  We present the first exploration of relativistic gas dynamics in the
immediate vicinity of binary black holes as the system inspirals close to
merger in the gravitational radiation-driven regime. We focus on 2D
hydrodynamical studies of comparable-mass, non-spinning systems. Relativistic
effects alter the dynamics of gas in this environment in several ways. Because
the gravitational potential between the two black holes becomes shallower than
in the Newtonian regime, the mini-disks stretch toward the L1 point and the
amount of gas passing back and forth between the mini-disks increases sharply
with decreasing binary separation. This "sloshing" is quasi-periodically
modulated at $2$ and $2.75$ times the binary orbital frequency, corresponding
to timescales of hours to days for supermassive binary black holes. In
addition, relativistic effects add an $m=1$ component to the tidally-driven
spiral waves in the disks that are purely $m=2$ in Newtonian gravity; this
component becomes dominant when the separation is $\lesssim 100$ gravitational
radii. Both the sloshing and the spiral waves have the potential to create
distinctive radiation features that may uniquely mark supermassive binary black
holes in the relativistic regime.
