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An interesting feature of the Davies-Unruh effect is that an uniformlyaccelerated observer sees an isotropic thermal spectrum of particles eventhough there is a preferred direction in this context, determined by thedirection of the acceleration g. We investigate the thermal fluctuations in theUnruh bath by studying the Brownian motion of particles in the bath, especiallyas regards to isotropy. We find that the thermal fluctuations are anisotropicand induce different frictional drag forces on the Brownian particle dependingon whether it has a drift velocity along the direction of acceleration g or ina direction transverse to it. Using the fluctuation-dissipation theorem, weargue that this anisotropy arises due to quantum correlations in thefluctuations at large correlation time scales.

Drift, Drag and Brownian motion in the Davies-Unruh bath