  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 R 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 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 is a consequence of the fact that a two-sphere of any shape always admits
precisely six conformal Killing vector fields, which generate an action of the
Lorentz group on the sphere. Here we review the previous work in flat spacetime
and extend it in three directions: (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) in terms of operationally-defined geometrical
quantities on the boundary; (3) We use this new gravitational (or
"geometrical") energy flux to resolve another apparent paradox, this one
involving electromagnetism in flat spacetime, which we discovered in the course
of this work.
