Swimming versus swinging effects in spacetime

Abstract

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...

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