  This paper is the fourth in a series exploring the physical consequences of
the solidity of highly viscous liquids. It is argued that the two basic
characteristics of a flow event (a jump between two energy minima in
configuration space) are the local density change and the sum of all particle
displacements. Based on this it is proposed that density fluctuations are
described by a time-dependent Ginzburg-Landau equation with rates in k-space of
the form $\Gamma_0+Dk^2$ with $D\gg\Gamma_0a^2$ where $a$ is the average
intermolecular distance. The inequality expresses a long-wavelength dominance
of the dynamics which implies that the Hamiltonian (free energy) may be taken
to be ultra local. As an illustration of the theory the case with the simplest
non-trivial Hamiltonian is solved to second order in the Gaussian
approximation, where it predicts an asymmetric frequency dependence of the
isothermal bulk modulus with Debye behavior at low frequencies and an
$\omega^{-1/2}$ decay of the loss at high frequencies. Finally, a general
formalism for the description of viscous liquid dynamics, which supplements the
density dynamics by including stress fields, a potential energy field, and
molecular orientational fields, is proposed.
