  Unlike traditional electromagnetic measurements, gravitational-wave
observations are not affected by crowding and extinction. For this reason,
compact object binaries orbiting around a massive black hole can be used as
probes of the inner environment of the black hole in regions inaccessible to
traditional astronomical measurements. The orbit of the binary's barycenter
around the massive black hole will cause a Doppler shift in the gravitational
waveform which is in principle measurable by future space-based
gravitational-wave interferometers, such as the Laser Interferometer Space
Antenna (LISA). We investigate the conditions under which these Doppler shifts
are observable by LISA. Our results imply that Doppler shift observations can
be used to study the central region of globular clusters in the Milky Way, as
well the central environment of extragalactic massive black holes.
