I learned that dust is a collection of particles at rest with repect to each other. Thus the four velocity field $U^\mu$ of dust will be a constant and it is easy to think of what the rest frame of dust means. It is simply the frame that every individual particle is at rest and is applicable to every point in space.
But what does the rest frame of a perfect fluid mean? A perfect fluid would not have a constant four velocity field $U^\mu$ so how do we choose the rest frame?
On pg. 34-35 of Sean Carroll's general relativity textbook, it was also written that to guess the energy momentum tensor for perfect fluid, we start with that of dust which is $$T^{\mu\nu}=\rho U^\mu U^\nu$$ where $\rho $ is the rest frame energy density. He then said that using this expression for a perfect fluid meant that $T^{ii}=0$, which I understand is because $U^i=0$.
That means that the four velocity of a perfect fluid in its rest frame is zero throughout space. But how is that possible if a perfect fluid does not have a constant four velocity in the first place?
