I have always been a little confused by the meaning of the "$t$" which appears in spacetime intervals or metrics in general relativity. I concluded that $t$ was just a mathematical thing which allow to label the "spacetime manifold" and only proper time $\tau$ had a physical meaning. On wikipedia I also found:
"But the coordinate time is not a time that could be measured by a clock located at the place that nominally defines the reference frame." (http://en.wikipedia.org/wiki/Coordinate_time)
I don't know if my point of view is correct or not, but the following two computations made me think that I'm missing something:
1) Time dilation:
$$ \frac{dt}{d\tau} = \gamma $$
If $t$ is non-physical, what are the two clocks which one should "compare" to see this time dilation?
2) Gravitational redshift:
This is a computation I have seen in my lectures which in my opinion has no physical sense.
Consider an atomic transition at the surface of the earth, at $x^{\mu}_1$. The time interval measured by a stationary observer close to the atom is given by:
$$d\tau_1 = \sqrt{g_{00}(x_1)} dx^0_1$$
Imagine now the same atomic transition but, say, 100 km above the surface of the earth at $x^{\mu}_2$. The time interval measured by an observer near the atom is:
$$d\tau_2 = \sqrt{g_{00}(x_2)} dx^0_2$$
Since the physics of atomic transitions is the same in $x_1$ and $x_2$ then one should have:
$$d\tau_1=d\tau_2$$
$$\frac{dx^0_1}{dx^0_2} = \frac{\sqrt{g_{00}(x_2)}}{\sqrt{g_{00}(x_1)}}$$
But what is the physical meaning of the quantity $\frac{dx^0_1}{dx^0_2}$? In my opinion the only way to compute gravitational redshift is to compare the proper interval measured by an observer in $x_1$ and one in $x_2$ for an atomic transition happening in $x_1$.
Best Answer
Proper time represents the physical aging of a massive particle, and by this it is the only time which is to take into account for the physical description of a particle.
But coordinate time is not without physical meaning: There would be no detection of events without coordinate time. When two particles are traveling through the same place in space, their proper time will not provide the information if it happened simultaneously, i.e. that they encountered, i.e. that there is an event. For this information you need the Minkowski diagram of at least one of both particles, and by the way the Minkowski diagram of any observer includes the coordinates of both particles, providing the information if they did encounter or if they did not.
Minkowski diagrams are showing the coordinate time of all particles (with different simultaneities). In contrast, it is not possible to represent the proper time of two different frames in one diagram.