  Entropy bounds in black hole physics, based on a wide variety of different
approaches, have had a long and distinguished history. Recently the current
authors have turned attention to uncollapsed systems and obtained a robust
entropy bound for uncollapsed static spherically symmetric configurations. In
the current article we extend this bound to rotating systems. This extension is
less simple than one might at first suppose. Purely classically, (using only
classical general relativity and basic thermodynamics), it is possible to show
that the entropy of uncollapsed matter inside a region enclosed by a surface of
area A is bounded from above by S <= kappa(surface) A / (4 pi T). Here
kappa(surface) is a suitably defined surface gravity. By appealing to the Unruh
effect, which is our only invocation of quantum physics, we argue that for a
suitable class of fiducial observers there is a lower bound on the temperature
(as measured at spatial infinity): T >= max kappa(FIDOs) / (2 pi). Thus, using
only classical general relativity, basic thermodynamics, and the Unruh effect,
we are able to argue that for uncollapsed matter S <= {1/2} A.
