# Comoving coordinate system

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.

• 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.