Rigid motion revisited: rigid quasilocal frames

Abstract

In this first of a series of papers we will introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a ‘system’ in the context of the dynamical spacetime of general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary (topologically ) of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (the ‘size’ of the system is not changing) and shear-free (the ‘shape’ of the system is not changing). This definition of a system is anticipated to yield simple, exact geometrical insights into the problem of motion in general relativity. It begins by answering, in a precise way, the questions of what is in motion (a rigid two-dimensional system boundary with topology S2, and whatever matter and/or radiation it happens to contain at the moment), and what motions of this rigid boundary are possible. Nearly a century ago Herglotz and Noether showed that a three-parameter family of timelike worldlines in Minkowski space satisfying Born's 1909 rigidity conditions does not have the 6 degrees of freedom we are familiar with from Newtonian mechanics, but a smaller number—essentially only 3. This result curtailed, to a large extent, subsequent study of rigid motion in special and (later) general relativity. We will argue that in fact we can implement Born's notion of rigid motion in both flat spacetime (this paper) and arbitrary curved spacetimes containing sources (subsequent papers)—with precisely the expected 3 translational and 3 rotational degrees of freedom (with arbitrary time dependence)—provided the system is defined quasilocally as the two-dimensional set of points comprising the boundary of a finite spatial volume, rather than the three-dimensional set of points within the volume.