Chaos and Rotating Black Holes with Halos

The occurrence of chaos for test particles moving around a slowly rotating black hole with a dipolar halo is studied using Poincaré sections. We find a novel effect, particles with angular momentum opposite to the black hole rotation have larger chaotic regions in phase space than particles initially moving in the same direction. 04.20.Jb, 04.70.Bw, 05.45.+b. Typeset using REVTEX ∗e-mail: [email protected] †e-mail: [email protected] 1 The main paradigm in the study of the motion of stars in a galaxy is a model of a central bulge surrounded by a halo [1]. In this context, for the particular axially symmetric case, arose the celebrated Hénon-Heiles model [2] whose study has been the source of inspiration of many researches on chaotic behavior [3]. The underlying theory in this case is the usual Newtonian Gravitation that for large masses and velocities is known to be less appropriate than Einsteinian General Relativity. In the late case the Newtonian potential is replaced by the spacetime metric and Newton motion equations by geodesics. This change of dynamics can produce dramatic effects, for instance, test particles moving in the presence of systems of masses that are integrable in Newtonian theory are chaotic in General Relativity, examples are: the fixed two body problem [4,5], and particles moving in a monopolar center of attraction surrounded by a dipolar halo [6]. Also gravitational waves, a non existing phenomenon in the Newtonian realm, can produce irregular motion of test particles orbiting around a static black hole [7,8]. Another distinctive feature of general relativity is the dragging of inertial frames due to mass rotation. This fact is observed, for instance, in the impressive differences of the geodesic motion in Schwarzschild and Kerr geometries [9]. In this Letter we study the effects of rotation in the motion of a particle orbitating around a slowly rotating black hole surrounded by a dipolar halo. The non rotating case is chaotic, so we shall study mainly the change in the chaotic behavior due to the rotation of the center of attraction. A typical situation is represented by a galaxy with a rapid rotating center surrounded by a distant massive halo, ring or other shell-like distributions of matter. We study the motion of particles moving between the center and the halo whose first contribution is dipolar. This contribution is always present whenever the halo does not possess reflection symmetry with respect to the black hole equatorial plane. The metric that represents the superposition of a Kerr black hole and a dipole along the rotation 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 [10]. Techniques to actually find the solutions are Bäcklund transformations and the inverse scattering method, also a third method constructed with 2 elements of the previous two is the “vesture method”, all these methods are closely related [10]. The general metric that represents the nonlinear superposition of a Kerr solution with a Weyl solution, in particular, with a multipolar expansion can be found by using the “inverse scattering method” [11]. We find ds = gtt(r, z)dt 2 + 2gtφ(r, z)dtdφ+ gφφ(r, z)dφ 2 + f(r, z)(dz + dr), (1)