Difference between coordinate time and proper time in general relativity

Question

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?

ohneVal · Accepted Answer

I honestly believe this sort of questions require some formulas. First of all, let us agree on the setting. In general relativity (GR) the metric $g_{munu}$ is a dynamical tensor, meaning it is a tensor which is not constant. The metric encodes how one measures distances, time intervals or better, space-time intervals. This metric will depend on the coordinates you choose for the patch of the space-time you are considering, with out loss of generality call them as follows:
$$g_{munu} = g_{munu}(t,x_1,x_2,x_3)$$
The important thing is that locally, let us say if we are studying a small enough patch, things are like in special relativity and this means that there is one coordinate, namely $t$ in this example, to which a diagonal term $g_{tt}$, with an opposite relative sign is associated. This coordinate is usually called the coordinate time, or at least is responsible of defining what time-like is. Different coordinates and metrics have different behaviors, names but they all share the fact that the signature of the metric (realistic metrics, non-euclidean) is the same and this special coordinate always exist.
So far we have only chosen a set of coordinates for our patch of the "Universe" and recognized that one of them behaves slightly different. Now let us speak about proper time. Over this chosen coordinates let us consider some geodesics, that is paths that experience no acceleration. Mathematically in this coordinates, a path in space-time is just some function depending on some parameter $s$, that returns a point in space-time:
$$gamma(s)=(t(s),x_1(s),x_2(s),x_3(s))$$
As you might know there are infinitely many ways to parameterize a curve, in other words $s$ can be changed for some other parameter. But again for the sake of comparison one looks for a "standard", this natural choice is the arc-length of the path itself. Assuming this path is time-like (meaning simply, its velocity is always lower than the speed of light) the arc-length of this path in 4 dimensions is what we call proper time, mathematically:
$$gamma(tau)=(t(tau),x_1(tau),x_2(tau),x_3(tau))Leftrightarrow bigg|frac{dgamma}{dtau}bigg|^2=1$$
it has the units of time, and has the interpretation of being what a clock traveling along that geodesic would display. It is the parameterization that ensures a constant speed of 1 w.r.t. the parameter $tau$.
Above I presented just the definitions as best as I could without going full math mode. Let us make contact with observers, and what has been mentioned in the post. Asymptotic observers are thought of experiencing a flat metric, (so Minkowski if you will), and it simply happens that their proper time might coincide with the coordinate time as defined above, therefore the terminology and the usage. Notice how coordinate time does not depend on any geodesic, it is only dependent on our coordinate choice, while proper time is different for every geodesic but its intervals will not depend on our choice of coordinates, it is an intrinsic property of the geodesic.
To address the last part of your question. Events are points in space-time, for example
$$(t_1,x_1^1,x_1^2,x_1^3)$$
$$(t_2,x_2^1,x_2^2,x_2^3)$$
where I have used the same name for the coordinates as before. This points as they are written have coordinate times $t_1$ and $t_2$ and you may subtract them to find the coordinate time interval. Nonetheless I can speak about the same points in many different ways, I can change the coordinates all together, or if I happen to have geodesics that go through them, one could describe them by the value of the parameter of the geodesic when it goes through those points. Take this just as an invitation to think about the geometry of the situation. To close, one could say that for certain space-time metrics that are asymptotically flat, the time in a clock of a far away observer (its proper time) coincides with coordinate time, so the time intervals he measure will be intervals of coordinate time as well.

Adam Herbst · 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.

[EDIT: Here is a picture to illustrate the concept.  You can imagine that these geodesics run along a curved surface which represents the spacetime coordinates of the observer.  Just keep in mind that this isn't "to scale", not only because real spacetime is 4D, but because the metric itself is hyperbolic.  For example, if you drew a path representing a photon, it's proper time between any two events would be zero.]
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.

benrg · Answer

Coordinate time is like coordinate position: it's a coordinate. Coordinates in relativity are like coordinates in ordinary geometry, and you can carry over a lot of your intuition from ordinary geometry.
Sometimes there may not be a time coordinate. For example, in Eddington-Finkelstein coordinates for a Schwarzschild black hole, all of the coordinate axes (including the one called "$t$") point in a spacelike direction inside the event horizon. This doesn't have any physical significance. There are still timelike directions inside the event horizon, it's just that none of the coordinates of this particular arbitrary coordinate system happen to point that way. You can still talk about $Δt$ inside the horizon as long as you understand that it's spacelike.
Proper time is the length of a worldline. This is the elapsed time recorded by a stopwatch with that worldline, or the amount you'll age if it's your worldline. Unlike coordinate time, it's always a physically meaningful quantity (at least if some actual object has that worldline).

the coordinate time $Δt$ along a path between two events is the time between the two events measured by a faraway observer

$Δt$ is just the $t$ coordinate  of one event minus the $t$ coordinate of another. It's independent of any path between them. Generally, this is as meaningless as the $x$ coordinate of one point minus the $x$ coordinate of another point would be in Euclidean geometry. If it does have some meaning, it's usually because it happens to be equal to some proper time.
It's definitely not true in general that $Δt$ is "the time measured by a faraway observer". It can be true for specific coordinate systems in specific experiments. For example, if there are two rocket ships at rest relative to a Schwarzschild black hole and the first one emits two light pulses and the second detects them, in the limit that the second ship is infinitely far from the hole, the proper time between the detection events will equal the $Δt$ of the emission events if you're using the $t$ coordinate of Schwarzschild or Eddington-Finkelstein coordinates. It won't be equal if the ships are moving, or if you're using Kruskal-Szekeres coordinates.

Difference between coordinate time and proper time in general relativity

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