  We consider the relativistic deformation of quantum waves and mechanical
bodies carrying intrinsic angular momentum (AM). When observed in a moving
reference frame, the centroid of the object undergoes an AM-dependent
transverse shift. This is the relativistic analogue of the spin Hall effect,
which occurs in free space without any external fields. Remarkably, the shifts
of the geometric and energy centroids differ by a factor of 2, and both
centroids are crucial for the correct Lorentz transformations of the AM tensor.
We examine manifestations of the relativistic Hall effect in quantum vortices,
and mechanical flywheels, and also discuss various fundamental aspects of this
phenomenon. The perfect agreement of quantum and relativistic approaches allows
applications at strikingly different scales: from elementary spinning
particles, through classical light, to rotating black-holes.
