Type N Einstein space Time Machine spacetime

Annals of Physics 382 (2017) 127–135

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Type N Einstein space Time Machine spacetime Faizuddin Ahmed Hindustani Kendriya Vidyalaya, Dinesh Ojah Road, Guwahati-781005, India

highlights • A vacuum solution of the Einstein field equations with a Cosmological constant is presented. • The spacetime admits closed timelike curves which appear after a certain instant of time in a causally wellbehaved manner.

• The spacetime is a 4D generalization of flat Misner space in curved spacetime and the spacetime is axially symmetric.

• The spacetime admits a non-expanding, non-twisting, and shear-free geodesic null congruence and is of type N in the Petrov classification scheme.

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Article history: Received 24 September 2016 Accepted 9 April 2017 Available online 15 May 2017 Keywords: Vacuum spacetime Closed timelike curves Misner space Cosmological constant

a b s t r a c t We present an Einstein space axially symmetry spacetime admitting closed timelike curves (CTCs) which appear after a certain instant of time, i.e., a time machine spacetime. The spacetime is a four-dimensional generalization of flat Misner space in curved spacetime, free-from curvature divergences and belongs to type N in the Petrov classification. The spacetime admits a non-expanding, non-twisting, and shear-free null geodesic congruence. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The Einstein field equations of the Theory of General Relativity are a set of non-linear partial differential equations. It is very hard to find the exact solutions of the field equations without simplifying assumptions. Some known solutions of the field equations admit closed causal curves (CCCs) in the form of closed timelike curves (CTCs), closed timelike geodesics (CTGs) and closed null geodesics (CNGs). The presence of such curves in a spacetime violates the causality condition in General Relativity. The first solution of the field equations with causality violating curves, namely, E-mail address: [email protected]. http://dx.doi.org/10.1016/j.aop.2017.04.012 0003-4916/© 2017 Elsevier Inc. All rights reserved.

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CTCs was the Gödel rotating-dust Cosmological model [1]. Another spacetime, that of van Stockum [2], which pre-dates the one given by Gödel, was shown later to have CTCs [3]. Other examples including wormhole models by Morris and collaborators [4,5], Gott’s solution [6] of two infinitely-long cosmic string, Bonnor’s rotating dust spacetime [7] and a pure radiation metric [8] have CTCs. Some well known vacuum spacetimes admitting CTC are the NUT-Taub metric [9] (see also [10]), Kerr and Kerr–Newmann black holes solution [11,12] (see also [13]), spacetime found by Ori [14,15], Bonnor metric [16,17], a maximally symmetric locally AdS metric [18] and an axially symmetric vacuum metric with naked singularity [19]. Some other spacetimes with CTC would be in [20–39]. There are other causality violating spacetimes that, in addition to CTCs, admit closed timelike geodesics and closed null geodesics. There have only been a handful of solutions with closed timelike geodesics found previously. These were discussed in detail in [39–45]. Another recent example shows the existence of closed null geodesics in a Ricci flat spacetime [46]. One way of classifying such causality violating spacetimes would be to categorize as either eternal time machine spacetime in which CTCs always exist (e.g. [1,2,11] etc.) or as time machine spacetime in which CTCs appear after a certain instant of time. In the latter category, the work of Ori and his collaborator [14,15,31,32] deserves special mention. However, many of the above models suffer from some severe physical problems for time machine spacetimes. For instance, in some of them, the Weak energy condition (WEC) is violated indicating unrealistic matter-energy source (e.g. [4,5,20,33,34,47]) and/or there is a curvature singularity [3,15,35,48]. The WEC states that for any physical (timelike) observer the energy density is non-negative, which is the case for all known types of (classical) matter fields. The time machine models discussed in [31,32,37,49] violate the Strong energy condition (SEC). An eternal time machine spacetime is the one where CTCs form everywhere for some values of the radial coordinate and the spacetime does not admit a partial Cauchy surface and/or violate the Weak energy condition. The example of this category is Gödel’s Cosmological model [1] where CTCs form √ everywhere for the radial coordinate r > r0 = ln(1+ 2) and do not admit a partial Cauchy surface (an initial spacelike hypersurface). Tipler’s rotating solution [3] and Mallett’s solution [35] do not admit a partial Cauchy surface, and wormhole models [4,5] including some other models mentioned above violate the Weak energy condition. In time machine spacetimes, CTCs appear after a certain instant of time and are confined within some region called non-chronal region. There exists another region called chronal region where there are no closed causal (timelike or null) curves. The chronal region without CTCs of the spacetime is separated from a non-chronal (or bounded) region with CTCs by a Chronology horizon. Additionally, the time machine spacetimes should satisfy the basic requirements, namely, that the spacetime (i) admits an initial spacelike hypersurface, (ii) content known types of matter fields, (iii) obeys the Weak energy condition (also the other energy conditions) [13], (iv) is a four dimensional curved spacetime (not flat-space), and (v) is free-from curvature divergence. The time machine spacetime presented in [37] satisfies all the basic requirements, except the condition (ii), i.e., the model does not fit any known types of matter fields and hence is not acceptable. A spacetime satisfies the above mention condition, and in addition, CTCs appear after a certain instant of time, would be physically acceptable as time machine spacetime than an eternal one. The possibilities that a naked curvature singularity gives rise to a Cosmic Time Machine have been discussed by Clarke and Felice [50] (see also [51–53]). A Cosmic Time Machine is a spacetime which is asymptotically flat and admits closed non-spacelike curves which extend to future infinity. Recently, the author and collaborators [19,54] constructed time machine spacetimes which may represent such Cosmic Time Machines. In General Relativity, Chronological violating set in a spacetime (M , g ) is a set of points through which closed timelike curves pass. The boundary of Chronology violating set is called the Cauchy horizon. Thus Cauchy horizon is a light-like boundary of the domain of dependence for which every causal curve passing through intersects the hypersurface exactly once. It can be caused either by the closed timelike curves, i.e., the closed causal (timelike or null) curves might intersect the hypersurface more than once or by singularities. On one side of the Cauchy horizon, events in a spacetime cannot be causally connected. On the other side, events can be causally connected but the causal connection is Chronology violating (an event can come before its cause). A Chronology horizon is a special type of Cauchy horizon which separates spacetime a chronal region without CTCs to a non-chronal region with CTCs. A detailed discussion of the Cauchy horizon would be in [4,13,20,55–60].

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We discuss below the Misner space, a two dimensional metric (flat-space) with peculiar characteristics. The Misner space is interesting in the context of CTCs as it is a prime example of spacetime where CTCs evolve from causally well-behaved initial conditions. The metric for the 2D Misner space [61] is ds2Misn = −2 dT dψ − T dψ 2 ,

(1)

where ψ is a periodic coordinate and −∞ < T < ∞. The metric (1) is regular everywhere as detg = −1 including at T = 0. The curves T = T0 , where T0 is a constant, are closed since ψ is periodic. The curves T = T0 < 0 are spacelike, but become timelike for T = T0 > 0 and null curves T = T0 = 0 form a Chronology horizon. The second type of curves, namely, T = T0 > 0 is closed timelike curves. The hypersurface T = const = T0 < 0 is spacelike (since g TT < 0) and can be chosen as initial hypersurface over which the initial data may be specified. There is a Cauchy horizon at T = T0 = 0 for any such spacelike hypersurface T = const = T0 < 0. This spacetime evolves from an initial spacelike hypersurface T = const = T0 < 0 in a causally well-behaved manner and the formation of CTCs takes place from causally well-behaved initial conditions. The properties of the Misner space have been analyzed in detail in [13,55,61–63]. A few of them are (i) the Misner space metric in 2D is regular everywhere, (ii) the Misner space metric in 2D has flat-space, (iii) the Misner space metric in 2D has topologically non-trivial (R1 × S 1 ) character, (iv) the Misner space metric in 2D is multiply-connected since there is no axis of symmetry, (v) the Misner space metric in 2D is a vacuum solution of the field equations, and the solution evolves from an initial spacelike hypersurface in a causally well-behaved manner, (vii) CTCs of the Misner space metric are confined within a bounded (non-chronal) region, (viii) the Misner space metric admits a partial Cauchy surface (an initial spacelike hypersurface), and (ix) the Misner space has a single closed null geodesics called generator of the Cauchy horizon. The Misner space metric has been the subject of intense study and many authors studied the Misner space in flat-space as well as in curved spacetime. Grant space [38] in 4D is a generalization of Misner space. Hiscock and Konkowski have calculated the vacuum stress–energy of a massless scalar field in Misner space [64] (see also [65,66]). Li and Gott [67] have found a self-consistent vacuum for a massless conformally coupled scalar field in Misner space. Li [68] constructed a Misnerlike anti-de Sitter spacetime as a time machine in 4D. Levanony and Ori [63] have studied the motion of extended bodies in 2D Misner space and its flat-space 4D generalizations. Berkooz and collaborators [69] have studied the closed strings in Misner space. The author and collaborators have studied 4D generalizations of Misner space in curved spacetime [18,19,30,54], quite recently. We have already mentioned above that a spacetime admitting CTCs which are confined within a bounded (non-chronal) region and satisfying the basic requirement is termed as time machine spacetime. Eternal time machine spacetime always fails to satisfy one or more of these basic requirements, and in addition, CTCs form everywhere (or pre-existing) and hence are unphysical. Therefore, a four dimensional curved spacetime, satisfying all the energy conditions, especially a vacuum spacetime, but with causality violating properties of the Misner space, primarily that CTCs evolve smoothly from an initially causally well-behaved manner, would be physically acceptable as time machine spacetime than eternal time machine spacetime. In this letter, we attempted to write down an axially symmetric spacetime, a four dimensional extension of the Misner space in curved spacetime. Additionally, in the spacetime studied here, closed timelike curves form at a definite instant of time in a causally well-behaved initial conditions and therefore a time machine spacetime. 2. Analysis of the spacetime with CTCs Consider the following line element ds2 = a2 dr 2 − b2 f ′ dφ 2 − 2 b2 f dt dφ + 2 β h dr dφ + b2 dz 2 ,

(2)

where the different metric functions are a = a(r) = coth(α r), f = f (t) = cosh t ,

b = b(r) = sinh(α r), a(r) h = h(r) = b(r)

(3)

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and prime denotes differentiation with respect to time, f ′ = dt . Here φ coordinate is assumed periodic φ ∈ [0, 2 π ), where α > 0 and β > 0 are real number. We have used coordinates x1 = r, x2 = φ , x3 = z and x4 = t. The ranges of the other coordinates are 0 ≤ r < ∞, −∞ < z < ∞ and −∞ < t < ∞. The metric has signature (+, +, +, −) and the determinant of the corresponding metric tensor gµν is df

det g = −cosh2 (α r) sinh4 (α r) cosh2 t .

(4)

The main features of the spacetime geometry are (i) the spacetime is a Λ-vacuum solution of the field equations, i.e., the Ricci tensor is proportional to the metric tensor, (ii) the spacetime has coordinate singularity at r = 0, since the metric components grr and gr φ diverge, (iii) the metric tensor gµν is degenerated at r = 0, since its determinant vanishes, (iv) the spacetime has topologically non-trivial (R3 × S 1 ) character, (v) the spacetime is free-from curvature divergence and has constant curvature invariants, (vi) the spacetime is axially symmetric, and hence simply-connected, (vii) the spacetime near to the origin axis r → 0 is not locally flat, i.e., the symmetry axis is not regular, it is singular, (viii) the spacetime admits CTCs which appear after a certain instant of time and are confined within a non-chronal region of the spacetime, (ix) the spacetime admits an initial spacelike hypersurface, and (x) the spacetime provides an example of algebraically special type N metrics, i.e., the metric admits a non-expanding, non-twisting, and shear-free geodesic null congruence. The spacetime (2) is a vacuum solution of the field equations with the diagonal Einstein tensor 2 Gµ µ = 3α .

(5)

The non-zero components of the Ricci tensor Rµν for the metric (2) are Rr r = −3 α 2 coth2 (α r), Rr φ = −3 α 2 β coth(α r) csch(α r), Rφ φ = 3 α 2 sinh t sinh2 (α r), Rφ t = 3 α 2 cosh t sinh2 (α r), Rz z = −3 α 2 sinh2 (α r),

(6)

where csch = cosech and the Ricci scalar R = −12 α 2 . The Einstein field equations in vacuum with Cosmological constant Λ are given by Gµν + Λ gµν = 0

or Rµν −

1 2

gµν R + Λ gµν = 0,

(7)

where units are chosen such that c = 1, 8 π G = 1, and µ, ν = 1, 2, 3, 4. Taking trace of the field equations (7) we get R = 4 Λ.

(8)

Substituting this into the field equations we get Rµν = Λ gµν .

(9)

The presented spacetime (2) satisfies the above relations (8) and (9) provided Λ = −3 α < 0. For the spacetime (2), the curvature scalar invariants are 2

Rµµ = R = −12 α 2 = 4 Λ, Rµν Rµν = 36 α 4 = 4 Λ2 , 8 Rµνρσ Rµνρσ = 24 α 4 = Λ2 , (10) 3 16 3 6 Rµνρσ Rρσ τ κ Rµν Λ τ κ = −48 α = 9 which are non-vanishing constants. Thus the presented spacetime is free-from curvature divergence.

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Consider an azimuthal curve γ defined by r = r0 , z = z0 and t = t0 , where r0 > 0, z0 , t0 are constants. From the metric (2) we have ds2 = − sinh t sinh2 (α r) dφ 2 .

(11)

These curves are null for t = t0 = 0, spacelike throughout for t = t0 < 0, but become timelike for t = t0 > 0, which indicates the presence of CTCs. Hence CTCs form at a definite instant of time satisfying t = t0 > 0. We need to analyze that the CTCs evolve from an initial spacelike t = const hypersurface (and thus t is a time coordinate) [14]. This can be ascertained by calculating the norm of the vector ∇µ t [70,71] (or by determining the sign of the component g tt in the inverse metric tensor g µν [14]). From the metric (2), we find that g tt =

sech2 t sinh (α r) 6

[ 2 ] β + sinh t sinh4 (α r) .

(12)

A hypersurface t = const is spacelike (r ̸ = 0) provided g tt < 0 for t = t0 < 0. We choose here r = r0 (r0 ̸ = 0, a constant) and β sufficiently small positive number such that t = const hypersurface is spacelike (this ensures that the bracket factor in the above expression (12) is negative). The above condition is satisfied for t = t0 = −T0 < 0, where T0 > 0 and the conditions between r = r0 and β are sinh4 (α r0 ) sinh T0 > β 2 . Thus spacelike t = const < 0 hypersurface can be chosen as initial conditions over which the initial may be specified. The metric (2) reduces to 2D Misner space for constant r and z ds2 = −sinh2 (α r) sinh t dφ 2 + 2 cosh t dt dφ ,

(

)

(13)

indicating that the presented metric is a four dimensional generalization of the Misner space in curved spacetime. Note that if one takes β = 0, then the spacetime represented by (2) is a maximally symmetric solution of the field equations and locally isometric anti-de Sitter space in four dimension. Recently, the author and collaborators [18] constructed a maximally symmetric locally anti-de Sitter spacetime with CTCs. That spacetime (2) is axisymmetric is clear from the following. Consider the axial Killing vector ξ = ∂φ which has the normal form

ξ µ = (0, 1, 0, 0) .

(14)

Its co-vector form is

( ) ξµ = β coth(α r) csch(α r), − sinh t sinh2 (α r), 0, − cosh t sinh2 (α r) .

(15)

Eq. (15) satisfies the Killing equation ξµ ; ν + ξν ; µ = 0. For a cyclically symmetric metric, the norm of the axial Killing vector is spacelike, closed orbits [72–77]. A further step is the axial symmetry which means that the norm of the axial Killing vector must vanish on the origin axis [75,77–81]. In our case, the norm of the axial Killing vector ξ µ is X = ξ µ ξµ = gµν ξ µ ξ ν = − sinh t sinh2 (α r),

(16)

which is spacelike for t < 0, closed orbits (since φ coordinate is periodic). Additionally, the norm of the axial Killing vector vanishes, i.e., X → 0 as r → 0+ , where we have chosen the radial coordinate r such that the origin axis is located at r = 0. The norm of the axial Killing vector ξ µ is timelike in the region t > 0 implying the formation of CTCs which we discussed earlier. The spacetime admits another spacelike Killing vector ∂z generator of translational symmetry along the cylinder and it has non-closed orbits. However, the spacetime (2) fails to satisfy the regularity condition (elementary flatness) [81], namely, (∇ µ X ) ( ∇ µ X ) 4X

→ 1,

(17)

as r → 0+ . That means, the spacetime defined by (2) near the symmetry axis is not locally flat. The symmetry axis is not regular, it is singular [82]. According to the Stephani et al. [81], there are conical singularities (rods or struts) on the axis.

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3. Classification and kinematical properties of the spacetime Physically, the Petrov classification is a way to characterize a spacetime by the number of principal null directions (PNDs) it admits. The various algebraically special Petrov types have some interesting physical interpretations in the context of gravitational radiation. The algebraically special metrics are defined by the algebraic degeneracy in their principal null vectors, which form (by the Goldberg– Sachs theorem [83]) a null congruence which is both geodesic and shear-free. Among the studied spacetimes are those referred to as algebraically special that possess two or more coinciding principal ¯ where k, l are real null direction. In Newmann–Penrose notations (null tetrad vectors (k, l, m, m), ¯ are complex conjugate of one another) [84,85], if the tetrad vector kµ is a principal null and m, m direction, then the algebraically special metrics automatically implies, Ψ0 = 0. For the algebraically special metrics, the special cases are:

Ψ0 Ψ0 Ψ0 Ψ0

= 0 = Ψ1 , = Ψ1 = Ψ2 = Ψ1 = Ψ2 = Ψ1 = Ψ3

Ψ2 ̸ = 0 : type II , = 0, Ψ3 ̸= 0 : type III , = Ψ3 = 0, Ψ4 ̸= 0 : type N , = Ψ4 = 0, Ψ2 ̸= 0 : type D.

Among the algebraically special Petrov types, specifically type N spacetimes (with or without Cosmological constant), there is a single principal null direction of multiplicity 4. The non-vanishing components of the Weyl scalar is Ψ4 and this corresponds to transverse waves. A comprehensive discussion of the Petrov classification would be in [81,84–90]. In General Relativity, optical scalars refer to a set of three scalar functions Θ (expansion), σ (shear) and ω (twist/rotation) describing the propagation of geodesic null congruences. A congruence is called geodesic null congruence if the tangent vector field at each point of the congruence is parallely propagated along it and that the congruence is affinely parametrized. The expansion represents how fast the congruence expands, the vorticity how fast it rotates, and the shear how fast its ellipticity is changing. The vanishing of one or more optical scalars represents some algebraically special metrics. A detailed discussion of geodesic null congruences would be in [81,84,85,87,89,91,92]. ¯ For classification of the spacetime (2), we can construct the following set of null tetrads (k, l, m, m). They are kµ = (0, 1, 0, 0) ,

(18)

) ( sinh t sinh2 (α r) lµ = −β coth(α r) csch(α r), , 0, cosh t sinh2 (α r) ,

(19)

1 mµ = √ (coth(α r), 0, i sinh(α r), 0) , 2

(20)

1 ¯ µ = √ (coth(α r), 0, −i sinh(α r), 0) , m 2

(21)

2

√

where i = −1. The set of null tetrads above is such that the metric tensor for the line element (2) can be expressed as

¯ν +m ¯ µ mν . gµν = −kµ lν − lµ kν + mµ m

(22)

The null vectors (18)–(21) satisfy the following relations

¯ µ = 1, − kµ lµ = mµ m

¯ µ, kµ kµ = 0 = kµ mµ = kµ m

¯ µ = mµ mµ = m ¯µm ¯ µ. lµ lµ = 0 = lµ mµ = lµ m

(23) µ

ν

We set up an orthonormal frame e(a) = {e(1) , e(2) , e(3) , e(4) }, e(a) · e(b) ≡ e(a) e(b) gµν = ηab = diag(+1, +1, +1, −1) which consists of three spacelike unit vectors e(i) , i = 1, 2, 3 and one timelike vector e(4) [40]. Notations are such that small latin indices are raised and lowered with the Minkowski metric ηab , ηab and greek indices are raised and lowered with g µν , gµν . The dual basis is e(i) = e(i) and

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133

e(4) = −e(4) . These frame components can conveniently be expressed using the corresponding natural null tetrad vectors (18)–(21) as 1 µ e(1) = √ 2 1 µ e(2) = √ 2 −i µ e(3) = √ 2 1 µ e(4) = √ 2

¯ µ ) = (tanh(α r), 0, 0, β csch3 (α r) sech t), (mµ + m 1 (kµ − lµ ) = √ 2

(

0, 1, 0, −csch2 (α r) sech t −

tanh t

)

2

,

¯ µ ) = (0, 0, csch(α r), 0), (mµ − m 1 (kµ + lµ ) = √ 2

(

0, −1, 0, −csch2 (α r) sech t +

(24) tanh t 2

)

.

The line element (2) can be written as µ where Θ a = e(a) µ dx .

ds2 = Θ a Θ b ηab ,

(25)

For the metric (2), the trace-less Weyl tensor Cµνρσ can be expressed in terms of the Riemann tensor and the Ricci scalar by Cµνρσ = Rµνρσ −

R (

gµρ gνσ − gµσ gνρ .

)

(26) 12 Since the spacetime (2) is not conformally flat (Cµνρσ ̸ = 0), and it satisfies the relations (8) and (9), the presented spacetime is called an Einstein spacetime with constant negative scalar curvature (it is not an anti-de Sitter space). Using the set of null tetrad vectors (18)–(21) we have calculated the five Weyl scalars of which, only

Ψ4 = −

αβ

(27)

2 sinh(α r)

is non-vanishing, while Ψ0 = Ψ1 = Ψ2 = Ψ3 = 0. The Weyl tensor satisfies the Bel criteria Cµνρσ kσ = 0. Physically, the non-zero Weyl scalar Ψ4 denotes a transverse wave component propagating in the principal null direction kµ . Thus the metric (2) is of type N in the Petrov classification scheme. Using the null tetrad (18) we have calculated the optical scalars the expnasion, the twist and the shear and they are Θ =

ω2 =

1 2 1 2 1

µ

k ; µ = 0, k[µ ; ν] kµ ; ν = 0,

(28)

k(µ ; ν ) kµ ; ν − Θ 2 = 0. 2 And the null vector (18) satisfies the geodesic equation

σ σ¯ =

kµ ; ν kν = 0.

(29) µ

Physically, the null tetrad vector k can be interpreted as the tangent vector field of a null congruence which is geodesic. The vanishing of the expansion, the twist and the shear indicate that the geodesic null congruence is non-expanding, non-twisting, and shear-free. Since the Newmann–Penrose spin coefficients are such that, the complex divergence ρ = −(Θ + i ω) = 0, and κ = 0 = σ , the gravitational field contains a non-diverging and shear-free null geodesic congruence which is hypersurface orthogonal. According to the Goldberg–Sachs theorem, the presented type N vacuum spacetime with Cosmological constant is algebraically special. 4. Conclusions In this paper, we presented an axially symmetric vacuum solution of the field equations with negative Cosmological constant. The presented spacetime is a 4D generalization of the Misner space

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in curved spacetime admitting closed timelike curves (CTCs). These curves evolve from an initial spacelike hypersurface which is free-from CTCs. The metric (2) satisfies two out of three conditions of a maximally symmetric anti-de Sitter space in four dimension and is therefore so called Einstein spacetime with constant negative scalar curvature. Our primary motivation was to write down a metric (not flat-space) in curved spacetime that incorporates the Misner space and its causality violating properties and to classify them. The model (2) serves as a time machine spacetime in the sense that CTCs appear after a certain instant of time from an initially causally well-behaved initial conditions and satisfies the basic requirement of time machine spacetime. Most of the CTC spacetimes violate one or more energy conditions or need unrealistic matter source and/or have singularities and hence are unphysical. Our model discussed here is free-from all these problems and may be termed so called time machine spacetime. The presented spacetime admits a non-expanding, non-twisting, and shear-free null geodesic congruence and is of type N in the Petrov classification scheme. Acknowledgments I would like to thank all the referees for their valuable comments and suggestions to improve this article. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

K. Gödel, Rev. Modern Phys. 21 (3) (1949) 447–450. W.J. van Stockum, Proc. Roy. Soc. Edinburgh Sect. A 57 (1937) 135–154. F.J. Tipler, Phys. Rev. D 9 (8) (1974) 2203–2206. M.S. Morris, K.S. Thorne, U. Yurtsever, Phys. Rev. D 61 (13) (1988) 1446–1449. M.S. Morris, K.S. Thorne, Amer. J. Phys. 56 (1988) 395–412. J.R. Gott, Phys. Rev. Lett 66 (9) (1991) 1126–1129. P. Collas, D. Klein, Gen. Relativity Gravitation 36 (2004) 2549–2557. D. Sarma, M. Patgiri, F. Ahmed, Gen. Relativity Gravitation 46 (1) (2014) 1633–1642. C.W. Misner, A.H. Taub, Sov. Phys. JETP 28 (1969) 122–133. C.W. Misner, J. Math. Phys. 4 (1963) 924–937. R.P. Kerr, Phys. Rev. Lett. 11 (1963) 237–238. B. Carter, Phys. Rev. 174 (5) (1968) 1559. S. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time, 1973. A. Ori, Phys. Rev. Lett. 95 (7) (2005) 021101–021107. A. Ori, Phys. Rev. D 76 (2007) 044002–044015. W.B. Bonnor, Classical Quantum Gravity 19 (23) (2002) 5951–5957. W.B. Bonnor, Classical Quantum Gravity 18 (7) (2001) 1381–1388. F. Ahmed, B.B. Hazarika, D. Sarma, Eur. Phys. J. Plus 131 (7) (2016) 230–233. D. Sarma, F. Ahmed, M. Patgiri, Adv. High Energy Phys. 2016 (2016) 2546186. J.L. Friedman, M.S. Morris, I.D. Novikov, F. Echeverria, G. Klinkhammer, K.S. Thorne, U. Yurtsever, Phys. Rev. D 42 (1990) 1915. F. Echeverria, G. Klinkhammer, K.S. Thorne, Phys. Rev. D 44 (1991) 1077. F. Lobo, P. Crawford, NATO Sci. Ser. II Math. Phys. Chem 95 (2003) 289–296. W.B. Bonnor, Internat. J. Modern Phys. D 12 (2003) 1705–1708. W.B. Bonnor, B.R. Steadman, Classical Quantum Gravity 21 (11) (2004) 2723. M. Gürses, A. Karasu, Ö. Sarioğlu, Classical Quantum Gravity 22 (9) (2005) 1527–1543. V.M. Rosa, P.S. Letelier, Internat. J. Theoret. Phys. 49 (2010) 316–323. S.V. Krasnikov, Classical Quantum Gravity 15 (1998) 997–1003. J. Evslin, T. Qiu, J. High Energy Phys. 2011 (2011) 032 arXiv Print: arXiv:hepth/1106.0570v2. E.V. Palesheva, A new stationary cylindrically symmetric solution of the Einstein equations, 2001. arXiv pre-Print: arXiv: gr-qc/0107095. F. Ahmed, Commun. Theor. Phys. 67 (2) (2017) 189–191. Y. Soen, A. Ori, Phys. Rev. D 54 (8) (1996) 4858–4861. A. Ori, Y. Soen, Phys. Rev. D 49 (8) (1994) 3990–3997. M. Alcubierre, Classical Quantum Gravity 11 (1994) L73. A.E. Everett, Phys. Rev. D 53 (1996) 7365. R.L. Mallett, Found. Phys. 33 (9) (2003) 1307–1314. A. Ori, Phys. Rev. D 44 (1991) R2214(R). A. Ori, Phys. Rev. Lett. 71 (1993) 2517. J.D.E. Grant, Phys. Rev. D 47 (1993) 2388. W.B. Bonnor, B.R. Steadman, Gen. Relativity Gravitation 37 (11) (2005) 1833–1844.

F. Ahmed / Annals of Physics 382 (2017) 127–135 [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89] [90] [91] [92]

135

I.D. Soares, J. Math. Phys. 21 (3) (1980) 521. B.R. Steadman, Gen. Relativity Gravitation 35 (2003) 1721–1726. Ø. Grøn, S. Johannesen, New J. Phys. 10 (2008) 103025. Ø. Grøn, S. Johannesen, Nuovo Cimento B 125 (2010) 1215–1221. M.S. Modgil, D. Sahdev, Recurrence metrics and the physics of closed timelike curves, 2001, arXiv pre-Print: arXiv: 0107055/gr-qc. F. Ahmed, Prog. Theor. Exp. Phys. 2017 (4) (2017) 043E02. D. Sarma, M. Patgiri, F.U. Ahmed, Ann. Physics 329 (2013) 179–184. A.E. Everett, Phys. Rev. D 53 (1996) 7365. K.D. Olum, A.E. Everett, Found. Phys. Lett. 18 (4) (2005) 379. K.D. Olum, Phys. Rev. D 61 (2000) 124022. C.J.S. Clarke, F. de Felice, Gen. Relativity Gravitation 16 (1984) 139–148. F. de Felice, in: Lecture Notes in Physics, vol. 455, 1995, pp. 99–102. F. de Felice, Nuovo Cimento B 122 (2007) 481. F. de Felice, Eur. Phys. J. Web Conf. 58 (2013) 01001, 4. F. Ahmed, Adv. High Energy Phys. 2017 (2017) 7943649, 6. S.W. Hawking, Phys. Rev. D 46 (2) (1992) 603–611. K.S. Thorne, in: R.J. Gleiser, C.N. Kozameh, O.M. Moreschi (Eds.), Proceedings of the 13th International Conference on General Relativity and Gravitation, 1993. J.L. Friedmann, in: P.T. Chrusciel, H. Friedrich (Eds.), The Cauchy problem on spacetime that are not globally hyperbolic, 2004, pp. 331–346 arXiv e-Print: arXiv:gr-qc/0401004. C.J.S. Clarke, in: P.G. Bergmann, V. de Sabbata (Eds.), Topological Properties and Global Structure of Space-time, 1986. R.P. Geroch, J. Math. Phys. 8 (1967) 782; Singularities in the Spacetime of General Relativity (Ph.D. dissertation), Princeton University, Princeton, 1968. J.D. Goodwin, The Cauchy Problem in Spacetime with Closed Timelike Curves (Ph.D. dissertation), Faculty of Mathematical studies, University of Southampton, 1998. C.W. Misner, in: J. Ehlers (Ed.), Relativity Theory and Astrophysics 1. Relativity and Cosmology, 1967, pp. 160–169. K.S. Thorne, Directions in General Relativity: Papers in Honor of Charles Misner, 1993, p. 333. D. Levanony, A. Ori, Phys. Rev. D 83 (2011) 044043. W.A. Hiscock, D.A. Konkowski, Phys. Rev. D 26 (1982) 1225. T. Tanaka, W.A. Hiscock, Phys. Rev. D 49 (1994) 5240. S.V. Sushkov, Classical Quantum Gravity 14 (2) (1997) 523. L.-X. Li, J.R. Gott III, Phys. Rev. Lett. 80 (1998) 2980. L.-X. Li, Phys. Rev. D 59 (1999) 084016. B.P. M.Berkooz, M. Rozali, J. Cosmol. Astropart. Phys. 2004 (08) (2004) 04. arXiv:hep-th/0405126v2. V.E. Hubeny, M. Rangamani, S.F. Ross, Phys. Rev. D 69 (2004) 024007. J. Dietz, A Dirmeier, M. Scherfner, in: M. Ortega, et al. (Eds.), Proceedings in VI international Meeting on Lorentzian Geometry, Facultad de Ciencias, Universidad de Granada, Spain, 2011 arXiv e-Print: arXiv:1201.1929/gr-qc. B. Carter, Commun. Math. Phys. 17 (1970) 233–238. B. Carter, J. Math. Phys. 10 (1969) 70–81. J. Bi˘cák, B.G. Schmidt, J. Math. Phys. 25 (1984) 600. M. Mars, J.M.M. Senovilla, Classical Quantum Gravity 10 (8) (1993) 1633–1647. M. Heusler, Black Hole Uniqueness Theorems, 1996. M. Mars, J.M.M. Senovilla, in: F.J. Chinea, L.M. Gonzalez-Romero (Eds.), in: Lecture Notes in Physics, vol. 423, 1993, pp. 141–146. J. Carot, J.M.M. Senovilla, R. Vera, Classical Quantum Gravity 16 (9) (1999) 3025. A. Barnes, Classical Quantum Gravity 17 (13) (2000) 2605. A. Barnes, Classical Quantum Gravity 18 (24) (2001) 5511–5520. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions to Einstein’s Field Equations, 2003. M.A.H. MacCallum, N.O. Santos, Classical Quantum Gravity 15 (6) (1998) 1627. J.N. Goldberg, R.K. Sachs, Acta Phys. Polon. Suppl. 22 (1962) 13–23. E. Newmann, R. Penrose, J. Math. Phys. 3 (1962) 566. E. Newmann, R. Penrose, J. Math. Phys. 4 (1963) 998. A.Z. Petrov, Uch. Zapiski Kazan. Gos. Univ. 114 (8) (1954) 55. J.B. Griffiths, J. Podolsky, Exact Space-Time in Einstein’s General Relativity, 2009. S. Chandrasekhar, The Mathematical Theory of Black Holes, 1998. G.S. Hall, Symmetries and Curvature Structure in General Relativity, 2004. R. Penrose, Ann. Physics 10 (1960) 171–201. R.K. Sachs, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 270 (1962) 103. Eric Poisson, A Relativist’s Toolkit: The Mathematics of Black-hole Mechanics, 2004.