Are clocks used in general relativity?

[nortonian]

In special relativity clocks are used to record events and proper time gives the time in an inertial frame so two times are used. In general relativity I only see proper time being used. Is there an implicit understanding about clocks?

 

[Passionflower]

In special relativity clocks are used to record events and proper time gives the time in an inertial frame so two times are used. In general relativity I only see proper time being used. Is there an implicit understanding about clocks?

All clocks always record proper time, unless they are defect.

 

[Chestermiller]

In special relativity clocks are used to record events and proper time gives the time in an inertial frame so two times are used. In general relativity I only see proper time being used. Is there an implicit understanding about clocks?

GR does not only use proper time. The time parameter in the Schwartzschild metric is not proper time, except in the limit of large distances from the central mass.

 

[bcrowell]

In special relativity clocks are used to record events and proper time gives the time in an inertial frame so two times are used. In general relativity I only see proper time being used. Is there an implicit understanding about clocks?

Proper time is the same thing as clock time in both SR and GR.

GR does not only use proper time. The time parameter in the Schwartzschild metric is not proper time, except in the limit of large distances from the central mass.

This is incorrect. Proper time plays the same role in GR that it does in SR. Your example of the time in the Schwarzschild metric (presumably expressed in Schwarzschild coordinates) is an example of a timelike coordinate in GR. Timelike coordinates exist in GR and are not the same thing as proper time.

 

[Passionflower]

Your example of the time in the Schwarzschild metric (presumably expressed in Schwarzschild coordinates) is an example of a timelike coordinate in GR. Timelike coordinates exist in GR and are not the same thing as proper time.

Then what is incorrect about Chestermiller's response?

 

[WannabeNewton]

Then what is incorrect about Chestermiller's response?

The clocks still measure proper time regardless (as you and Ben and others noted). Coordinate time is irrelevant. In fact coordinate time is present in SR too and is not the same as proper time there as well in general (unless of course we are in the rest frame of the observer whose worldline actually passes through the two events in between which the proper time is being measured) so there is no value in pointing out there are "other kinds of time" in GR as if this was special to GR. Clocks measure proper time - doesn't matter if it's GR or SR.

 

[Passionflower]

The clocks still measure proper time regardless (as you and Ben and others noted). Coordinate time is irrelevant. In fact coordinate time is present in SR too and is not the same as proper time there as well in general (unless of course we are in the rest frame of the observer whose worldline actually passes through the two events in between which the proper time is being measured) so there is no value in pointing out there are "other kinds of time" in GR as if this was special to GR. Clocks measure proper time - doesn't matter if it's GR or SR.

I must be missing something, what do you think is wrong about Chestermiller's response?

 

[WannabeNewton]

I must be missing something, what do you think is wrong about Chestermiller's response?

There is nothing wrong with it as it stands of course but it doesn't really relate to the OP's question. The notion / existence of coordinate time doesn't change the fact that the time measured on an observer's clock between two events in space - time is the proper time.

 

[nortonian]

Proper time is the same thing as clock time in both SR and GR.

The notion / existence of coordinate time doesn't change the fact that the time measured on an observer's clock between two events in space - time is the proper time.

All clocks always record proper time, unless they are defect.

So in a time dilation experiment we have an atomic clock in an airplane which has a proper time and as it moves it passes through events that use the Earth's proper time as a coordinate value in order to measure time dilation. The two times are not the same. So how can you have an event with two time coordinates?

GR does not only use proper time. The time parameter in the Schwartzschild metric is not proper time, except in the limit of large distances from the central mass.

You are saying that you can use either proper time or coordinate time depending on where you use it? Is there a convention to know when?

I realize that proper time is measured between two events, but Earth time is a continuum so that shouldn't matter when they are being compared.

 

[ghwellsjr]

So in a time dilation experiment we have an atomic clock in an airplane which has a proper time and as it moves it passes through events that use the Earth's proper time as a coordinate value in order to measure time dilation. The two times are not the same. So how can you have an event with two time coordinates?

Every coordinate system assigns a different time to each event. Just like you can refer to the rover landing on Mars in Eastern, Mountain, Pacific Time or any of many other coordinate systems. If a clock is stationary in an Inertial Reference Frame (IRF) then its Proper Time will tick at the same rate as the Coordinate Time for that IRF. The significance of Coordinate Time is that it ticks everywhere simultaneously whereas individual clocks, say those on earth, even if they tick at the same rate as the Coordinate Time only tick at the location where they are at.

You are saying that you can use either proper time or coordinate time depending on where you use it? Is there a convention to know when?

You always use Coordinate Time when specifying the time coordinate of an event in a particular IRF.

I realize that proper time is measured between two events, but Earth time is a continuum so that shouldn't matter when they are being compared.

You can only use the Proper Time on a real clock to measure the Coordinate Time between two events if those two events are at the same location in the selected IRF.

Another subject is the spacetime interval between two events. If that interval is timelike, then you can use a real clock that is inertial to measure the spacetime interval if it is present at those two events. In other words, in the IRF in which those two events are at the same location, a clock that is also stationary in that IRF and present at the same location will tick at the same rate as the Coordinate Time in that IRF between those two events.

 

[WannabeNewton]

Coordinate time is different from proper time. Let's say we are in the rest frame of an observer ##O## whose worldline passes through two events ##A## and ##B##. Furthermore let's say ##O## has set up his frame with coordinates ##(t,x,y,z)##. In the rest frame of ##O## he measures no spatial displacement when he goes from ##A## to ##B## of course so ##\Delta \tau_{AB} = \Delta t_{AB}## so the time measured on a clock held by ##O## coincides the coordinate time in the coordinates set up by ##O## with the proper time, between the two events.

Let's say now we have another observer ##O'## who is moving with velocity ##v## in the ##x## direction relative to ##O## and say ##O'## has set up in his frame coordinates ##(t',x',y',z')## (assume they have synchronized their clocks etc.). According to ##O'## the proper time for ##O## to go from ##A## to ##B## will be ##\Delta \tau'_{AB} = \sqrt{\Delta t_{AB}'^{2} - \Delta x_{AB}'^2}##.

We know how the coordinates ##(t,x,y,z)## of ##O## will related to the coordinates ##(t',x',y',z')## of ##O'## through the lorentz transformations ##t' = \gamma (t - vx), x' = \gamma (x - vt)##. We see that ##\Delta t_{AB}'^{2} = \gamma^{2} \Delta t_{AB}^{2}, \Delta x_{AB}' = \gamma ^{2}v^2\Delta t_{AB}^{2}## so ##\Delta \tau'_{AB} = \sqrt{\gamma ^{2}\Delta t_{AB}^{2}(1 - v^2)} = \sqrt{\gamma ^{2}\Delta t_{AB}^{2}\gamma ^{-2}} = \Delta t_{AB} = \Delta \tau_{AB}##. Indeed, both ##O## and ##O'##'s measurements yield the same proper time between events ##A## and ##B## even though ##O## has set up different coordinates in his frame from those of ##O'##. Proper time is a frame invariant quantity in this sense whereas the coordinate times ##t## and ##t'## are not.

 

[WannabeNewton]

You can only use the Proper Time on a real clock to measure the Coordinate Time between two events if those two events are at the same location in the selected IRF.

In order to avoid confusion that *might* arise, this is the same spatial location. There might be an ambiguity because "location" in SR can be interpreted as an event.

 

[ghwellsjr]

In order to avoid confusion that *might* arise, this is the same spatial location. There might be an ambiguity because "location" in SR can be interpreted as an event.

Well then, they are the same event, aren't they?

 

[WannabeNewton]

Well then, they are the same event, aren't they?

How do you mean? There is a temporal displacement between the two events even if there is no spatial displacement (I'm talking about two distinct events, both lying on the image of observer ##O##'s wordline and doing measurements in observer ##O##'s rest frame).

 

[ghwellsjr]

Coordinate time is different from proper time. Let's say we are in the rest frame of an observer ##O## whose worldline passes through two events ##A## and ##B##. Furthermore let's say ##O## has set up his frame with coordinates ##(t,x,y,z)##. In the rest frame of ##O## he measures no spatial displacement when he goes from ##A## to ##B## of course so ##\Delta \tau_{AB} = \Delta t_{AB}## so the time measured on a clock held by ##O##, the proper time, is just the coordinate time in the coordinates set up by ##O##.

Let's say now we have another observer ##O'## who is moving with velocity ##v## in the ##x## direction relative to ##O## and say ##O'## has set up in his frame coordinates ##(t',x',y',z')## (assume they have synchronized their clocks etc.). According to a clock held by ##O'## the proper time for ##O## to go from ##A## to ##B## will be ##\Delta \tau'_{AB} = \sqrt{\Delta t_{AB}'^{2} - \Delta x_{AB}'^2}##.

We know how the coordinates ##(t,x,y,z)## of ##O## will related to the coordinates ##(t',x',y',z')## of ##O'## through the lorentz transformations ##t' = \gamma (t - vx), x' = \gamma (x - vt)##. We see that ##\Delta t_{AB}'^{2} = \gamma^{2} \Delta t_{AB}^{2}, \Delta x_{AB}' = \gamma ^{2}v^2\Delta t_{AB}^{2}## so ##\Delta \tau'_{AB} = \sqrt{\gamma ^{2}\Delta t_{AB}^{2}(1 - v^2)} = \sqrt{\gamma ^{2}\Delta t_{AB}^{2}\gamma ^{-2}} = \Delta t_{AB} = \Delta \tau_{AB}##. Indeed, both ##O## and ##O'##'s clocks measure the same proper time between events ##A## and ##B## even though ##O## has set up different coordinates in his frame from those of ##O'##. Proper time is a frame invariant quantity in this sense whereas the coordinate times ##t## and ##t'## are not.

This, to me, is very confusing. I can't tell if there is just one clock whose Proper Time interval is being "measured" in two different frames or if there are two clocks that both "measure" the same Proper Time interval between a pair of events.

 

[WannabeNewton]

It's the first - a clock passing through the two events. Was the wording ambiguous? Perhaps: the measurements made by ##O## and ##O'## regarding the clock. I'll change it on account of what you said.

 

[nortonian]

The significance of Coordinate Time is that it ticks everywhere simultaneously whereas individual clocks, say those on earth, even if they tick at the same rate as the Coordinate Time only tick at the location where they are at.

You always use Coordinate Time when specifying the time coordinate of an event in a particular IRF.

You can only use the Proper Time on a real clock to measure the Coordinate Time between two events if those two events are at the same location in the selected IRF.

Another subject is the spacetime interval between two events. If that interval is timelike, then you can use a real clock that is inertial to measure the spacetime interval if it is present at those two events. In other words, in the IRF in which those two events are at the same location, a clock that is also stationary in that IRF and present at the same location will tick at the same rate as the Coordinate Time in that IRF between those two events.

Just to make sure I am on the same page, coordinate time refers to the euclidean space tangent to curved space-time, right? and I assume Euclidean space would also be used to describe the stress-energy.

 

[Passionflower]

Just to make sure I am on the same page, coordinate time refers to the euclidean space tangent to curved space-time, right?

No, coordinate time can be anything.

Imagine a 2 year old drawing horizontal and vertical wiggly (but not so much as to cross its direct neighbor) but smooth great circles on a football, then that is a perfectly valid coordinate system.

 

[WannabeNewton]

Just to make sure I am on the same page, coordinate time refers to the euclidean space tangent to curved space-time, right? and I assume Euclidean space would also be used to describe the stress-energy.

This is getting into the more advanced realm of differential geometry and general relativity (GR) and away from the physics of special relativity (SR) but we can take a crack at it.

There is no a prioi relationship between the coordinates set up by an observer in a neighborhood of space - time and the tangent space to space - time at some point in this neighborhood (by the way this tangent space is not Euclidean space itself but is isomorphic to it in the category of vector spaces).

Let ##M## be a space - time and let ##O## be an observer. On an open subset ##U\subseteq M##, observer ##O## can set up coordinates ##(t,x^1,x^2,x^3)## on ##U## so that he can represent events ##p\in U## in terms of his coordinates; this is a way of representing measurements made by ##O## using his measuring apparatus (three perpendicular meter sticks and a clock). The ##t:U\rightarrow \mathbb{R}## coordinate function is the coordinate time. As you can see, in its bare form it has nothing to do with the tangent space to ##M## at ##p## (denoted ##T_p(M)##) which is, as its name suggests, the set of all vectors tangent to ##M## at ##p##. I can go on to describe how to relate the coordinates set up by this observer to vectors in this tangent space but I fear it might get too abstract and it is not really related to the thread question anyways.

Your final sentence, "and I assume Euclidean space would also be used to describe the stress energy" doesn't make sense to me unfortunately.

 

[ghwellsjr]

You can only use the Proper Time on a real clock to measure the Coordinate Time between two events if those two events are at the same location in the selected IRF.

In order to avoid confusion that *might* arise, this is the same spatial location. There might be an ambiguity because "location" in SR can be interpreted as an event.

Well then, they are the same event, aren't they?

How do you mean? There is a temporal displacement between the two events even if there is no spatial displacement (I'm talking about two distinct events, both lying on the image of observer ##O##'s wordline and doing measurements in observer ##O##'s rest frame).

If you say that the word "location" can mean "event" and isn't specific enough because it might denote something more than "spatial location" then it seems that something more must be time. So if I say two events are at the same location, then it seems to me you are saying that my statement can be interpreted that the two events (both their spatial and temporal locations) are all the same.

 

[WannabeNewton]

So if I say two events are at the same location, then it seems to me you are saying that my statement can be interpreted that the two events (both their spatial and temporal locations) are all the same.

Yes, but I'm not saying everyone will of course - it just might be a possibility. What you said in your original post is true either way, trivial in the case of two identical events, but if someone does get mixed up then they might miss the point of your post, which is a crucial point in understanding the relationship between proper time and clocks. Cheers friend :)!

 

[ghwellsjr]

So if I say two events are at the same location, then it seems to me you are saying that my statement can be interpreted that the two events (both their spatial and temporal locations) are all the same.

Yes, but I'm not saying everyone will of course - it just might be a possibility.

Can you please provide an authoritative online reference for "location" that defines it as "time"? Thanks.

What you said in your original post is true either way, trivial in the case of two identical events, but if someone does get mixed up then they might miss the point of your post, which is a crucial point in understanding the relationship between proper time and clocks. Cheers friend :)!

Clocks are associated with Proper Time. Events are associated with Coordinate Time. That is the point of my post. Can someone get that mixed up?

 

[WannabeNewton]

Can you please provide an authoritative online reference for "location" that defines it as "time"? Thanks.

A location in space - time is an event, not just a spatial location. It's an elementary definition. If you are talking about observers in some neighborhood of space - time and talk about their "location" in space - time this is by definition an element on the image of their wordline, which is a subset of space - time, thus the element is an event. Do you actually need a reference for that? Take a look at any GR textbook.

 

[nortonian]

Imagine a 2 year old drawing horizontal and vertical wiggly (but not so much as to cross its direct neighbor) but smooth great circles on a football, then that is a perfectly valid coordinate system.

You are talking about curved space-time. I am talking about a measurement taken from the curved metric. In order to take a measurement (get a real number) you have to evaluate the metric on a locally Euclidean space. See Wikipedia below. Maybe I didn't use all the right words but that's what I wanted to say. How is that wrong?

Explicitly, the metric is a symmetric bilinear form on each tangent space of M which varies in a smooth (or differentiable) manner from point to point. Given two tangent vectors u and v at a point x in M, the metric can be evaluated on u and v to give a real number:

This can be thought of as a generalization of the dot product in ordinary Euclidean space. This analogy is not exact, however. Unlike Euclidean space — where the dot product is positive definite — the metric gives each tangent space the structure of Minkowski space.

 

[ghwellsjr]

Can you please provide an authoritative online reference for "location" that defines it as "time"? Thanks.

A location in space - time is an event, not just a spatial location. It's an elementary definition. If you are talking about observers in some neighborhood of space - time and talk about their "location" in space - time this is by definition an element on the image of their wordline, which is a subset of space - time, thus the element is an event. Do you actually need a reference for that? Take a look at any GR textbook.

OK, I looked in several relativity textbooks as well as online references and I did not find any that defined "location" as "time" which is what I asked for. I did find some that referred to a "spacetime location" as an "event" which is something that I was not previously aware of so I thank you for bringing this to my attention.

It's interesting that in the two editions of Taylor and Wheeler, they changed their terminology. The first edition, from the 60's when I had my formal education uses the term "location" in the sense that I used it, to mean "spatial position", a term they also use in both editions. They talk about the location of an event and never refer to "spacetime location". In the second edition, from 1992, they changed the phrase "coordinates of an event" to "spacetime location of an event".

I thought when you started this issue that you were saying that just like we have three spatial coordinates and one temporal coordinate for an event, a spacetime location (or event) is made up of three spatial locations and one temporal location. But now I think your point is that the confusion is not over spatial versus temporal, but over spatial location versus spacetime location (or event, as you said). However, I really don't see how anyone could be confused by my statement:

You can only use the Proper Time on a real clock to measure the Coordinate Time between two events if those two events are at the same location in the selected IRF.

They would have to think I meant:

You can only use the Proper Time on a real clock to measure the Coordinate Time between two events if those two events are at the same event in the selected IRF.

I think the context makes it abundantly clear that when I talk about the location of two events, I'm not saying they are the same event, otherwise, I would have said that.

 

[WannabeNewton]

It wasn't really a complaint George. It is just that the word "location" can be interpreted in different ways as you have noted from your texts i.e. spatial location vs event (I don't think I ever referred to location as being "time" - I checked back to my previous posts and couldn't find anything where I said location was "time"). If a person can interpret from context then great but it can't hurt to be too careful is all; it is a minor remark on my part of course, the main bulk of your post was spot on.

As a side note, PF is waaaaaay too distracting lol. I have an exam to study for and I'm still on the site :p. Cheers!

 

[Passionflower]

You are talking about curved space-time. I am talking about a measurement taken from the curved metric. In order to take a measurement (get a real number) you have to evaluate the metric on a locally Euclidean space. See Wikipedia below. Maybe I didn't use all the right words but that's what I wanted to say. How is that wrong?

The term "Euclidean" is sometimes used for a particular signature of a metric but it is also used to indicate that a (hyper) surface is flat. The two meanings have nothing to do with each other. So it often leads to confusion.

 

[ghwellsjr]

I have finally had a chance to study your modified post and I have some concerns now that I understand what you are talking about:

Coordinate time is different from proper time. Let's say we are in the rest frame of an observer ##O## whose worldline passes through two events ##A## and ##B##. Furthermore let's say ##O## has set up his frame with coordinates ##(t,x,y,z)##. In the rest frame of ##O## he measures no spatial displacement when he goes from ##A## to ##B## of course so ##\Delta \tau_{AB} = \Delta t_{AB}## so the time measured on a clock held by ##O## coincides the coordinate time in the coordinates set up by ##O## with the proper time, between the two events.

You apparently are equating the Proper Time on a clock with the Spacetime Interval between two events. This is what I was talking about in my post prior to yours where I said:

Another subject is the spacetime interval between two events. If that interval is timelike, then you can use a real clock that is inertial to measure the spacetime interval if it is present at those two events. In other words, in the IRF in which those two events are at the same location, a clock that is also stationary in that IRF and present at the same location will tick at the same rate as the Coordinate Time in that IRF between those two events.

Let's say now we have another observer ##O'## who is moving with velocity ##v## in the ##x## direction relative to ##O## and say ##O'## has set up in his frame coordinates ##(t',x',y',z')## (assume they have synchronized their clocks etc.). According to ##O'## the proper time for ##O## to go from ##A## to ##B## will be ##\Delta \tau'_{AB} = \sqrt{\Delta t_{AB}'^{2} - \Delta x_{AB}'^2}##.

We know how the coordinates ##(t,x,y,z)## of ##O## will related to the coordinates ##(t',x',y',z')## of ##O'## through the lorentz transformations ##t' = \gamma (t - vx), x' = \gamma (x - vt)##. We see that ##\Delta t_{AB}'^{2} = \gamma^{2} \Delta t_{AB}^{2}, \Delta x_{AB}' = \gamma ^{2}v^2\Delta t_{AB}^{2}## so ##\Delta \tau'_{AB} = \sqrt{\gamma ^{2}\Delta t_{AB}^{2}(1 - v^2)} = \sqrt{\gamma ^{2}\Delta t_{AB}^{2}\gamma ^{-2}} = \Delta t_{AB} = \Delta \tau_{AB}##. Indeed, both ##O## and ##O'##'s measurements yield the same proper time between events ##A## and ##B## even though ##O## has set up different coordinates in his frame from those of ##O'##. Proper time is a frame invariant quantity in this sense whereas the coordinate times ##t## and ##t'## are not.

And now you are showing how to calculate the spacetime interval between the same two events from a frame moving with respect to the first one using the new coordinates of the events (although you skipped a lot of steps so I think you are taking a lot of liberty when you say "We see that...").

That's all well and good but it's not related to the more general question of what is Proper Time? You can have two clocks that are both present at the same two events (such as in the initial and final states of the Twin Paradox), one of which remained inertial and the other of which experienced acceleration and they will have different Proper Times between the same two events, the exact opposite of what you presented.

I think, if you want to discuss how observer O' uses his coordinates to determine the Proper Time interval of an inertially moving clock, all he has to do is divide the coordinate time interval by gamma.

 

[Passionflower]

That's all well and good but it's not related to the more general question of what is Proper Time?

The length of a path between two points in spacetime.

 

[WannabeNewton]

That's all well and good but it's not related to the more general question of what is Proper Time? You can have two clocks that are both present at the same two events (such as in the initial and final states of the Twin Paradox), one of which remained inertial and the other of which experienced acceleration and they will have different Proper Times between the same two events, the exact opposite of what you presented.

Yes in the case of the Twin Paradox the proper time won't be the same because there is an acceleration involved. In SR, the proper time between two events is only preserved between inertial observers i.e those related by a lorentz transformation. So if you have an observer moving uniformly relative to the first then you can apply a lorentz transformation and the proper time will remain invariant. However if one is accelerating then yeah it won't work. There will be discrepancy in the measured proper time. As far as your first question goes, "what is proper time", you already gave a definition in your own post as the time read on a clock passing through the two events in between which the proper time is being measured.

I think, if you want to discuss how observer O' uses his coordinates to determine the Proper Time interval of an inertially moving clock, all he has to do is divide the coordinate time interval by gamma.

Sorry, I missed this on a first reading. The proper time, mathematically, is defined as the length of the worldline containing the two events in between which proper time is being measured. In SR, this is given by ##\Delta \tau _{AB} = \int_{A}^{B}\sqrt{dt^{2} - dx^{2} - dy^{2} - dz^{2}}##. But this is just [tex]\int_{A}^{B}\sqrt{dt^{2} - dx^{2} - dy^{2} - dz^{2}} = \int_{A}^{B}\sqrt{dt^{2}(1 - ((\frac{\mathrm{d} x}{\mathrm{d} t})^{2} +(\frac{\mathrm{d} y}{\mathrm{d} t})^{2} + (\frac{\mathrm{d} z}{\mathrm{d} t})^{2})} = \int_{A}^{B}dt\sqrt{1 - v^{2}} = \int_{A}^{B}\frac{dt}{\gamma }[/tex] and in the case of inertial motion this just becomes ##\Delta \tau _{AB} = \frac{\Delta t_{AB}}{\gamma }## so yes you think correctly.

 

[ghwellsjr]

Yes in the case of the Twin Paradox the proper time won't be the same because there is an acceleration involved. In SR, the proper time between two events is only preserved between inertial observers i.e those related by a lorentz transformation. So if you have an observer moving uniformly relative to the first then you can apply a lorentz transformation and the proper time will remain invariant. However if one is accelerating then yeah it won't work. There will be discrepancy in the measured proper time. As far as your first question goes, "what is proper time", you already gave a definition in your own post as the time read on a clock passing through the two events in between which the proper time is being measured.

I'm trying to disconnect the Proper Time of a clock from the Coordinate Time of events. We should not be thinking that there is such a thing as an invariant Proper Time between two events. We should make it clear that the invariant spacetime interval between two events is only the Proper Time on an inertial clock that is present at those two events but other clocks that accelerate between those two events will have a different Proper Time.

 

[WannabeNewton]

We should not be thinking that there is such a thing as an invariant Proper Time between two events.

Indeed it is only for the case of inertial observers that this is true. The proper time is not invariant amongst all frames, just the inertial ones.

We should make it clear that the invariant spacetime interval between two events is only the Proper Time on an inertial clock that is present at those two events but other clocks that accelerate between those two events will have a different Proper Time.

Agreed.

 

[ghwellsjr]

We should not be thinking that there is such a thing as an invariant Proper Time between two events.

Indeed it is only for the case of inertial observers that this is true. The proper time is not invariant amongst all frames, just the inertial ones.

You keep wanting to associate Proper Time with a pair of events and frames instead of with the time on a clock. The Proper Time on any clock is invariant as it has nothing to do with any frame. If we assign a single event to a single Proper Time on a clock, all frames will agree on the Proper Time on the clock but they will all assign different Coordinate times to that single event.

 

[WannabeNewton]

You keep wanting to associate Proper Time with a pair of events and frames instead of with the time on a clock.

How would you even make non-trivial sense of proper time for a single event? Proper time in SR is defined as ##\tau =\int_{\gamma }(-\eta _{ab}u^{a}u^{b})^{1/2}## where ##\gamma## is the worldline of an observer carrying a clock that passes through the two events in between which we are measuring the proper time; different worldlines result in different proper times even if it is between the same two events. The line integral has to have some pair of endpoints otherwise we are just dealing with sets of measure zero and get nothing useful. The infinitesimal version links nearby events on an observer's worldline.

If we assign a single event to a single Proper Time on a clock...

Again how would you even make non-trivial sense of proper time for a single event? Proper time is an elapsed time whether you are looking at the infinitesimal form or the integrated form.

 

[Passionflower]

Again how would you even make non-trivial sense of proper time for a single event? Proper time is an elapsed time whether you are looking at the infinitesimal form or the integrated form.

Exactly!