I Coordinate time between spatially separated events in Schwarzschild spacetime

Edit: I'm leaving the original post as is, but after discussion I'm not confused over coordinate time having a physical meaning. I was confused over a particular use of a coordinate time difference to solve a problem, in which a particular coordinate time interval for a particular choice of coordinates is taken to have a physical meaning that I found confusing and problematic.

I'm reading through Collier's A Most Incomprehensible Thing, and am getting tripped up thinking through the physical meaning of coordinate time between two spatially separated events. Any help would be much appreciated!

In the chapter on black holes, Collier asks what it would be like to watch something fall into a black hole from far away. To do this he uses the Schwarzschild metric to calculate the coordinate time separating two events: a signal emitted at ##r_{1}## and the signal being received by an observer at ##r_{2}##, where ##r## is the radial Schwarzschild coordinate, not a proper distance. He considers a signal along a radial line where ##\theta = \phi = 0##, so that the interval is defined by $$ds^2 = (1 - \frac{R_{s}}{r})c^2dt^2 - \frac{dr^2}{1 - \frac{R_{s}}{r}},$$ ##R_{s}## being the Schwarzschild radius, which corresponds to the event horizon. For a beam of light ##ds = 0##, so this reduces to $$0 = (1 - \frac{R_{s}}{r})c^2dt^2 - \frac{dr^2}{1 - \frac{R_{s}}{r}}.$$ Integrating leads to the relationship $$t_{2} - t_{1} = \Delta t = \frac{r_{2}-r_{1}}{c} + \frac{R_{s}}{c} \ln(\frac{r_{2}-R_{s}}{r_{1}-R_{s}}),$$ which gives the coordinate time separating the event at ##r_{1}## and the event at ##r_{2}##.

Collier lets ##r_{1}## approach ##R_{s}## to show that the coordinate time separating the emission of light at the event horizon of a black hole and the reception of that signal far away is infinite.

I understand that this indicates that a signal sent from the event horizon will never reach a point far away (nor, the math seems to suggest, any point with ##r_{2} > r_{1}##?). But I'm having trouble wrapping my head around the physical meaning of a coordinate time difference between two events that are separated in space. Am I correct in thinking that even though for points far from ##r = 0## the proper time will agree with coordinate time, here the coordinate time difference does not correspond to a proper time difference far away?

My reasoning is that it doesn't seem to make physical sense to think about a proper time difference between an event at ##r_{1}## and an event at ##r_{2}## for ##r_{1} \neq r_{2}## for an observer sitting at ##r_{2}##. If we want to talk about an actual physical period of time between these events, since they're spatially separated, wouldn't we need to talk about the interval of proper time on a world line that connects them? And then isn't the argument that ##\Delta t \to \infty## subtly different than "it takes the faraway observer an infinitely long time to receive the signal"? Collier writes "if we can find the coordinate time taken for a light signal, or for a change in position of the freely falling object, we have automatically found the distant observer's proper time measurement for those two events." But I'm having trouble squaring this with the way I'm thinking about the problem. Any advice?

I'm thinking that one sort of remedy might be to modify the thought experiment a bit so that I'm thinking about events that aren't spatially separated. For example, I could look for the coordinate time difference between an object in free fall emitting a photon near the event horizon and then emitting one at the event horizon. But because coordinate time corresponds to proper time far away, wouldn't this coordinate time difference be the same as the proper time difference between the faraway observer receiving each of those signals?

Thanks in advance for any assistance!
 
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Nugatory

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But I'm having trouble wrapping my head around the physical meaning of a coordinate time difference between two events that are separated in space.
That’s easy - a difference in coordinate time has no physical significance, so you can stop trying.
 
That’s easy - a difference in coordinate time has no physical significance, so you can stop trying.
Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?

If so, the question I posed is about a specific case of that: the text I'm reading seemed to imply that a particular difference in coordinate time corresponded to a difference in proper time as experienced by a particular observer, and I didn't follow the logic of the text's argument.
 

Orodruin

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Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?
There are cases where you can use it to compute and deduce things, yes. However, that does not mean that it holds any particular physical relevance. Ultimately, coordinate time is, well, coordinate dependent.

It also strikes me that you do not seem to grasp what proper time is. Proper time is the time elapsed along a particular world-line. It can be different for different world-lines even if they both start and end at proper times ##t_1## and ##t_2## respectively - even if they start and end at the same event.
 
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If we want to talk about an actual physical period of time between these events, since they're spatially separated, wouldn't we need to talk about the interval of proper time on a world line that connects them?
This is true. But I think the point Collier is making is that if there exists a finite coordinate time that the light reaches the distant observer, then depending on what that observer calls ##t=0## then you could find what time he also assigns to receiving the light.
 
It also strikes me that you do not seem to grasp what proper time is. Proper time is the time elapsed along a particular world-line. It can be different for different world-lines even if they both start and end at proper times ##t_1## and ##t_2## respectively - even if they start and end at the same event.
I understand that, my apologies if I wrote something that was unclear. This is why I was confused when the text I'm reading seemed to imply that the coordinate time difference I described, between events that are spatially separated, could be understood to correspond to some proper time experienced by a faraway observer. I mentioned that a proper time between those events would be defined on a world-line connecting the events, rather than being some globally defined concept by which we can talk about the time elapsed for an observer at ##r_{2}## between events that didn't both happen at ##r_{2}##. The fact that proper time is the time elapsed on a particular world-line is the basis of my question and of my confusion over the way I'm understanding what I've read. My apologies if that wasn't clear.
 

Ibix

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A clock that measures coordinate time in Schwarzschild coordinates in Schwarzschild spacetime is easy enough to make. It needs to have a rocket so that it hovers at constant ##r## and every time it ticks one second it needs to show ##(1-R_s/r)^{-1/2}## seconds elapsed. Also you need to zero clocks at different altitudes - you can do that by exchanging light signals in this case.

You can then measure the coordinate time difference between two spacelike separated events by noting the times on a coordinate clock at each event and taking the difference. However, my first paragraph should show you that the process of "synchronising" the clocks is pretty arbitrary, so the result depends on choices you made and hence doesn't really mean anything. It also shows you that you can't define time coordinates this way at the horizon - primarily because the rocket on the clock would have to be infinitely powerful, but also because the time multiplier is infinite.

Also, you can see that the time multiplier goes to one as ##r## goes to infinity. That means that there is (in the limit) a one-to-one relationship between a distant clock's proper time and the local coordinate time. But there's no relationship between that time and a "coordinate clock" closer to the black hole, except through the synchronisation process that I outlined. Or any other synchronisation process you prefer.
 
This is true. But I think the point Collier is making is that if there exists a finite coordinate time that the light reaches the distant observer, then depending on what that observer calls ##t=0## then you could find what time he also assigns to receiving the light.
Right, and that would have nothing to do with that observer assigning physical meaning to the difference in coordinate time between the light being emitted and the light being received, right?

I think this is the crux of the question I'm trying to ask. You could use the coordinate time difference between those events to deduce when the observer should receive the signal, but that difference does not correspond to an actual physical difference in time, such as "the time between the light being emitted and the light being received." Correct? That physical difference (i.e. in some proper time along some corresponding world-line) seems to be what the text implies, which is what threw me for a loop.

To be as clear as possible, the exact phrasing of the text is: "We saw when looking at gravitational time dilation that the coordinate time ##t## of an event in Schwarzschild spacetime is the same as the proper time ##\tau## measured by a stationary distant observer, i.e. ##d\tau_{\infty} = dt##. This means that if we can find the coordinate time taken for a light signal, or for a change in position of the freely falling object, we have automatically found the distant observer's proper time measurement for those two events." Emphasis is mine, to highlight exactly what's causing confusion.

"The coordinate time ##t## of an event in Schwarzschild spacetime is the same as the proper time ##\tau## measured by a stationary distant observer" actually seems fine to me as long as you're strictly talking about events happening infinitely far from ##r = 0##. Unless I'm seriously misreading, the text seems to argue, en route to the claim that ##t_{2}- t_{1}## can be understood as a proper time as measured by the observer receiving the signal, that the coordinate time ##t_{1}## at which the signal is emitted corresponds to the proper time at which it is emitted for the observer who is located somewhere else. The text also states "Because coordinate time is the same as a distant observer's proper time, such an observer will never quite see the object reach the event horizon." Maybe he means to imply only that ##t_{2}##, the time at which the observer receives the signal, can be understood as equivalent to a proper time on a clock located infinitely far away? In fact the more I reread what's written, the more I think this must be what he means.

To everyone, sorry about getting hung up on "physical significance." The text I'm reading seems to imply that in the thought experiment coordinate time is identical to a proper time in a circumstance where I don't understand that as being possible. Regardless of how we philosophically understand "physically significant," my problem was that I couldn't square that implication with the definition of proper time.
 
That means that there is (in the limit) a one-to-one relationship between a distant clock's proper time and the local coordinate time. But there's no relationship between that time and a "coordinate clock" closer to the black hole, except through the synchronisation process that I outlined.
Fantastic, this is what I was looking for as a sanity check - thank you.

As I suggested in my last post, I may very possibly just be misreading the text, but regardless I'm glad I'm not totally off base about my (possibly mistaken) interpretation of it being problematic.
 

Ibix

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The text I'm reading seems to imply that in the thought experiment coordinate time is identical to a proper time in a circumstance where I don't understand that as being possible.
For a distant observer at rest with respect to the black hole and using Schwarzschild coordinates, the coordinate time between event A, on the observer's worldline and declared to be simultaneous with the event at ##r_1##, and event B, also on the observer's worldline and declared to be simultaneous with the event at ##r_2##, is equal to the observer's elapsed proper time.

This tells you a fact about Schwarzschild coordinates: far from the black hole they are based on the "natural" coordinates of an inertial observer at rest with respect to the black hole. It doesn't really tell you anything about the physics.
 

Nugatory

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Surely there are at least cases in which coordinate time can inform us about physical reality, or else it wouldn't be useful to talk about it? Can we think about something like a map between it and proper time?
Yes, and indeed that is how we use it. Coordinate time has no physical significance because the relationship between coordinate time and proper time (which does have physical significance) is pretty much completely arbitrary; as long as we respect certain mathematical niceties, we can choose just about any rule for associating time coordinate values with events.

This freedom to choose a mapping allows us to choose a mapping that works well in whatever problem we’re working with at the moment; in practice we generally choose a mapping to make it easy to do exactly that.
For example, if I am floating in empty space it is very natural (so natural that we usually don’t stop to consider that I have other options and am making a choice) to consider myself to be at rest in space and choose to assign time coordinates in such a way that if light from an event at a distance DD away from me reaches my eyes at time TT, then the time coordinate of that event is T−(D/c)T−(D/c). Using this convention, I can easily calculate the proper time between two events on a distant spaceship: if the spaceship is at rest relative to me the proper time between the events is equal to the difference in the time coordinates I have assigned to them, and if it is moving relative to me the proper time will be that difference divided by √1−v2/c21−v2/c2. But this doesn’t mean that the coordinate times have any physical significance; it means that I’ve chosen them in a way that makes it easy to calculate the proper time between events on the distant spaceship.

Of course for this problem it would be perverse of me to choose any other rule for assigning time coordinates, but that’s just because this is an unusually simple problem with an obvious convention right under our noses. Your more complicated black hole example isn’t that way.
 
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physical meaning of coordinate time
In general there is no physical meaning of coordinate time.

The only exception is if the meaning is built in to a specific coordinate chart. Then that information is contained in the derivation or the description of the coordinates.
 
How would you all propose interpreting a claim like "we know that in Schwarzschild spacetime, coordinate time ##t## is the same as proper time ##\tau## as measured by a distant stationary observer."?

This is stated shortly after the other quote I mentioned above. It's the sort of claim that's giving me trouble, because it doesn't seem true unless we're talking specifically about an event on the world-line of an observer who is at rest very far from the central mass. Can we choose a convention that makes it "true" generally, similar to what Nugatory suggested earlier?

For example, if I am floating in empty space it is very natural (so natural that we usually don’t stop to consider that I have other options and am making a choice) to consider myself to be at rest in space and choose to assign time coordinates in such a way that if light from an event at a distance ##D## away from me reaches my eyes at time ##T##, then the time coordinate of that event is ##T−(D/c)##. Using this convention, I can easily calculate the proper time between two events on a distant spaceship
Can we use a similar conventional tactic to say something meaningful about the time, for the observer, at which an event near the black hole occurs? And then would Collier's claim above be made true by convention?
 
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How would you all propose interpreting a claim like "we know that in Schwarzschild spacetime, coordinate time ttt is the same as proper time ττ\tau as measured by a distant stationary observer."?
I would interpret it as wrong. The coordinate time is defined throughout spacetime (except at the horizon). The proper time of a distant stationary observer is only defined along the worldline of the distant stationary observer. Since they are defined over different regions they are clearly not the same, although where both are defined they do agree.
 
Since they are defined over different regions they are clearly not the same, although where both are defined they do agree.
Thank you. That's what I thought. It was starting to drive me insane.
 

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