Axial Symmetry Cosmological Constant Vacuum Solution of Field Equations with a Curvature Singularity, Closed Time-Like Curves, and Deviation of Geodesics

In this paper, we present a type D, nonvanishing cosmological constant, vacuum solution of Einstein’s field equations, extension of an axially symmetric, asymptotically flat vacuummetric with a curvature singularity. The space-time admits closed time-like curves (CTCs) that appear after a certain instant of time from an initial space-like hypersurface, indicating it represents a time-machine space-time. We wish to discuss the physical properties and show that this solution can be interpreted as gravitational waves of Coulomb-type propagate on anti-de Sitter space backgrounds. Our treatment focuses on the analysis of the equation of geodesic deviations.

In literature, only a handful of solutions of Einstein's field equations with the stress-energy tensor in [1,33,34] and type N Einstein space-time in [30] have a negative cosmological constant. In this work, we try to construct a type D Einstein space-time with a negative cosmological constant which was not studied earlier. The cosmological constant plays a vital role in explaining the dynamics of the universe. A tiny positive cosmological constant neatly explains the late-time accelerated expansion of the universe. Indeed, our universe is observed to be undergoing a de Sitter (dS) type expansion in the present epoch. For a negative cosmological constant, space-time is labelled as an anti-de Sitter (AdS) space. The AdS space has been a subject of intense study in recent times on account of the celebrated AdS/CFT correspondence [54], which provides a link between a quantum theory of gravity on an asymptotically AdS space and a lower-dimensional conformal field theory (CFT) on its boundary.

Review of a Type D Vacuum Space-Time with
a Curvature Singularity and CTCs [51] In Ref. [51], a type D axially symmetric, asymptotically flat vacuum solution of the field equations with zero cosmological constant, was constructed. This vacuum metric is as follow After doing a number of transformations into the above metric, we arrive at the following The Kretschmann scalar of the above metric is For constant r, z, the metric (3) reduces to conformal Misner metric in 2D where Ω = sinh 2 r is the conformal factor.
In the context of CTCs, the Misner space metric in 2D is interesting because CTCs appear after a certain instant of time from causally well-behaved conditions. The metric for the Misner space in 2D [55] is given by where −∞ < T < ∞ but the coordinate X is periodic locally. The metric (6) is regular everywhere as det g = −1 including at T = 0. The curves T = T 0 , where T 0 is a constant, are closed since X is periodic. The curves T < 0 are spacelike, T > 0 are time-like, while the null curves T = 0 form the chronology horizon. The second type of curves, namely, T = T 0 > 0, are closed time-like curves. Therefore, the metric (2) or (3) is a four-dimensional generalization of Misner space in curved space-time. Note that the above space-time is the vacuum solution of field equations, a Ricci flat, that is, R μν = 0. Li [56] constructed a Misner-like AdS space-time, a timemachine model. Levanony and Ori [57] constructed a three-, four-dimensional generalization of flat Misner space metric.
In this paper, we extend the above Ricci flat space-time (2) to the Einstein space-times of Petrov type D, which satisfy the following conditions It is an anti-de Sitter-like space if Λ < 0 and de Sitter like if Λ > 0. The extended space-time satisfies all the basic requirements (see details in Ref. [30]) for a time machine space-time except one, that is, this new model is not free from curvature singularity.

Analysis of a Cosmological Constant Vacuum Space-Time
Consider the following line element, a modification of the metric (2) given by Here, α is a positive constant, and β is real. The coordinates are labelled x 0 = t, x 1 = r, x 2 = ϕ, and x 3 = z. The ranges of the coordinates are and ϕ is a periodic coordinate ϕ~ϕ + ϕ 0 , with ϕ 0 > 0. The metric is Lorentzian with signature ð−, + , + , + Þ and the determinant of the corresponding metric tensor g μν , det g = − cosh 2 r sinh 4 r cosh 2 t: 2 Advances in High Energy Physics Now, we have evaluated the Ricci tensor R μν of the spacetime (8) as follows: The Ricci scalar is given by Using the metric tensor components of the above spacetime, we have found that the Ricci tensor and the Einstein tensor G μν are From the Einstein's field equations G μν + Λg μν = 0 and from eq. (14), we have Thus, from the above analysis, it is clear that the spacetime considered by (8) is an example of the class of Einstein space of anti-de Sitter-type and satisfies eq. (7) for a negative cosmological constant. We have shown later that the spacetime possesses a curvature singularity at r → 0.
An interesting property of the metrics (8) is that it reduces to 2D Misner space metric [55] for constant r, z. For that, we do the following transformations into the metric (8) (replacing β 2 → −Λ/3), we arrive at the following line element For constant r = r 0 > 0 and z = z 0 , the metric (17) becomes a conformal Misner space metric in 2D where Ω is the conformal factor. Therefore, the space-time admits CTC for t = t 0 > 0 similar to the Misner space discussed earlier.
We check whether the CTCs evolve from an initially space-like t = constant hypersurface (and thus t is a time coordinate). This is determined by calculating the norm of the vector ∇ μ t [37] (or alternately from the value of g tt in the inverse metric tensor g μν ). A hypersurface t = constant is space-like when g tt < 0 at t < 0, time-like when g tt > 0 for t > 0, and null g tt = 0 for t = 0. For the metric (8), we have Thus, a hypersurface t = constant is spacelike for t < 0, time-like for t > 0, and null at t = 0. We restrict our analysis to r > 0; otherwise, no CTCs will be formed. Thus, the space-like t = constant < 0 hypersurface can be chosen as initial hypersurface over which initial data may be specified. There is a Cauchy horizon at t = t 0 = 0 called the chronology horizon, which separates the causal past and future in a pastdirected and future-directed manner. Hence, the space-time evolves from a partial Cauchy surface (i.e., initial space-like hypersurface) in a causally well-behaved, up to a moment, i.e., a null hypersurface t = t 0 = 0 and the formation of CTCs takes place from causally well-behaved initial conditions. The evolution of CTC is thus identical to the case of the Misner space.
That the space-time represented by (8) satisfies the requirements of axial symmetry is clear from the following. Consider the Killing vector η = ∂ ϕ having the normal form Its covector form The vector (22) satisfies the Killing equation η μ;ν + η ν;μ = 0. The space-time is axial symmetry if the norm of the Killing vector η μ vanishes on the axis i.e., at r = 0 (see [58,59] and references therein). In our case as r → 0. The metric has a curvature singularity at r = 0. We find that the Kretschmann scalar is

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We can see that the scalar curvature diverges at r → 0, which indicates that the space-time possesses a curvature singularity. In addition, the Kretschmann scalar becomes K → 8Λ 2 /3 for r → ∞, indicating that the metric (8) is asymptotically anti-de Sitter-like space radially [60].

Classification and Physical
Interpretation of the Space-Times. Here, we first classify the space-time according to the Petrov classification scheme and then analyze the effect of local fields of the solution. We construct a set of null tetrad ðk, l, m, mÞ [61] for the space-time (8). Explicitly, these covectors are The set of null tetrad above is such that the metric tensor for the line element (8) can be expressed as The vectors (25), (26), (27), and (28) are null vector and orthogonal, except for k μ l μ = −1 and m μ m μ = 1. We calculate the five Weyl scalars, of these only is nonvanishing, while the rest are vanish. Thus, the metric is clearly of type D in the Petrov classification scheme. We set up an orthonormal frame e ðaÞ = fe ð0Þ , e ð1Þ , e ð2Þ , e ð3Þ g, e ðaÞ · e ðbÞ ≡ e μ ðaÞ e ν ðbÞ g μν = η ab = diag ð−1,+1,+1,+1Þ, which consists of three space-like unit vectors e ðiÞ , i = 1, 2, 3 and one time-like vector e ð0Þ [62]. Notations are such that small Latin indices are raised and lowered with Minkowski metric η ab , η ab , and Greek indices are raised and lowered with metric tensor g μν , g μν . The dual basis is e ðiÞ = e ðiÞ and e ð0Þ = −e ð0Þ . These frame components in terms of tetrad vector can be expressed as In order to analyze the effect of local gravitational fields of these solutions, we have used the equations of geodesic deviation [25,33,52,[63][64][65][66] which in terms of orthonormal frame e ðaÞ are where e ð0Þ = u is a time-like four-velocity vector of the free test particles. We set here Z ð0Þ = 0 such that all test particles are synchronized by the proper time. From the standard definition of the Weyl tensor and the Einstein's field equation for zero the stress-energy tensor, we get (see Eq. (4) in [66]) where C ðiÞð0ÞðjÞð0Þ ≡ e μ ðiÞ u ν e ρ ðjÞ u σ C μνρσ are the components of the Weyl tensor.
The only nonvanishing Weyl scalars are given by (30) so that Therefore, the equations of geodesic deviation (32) take the following form In the limit α → 0, all the Weyl scalars including Ψ 2 vanishes. In this limit, the space-time (8) becomes anti-de Sitter (AdS) space. So the equations of geodesic deviation (35) in this limit reduces to Advances in High Energy Physics with the solutions Z i ð Þ = a i τ + b i for Λ = 0, where a i , b i , i = 1, 2, 3 are the arbitrary constants. Again, in the limit Λ → 0, that is, β → 0, the only nonvanishing Weyl scalars is Ψ 2 given by (30). The space-time (8) reduces to type D vacuum space-time of zero cosmological constant with a curvature singularity which we discussed, in detail in Ref. [51]. In this limit (Λ → 0), the equations of geodesic deviation (35) becomes