General theory of swimming in curved spacetimes

Abstract

Swimming in curved spacetimes is a phenomenon whereby free bodies in curved spacetimes are able to propel themselves by performing cyclic internal motions. When originally proposed, it was further suggested that, in the limit of fast internal cycles, the net motion would display a simple geometric-phase character, in which the displacement per cycle would not depend on the time progression of the internal motions but only on the sequence of shapes assumed by the body, like a swimmer in a non-turbulent viscous fluid (low Reynolds number). In this paper we develop a general, covariant theory of swimming in curved spacetimes, describing a technique to study the motion of free, small, light, articulated bodies in general relativity by mapping the problem to an analogue in special relativity. We give considerable attention to the limit of fast cycles and investigate the conditions in which the overall motion could display such geometric-phase behavior. The conclusion, however, is that this simple behavior is only realized in very specific circumstances, depending on the structure of the body, characteristics of internal motions, initial conditions, and symmetries of the spacetime; whereas, in general, our formulas predict a more complicated dynamics.