  The violation of Stokes--Einstein (SE) relation $D\sim (\eta/T)^{-1}$ between
the shear viscosity $\eta$ and the translational diffusion constant $D$ at
temperature $T$ is of great importance for characterizing anomalous dynamics of
supercooled water. Determining which time scales play key roles in the SE
violation remains elusive without the measurement of $\eta$. Here we provide
comprehensive simulation results of the dynamic properties involving $\eta$ and
$D$ in TIP4P/2005 supercooled water. This enabled the thorough identification
of the appropriate time scales for SE relation $D\eta/T$. In particular, it is
demonstrated that the temperature dependence of various time scales associated
with structural relaxation, hydrogen bond breakage, stress relaxation, and
dynamic heterogeneities can be definitely classified into only two classes.
That is, we propose the generalized SE relations that exhibit "violation" or
"preservation". The classification depends on the examined time scales that are
coupled or decoupled with the diffusion. On the basis of the classification, we
explain the physical origins of the violation in terms of the increase in the
plateau modulus and the nonexponentiality of stress relaxation. This implies
that the mechanism of SE violation is attributed to the attained solidity upon
supercooling, which is in accord with the growth of non-Gaussianity and
spatially heterogeneous dynamics.
