Did you know that its possible to swim in the cold, hard vacuum of space, even though you have nothing to push against? Well according to Jack Wisdom from M.I.T. you can; admittedly not very far, and the space-time has to be curved. But even so its the principle here that counts and this unlikely result appears correct.
This "thrust" doesn't arise from some wacky warping of space, gravity waves or manipulation of zero-point energy, but is due to conservation of momentum and the geometry of the whole system. Although it takes some maths thats way above my head to prove it, Jack starts simply considering two concentric spheres in flat space, connected at their centres. When one is rotated, the whole orientation of the system changes, due to conservation of angular momentum. You see this effect all the time, if you hold a heavy spinning disk by its axle, whilst sat on a office swivel chair, then by changing the angle of the disk, you can rotate the chair. (Waggling your arms around furiously around just sat on the chair, with no spinning disks works to some extent of course...)
Jack moves through various examples, firstly spinning caps on a frictionless sphere, joined together, can move across the surface of the sphere by repeating a cycle of movement. Also by making this system small compared to the size of the sphere it sits on, shows that the movement is not due to the fact that the body is rigid, and somehow constrained to move like that, but really is moving due to how momentum must be conserved on a curved surface.
Next he considers a simple swimmer, again on a frictionless sphere; made from two equal masses (legs, if you will) each joined to a central, third mass by equal length struts. He shows by moving the legs, in time with each other up and down, the body moves over the surface of the sphere.
### /) Angle α
--------------###----------------------Equator of sphere/Axis of motion
### \) Angle α
During this proof, as an aside he shows that the curvature must be intrinsic, for this to work. On a cylinder with extrinsic curvature but no intrinsic curvature nothing happens, nor on a plane , which has no curvature whatsoever. On a sphere which does have both intrinsic and extrinsic curvature, the swimming cycle does produce motion.
Lastly he goes through the behaviour of a swimmer like the above, except with three equaly spaced leg masses on a body in a tripod configuration, and swimming in the kind of curved space-time you'd find around a spherical mass. He does make a couple of assumptions, that the swimmer is small compared to the curvature, slow compared to the speed of light and the body isn't distorted by the curvature of the space time. The latter is to counteract the lorentz contraction; he assumes the body has been manufactured to be 'quasi-rigid', that is able to change it's size so the masses stay an equal distance from the central mass, as it moves through different amounts of curvature. However he thinks you probably won't need this for the principle to still be sound. The equation he came up with to describe how far a space time swimmer moves with each stroke is as follows :-
Δr = - ----------l2-------sinα Δl Δα
Where r=distance from centre of gravity well to swimmer.
c=Speed of light
M=Mass making the gravity well
l, m0, m1, α are as noted on the above diagram.
Taking a 1 metre sized swimmer, performing 1 metre sized leg-swings, on the surface of the earth; each stroke moves the swimmer outwards by about 10-23 metres! An important point to make is that this is not 10-23m worth of acceleration; you won't increase your speed, its only your position that changes. Although the effect is tiny, it is perhaps something that could be tested by a satelite in orbit; if it tried swimming, eventually its orbit would be disturbed.
S. K. Blau Physics Today 06/03,20
J. Wisdom, Science 299, 1865 (2003)