Rigid motion revisited: rigid quasilocal frames

TL;DR

This paper introduces rigid quasilocal frames (RQFs) in general relativity, defining a finite system boundary with rigidity conditions to analyze motion, extending classical rigidity concepts to curved spacetimes with six degrees of freedom.

Contribution

It proposes the concept of RQFs as a natural, geometrical way to define and analyze rigid motion boundaries in general relativity, extending classical notions to curved spacetimes.

Findings

RQFs have six degrees of freedom, matching Newtonian rigid body motion.

The boundary-based definition simplifies understanding of motion in curved spacetime.

The approach generalizes Born's rigidity conditions to arbitrary spacetimes.

Abstract

We introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a "system" in general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (constant size) and shear-free (constant shape). 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 the questions what is in motion (a rigid two-dimensional system boundary), 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 has only three degrees of freedom instead of the six we are familiar with from Newtonian mechanics. We 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 three translational and three rotational degrees of freedom - 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.