Swimming in Spacetime - apparent conserved quantity violation
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?
That is precisely the case. No error in your reasoning. In the case of a curved spacetime the "center of mass" of an extended body is no longer well-defined w.r.t external - i.e. located in an asymptotically flat region - observers.
In order to "swim" through spacetime one exploits the inhomogeneities of the gravitational field. The presence of these inhomogeneities breaks local Lorentz symmetry which is necessary for the mechanism to work.
In particular the scale of the swimmer and the inhomogeneities should be comparable. This is one reason why, at present, the construction of an actual swimmer is far beyond our technological means.
Edit: For those interested on extended body effects in GR there is are classic papers by Dixon. More recently Abraham Harte has done some amazing work along these lines Extended-body effects in cosmological spacetimes.
Well, I hope that my primitive understanding of GR makes for a good non-expert explanation... In GR, the Lorentz group symmetries are generally only valid locally, that is for a given space-time point. If you want to translate a vector to another space-time point, you need to do a parallel transport, which usually introduces correction terms depending on the curvature
Hard to tell exactly what the scenario is from that article. From what is shown, my guess is that the swimmer is doing work in deforming the object, which then moves the object. Then, the object having moved, the swimmer deforms the object BACK.
While this cycle would create zero work classically, in the case of relativity, you are now at a point where the gravitational potential has a different value, and therefore, the work you do to restore the object has been "redshifted" to a different value. In essence, the 'swimming' scheme converts gravitational potential energy into kinetic energy.
But that might not be quite what they're doing in this article.