  We evaluate the ratio of the shear viscosity to the relaxation time of the
shear flux above but near the critical temperature $T_c$ in SU(3) gauge theory
on the lattice. The ratio is related to Kubo's canonical correlation of the
energy-momentum tensor in Euclidean space with the relaxation time
approximation and an appropriate regularization. Using this relation, the ratio
is evaluated by direct measurements of the Euclidean observables on the
lattice. We obtained the ratio with reasonable statistics for the range of
temperature $1.3T_c \lesssim T \lesssim 4T_c$. We also found that the
characteristic speed of the transverse plane wave in gluon media is almost
constant, $v \simeq 0.5$, for $T \gtrsim 1.5T_c$, which is compatible with the
causality in the second order dissipative hydrodynamics.
