  We investigate the microscopic origin of the relaxation time coefficient in
relativistic fluid dynamics. We show that the extraction of the shear viscosity
relaxation time via the gradient expansion is ambiguous and in general fails to
give the correct result. The correct value for the shear viscosity relaxation
time is extracted from the slowest non-hydrodynamic pole of the corresponding
retarded Green's function, if such a pole is purely imaginary. According to the
AdS/CFT correspondence, in strongly-coupled $\mathcal{N}=4$ SYM the
non-hydrodynamic poles of the shear stress tensor nearest to the origin have a
nonzero real part, which implies that the transient fluid-dynamical equations
for this gauge theory are not equivalent to the well-known Israel-Stewart
equations.
