  Extreme mass ratio bursts (EMRBs) have been proposed as a possible source for
future space-borne gravitational wave detectors, such as the Laser
Interferometer Space Antenna (LISA). These events are characterized by
long-period, nearly-radial orbits of compact objects around a central massive
black hole. The gravitational radiation emitted during such events consists of
a short burst, corresponding to periapse passage, followed by a longer, silent
interval. In this paper we investigate the impact of including relativistic
corrections to the description of the compact object's trajectory via a
geodesic treatment, as well as including higher-order multipole corrections in
the waveform calculation. The degree to which the relativistic corrections are
important depends on the EMRB's orbital parameters. We find that relativistic
EMRBs (v_{max}}/c > 0.25) are not rare and actually account for approximately
half of the events in our astrophysical model. The relativistic corrections
tend to significantly change the waveform amplitude and phase relative to a
Newtonian description, although some of this dephasing could be mimicked by
parameter errors. The dephasing over several bursts could be of particular
importance not only to gravitational wave detection, but also to parameter
estimation, since it is highly correlated to the spin of the massive black
hole. Consequently, we postulate that if a relativistic EMRB is detected, such
dephasing might be used to probe the relativistic character of the massive
black hole and obtain information about its spin.
