In Baumann's cosmology lecture, chapter 4, page 89, he defines adiabatic perturbation as:
Adiabatic perturbations have the property that the local state of matter (determined, for example, by the energy density and the pressure P) at some spacetime point $(\tau, x)$ of the perturbed universe is the same as in the background universe at some slightly different time $\tau + \delta \tau(x).$ (Notice that the time shift varies with location x!) We can thus view adiabatic perturbations as some parts of the universe being “ahead” and others “behind” in the evolution. If the universe is filled with multiple fluids, adiabatic perturbations correspond to perturbations induced by a common, local shift in time of all background quantities.
What is the relation of this definition with the usual definition of adiabatic processes?
Another definition of adiabatic perturbation according to Kamionkowski's lecture, page 1, is:
The total density of universe may vary from one point to another but the ratio of various types of stuff (e.g. the photon to baryon ratio and dark matter to baryon ratio) are the same everywhere.
How can we relate these definitions?
Thanks in advance for any help.