  We analyze the linear causality and stability of third-order fluid dynamics
considering perturbations around a global equilibrium state. We investigate the
formulation derived from kinetic theory, using the Chapman-Enskog expansion, in
PRC 88, 021903, which was shown to be in excellent agreement with solutions of
the microscopic theory. From this analysis, we demonstrate that this theory is
linearly acausal and unstable and that such instabilities cannot be corrected
by tuning the transport coefficients. We then propose a modification of this
theory, valid only in the linear regime, that can be constructed to be linearly
causal and stable and obtain the conditions the transport coefficients must
satisfy in order for this to be the case.
