# Axially Symmetric Null Dust Space-Time, Naked Singularity, and Cosmic Time Machine

Authors: Faizuddin Ahmed
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We present a gravitational collapse null dust solution of the Einstein field equations. The space-time is regular everywhere except on the symmetry axis where it possesses a naked curvature singularity and admits one parameter isometry group, a generator of axial symmetry along the cylinder which has closed orbits. The space-time admits closed timelike curves (CTCs) which develop at some particular moment in a causally well-behaved manner and may represent a Cosmic Time Machine. The radial geodesics near the singularity and the gravitational lensing (GL) will be discussed. The physical interpretation of this solution, based on the study of the equation of the geodesic deviation, will be presented. It was demonstrated that this solution depends on the local gravitational field consisting of two components with amplitudesΨ2 and Ψ4.


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## Body

## 1. Introduction

For algebraically special metrics, the Petrov classification is a way to characterize the space-time by the number of times a principal null direction (PND) admits. The various algebraically special Petrov types have some interesting physical interpretations in the context of gravitational radiation. In the Newmann-Penrose notations [1, 2] (null tetrad k,l,m,m¯, where k,l are real and m,m¯ are complex conjugate of each other), if the tetrad vector kμ is a principal null direction, then the algebraically special metrics automatically give Ψ0=0. For the algebraically special metrics, the special cases are as follows:Type  II:(1)Ψ0=0=Ψ1,Ψ2≠0.Type  III:(2)Ψ0=Ψ1=Ψ2=0,Ψ3≠0.Type  N:(3)Ψ0=Ψ1=Ψ2=Ψ3=0,Ψ4≠0.Type  D:(4)Ψ0=Ψ1=Ψ3=Ψ4=0,Ψ2≠0.Penrose proposed a Cosmic Censorship Conjecture (CCC) [3–5] to forbid the occurrence of naked singularities in a solution of the field equations. According to the Weak Cosmic Censorship, singularities have no effect on distant observers; that is, they cannot communicate with far-away observers. Till date, there are no mathematical theorems or proofs that support (or counter) this conjecture. On the contrary, there is no mathematical reason that a naked singularity cannot exist in a solution of the field equations. Therefore, the formulation and proof (or disproof) of this conjecture remain to be one of the unsolved problems in General Relativity. 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). Some significant works in this direction are the study of strong gravitational lensing in the Janis-Newman-Winicour space-time [6, 7] and its rotating generalization [8] and notably the work in [9–12]. Other workers have shown that naked singularities and black holes are differentiated by the properties of the accretion disks that accumulate around them. Consequently, the study of naked singularities and space-time with such objects is of considerable current interest. In [13], the authors have enumerated three possible end states of gravitational collapse. There are a number of examples of gravitational collapse in spherically symmetric that formed naked singularities known. An earliest model that admits both naked and covered singularities is the Lemaitre-Tolman-Bondi (LTB) [14–16] solutions, a spherically symmetric inhomogeneous collapse of dust fluid. Papapetrou [17] pointed out the formation of naked singularities in Vaidya [18] radiating solution, a null dust fluid space-time generated from Schwarzschild vacuum solution. A small sample of spherically symmetric gravitational collapse solutions with a naked singularities are in [19–30]. Thus, the theoretical existence of naked singularities would mean that the gravitational collapse may be observable from the rest of the space-time.In the present article, an axially symmetric solution of the field equations with a naked curvature singularity will be presented. The gravitational collapsing of cylindrically symmetric models that formed a naked singularity has been discussed in [31, 32]. Apostolatos and Thorne [33] investigated the collapse of counter-rotating dust shell cylinder and showed that rotation, even if it is infinitesimally small, can halt the gravitational collapse of the cylinder. Echeverria [34] studied the evolution of a cylindrical dust shell analytically at late times and numerically for all times. Guttia et al. [35] studied the collapse of nonrotating, infinite dust cylinders. Nakao and Morisawa [36] studied the high-speed collapse of cylindrically symmetric thick shell composed of dust and perfect fluid with nonvanishing pressure [37]. Recent work describes the cylindrically symmetric collapse of counter-rotating dust shells [38–40], self-similar scalar field [41], axially symmetric vacuum space-time [42], and a cylindrically symmetric anisotropic fluid space-time [43]. Some other examples of nonspherical gravitational collapse would be discussed in [44–53].The Einstein field equations (taking cosmological constantΛ=0) are given by(5)Rμν-12gμνR=Tμν,μ,ν∈0,1,2,3,where Rμν is the Ricci tensor, R is the Ricci scalar, and Tμν is the energy-momentum tensor.Pure radiation or null dust fields are the fields of massless radiation which is considered as the incoherent superposition of waves with random phases and different polarization but with the same propagation direction. The radiation can arise from fields of different types, from electromagnetic null fields, massless scalar fields, and neutrino fields, or from the high frequency limit of gravitational waves. The energy-momentum tensor of pure radiation field [54] is(6)Tμν=ρkμkν=2Φ22kμkν,kμkμ=0,where ρ is the energy density of null dust (pure radiation field) and kμ is the null vector.


## 2. Analysis of the Null Dust Space-Time

Consider the following axially symmetric metric int,r,ϕ,z coordinates:(7)ds2=grrdr2+gϕϕdϕ2+2gtϕdtdϕ+2gzϕdzdϕ+gzzdz2,where the different metric functions are(8)grr=sech4⁡rtanh⁡r,gϕϕ=-tanh2⁡rsinh⁡t,gϕt=gtϕ=gϕϕ′,gϕz=gzϕ=ztanh2⁡r,gzz=coth⁡r,where prime denotes derivate with respect to time, gϕϕ′=dgϕϕ/dt. Here ϕ coordinate is assumed periodic ϕ∈0,2π. We have labelled the coordinates x0=t, x1=r, x2=ϕ, and x3=z. The ranges of the other coordinates are -∞<t<∞, 0≤r<∞, and -∞<z<∞. The metric is Lorentzian with signature +,+,+,- and the determinant of the corresponding metric tensor gμν is(9)det⁡g=-tanh4⁡rsech4⁡rcosh2⁡t.For space-time (7), the Ricci scalar R=0 and the nonzero component of the Ricci tensor Rμν is(10)Rϕϕ=12tanh3⁡r.For space-time (7), the null vector is defined by kμ=0,0,1,0. Therefore, the nonzero component of the energy-momentum tensor (6) is(11)Tϕϕ=ρ.From the field equation (5) using (10) and (11), we get(12)ρr=12tanh3⁡r.The energy density of null dust satisfies the null energy condition (NEC).The curvature scalar invariants for metric (7) are(13)RμνρσRμνρσ=12coth6⁡r,Rμνρσ;τRμνρσ;τ=180coth9⁡r.From the above equation, it is clear that the scalar curvature invariants and first-order differential invariants diverge (blow up) on the symmetry axis r=0, indicating the existence of a naked curvature singularity. Since the naked curvature singularity occurs without an event horizon, the Cosmic Censorship Conjecture has no physical interest.


### 2.1. Geodesics in the Neighborhood of the Singularity

To discuss the geodesics motion of free test particles which necessarily hit the singularityr=0, one needs to derive expression for their paths. Here we focus on radial geodesics on the symmetry axis r=0 [55].The Lagrangian for the metric (7) is given by(14)£=12gμνx˙μx˙ν=12grrr˙2+gzzz˙2+gϕϕϕ˙2+2gtϕt˙ϕ˙+2gzϕz˙ϕ˙,where dot stands for derivative with respect to an affine parameter. From (7) and (14), it is clear that ϕ is a cyclic coordinate. There exists constant of motion corresponding to this cyclic coordinate; that is, the azimuthal angular momentum pϕ is a constant given by(15)pϕ=const=gzϕz˙+gϕϕϕ˙+gtϕt˙.For metric (7), geodesics equations for t and r coordinates are(16)t¨=sech⁡t-12z2tanh3⁡rϕ˙2+z˙3zcsch⁡rsech⁡rr˙+z˙-2csch⁡rsech⁡rr˙+ϕ˙t˙-12tanh⁡tϕ˙2+2t˙2,r¨=12-sech⁡rcsch⁡rr˙2+4tanh⁡rr˙2-cosh2⁡r2sinh⁡tϕ˙2-4zϕ˙z˙+coth3⁡rz˙2+4cosh⁡tϕ˙t˙.For radial geodesics we havez˙=0=ϕ˙. From (16), we get(17)t¨=-2csch⁡rsech⁡rr˙t˙-tanh⁡tt˙2,r¨=12-csch⁡rsech⁡r+4tanh⁡rr˙2.The solution of (17) is as follows:(18)t˙s=c1sech⁡tcoth2⁡r,c1>0,r˙s=c2cosh2⁡rcoth1/2⁡r,c2>0.Again solving (18) yields(19)ts=sinh-1⁡c4-3c1c2234/3c3+c2s-1/3,rs=tanh-1⁡322/3c3+c2s2/3,where ci, i=1,…,4, are arbitrary constants. Thus from (19), it is clear that the geodesic path t is complete for finite value of the affine parameter s except at s=0 (taking c3=0), where it is unbounded. Therefore, the presented space-time is radially geodesically incomplete.


### 2.2. Strength of Naked Singularities

The strength of singularity, which is the measure of its destructive capacity, is the most important feature. To determine the strength of naked singularities as strong and weak types, we consider the criteria developed by Tipler [56] and Krolak [57], which provide insights into the magnitude of tidal forces experienced by an in-falling detector (or an observer) towards the singularity [58]. It is widely believed that the analytical extension of the space-time through a singularity is not possible, if it satisfies the strong curvature condition.A naked singularity (NS) is said to be strong if the collapsing objects do get crushed to a zero volume at the singularity and a weak one if they do not [21, 59]. Following Clarke and Krolak [58], a sufficient condition for a singularity to be strong in the sense of Tipler [56] is that at least along one radial geodesic we must have(20)lims→0⁡s2Rμνdxμdsdxνds≠0>0,where dxμ/ds is the tangent vector to the radial geodesics and Rμν is the Ricci tensor. The weaker condition, which we called thelimiting focusing condition [57], is defined by(21)lims→0⁡sRμνdxμdsdxνds≠0.Hence from (20), we have the strong curvature condition for space-time (7)(22)lims→0⁡s2Rrrdrds2+Rttdtds2=lims→0⁡s20×drds2+0×dtds2=0.Similarly, one can calculate thelimiting focusing condition, and it also becomes zero. Thus the naked singularity (NS) which is formed due to the curvature singularity satisfies neither the strong curvature condition nor thelimiting focusing condition. Therefore, the analytical extension of the space-time through the singularity is possible.


### 2.3. Equation of Orbit and Gravitational Lensing

We calculate first the equation of orbit forrϕ. Here we restrict our discussion to the orbital motion of the free test particle which moves in the z=const-plane. To get an equation for rϕ we start with(23)r′ϕ=drdϕ=r˙ϕ˙.To work out the lens equation we have to calculate the null geodesics in z=const-planes. From the null geodesics ds2=0 using space-time (7) we have(24)grrr˙2+gϕϕϕ˙2+2gtϕt˙ϕ˙=0,where we have chosen z=z0 (where z0 is a constant equal to zero).We define angular velocityΩ of a zero angular momentum particle as measured by an observer (ZAMO) in z=const-planes. The angular velocity Ω (the angular velocity of the frame dragging) is defined by [60, 61](25)Ω=ϕ˙t˙=-gtϕgϕϕ=-gϕϕ′gϕϕ=-coth⁡t,t<0.Therefore, from (24) using (25), one can obtain the following equation for the photon orbit:(26)grrr˙2-gϕϕϕ˙2=0⟹r˙ϕ˙2=gϕϕgrr⟹r˙ϕ˙=±gϕϕgrr⟹drdϕ=±gϕϕgrr=±Hr,t.Substituting gϕϕ and grr and integrating (26) immediately give the azimuthal angle ϕ as a function of r(27)ϕ=±2-csch⁡ttanh1/2⁡r=±2csch1/2⁡Ttanh1/2⁡r,where t=-T<0, T>0. Here we have set ϕr=∞=0. Therefore, the Einstein deflection angle α^ is(28)α^r0=2∫r=r0∞drHr,t-π=4csch1/2⁡T1-tanh1/2⁡r0-π.


### 2.4. Closed Timelike Curves of the Space-Time: A Cosmic Time Machine

The presented space-time admits closed timelike curves which appear after a certain instant of time. There are many solutions of the field equations admitting closed timelike curves known (see references in [62]).Consider an azimuthal curveγ defined by r=r0, z=z0, and t=t0, where r0>0, z0, and t0 are constants. From the line element (7) we have(29)ds2=-sinh⁡ttanh2⁡rdϕ2.These curves are null at t=t0=0 and spacelike throughout t=t0<0 but become timelike for t=t0>0, which indicates the presence of closed timelike curves (CTCs). Hence CTCs form at a definite instant of time satisfying t=t0>0. The above analysis is valid provided that the CTCs evolve from an initial spacelike t=const hypersurface (thus t is a time coordinate) [63]. This can be determined by calculating the norm of the vector ∇μt (or by determining the sign of gtt in the metric tensor gμν [63]). A hypersurface t=const is spacelike provided gtt<0 for t<0, timelike provided gtt>0 for t>0, and null hypersurface provided gtt=0 for t=0. In our case, from metric (7) we found that(30)gtt=∇μt∇μt=sech2⁡tcoth2⁡rz2tanh3⁡r+sinh⁡t.Here we have chosen constant z-planes defined by z=z0, where z0 is a constant equal to zero. Thus, the hypersurface t=const=t0<0 is spacelike (r≠0) and can be chosen as initial conditions over which the initial data may be specified. There is a Cauchy horizon at t=t0=0 for any such initial spacelike hypersurface t=t0<0. The null curve t=t0=0 serves as the Chronology horizon which divided the space-time into a nonchronal regionwith CTCs and a chronal regionwithout CTCs. The Chronology horizon is a special type of Cauchy horizon which has been discussed in detail in [64, 65]. Hence the space-time evolves from an initial spacelike hypersurface in a causally well-behave mannerup to a moment, that is, a null hypersurface t=t0=0, and the formation of CTCs takes place from a causally well-behaved initial condition.The possibility that a naked curvature singularity gives rise to a Cosmic Time Machine has been discussed by Clarke and de Felice [66] (see also [67–69]). A Cosmic Time Machine is a space-time which is asymptotically flat and admits closed nonspacelike curves which extend to future infinity. The author and collaborators constructed such Cosmic Time Machines [42, 43], quite recently. The presented time-dependent space-time may represent such a Cosmic Time Machine.


## 2.1. Geodesics in the Neighborhood of the Singularity

To discuss the geodesics motion of free test particles which necessarily hit the singularityr=0, one needs to derive expression for their paths. Here we focus on radial geodesics on the symmetry axis r=0 [55].The Lagrangian for the metric (7) is given by(14)£=12gμνx˙μx˙ν=12grrr˙2+gzzz˙2+gϕϕϕ˙2+2gtϕt˙ϕ˙+2gzϕz˙ϕ˙,where dot stands for derivative with respect to an affine parameter. From (7) and (14), it is clear that ϕ is a cyclic coordinate. There exists constant of motion corresponding to this cyclic coordinate; that is, the azimuthal angular momentum pϕ is a constant given by(15)pϕ=const=gzϕz˙+gϕϕϕ˙+gtϕt˙.For metric (7), geodesics equations for t and r coordinates are(16)t¨=sech⁡t-12z2tanh3⁡rϕ˙2+z˙3zcsch⁡rsech⁡rr˙+z˙-2csch⁡rsech⁡rr˙+ϕ˙t˙-12tanh⁡tϕ˙2+2t˙2,r¨=12-sech⁡rcsch⁡rr˙2+4tanh⁡rr˙2-cosh2⁡r2sinh⁡tϕ˙2-4zϕ˙z˙+coth3⁡rz˙2+4cosh⁡tϕ˙t˙.For radial geodesics we havez˙=0=ϕ˙. From (16), we get(17)t¨=-2csch⁡rsech⁡rr˙t˙-tanh⁡tt˙2,r¨=12-csch⁡rsech⁡r+4tanh⁡rr˙2.The solution of (17) is as follows:(18)t˙s=c1sech⁡tcoth2⁡r,c1>0,r˙s=c2cosh2⁡rcoth1/2⁡r,c2>0.Again solving (18) yields(19)ts=sinh-1⁡c4-3c1c2234/3c3+c2s-1/3,rs=tanh-1⁡322/3c3+c2s2/3,where ci, i=1,…,4, are arbitrary constants. Thus from (19), it is clear that the geodesic path t is complete for finite value of the affine parameter s except at s=0 (taking c3=0), where it is unbounded. Therefore, the presented space-time is radially geodesically incomplete.


## 2.2. Strength of Naked Singularities

The strength of singularity, which is the measure of its destructive capacity, is the most important feature. To determine the strength of naked singularities as strong and weak types, we consider the criteria developed by Tipler [56] and Krolak [57], which provide insights into the magnitude of tidal forces experienced by an in-falling detector (or an observer) towards the singularity [58]. It is widely believed that the analytical extension of the space-time through a singularity is not possible, if it satisfies the strong curvature condition.A naked singularity (NS) is said to be strong if the collapsing objects do get crushed to a zero volume at the singularity and a weak one if they do not [21, 59]. Following Clarke and Krolak [58], a sufficient condition for a singularity to be strong in the sense of Tipler [56] is that at least along one radial geodesic we must have(20)lims→0⁡s2Rμνdxμdsdxνds≠0>0,where dxμ/ds is the tangent vector to the radial geodesics and Rμν is the Ricci tensor. The weaker condition, which we called thelimiting focusing condition [57], is defined by(21)lims→0⁡sRμνdxμdsdxνds≠0.Hence from (20), we have the strong curvature condition for space-time (7)(22)lims→0⁡s2Rrrdrds2+Rttdtds2=lims→0⁡s20×drds2+0×dtds2=0.Similarly, one can calculate thelimiting focusing condition, and it also becomes zero. Thus the naked singularity (NS) which is formed due to the curvature singularity satisfies neither the strong curvature condition nor thelimiting focusing condition. Therefore, the analytical extension of the space-time through the singularity is possible.


## 2.3. Equation of Orbit and Gravitational Lensing

We calculate first the equation of orbit forrϕ. Here we restrict our discussion to the orbital motion of the free test particle which moves in the z=const-plane. To get an equation for rϕ we start with(23)r′ϕ=drdϕ=r˙ϕ˙.To work out the lens equation we have to calculate the null geodesics in z=const-planes. From the null geodesics ds2=0 using space-time (7) we have(24)grrr˙2+gϕϕϕ˙2+2gtϕt˙ϕ˙=0,where we have chosen z=z0 (where z0 is a constant equal to zero).We define angular velocityΩ of a zero angular momentum particle as measured by an observer (ZAMO) in z=const-planes. The angular velocity Ω (the angular velocity of the frame dragging) is defined by [60, 61](25)Ω=ϕ˙t˙=-gtϕgϕϕ=-gϕϕ′gϕϕ=-coth⁡t,t<0.Therefore, from (24) using (25), one can obtain the following equation for the photon orbit:(26)grrr˙2-gϕϕϕ˙2=0⟹r˙ϕ˙2=gϕϕgrr⟹r˙ϕ˙=±gϕϕgrr⟹drdϕ=±gϕϕgrr=±Hr,t.Substituting gϕϕ and grr and integrating (26) immediately give the azimuthal angle ϕ as a function of r(27)ϕ=±2-csch⁡ttanh1/2⁡r=±2csch1/2⁡Ttanh1/2⁡r,where t=-T<0, T>0. Here we have set ϕr=∞=0. Therefore, the Einstein deflection angle α^ is(28)α^r0=2∫r=r0∞drHr,t-π=4csch1/2⁡T1-tanh1/2⁡r0-π.


## 2.4. Closed Timelike Curves of the Space-Time: A Cosmic Time Machine

The presented space-time admits closed timelike curves which appear after a certain instant of time. There are many solutions of the field equations admitting closed timelike curves known (see references in [62]).Consider an azimuthal curveγ defined by r=r0, z=z0, and t=t0, where r0>0, z0, and t0 are constants. From the line element (7) we have(29)ds2=-sinh⁡ttanh2⁡rdϕ2.These curves are null at t=t0=0 and spacelike throughout t=t0<0 but become timelike for t=t0>0, which indicates the presence of closed timelike curves (CTCs). Hence CTCs form at a definite instant of time satisfying t=t0>0. The above analysis is valid provided that the CTCs evolve from an initial spacelike t=const hypersurface (thus t is a time coordinate) [63]. This can be determined by calculating the norm of the vector ∇μt (or by determining the sign of gtt in the metric tensor gμν [63]). A hypersurface t=const is spacelike provided gtt<0 for t<0, timelike provided gtt>0 for t>0, and null hypersurface provided gtt=0 for t=0. In our case, from metric (7) we found that(30)gtt=∇μt∇μt=sech2⁡tcoth2⁡rz2tanh3⁡r+sinh⁡t.Here we have chosen constant z-planes defined by z=z0, where z0 is a constant equal to zero. Thus, the hypersurface t=const=t0<0 is spacelike (r≠0) and can be chosen as initial conditions over which the initial data may be specified. There is a Cauchy horizon at t=t0=0 for any such initial spacelike hypersurface t=t0<0. The null curve t=t0=0 serves as the Chronology horizon which divided the space-time into a nonchronal regionwith CTCs and a chronal regionwithout CTCs. The Chronology horizon is a special type of Cauchy horizon which has been discussed in detail in [64, 65]. Hence the space-time evolves from an initial spacelike hypersurface in a causally well-behave mannerup to a moment, that is, a null hypersurface t=t0=0, and the formation of CTCs takes place from a causally well-behaved initial condition.The possibility that a naked curvature singularity gives rise to a Cosmic Time Machine has been discussed by Clarke and de Felice [66] (see also [67–69]). A Cosmic Time Machine is a space-time which is asymptotically flat and admits closed nonspacelike curves which extend to future infinity. The author and collaborators constructed such Cosmic Time Machines [42, 43], quite recently. The presented time-dependent space-time may represent such a Cosmic Time Machine.


## 3. Classification of the Space-Time and Effects on the Test Particles

For classification of metric (7), one can construct a set of null tetrad vectors k,l,m,m¯. The metric tensor can be expressed as(31)gμν=-kμlν-lμkν+mμm¯ν+m¯μmν,where the tetrad vectors are orthogonal except for kμlμ=-1 and mμm¯μ=1. The nonzero Weyl scalars using the set of null tetrad are(32)Ψ2=Cμνρσkμmνm¯ρlσ=12tanh3⁡r,Ψ4=Cμνρσlμm¯νlρm¯σ=-14tanh3⁡r-i3z4tanh⁡r,while others are Ψ0=0=Ψ1=Ψ3.We set up an orthonormal frameea=e0,e1,e2,e3, ea·eb≡eaμebνgμν=ηab=diag⁡-1,+1,+1,+1, which consists of one timelike vector e0 and three spacelike unit vectors ei, i=1,2,3. Notation is such that small Latin indices are raised and lowered with ηab, ηab, and Greek indices are raised and lowered with gμν and gμν. These frame components can conveniently be expressed using the corresponding null tetrad vectors as(33)e0=12k+l,e1=12m+m¯,e2=12k-l,e3=-i2m-m¯.In order to analyze the effects of the gravitational field of the above vacuum solution, it is natural to investigate the specific influence of various components of these fields on the relative motion of the free test particles. Such a local characterization of space-time, based on the equation of geodesic deviation frame, was described by Pirani [70, 71] and Szekeres [72, 73] (see also [74, 75])(34)D2Zμdτ2=-RνρσμUνZρUσ,where U·U=-1 is the timelike four-velocity of the free test particle (observer) and Zτ is the displacement vector, where τ is the proper time. By projecting (34) onto the orthonormal frame ea given by (33), we get(35)Z¨i=-R0j0iZj,where i,j=1,2,3, e0=U, and(36)R0j0i=eiμUνejρUσRμνρσ.The frame components of the displacement vector Zj≡eμiZμ determine directly the distance between close test particles. Their physical relative acceleration is given by(37)Z¨i=eμiD2Zμdτ2.Equation (34) also implies(38)Z¨0=-UμD2Zμdτ2=RμνρσUμUνZρUσ=0,so that Z0=a0τ+b0, a0,b0 are constants. Setting Z0=0, all test particles are synchronized by the proper time. From the standard definition of the Weyl tensor using the metric (7), we get(39)Ri0j0=Ci0j0+12δijT00-Tij.For space-time (7), the only nonvanishing scalars are given by (32) so that(40)C1010=12ReΨ4-ReΨ2,C2020=-12ReΨ4-ReΨ2,C1020=-12ImΨ4,C3030=2ReΨ2,T00=T22≡Φ22.Therefore, the equation of geodesic deviation (20) takes the form (41)Z¨1=-R0j01Zj=-C1010+12T00Z1-C1020Z2=-12ReΨ4Z1+12ImΨ4Z2+ReΨ2Z1-12Φ22Z1=-A+Z1+A×Z2+CZ1-12Φ22Z1,Z¨2=-R0j02Zj=-C2020Z2=12ReΨ4Z2+ReΨ2Z2=A+Z2+CZ2,Z¨3=-R0j03Zj=-C3030+12T00Z3=-2CZ3-12Φ22Z3,where A+=(1/2)ReΨ4, A×=(1/2)ImΨ4, C=ReΨ2, and Re and Im stand for real and complex, respectively. The above equations are well suited for physical interpretation. Clearly, the relative motions of nearby test particles depend on the following:(
1
) The local free gravitational field, consisting of two components. There are a Coulomb-component with amplitude C and transverse wave components with amplitudes A+ and A× of two polarization modes “+” and “×” representing the effect of gravitational waves on the particles in the presented type II space-time.(
2
) The terms describing the matter-content pure radiation field Φ22 which affects the motion only in the transverse plane spanned by the vectors e1 and e3.


## 4. Conclusions

We presented an axially symmetric time-dependent solution of the field equations which possesses a naked curvature singularity on the symmetry axis. The naked singularity occurs without an event horizon; therefore the Cosmic Censorship has no physical interest. The space-time satisfies null dust as that for matter-energy content with positive energy density which is finite on the symmetry axis and obeys the null energy condition. The radial geodesics of the space-time near the singularity were discussed and it was found that these are incomplete. The analytical extension of the space-time through the singularity is possible since the strength of the naked curvature singularity fails to satisfy the strong curvature condition. We obtained the equation for the photon orbit and subsequently the deflection angleα^ which is found time-dependent. The presented space-time admits closed timelike curves which develop at some particular moment from an initial spacelike hypersurface in a causally well-behaved manner. The possibility that a naked curvature singularity gives rise to a Cosmic Time Machine has been discussed by Clarke and de Felice [66]. The presented space-time may represent such a Cosmic Time Machine. Finally, the physical interpretation of the presented solution, based on the study of the equation of the geodesic deviation, was presented. It was demonstrated that the solution represents exact gravitational waves consisting of two components, transverse wave components with amplitudes A+ and A× of two polarization modes “+” and “×” and Coulomb-like components of amplitude C. There is a matter-energy null dust field interaction with the test particles.



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