S1 × S1. The topological black holes in four- ically symmetric in-homogeneous collapse of dust fluid that dimensional non-static space-time with non-zero cosmolog- admits both naked and covered singularity. Papapetrou [37] ical constant, were investigated in [14]. Wu et al. [15] con- pointed out the formation of naked singularities in Vaidya structed a topologically charged static black hole solution [38] radiating solution, a null dust fluid space-time gener- in four-dimensions. Once the energy condition is relaxed, a ated from Schwarzschild vacuum solution. The examples black hole can have quite different topology. Such examples of spherically symmetric gravitational collapse space-times can occur even in four-dimensional space-times where the with naked singularities would be [39Ð49] and many more. cosmological constant is negative [8,16Ð21]. To counter the occurrence of naked singularity in a solution The curvature singularities of matter-filled as well as vac- of the Einstein’s field equations, Penrose proposed a Cosmic uum space-times are recognized from the divergence of the Censorship Conjecture (CCC) [50Ð52]. However, the gen- energy-density and/or the scalar , such eral and/or detail proofs of this Conjecture has not yet been μνρσ as the zeroth order scalar curvature Rμνρσ R (so called given. On the contrary, there is no mathematical details yet ρσλτ μν the Kretschmann scalar), and Rμνρσ R Rλτ . In addition, known which forbid the appearance of naked singularity in by analysing the outgoing radial null geodesics of a space- a solution of the field equations. time containing space-time singularity, one can determine The Gravitational collapsing of cylindrically symmetric whether the curvature singularity is naked or covered by an models that is formed a naked singularity has been discussed event horizon (see Refs. [22,23] for detail discussion). For the in [53,54]. Apostolatos and Thorne [55] investigated the col- strength of curvature singularities, two conditions are gen- lapse of counter-rotating dust shell cylinder. Echeverria [56] erally used: first one being the strong curvature condition has studied the evolution of cylindrical dust shell analyti- (SCC) given by Tipler [24], and second one is the limiting cally at late times and numerically for all times. Guttia et focusing condition (LFC) given by Krolak [25]. Meanwhile, al. [57] have studied the collapse of non-rotating, infinite the curvature singularities are basically of three types: first dust cylinders. Nakao and Morisawa [58] have studied the one being a space-like singularity (e.g. Schwarzschild sin- high-speed collapse of cylindrically symmetric thick shell gularity), second one is the time-like singularity, and third composed of dust, and perfect fluid with non-vanishing pres- one is the null singularity. In time-like singularity, two pos- sure [59]. Recent work describes the cylindrically symmetric sibilities arise: (i) there is an event horizon around a time- collapse of counter-rotating dust shell [60Ð62], self-similar like singularity (e.g. RN black holes). Here an observer can- scalar field [63], axially symmetric vacuum space-time [64], not see a time-like singularity from an outside region of the cylindrically symmetric anisotropic fluid space-time [65], space-time, that is, the singularity is covered by an event axially symmetric null dust space-time [66], cylindrically horizon; (ii) there is no event horizon around a time-like symmetric type N anisotropic and null fluid space-time [67], singularity which we called naked singularities (NS), and cylindrically symmetric vacuum space-time [68], and cylin- it would observable for far away observers. Therefore exis- drically symmetric and static anisotropic fluid space-time tence of a naked singularity would present opportunities to [69]. Some other examples of non-spherical gravitational col- observe the physical effects near the very dense regions that lapse with naked singularity would be discussed [70Ð79]. formed in the very final stages of a gravitational collapse. The Einstein field equations with cosmological constant But in a black hole scenario, such regions are necessarily  are given by hidden within the event horizon. Attempts have been made to provide the theoretical framework to devise a technique Gμν +  gμν = κ Tμν, (1) to distinguish between black holes and naked singularities from astrophysical data mainly through gravitational lens- where Tμν is the stressÐenergy tensor. For our present work, ing (GL) method. Some significant works in this direction we have chosen the following stressÐenergy tensors: are the study of strong gravitational lensing in the JanisÐ (i) Perfect fluid: NewmanÐWinicour space-time [26,27], and its rotating gen- eralization [28], and notable work in [29Ð32]. Other workers 0 =−ρ, 1 = 2 = 3 = , T0 T1 T2 T3 p (2) have shown that naked singularities and black holes can be differentiated by the properties of the accretion disks that where ρ is the energy-density and p is the isotropic pressure. accumulate around them. Consequently, the study of naked (ii) Electromagnetic field: singularities and space-time with such objects are of consid-   erable current interest. In [33], the authors have enumerated 2 αβ 1 αβ Tμν = g Fαμ Fβν − gμν Fαβ F , (3) three possible end states of gravitational collapse. There are κ 4 123 Eur. Phys. J. C (2019) 79 :493 Page 3 of 11 493 where Fμν = ∂μ Aν − ∂ν Aμ is the electromagnetic field The time-like four-velocity vector when t is time-like and tensor satisfying the following conditions r is space-like defined by μ − / μ μ μν = ( ) 1 2 δ , =− , u f r t u uμ 1 (11) F ;ν = 0, F[μν;α] = 0, (4) such that the four-acceleration vector for the source-free regions, where Aμ is the four-vector μ μ ; ν μ 1  potential. a = u uν = k(r)δ , k(r) = f (r). (12) r 2 (ii) Anisotropic fluid: Before proceeding, we do the following transformation 0 =−ρ, 1 = , 2 = = 3,  T0 T1 pr T2 pt T3 (5) dr t → v − ( ) (13) where ρ is the energy-density, and pr , pt are the radial and f r tangential pressures, respectively. into the metric (6), one will get In the present work, taking into account the energy condi- ds2 =−f (r) dv2 + 2 dv dr + r 2 (H(v, r) dφ2 tions we attempt to generalize a four-dimensional cylindri- 1 + dz2), (14) cally symmetric black hole into non-static case which may H(v, r) represent model of black holes and/or naked singularities under various parameters condition. where  − γv 2 γ dr H(v, r) = e 2 e f (r) . (15) Now, we study the following cases of the above space-time 2 A four-dimensional non-static space-times in details. Case 1: β →−β, b2 > 0,α= 0,δ= 0,γ= 0. The four dimensional time-dependent cylindrical symmetry space-time is given by The function f (r) and the quantity k(r) under this case 2 μ ν are ds = gμν dx dx       β β 3 2 2 2 2 r 2 1 2 2 2 1 2 f (r) = b r − = − 1 , =−f (r) dt + dr + r Adφ + dz , 3 ( ) r r r f r A   0 (6) β k(r) = b2 r + . (16) r 2 where the ranges of the coordinates are −∞ < t < ∞, r ∈[0, +∞[, −∞ < z < ∞, φ ∈[0, 2 π) and The quantity k(r) for f (r ) = 0isk(r = r ) = 3 b2 r > 0   0 0 2 0 2 2 β 1 2 β α which is space-like where, r = r = ( ) 3 .Herev is time- f (r) = + + b2 r 2 + δ r , A = A(t) = e−2 γ t . 0 b2 r r 2 like as long as r > r0 and k(r) is space-like for all r > (7) 0. One can easily check that the scalar curvature invariant constructed from the Riemann tensor is singular at r = 0 β α 2 Here represents mass parameter, represents charge, b which is covered by an event horizon r = r and vanish  δ γ 0 is related to cosmological constant ( ), and are integer. rapidly at r →∞. The metric functions with its inverse are Consider the stressÐenergy tensor the perfect fluid (2), 1 1 1 g =−f (r) = , g = = , from the field equations (1)using(10), we get 00 g00 11 f (r) g11 γ 2 2 2 −2 γ t 1 2 2 γ t 1  =−3 b < 0,κρ= κ p =−  . (17) g22 = r e = , g33 = r e = . (8) g22 g33 2 β r3 − r 3 1 r0 The metric is Lorentzian with signature (−, +, +, +) and the determinant of the corresponding metric tensor gμν is The energy-density violate the weak energy conditions in the exterior region r > r0. Inside the black hole region 0 < =− 4. det g r (9) r < r0, the energy-density satisfy the energy conditions. If one takes γ = 0, this corresponds to the well-known Lemos The non-zero components of the Einstein tensor Gμν are cylindrical symmetric and static black hole solution [12]. α2 2 δ γ 2 Thus from the above analysis we have seen that for γ = G0 = 3 b2 − + + , 0 r 4 r f (r) 0, the solution represent a non-static stiff fluid black holes α2 δ γ 2 model in the background of anti-de Sitter (AdS) spaces with 1 2 2 G = 3 b − + − , black holes region r ≤ r , a generalization of Lemos static 1 r 4 r f (r) 0 black hole solution into non-static one. α2 δ γ 2 2 3 2 G = G = 3 b + + − . (10) : β = ,α2 →−α2, 2 > ,γ= ,δ= . 2 3 r 4 r f (r) Case 2 0 b 0 0 0 123 493 Page 4 of 11 Eur. Phys. J. C (2019) 79 :493

The function f (r) and the surface gravity k(r) are given where hμν is the spatial metric in the two-space defined by by = + + ,     hμν gμν kμ lν lμ kν (26) α2 α2 4 2 2 r ν ν μ μ f (r) = b r − = − 1 , satisfy the conditions hμν k = hμν l = 0, hμ = 2 and hν r 2 r 2 r 4 0 is the projection operator.     1 α2 α2 4 Once we have the expansions, following Penrose defini- 2 k(r) = b r + , r0 = . (18) S v r 3 b2 tion [82] one can define that the two-surface, , of constant and r is trapped, marginally trapped,orun-trapped, accord- The Kretschmann scalar given by ing to whether l k > 0, l k = 0, or l k < 0. α4 An apparent horizon,ortrapping horizon in Hayward’s ter- μνρσ 56 4 Rμνρσ R = + b minology [54,83] is defined as a hypersurface foliated by 8 24 (19) r marginally trapped surfaces. However, this need not be the is singular at r = 0 which is covered by an event horizon case in regions of strong curvature. r = r0 and becomes finite at r →∞. The surface gravity In our case, we have 2 for f (r0) = 0isk(r0) = 2 b r0 > 0 which is space-like.  > > Here t is time-like and r is space-like for r > r0 and vice- l 0forr r0 versa in the region r < r0. = 0forr = r0 Considering anisotropic fluid (5) as the stressÐenergy ten- < 0forr < r0. (27) sor, from the field equations (1)using(10)wehave Thus, in the quotient manifold with metric g =−f (r) dv2 α2 α2  =−3 b2,κρ= κ p =− ,κp = . (20) + 2 dv dr,wehave t r 4 r r 4 , ={(v, ) : > }, We have seen from above that at the point of curvature sin- Regular region R r r r0 gularity r = 0, the stressÐenergy tensor diverge and vanish Trapped region, T ={(v, r) : r < r0}, →∞ rapidly at r . Thus the solution is radially asymptoti- Apparent region, A ={(v, r) : r = r0}. (28) cally AdS. For stationary black hole solution case, we have from (14) The induced metric on A is   1 ds2 =−f (r) dv2 + 2 dv dr + r 2 (dφ2 + dz2). (21) h = 2 dv dr = 2 f (r) dv dv = f (r) dv2, (29) 2 For light-like curves, from the metric (21) the null condi- vanish at r = r , a null hypersurface condition, where we tion implies (radial condition φ = const, z = const) 0 have used Eq. (22). So A is a null hypersurface. dr The 2-surface S with  < 0[80Ð84] − f (r) dv + 2 dr = 0 ⇒ 2 = f (r). (22) k dv S is called untrapped if l > 0 Thus we see that   marginally trapped if l = 0 dr 1 α2 r 4  < = f (r) = − 1 > 0forr > r trapped if l 0(30) v 3 4 0 d 2 r r0 and we can rephrase as < 0forr < r0. (23) Sv, is untrapped for r > r Let us consider a closed two-surface, S, of constant v and r 0 = r, from the metric (21) we have two null vector fields marginally trapped for r r0 < . μ trapped for r r0 (31) k =−∂r (the inner null normal) μ 1 A future outer trapping horizon (FOTH) is the closure of a l = ∂v + f (r)∂r (the outer null normal). (24) 2 surface foliated by marginal surfaces, such that μ k ∇μ  | = < 0,  = 0 and  < 0. In our case, we We compute the expansion scalar associated with k, l for l r r0 l k have two-surface [80,81]   3 1 μν 2 μ ∇  =−α2 + < ,  = h ∇ν kμ =− k μ l 0 k r 4 4 r   r0 f (r) α2 r 4 α2 μν μ 4 2 4 l = h ∇ν lμ = = − 1 , (25) k ∇μ  | = =− =−4 b = <0. (32) r r 3 r 4 l r r0 4 0 r0 3 123 Eur. Phys. J. C (2019) 79 :493 Page 5 of 11 493

To show that the region r ≤ r0 represent a black hole the temperature Th known as Hawking temperature, given region for the metric (6) under this case with the event horizon by, = exist at r r0, we apply the theorem (theorem 1.2.5) [85]. k From (21), we get Th = . (42) 2 π 2 2 gvv =−f (r), gφφ = r , gvr = 1 = grv, gzz = r . In our case we have the black holes temperature √  (33) 2 3 2 b r0 α  4 Th(r ) = = − . (43) The inverse metric tensor for the metric (21) in this case can 0 2 π π 3 be express as Thus from above analysis we have seen that the solution μν ∂ ∂ = ∂ ∂ + ( )∂2 + −2 ∂2 + −2 ∂2. g μ ν 2 v r f r r r φ r z (34) (6) in this case represent a non-rotating cylindrical sym- vv metry and static charged singular black hole solution with Since the metric component g from (21)is anisotropic fluid as the stressÐenergy tensor violating the vv g = 0 ⇒ g(∇v,∇v) = 0, (35) weak energy condition (WEC) in the backgrounds of anti- de Sitter (AdS) spaces. which implies that the integral curves of : β = ,α= , 2 = ,γ= ,δ= . μν Case 3 0 0 b 0 0 0 ∇v = g ∂μ v∂ν = ∂r , (36) The metric under this case is given by are null, affinely parameterized geodesic, that means, the ν 2 curves are the null geodesic (since the conditions X 2 =− ( ) 2 + dr + 2 ( φ2 + 2), μ ds f r dt r d dz (44) ∇ν X = 0 hold where, we have defined X =∇v). f (r) Again from metric (21), we have where the function f (r) is given by μν   ∇r = g ∂μ r ∂ν = ∂v + f (r)∂r = ∂v (37) α2 2 β f (r) = + + δ r . (45) α2 1 r 2 r f (r) = ⇒ r = r = ( ) 4 r = r provided 0 0 b2 . Since at 0,the rr metric function f (r) = 0 which implies g = f (r = r0) = The Kretschmann scalar given by 0, the null hypersurface condition. Therefore the curve γ(λ) μνρσ 8 4 2 2 2 {v = v(λ), = ,φ = , = } Rμνρσ R = [7 α + 12 r α β + 6 r β defined by r r0 const z const are r 8 the null geodesics where, λ is an affine parameter. The null −r 3 α2 δ + r 6 δ2] (46) geodesics are the generators of the event horizon. Now we study the geodesic motion of test particles. The is singular at r = 0 and tends to zero, as r →∞. Lagrangian of the metric (6) in the constant z-planes in this The non-zero components of the Einstein’s tensor are case is given by α2 2 δ α2 2 δ G0 = G1 =− + , G2 = G3 = + . (47) r˙2 0 1 r 4 r 2 3 r 4 r − f (r) t˙2 + + r 2 φ˙2 =−, (38) f (r) Considering the stressÐenergy tensor the electromagnetic where  = 0, 1 for photon and massive particles, respec- field given by tively. There are two constants of motion which are given B2 by − T 0 =−T 1 = T 2 = T 3 = , (48) 0 1 2 3 κ r 4 E L = t˙ = , φ˙ = . (39) where we have chosen the electromagnetic field tensor F32 ( ) 2 1 f r r −F23 = B = B and others are zero. The stressÐenergy Therefore from (38)wehave tensor of anisotropic fluid is given by (5).   Therefore for zero cosmological constant, from the field 2 1 dr 1 2 equations Gμν = κ Tμν = κ(Tμν(e.m.) + Tμν(f)), we get + Vef f = E , (40) 2 dλ 2 2 δ δ B = α, −κρ= κ p = ,κp =− . (49) where the effective potential is given by r r t r  2 f (r) L From above it is clear that the physical parameters ρ, pr , pt Vef f (L, r) = +  . (41) 2 r 2 are singular at r = 0 and tends to zero at r →∞.The matter-energy tensor the anisotropic fluid satisfy the follow- This potential equals 0 at the surface of the black holes. ing energy conditions provided δ>0 According to the classical black hole (BH) thermodynam- ics [86Ð89], the surface gravity of BH, k, is connected with WEC: ρ<0, 123 493 Page 6 of 11 Eur. Phys. J. C (2019) 79 :493

WECr : ρ + pr = 0, The C-energy is defined in terms of the C-energy flux vector μ μ P which satisfies the conservation law ∇μ P = 0. The WECt : ρ + pt < 0, C-energy flux vector Pμ is defined by SEC: ρ + pi < 0. (50) μ μνρσ i P = 2 π U,ν ξ(z)ρ ξ(φ)σ , (56) And for δ<0wehave μνρσ where  is the Levi-Civita skew tensor, and the C-energy WEC: ρ>0, flux vector is 4  WEC : ρ + p = 0, μ r f (r) r r P =− (1, 0, 0, 0). (57) : ρ + > , 4 π WECt pt 0 SEC: ρ + p > 0. (51) The C-energy density measured by an observer whose world- i μ i line has a tangent vector u is √ 4  A cylindrically symmetric space-time locally is defined by μ r f (r) f (r) E(r) =−P uμ = , where the existence of two commuting, spacelike Killing vectors, 4 π whose one orbits are closed and other is open. According to μ 1 μ u = √ δ . (58) the definition in Ref. [90], a space-time (M, g) is cylindri- f (r) t cally symmetric if and only if it admits a G on S group 2 2 β = = δ of isometries containing an axial symmetry (see also, Refs. Sub-case (i): If 0 . [91Ð93]). For the space-time (44), the metric tensor is inde- The metric in this case reduces to (φ, ) pendent of the coordinates z so that the Killing vectors α2 r 2 (ξ ,ξ ) = (∂ ,∂ ) ds2 =− dt2 + dr2 + r 2 (dφ2 + dz2). (59) are (φ) (z) φ z . The existence of axial symmetry r 2 α2 about z-axis (since |gφφ|→0, as r → 0) implies that the r3 ∂φ By applying the transformations t → v − into the above orbits of one of these Killing vectors, namely, are closed 3 α2 but those of the other (∂z) is open. In addition, these spacelike metric, one will get Killing vectors generates a two-dimensional isometry group α2 2 2 2 2 2 G2 and they commute. Here the assumption on the existence ds =− dv + 2 dv dr + r (dφ + dz ). (60) r 2 of two-surfaces (S2) orthogonal to the group orbits is not nec- essary for the definition of cylindrical symmetry. So in the The above solution represents a Petrov type D static and astrophysical context, the orthogonal transitivity in the def- cylindrical symmetry model with non-null electromagnetic inition of cylindrical symmetry is removed and we assume field possesses a naked singularity. the space-time is cylindrically symmetric. The Lagrangian of metric (59)is The norm of the spacelike Killing vectors (∂φ,∂ ) are  z 1 α2 r 2 invariant, namely, the circumferential radius L = − t˙2 + r˙2 + r 2 (φ˙2 +˙z2) . (61) 2 r 2 α2 μν ζ = |ξ(φ)μ ξ(φ)ν g |, (52) There are three constant of motions corresponding to three and the specific length cyclic coordinates t, φ and z. These are

μν 2 l = |ξ( )μ ξ( )ν g |. (53) r 2 ˙ 2 z z t˙ = E, pφ = r φ, p = r z˙. (62) α2 z The gravitational energy per unit specific length in cylindrical symmetry system (also known as C-energy) is defined as Substituting (62)into(61) for null geodesics, we have [53,72] ˙ r = , 2 = α2 ( 2 + 2).   E a pφ pz (63) a2 1 1 −2 μ 1 − U = 1 − l ∇ A ∇μA , (54) r4 8 4 π 2 For the radial null geodesic φ˙ = 0 in the constant z-plane, where A = ζ l is the areal radius (area of two-dimensional we have from (61) cylindrical surface where the cylindrical space-time lies), and U is the C-energy scalar. α2 r˙ = t˙ = E ⇒ r(λ) = E λ + c , (64) For the metric (44), ζ = r and l = r so that area of r 2 1 the two-dimensional cylindrical surface A = r 2. Hence the where c1 is arbitrary constant. Therefore from (62) for time C-energy scalar U is  coordinate 1 f (r) E 1 U = 1 − . (55) t˙ = (E λ + c )2 ⇒ t(λ) = (E λ + c )3. (65) 8 π 2 α2 1 3 α2 1 123 Eur. Phys. J. C (2019) 79 :493 Page 7 of 11 493

From Eqs. (64) and (65) it is clear that the radial null The C-energy density in this sub-case is given by geodesics path t and r are bounded for finite value of the α3 affine parameter λ, and thus the presented space-time is radi- E(r) =− < π 0(72) ally null geodesically complete but incomplete for non-radial 2 null geodesics. which may positive or negative depending on the sign of α. Consider a congruence of radial null geodesics having the Sub-case (ii): If β = 0 = α. tangent vector K μ(r) = (K t , K r , 0, 0), where K t = dt and dλ The metric in this case reduces to r = dr K dλ . The geodesic expansion is defined by [94] 2 =−δ 2 + 1 2 + 2 ( φ2 + 2). ∂ √ ds rdt dr r d dz (73) μ 1 i δ r  = K ; μ = √ ( −gK ). (66) −g ∂ xi 1 And doing transformation t → v − δ lnr into the above For the metric (59) we get metric, one will get r r 2 2 2 2 2 ∂ K K √  ds =−δ rdv + 2 dv dr + r (dφ + dz ). (74)  = + √ ( −g) , (67) ∂r −g This solution represents a cylindrical symmetry and confor- where prime denotes derivative w. r. t. r. We proceed by noting mally flat static solution of only anisotropic fluid (49). It is that worth mentioning the conformally flat condition for static r ∂ r ∂ and cylindrically symmetric case with anisotropic fluids was dK = K r , dλ ∂r ∂λ obtained by Herrera et al. [96]. They had shown that any dKt ∂ K t ∂r conformally flat and cylindrically symmetric static source = . (68) cannot be matched through Darmois conditions to the Levi- dλ ∂r ∂λ Civita space-time, satisfying the regularity conditions. Fur- Therefore the final expression of the geodesic expansion from thermore, all static, cylindrical symmetric solutions (confor- (67)is mally flat or not) for anisotropic fluids have been found in 1 dKr 2 [97]. Very recently, a conformally flat cylindrical symmetry (r) = + K r K r dλ r and static anisotropic solution of the field equations with 1 2 naked singularities appeared in [69]. Therefore the study = [−r (K t )2 − r (K r )2]+ K r K r tt rr r static and cylindrical symmetry space-time (73) is a special (K t )2 2 case of these known solutions. =− r − r K r + K r K r tt rr r The C-energy density in this sub-case is given by (K t )2 α4 1 2 3 9 = − K r + K r δ 2 r 2 r 5 E(r) = . (75) K r r r 4 π 2 E r 2 = , K t = E Sub-case (iii): For δ = 0 = α. r α2 > 0, (69) The metric in this case reduces to β 2 2 2 r 2 2 2 2 which is positive, a fact that indicates the nature of singular- ds =− dt + dr + r (dφ + dz ). (76) r 2 β ity which is formed due to scalar curvature is naked. One can easily check that the naked singularity (NS) which is formed → v − r2 And by doing transformation t 4 β into the metric, due to the scalar curvature satisfies neither the strong curva- one will get ture condition developed by Tipler [24](seealso,[95]) nor 2 β the limiting focusing condition developed by Krolak [25]. ds2 =− dv2 + 2 dv dr + r 2 (dφ2 + dz2). (77) r These two criteria are as follows: μ ν Themetric(84) represents a cylindrical symmetry type D 2 dx dx lim λ Rμν = 0 (> 0), (70) vacuum solution of the field equations since Rμν = 0. This λ→ λ λ 0 d d type D vacuum solution possess a naked curvature singularity dxμ at r = 0 not covered by an event horizon. where dλ is the tangent vector field to the radial null geodesics. And We determine the fourteen scalar invariants of the RiemannÐChristoffel curvature tensor for the vacuum space- dxμ dxν lim λ Rμν = 0. (71) time from the list provided in [98] which is based on the one λ→ λ λ 0 d d in [99]. Of these fourteen, only four do not vanish identi- Therefore, the analytical extension of the space-time through cally for a vacuum spacetime (Ricci flat). These are (using the singularity is possible. the notation of [98,99]) 123 493 Page 8 of 11 Eur. Phys. J. C (2019) 79 :493

hj ik J1 = Ahijk g g , Consider the stressÐenergy tensor the perfect fluid (15), hj ik from the field equations (6) and (10), we get J2 = Bhijk g g , 1 γ 2 = cd ab − 2, 2 J3 Aab Acd J1  =−3 b ,κρ= κ p =− ,<0or> 0. 2 b2 r 2 5 (84) J = A cd B ab − J J , (78) 4 ab cd 12 1 2 There are two possible sub-cases arise: where the tensors Ahijk and Bhijk are obtained from the Weyl Sub-case (i): For b2 < 0, that is, >0, t is space-like tensor C (which is identical to the Riemann tensor in empty and r is a time-like coordinate. The matter-energy sources the space) via stiff perfect fluid satisfies the energy condition. The physical parameter ρ(= p) is singular at r = 0 and tends to zero at = pr qs Ahijk Chipq Crsjk g g spatial infinity along the radial distances i.e., r →∞. Thus pr qr Bhijk = Chipq Arsjk g g . (79) the non-static type D solution corresponds to a stiff perfect fluid in the backgrounds of de-Sitter (dS) spaces with naked For the metric (76), we find that singularities. b2 > < t 48 β2 576 β4 Sub-case (ii): For 0, that is, 0, is time- J = J = , J =− and like and r is a space-like coordinate. In that case, the stiff 1 2 r 6 3 r 12 perfect fluid violates the weak energy condition. Therefore 384 β2 J =− . the non-static type D solution corresponds to a stiff fluid in 4 12 (80) r the background of anti-de Sitter (AdS) spaces with naked From above analysis with the Kretschmann scalar, we have singularities violating the weak energy condition. seen that the curvature invariants diverge at r = 0 not covered Noted that if one takes γ = 0, then the static space-time by an event horizon. represent a de-Sitter or anti-de Sitter spaces, depending on the The C-energy density in this sub-case is given by signs of the cosmological constant  whose global structure were studied extensively in [100]. 3 (2 β r) 2 E(r) =− < 0. (81) Case 5: β = 0,α= 0, b2 = 0,γ= 0,δ= 0. 4 π The function f (r) and the surface gravity k(r) is given by By doing similar analysis as done in sub-case (i) earlier, one   can show that the singularity which is formed due to scalar α2 f (r) = + b2 r 2 + δ r , curvature is naked which satisfies neither the strong curvature r 2   condition nor the limiting focusing condition. Recently, the α2 δ author with other [68] constructed a cylindrical symmetric k(r) = − + b2 r + . (85) r 3 2 type D vacuum solution of the field equations which is free- from causality possesses a naked singularity. In that case, the space-time represents a cylindrical symmet- ric and static solution with naked singularities which we dis- Case 4: β = 0,α= 0, b2 = 0,δ= 0,γ= 0. cuss graphically i.e. one can see that no horizon exists for spe- cific values of the parameters. Figure 1 indicates that naked The metric is given by singularities exist for positive values of α and δ with both de- Sitter and anti de-Sitter spaces. Figure 2 indicates that naked 1 dr2 ds2 =−b2 r 2 dt2 + + r 2 (e−2 γ t dφ2 singularities exist for negative values of α and positive values 2 2 b r of δ with both de-Sitter and anti de-Sitter spaces. + 2 γ t 2). e dz (82) One can easily check that the stressÐenergy tensor cor- responds to non-null electromagnetic field coupled with The Kretschmann scalar given by anisotropic fluid. The different physical parameters are as γ 4 γ 2 follows: μνρσ 4 12 8 R Rμνρσ = 24 b + − (83) b4 r 4 r 2 B = α>0,=−3 b2 < 0or > 0, δ δ δ is singular at r = 0 not clothed and thus a naked singularity 2 2 κρ=− ,κpr = ,κpt = . (86) is formed. At spatial infinity along the radial direction, that r r r μνρσ is, at r →∞, the curvature scalar R Rμνρσ → 24 b4 The anisotropy difference along the radial and tangential which indicates that the solution is asymptotically de-Sitter directions, respectively are or anti-de Sitter spaces. One can easily check the solution is 4 δ 3 δ κ(ρ− p ) =− ,κ(ρ− p ) =− . (87) of Petrov type D metric with naked singularities. r r t r 123 Eur. Phys. J. C (2019) 79 :493 Page 9 of 11 493

50

15 40 α = 0.01 ; δ =3.0 ; =0.01 α = 0.01 ; δ =1.0 ; =−0.01

30 10 f(r) 20 f(r) 5 10

r r 2 4 6810 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10

Fig. 1 Variation of f(r) with respect to r for positive values of α, δ. (Left panel) de-Sitter spaces (right panel) anti de-Sitter spaces

150 0.1; 10.0; 0.01

100

f(r)

50

r 2 4 6 810 2 4 6 8 10

Fig. 2 Variation of f(r) with respect to r for negative values of α and positive values of δ. (Left panel) de-Sitter spaces (right panel) anti de-Sitter spaces

Therefore the static solution corresponds to an anisotropic lapse solution with naked singularities exist (For a review, fluid coupled with non-null electromagnetic field in the back- see, [23]). For regular or non-singular black hole solution, grounds of de-Sitter and anti-de Sitter spaces with naked sin- Bardeen proposed the first static and spherically symmetric gularities. regular black hole solution [102]. After that several authors investigated this type black hole solution. For a comprehen- sive review of this type black hole solution, see [103]. In the present work, we are mainly interested on singular black hole 3 Conclusions solution where the central singularity is covered by an event horizon. The General Theory Relativity (GTR) is the successful the- A four dimensional non-static or static space-time with the ory not only because of the predictions of black holes but matter-energy sources the stiff fluid, anisotropic fluid and also its capability of predicting its shortcomings. One of the an electromagnetic field, is analyzed here. The space-time apparent shortcomings of General Relativity is that it predicts represents naked singularity models and/or singular black the existence of singularities, space-time geometrical points hole solutions depending on the various parameters condi- where curvature becomes infinite and laws of physics are no tions on the metric functions in the background of de-Sitter longer working. Interestingly the PenroseÐHawking singu- (dS) or anti de-Sitter (AdS) spaces. In Case 1,wehavepre- larity theorems do not say much about geometrical locations sented a generalization of cylindrical symmetric and static of singularities [100]. Since the introduction of cosmic cen- Lemos black hole solution into non-static one with a perfect sorship conjecture by Penrose, many endeavours attempted fluid satisfying the weak energy condition (WEC) within to argue in favour (please see, [101]) or against the weak and the black hole region. In Case 2, a cylindrical symmetric strong versions of cosmic censorship conjecture. None came and static black hole solution with a negative cosmological with conclusive definitive proof whether naked singularities constant and anisotropic fluid as the stressÐenergy tensor, is could or could not physically exist. On the contrary, there are discussed in details. In Case 3, type D cylindrical symmet- a number of spherical and non-spherical gravitational col- 123 493 Page 10 of 11 Eur. Phys. J. C (2019) 79 :493 ric and non-static solutions of non-null electromagnetic field 2. S.W. 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