  The first-order relativistic fluid theories of dissipation proposed by Eckart
and Landau-Lifshitz have been proved to be unstable. They admit solutions which
start in proximity of equilibrium and depart exponentially from it. We show
that this behaviour is due to the fact that the total entropy of these fluids,
restricted to the dynamically accessible states, has no upper bound. As a
result, these systems have the tendency to constantly change according to the
second law of thermodynamics and the unstable modes represent the directions of
growth of the entropy in state space. We, then, verify that the conditions of
stability of Israel and Stewart's theory are exactly the requirements for the
entropy to have an absolute maximum. Hence, we explain how the instability of
the first-order theories is a direct consequence of the truncation of the
entropy current at the first order, which turns the maximum into a saddle point
of the total entropy. Finally, we show that recently proposed first-order
stable theories, constructed using more general frames, do not solve the
instability problem by providing a maximum for the entropy, but, rather, are
made stable by allowing for small violations of the second law.
