My question is about the article Swimming in Spacetime.
My gut reaction on first reading it was "this violates conservation of momentum, doesn't it?". I now realize, however, that this doesn't allow something to change its momentum; it only allows something to move (change position) without ever having a nonzero momentum. Since this is relativity, there is no simple relationship between momentum and velocity like p = mv, so this is all well and good. An object can move, with a constant momentum of zero, by changing its shape in a nontrivial cycle.
However, now I'm thinking about a different conservation law and I can't see how "swimming through spacetime" is possible without violating it. The conserved quantity I'm thinking of is the Noether charge associated with Lorentz boosts, which is basically x - (p/E)t, that is, the position of the center of mass projected back to time t=0. If p = 0, then the conserved quantity is simply x, the position of the center of mass. This obviously contradicts the whole swimming idea.
What's going on here? Is swimming through spacetime only possible if the spacetime is curved in some way that breaks the symmetry under Lorentz boosts? Or is there some error in my reasoning?