Comoving coordinate system

Question

In Ta Pei Cheng's GR book, the comoving coordinate system is defined as $$t=\text{the proper time of each fluid element}$$ $$x^i=\text{the spatial coordinates carried by each fluid element}$$ The comoving observer flows with a fluid element. The comoving coordinate time can be synchronized over the whole system.

I have a few questions about this definition:

• If the comoving observer flows with a fluid element, shouldn't coordinate $t$ be the proper time of that particular fluid element (i.e. proper time of the comoving observer), instead of each fluid element?
• How can $t$ be synchronized if it is the proper time of each fluid element? Each fluid element will move at different velocities and hence elapse different proper times.
• What does the author mean by "spatial coordinates carried by each fluid element"? What does it mean to carry a spatial coordinate? Isn't the spatial coordinate just as measured by the observer flowing with a particular fluid element?

Answer 1

Rather than worry about the exact definition of 'each' and the synchronization issues, perhaps it is simpler, to say that the time coordinate is something like "the proper time for the fluid element since the big bang", or "since radiation decoupled from matter", which ties the synchronization point to something physical.

As far as the bit about the spatial coordinates, define a meter using physical characteristics, like "a second is X vibrations of cesium, and a meter is the distance travelled in Y seconds", where X and Y are pure numbers. All local observers can calibrate equipment to these standards, and don't need to communicate anything to each other to do so, other than the numbers X and Y.