Chaos and Taub-NUT related spacetimes

The occurrence of chaos for test particles moving in a Taub-NUT spacetime with a dipolar halo perturbation is studied using Poincaré sections. We find that the NUT parameter (magnetic mass) attenuates the presence of chaos. PACS numbers: 04.20.Jb, 04.70.Bw, 05.45.+b. Typeset using REVTEX ∗e-mail: [email protected] †e-mail: [email protected] 1 The Taub-NUT (Newman, Tamburino and Unti) spacetime [1,2] is one of the most bizarre solutions of the vacuum Einstein equations. Because of its many unusual properties it has been described as a “counterexample to almost anything” [3]. It has closed timelike curves, is nonsingular in a meaningful mathematical sense but is not geodesically complete, etc. For certain range of the coordinates, it can be seen as a Schwarzschild monopole endowed with a “magnetic mass” [4]. The Euclidean version of this metric has recently received some attention due to the fact that it is closely related to the dynamics of two non-relativistic Bogomol’nyi-PrasadSommerfield (BPS) monopoles [5]. The asymptotic motion of monopoles corresponds to geodesic motion in Euclidean Taub-NUT space, this motion is integrable. This fact has motivated the study of geodesics in Euclidean Taub-NUT and related spaces [6]. Examples of chaotic motion in General Relativity are the geodesic motion of a test particle moving in the geometry associated with : a) Fixed two body problem [7], b) A monopolar center of attraction surrounded by a dipolar halo [8,9] (in Newtonian theory this system is integrable), c) A monopolar center of attraction surrounded by a quadrupole plus octupole halo [10], d) Multi-Curzon and multi-Zipoi-Vorhees solutions [11], and e) A rotating black hole (Kerr geometry) with a dipolar halo [9]. Also gravitational waves can produce irregular motion of test particles orbiting around a static black hole [12,13]. In this Letter we consider the geodesic motion of test particles moving in a Taub-NUT spacetime perturbed by a distant distribution of matter that can be represented by a dipole. The case of a center of attraction (without NUT parameter) perturbed by a dipolar halo was studied in [8], the combined relativistic effects and the breakdown of the reflection symmetry in this case produces a non integrable motion. Our main goal in this letter is to study the effect that the magnetic mass has on the chaotic motion of test particles. The geodesic motion in a pure Taub-NUT spacetime was studied in [14], this case is completely integrable. Also generalizations and perturbations of this spacetime has been considered from the view point of spacetime dynamics [15]. The metric that represents the superposition of a Taub-NUT metric and a dipole along 2 the z-axis is a stationary axially symmetric spacetime. The vacuum Einstein equations for this class of spacetimes is an integrable system of equations that is closely related to the principal sigma model [16]. Techniques to actually find the solutions are Bäcklund transformations and the inverse scattering method, also a third method constructed with elements of the previous two is the “vesture method”, all these methods are closely related [16]. The general metric that represents the nonlinear superposition of a Kerr-NUT solution with a Weyl solution, in particular, with a multipolar expansion can be found by using the “inverse scattering method” [17]. For the particular case of a Taub-NUT metric with a dipolar halo we find ds = gtt(r, z)dt 2 + 2gtφ(r, z)dtdφ+ gφφ(r, z)dφ 2 + f(r, z)(dz + dr), (1)