  A brief overview is presented of a new Caltech/Cornell research program that
is exploring the nonlinear dynamics of curved spacetime in binary black hole
collisions and mergers, and of an initial project in this program aimed at
elucidating the flow of linear momentum in black-hole binaries (BBHs). The
"gauge-dependence" (arbitrariness) in the localization of linear momentum in
BBHs is discussed, along with the hope that the qualitative behavior of linear
momentum will be gauge-independent. Harmonic coordinates are suggested as a
possibly preferred foundation for fixing the gauge associated with linear
momentum. For a BBH or other compact binary, the Landau-Lifshitz formalism is
used to define the momenta of the binary's individual bodies in terms of
integrals over the bodies' surfaces or interiors, and define the momentum of
the gravitational field (spacetime curvature) outside the bodies as a volume
integral over the field's momentum density. These definitions will be used in
subsequent papers that explore the internal nonlinear dynamics of BBHs via
numerical relativity. This formalism is then used, in the 1.5PN approximation,
to explore momentum flow between a binary's bodies and its gravitational field
during the binary's orbital inspiral. Special attention is paid to momentum
flow and conservation associated with synchronous spin-induced bobbing of the
black holes, in the so-called "extreme-kick configuration" (where two identical
black holes have their spins lying in their orbital plane and antialigned).
