I was watching a video on relativity on YouTube that talked about the difference between coordinate time $t$ and proper time $\tau$ and I have a couple of questions.
As I understand it, the video said that the coordinate time $\Delta t$ along a path between two events is the time between the two events measured by a faraway observer. The proper time $\Delta \tau$ along a path between two events is the time measured by an observer traveling along that path. I understand this in the context of special relativity.
However, in the context of general relativity what would a faraway observer entail? Since the definition of coordinate time says it's the time measured by a faraway observer.
For example, consider a case where we are comparing the amount of time measured between two events in a strong gravitational field by two different observers. One observer is traveling through the gravitational field and the other observer is not in the gravitational field.
Would the coordinate time be the time the observer far away from the location of the two events occurring in the gravitational field (i.e. an observer in a flat Minkowski spacetime) measures? In general, how does the distinction between coordinate time and proper time work in general relativity? Is the coordinate time the time measured between two events by an observer in flat Minkowski spacetime?
Best Answer
Really, the coordinate time between two events could be that measured by any observer, not necessarily far away. As you said, for the person who actually passes through both events, their coordinate time happens to be the proper time. For someone who passes through the first event but not the second, we can just apply the hyperbolic rotation of special relativity to switch between the proper and observed coordinates, if the events are close.
But if the observer is far away from either event, you need a way to figure out which point on the observer's path is "simultaneous" with the event. The key idea here is that, in spacetime, whether special or general relativity, a direction that you perceive as a spatial separation is always orthogonal to the direction you perceive as time.
So, you trace a path which is orthogonal to the observer's worldline and passes through the event. It should be a "straight line", which means it is a geodesic. We can say the point where that geodesic intersects the observer's worldline represents the time at which they perceive the event.
Do that for both events, take the difference of the two observer times, and that will be the perceived (coordinate) time difference.
Incidentally, by taking a family of geodesics orthogonal to the worldline, and picking the point on each of them a certain distance out, you can construct a path which is "comoving" with the observer, that is, it maintains the same spatial separation. By assigning each such point the same spatial coordinates, and the time that matches the observer's time, you create a comoving coordinate system, which gives the observer time for every possible event. I think that's the system they're referring to when they call it coordinate time. There may be cases where this isn't globally possible though.