  An interesting feature of the Davies-Unruh effect is that an uniformly
accelerated observer sees an isotropic thermal spectrum of particles even
though there is a preferred direction in this context, determined by the
direction of the acceleration g. We investigate the thermal fluctuations in the
Unruh bath by studying the Brownian motion of particles in the bath, especially
as regards to isotropy. We find that the thermal fluctuations are anisotropic
and induce different frictional drag forces on the Brownian particle depending
on whether it has a drift velocity along the direction of acceleration g or in
a direction transverse to it. Using the fluctuation-dissipation theorem, we
argue that this anisotropy arises due to quantum correlations in the
fluctuations at large correlation time scales.
