Stability of Closed Timelike Geodesics

The existence and stability under linear perturbations of closed timelike geodesics (CTG) in Bonnor-Ward spacetime is studied in some detail. Regions where the CTG exist and are linearly stable are exhibited. In 1949 Gödel found a solution to the Einstein field equation with nonzero cosmological constant that admits closed timelike curves (CTC) [1]. It could be argued that the Gödel solution is without physical significance, since it corresponds to a rotating, stationary cosmology, whereas the actual universe is expanding and apparently non rotating. The van Stockum solution [2], that also contain CTC, is physically inadmissible since refers to an infinitely long cylinder. But there exist examples of solutions of vacuum Einstein’s equations which contains CTC that can represent the exterior of physically admissible sources [3] [4]. In [3] it is described the case of a massless spinning rod of finite length. In [4] it is analyzed the CTC in Kerr-Newman spacetime and in a solution of the Einstein equations for a source named Perjeon, due to Perjés [5], which represents a single charged, rotating, magnetic object. This solution was also studied independently by Israel and Wilson [6], and it is referred as a PIW spacetime. In these three cases one expect that the CTC region be covered by the source. The same does not happen when we work with two Perjeons [4]. The PIW metric is given by ds = − f hmndxdx + f (ωmdx + dt), (1) where the three dimensional positive definite tensor hmn has zero Ricci tensor and it will be taken as the usual three dimensional Euclidean metric in cylindrical coordinates, the electromagnetic field is given in terms of two scalar potentials: F4n = Φ,n, F ab = η fψ,m, (2) ∗[email protected] †[email protected] 1 ηabm being the Levi-Civita tensor related to hmn, and ( ),n = ∂/∂xn. The entire solution is generated by two functions L and M that are harmonic with respect to hmn by means of the equations, f = 1 4L2+4M2 , ψ = 2 ǫ M L2+M2 , Φ = − 1 2 ǫ L L2+M2 , ωa,b − ωb,a = 8(ML,c − LM,c)ηabmh. (3) The Bonnor-Ward (BW) solution [7] refers to two Perjeons, with masses m1 and m2, placed on the z-axis at z = ±a, (a > 0), with magnetic moments (μ1 and μ2) also parallel to the z-axis, and L = (1 + m1/r1 + m2/r2)/2 M = (μ1(z − a)/r 1 + μ2(z + a)/r 3 2)/2 r1 = √ ρ2 + (z − a)2 r2 = √ ρ2 + (z + a)2 ω = (−ρφ)Ω. (4) We shall consider the particular case of BW solution [4], Ω = μ1 r3 1 (