We present a purely relativistic effect according to which asymmetric oscillations of a quasi-rigid body slow down or accelerate its fall in a gravitational background.

The exact solution to the Einstein equations that represent a static axially symmetric source deformed by an internal quadrupole is considered. The Poincaré section method is used to study numerically the geodesic motion of test particles, for prolate quadrupole deformations, we find chaotic motions contrary to the oblate case where only regular motion is found. We also consider the metric that represents a rotating black hole deformed by a quadrupole term. This metric is obtained as a two-soliton solution in the context of Belinsky-Zakharov inverse scattering method. The stability of geodesics depends strongly on the relative direction of the spin of the center of attraction and the angular momentum of the test particle.

Abstract. Newtonian as well as special relativistic dynamics are used to study the stability of orbits of a test particle moving around a black hole with a dipolar halo. The black hole is modeled by either the usual monopole potential or the Paczyńki-Wiita pseudo-Newtonian potential. The full general relativistic similar case is also considered. The Poincaré section method and the Lyapunov characteristic exponents show that the orbits for the pseudo-Newtonian potential models are more unstable than the corresponding general relativistic geodesics.

The motion of particles in the field of forces associated to an axially symmetric attraction center modeled by a monopolar term plus a prolate quadrupole deformation are studied using Poincaré surface of sections and Lyapunov characteristic numbers. We find chaotic motion for certain values of the parameters, and that the instability of the orbits increases when the quadrupole parameter increases. A general relativistic analogue is briefly discussed.PACS numbers: 05.45.+b, 95.10.Fh, 95.10.Ce, 04.20.Jb, 03.20.+i Attraction forces represented by a monopolar plus a prolate quadrupolar distribution of masses (charges) are a good approximation for elongated massive (charged) bodies. Examples range from astrophysics to nuclear physics. There are many observed galaxy clusters with a cigar like shape [1]. Also, the nuclear charge of light gold atoms has been reported as having a large prolate deformation [2]. Most of the Dwarf Galaxies in the Virgo Cluster may obey the "prolate hypothesis", i.e., they probably have a prolate spheroidal shape [3]. Asteroids also have a prolate shape, but usually they are not axisymmetric. Merrit [4] found, from detailed modeling of triaxial galaxies, that most of the galaxies must be nearly axisymmetric, either prolate or oblate.Classical, as well as, quantum chaos have been studied in a variety of axially symmetric fields of forces. In particular, attraction centers described by potentials that are the sum of two terms: a monopolar term and a quadrupolar deformation. Furthermore this center is "perturbed" by an external distribution of masses (charges) represented by its external multipolar moments, i.e.,Sometimes the monopolar term is changed by the potential of a spring [5]. In general, in all these cases the terms that originate the chaos are the external multipolar moments. We shall consider the simplest, albeit, important case of a particle moving in the field of a monopole plus a quadrupole deformation. This deformation is usually considered to be the major deviation from spherical symmetry. In cylindrical coordinates, (r, ϕ, z), the field takes the generic form,where α is a constant that may be associated with the central body mass (charge). It is instructive to have a special model in mind, consider two equal masses located on the z-axis symmetrically, at z = −a and z = +a. The gravitational potential of the above mass configuration up to the order a 3 is (3) with q = 2αa 2 . We shall use α = 1 without loss of generality. Note that we are not considering external multipolar moments (V P = 0), i.e., only deformed cores will be studied.We can distinguish two cases depending on the sign of q. The oblate deformation case, q < 0. This is the common case for bodies deformed by rotation and has been analyzed in astronomy for more than two hundred years. The integrability of the Newton equations for a particle moving in the gravitational field of an axially symmetric oblate body is an unsolved problem. It is known as the classical problem of the existence of the third isolating inte...

Wisdom has recently unveiled a new relativistic effect, called ''spacetime swimming'', where quasirigid free bodies in curved spacetimes can ''speed up'', ''slow down'' or ''deviate'' their falls by performing local cyclic shape deformations. We show here that for fast enough cycles this effect dominates over a nonrelativistic related one, named here ''space swinging'', where the fall is altered through nonlocal cyclic deformations in Newtonian gravitational fields. We expect, therefore, to clarify the distinction between both effects leaving no room to controversy. Moreover, the leading contribution to the swimming effect predicted by Wisdom is enriched with a higher order term and the whole result is generalized to be applicable in cases where the tripod is in large redshift regions. Fig. 1). This is a full general-relativistic geometrical phase effect [3], which vanishes in the limit where the gravitational constant G ! 0 or the light velocity c ! 1. Similarly to the displacement attained by swimmers in low Reynolds number fluids [4,5], the displacement attained by swimmers in some given spacetime only depends on their local stroke.The fact that the swimming effect is purely relativistic has caused some perplexity [6,7], since it has been known for a long time that there is a similar classical effect in nonuniform Newtonian gravitational fields, which is present when c ! 1. For example, an orbiting dumbbell-shaped body can modify its trajectory by contracting the strut connecting the two masses at one point and expanding it at another one [8]. We stress here that this is a nonlocal effect, which appears due to the fact that the work performed by the dumbbell engine against the gravitational tidal force during the contraction differs from the one during the expansion. It is the resulting net work what allows the dumbbell to change from, say, a bounded to an unbounded orbit (see Fig. 2). The shorter is the period of the whole contraction-expansion process, the smaller is the change of the trajectory, although this cannot be made arbitrarily small if one requires that the deformation velocity does not exceed c. This is in analogy with playground swings, where the oscillation amplitude is modified by an individual through standing and squatting in synchrony with the swing motion [9].Here we perform a direct numerical simulation for a falling tripod to show that for fast enough cyclic deformations the swimming effect dominates over the swinging effect, while for slow enough cycles the opposite is true. We expect, thus, to set down any confusion concerning the independency of both effect. In addition, we calculate and discuss the idiosyncratic features of a higher order term beyond the leading one obtained by Wisdom and extend the whole result to be applicable in cases where the tripod is in large redshift regions.Let us begin considering a tripod falling along the radial axis in the Newtonian gravitational field of a spherically symmetric static body with mass M. The three tripod endpoint masses m i (i 1; 2; 3) a...