What does the coordinate of time really mean?

Question

The meaning of the $t$ which appears in space-time intervals or metrics in general relativity. I concluded that $t$ was just a mathematical thing which allows us to label the "space-time manifold" and only proper time τ had a physical meaning too. On Wikipedia I also found that the coordinate time is not a time that could be measured by a clock located at the place that nominally defines the reference frame. What does it really mean?

Answer 1

First, imagine that you're in an inertial reference frame. We can think of this as a sort of lattice throughout space, with a synchronized clock at each lattice point in space. Now imagine two events, $A$ and $B$, which occur at space-time coordinates $(t_A, x_A, y_A, z_A)$ and $(t_B, x_B, y_B, z_B)$. The reference frame has (again, synchronized) clocks at $(x_A, y_A, z_A)$ and $(x_B, y_B, z_B)$. The coordinate time $\Delta t_{AB}$ is the difference in time measured by the clock at $B$ and the clock at $A$, so that $\Delta t_{AB}=t_B-t_A$. If $\Delta t_{AB}=0$, the events are simultaneous in that reference frame. However, an observer in a different inertial frame (say, in one with a boost $\vec{\beta}$ in the $x$-direction) might measure a different $\Delta t_{AB}$. This means that coordinate time is frame-dependent - it depends on the reference frame you're in.

Proper time is sometimes called "wristwatch time". Now, there's a single clock, and it measures the time between two events $A$ and $B$ on a single worldline. This proper time $\tau_{AB}$ does not depend on your reference frame, but it does depend on the path you take from $A$ to $B$ - that is, it's path-dependent.

One way to think of the difference is that coordinate time tells you something about one reference frame relative to another (because it's frame-dependent), while proper time tells you something about the worldline of the observer (because it's path dependent).