Swimming versus swinging effects in spacetime

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.

Figure 1

Five snapshots of two tripods designed to have legs with length $l$ and angle $\alpha$ with the radial axis, along which they fall down, are shown with and without cyclic deformations, respectively. The swimming effect consists in realizing that local cyclic deformations lead, in general, to displacements of order $G/c^2$ in the quasirigid tripod trajectory when compared with the rigid one.

Figure 2

The top figure carries the information that the swimming effect is a result of the nonzero area of the square (in the shape space diagram) associated with the body deformation. Were the deformation such that the area were null, the swimming effect would vanish. The bottom figure illustrates precisely this situation. Space displacements can be achieved in this case, however, through the swinging effect, i.e. by expanding and contracting the legs at different points of the trajectory.

Figure 3

The positions of the masses $m_a$ $(a=0,1,2,3)$ are given through usual spherical coordinates $r_a,{\theta }_a,{\varphi }_a$ with origin at the central mass $M$.

Figure 4

The full and dashed lines show ${\Delta }^{\mathrm{C}}{r}_0\equiv (r_0^{\mathrm{quasi}-\mathrm{rigid}}-r_0^{\mathrm{rigid}}{)}_{\mathrm{clas}}$ and ${\Delta }^{\mathrm{R}}{r}_0\equiv (r_0^{\mathrm{quasi}-\mathrm{rigid}}-r_0^{\mathrm{rigid}}{)}_{\mathrm{rel}}$, i.e. how much a free-falling quasirigid tripod fails to follow a rigid one at the end of a complete cycle assuming a Newtonian gravitational field and a Schwarzschild spacetime characterized by a central mass $GM=1$, respectively. Here the tripod is assumed to change its shape as shown in Fig. 1 and $\omega \equiv 1/T$ is the cycle frequency. The rigid and quasirigid tripods are set free simultaneously with $G{m}_a=0.1$ and $r_0=100$ ($a=0,1,2,3$). Initially $\alpha =1$ and $l=1$ and they vary as much as $\Delta \alpha =-0.01$ and $\Delta l=0.01$ along the cycle. Each quarter of the whole cycle takes as long as $T/4$. (Here $c=1$.)