On the Existence and Utility of Rigid Quasilocal Frames

摘要

The notion of a rigid quasilocal frame (RQF) provides a geometrically natural way to define a system in general relativity, and a new way to analyze the problem of motion. An RQF is defined as a two-parameter family of timelike worldlines comprising the boundary (topologically ? x S~2) of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines be expansion- and shear-free. In other words, the size and shape of the system do not change. In previous work, such quasilocally defined systems in Minkowski space were shown to admit precisely the same six degrees of freedom of rigid body motion that we are familiar with in Newtonian space-time, without any constraints, circumventing a century-old theorem due to Herglotz and Noether. This surprising result is a consequence of the fact that any two-surface with S~2 topology always admits precisely six conformal Killing vectors, which generate an action of the Lorentz group on the sphere. Several representative examples of RQFs were constructed, including one that provides a quasilocal resolution to the Ehrenfest paradox, also a century old. Here we review the previous work in flat spacetime and extend it in three direc- tions: (1) Using a Fermi normal coordinates approach, we explicitly construct, to the first few orders in powers of areal radius, the general solution to the RQF rigidity equations in a generic curved spacetime, and show that the resulting RQFs possess exactly the same six motional degrees of freedom as in flat spacetime; (2) We discuss how RQFs provide a natural context in which to understand the flow of energy, momentum and angular momentum into and out of a system; in particular, we derive a simple, exact expression for the flux of gravitational energy (a gravitational analogue of the Poynting vector) across the boundary of an RQF in terms of operationally-defined geometrical quantities on the boundary; (3) We use this new gravitational (or "geomet- rical")energy flux to resolve another paradox, this one involving electromagnetism in flat spacetime, a paradox we discovered in the course of this work.