Are clocks used in general relativity?

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!

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
Gold Member
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. Lets say we are in the rest frame of an observer ##O## whose worldline passes through two events ##A## and ##B##. Furthermore lets 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.

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 $$\int_{A}^{B}\sqrt{dt^{2} - dx^{2} - dy^{2} - dz^{2}} = \int_{A}^{B}\sqrt{dt^{2}(1 - ((\frac{\mathrm{d} x}{\mathrm{d} t})^{2} +(\frac{\mathrm{d} y}{\mathrm{d} t})^{2} + (\frac{\mathrm{d} z}{\mathrm{d} t})^{2})} = \int_{A}^{B}dt\sqrt{1 - v^{2}} = \int_{A}^{B}\frac{dt}{\gamma }$$ and in the case of inertial motion this just becomes ##\Delta \tau _{AB} = \frac{\Delta t_{AB}}{\gamma }## so yes you think correctly.

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ghwellsjr
Gold Member
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
Gold Member
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.

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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!

ghwellsjr
Gold Member
I'm just re-emphasizing what both of you stated early on in this thread:
All clocks always record proper time...
The clocks still measure proper time regardless...
... to answer the OP's question:
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?
... and in contrast to his statement:
I realize that proper time is measured between two events...
His statement has no meaning apart from an additional specification of the arbitrary history of a clock that spans between those two events, if there is one, which there doesn't have to be.
How would you even make non-trivial sense of proper time for a single event?
Every tick of every clock can be considered 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.
You are talking about how to calculate the difference between two Proper Times on a particular clock--a delta Proper Time or a Proper Time interval.
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.
We don't have to start with a frame and describe the worldline of a clock and then calculate the advance of its Proper Time, we can start with a description of the Proper Time and then calculate what the Coordinate Time is for any arbitrary frame.
Again how would you even make non-trivial sense of proper time for a single event?
We often talk about two clocks with a relative speed between them and at the moment of their colocation, we synchronize them. That's a single event concerning the Proper Time on two clocks.
Proper time is an elapsed time whether you are looking at the infinitesimal form or the integrated form.
To me, this is no different than the terminology we apply to Coordinate Time. If we talk about Coordinate Time, we mean the time that is advancing throughout the reference frame with respect to its origin. If we want to talk about how long it takes for something to get from event A to event B, we don't say that is a Coordinate Time, we say it is a Coordinate Time interval or delta or an elapsed time or something similar.

I think the same thing should apply to avoid confusion with regard to Proper Time. Unlike Coordinate Time which is the same everywhere throughout an IRF, each individual clock can have a different Proper Time on it. We can consistently talk about the Proper Time on each clock at each event and if we care about an elapsed time between two events involving a single clock, then we subtract the two Proper Times at those two events, just like we do for Coordinate Time and we call it something like an elapsed time or a delta time or a time difference or an accumulated time or an amount of aging or something similar but we should not call it simply the Proper Time any more than we would call it the Coordinate Time. But unlike for Coordinate Time, we cannot talk about the delta time between any two events unless there happens to be a clock that is present at those two events and we know its history. And if there are two or more such clocks, then there are two or more delta Proper Times.

Every tick of every clock can be considered a single event.
But a single event is not proper time.
However there is proper time between two ticks of a clock, again proper time it is a path between two events.

You are talking about how to calculate the difference between two Proper Times on a particular clock--a delta Proper Time or a Proper Time interval.
Nonsense, proper time is always a path between two events.

ghwellsjr
Gold Member
The OP has stated some erroneous concepts in his first post:
In special relativity clocks are used to record events and proper time gives the time in an inertial frame so two times are used.
Suppose he had said:

In special relativity rulers are used to record events and proper length gives the distance in an inertial frame so two distances are used.

Wouldn't you feel the need to point out where he is confused and to make clear the difference between Coordinate Length and Proper Length? Would you state uncategorically that Proper Length is always associated between two events?

My point is that you can pick any two events and in every reference frame there will always be a Coordinate Time associated with them (and a Coordinate Length) but you cannot say that there will always be a Proper Time (or a Proper Length) associated with them. And you can always say that wherever there is a Proper Time (or a Proper Length), it is never associated with any reference frame.

The OP has stated some erroneous concepts in his first post:

Wouldn't you feel the need to point out where he is confused and to make clear the difference between Coordinate Length and Proper Length? Would you state uncategorically that Proper Length is always associated between two events?

My point is that you can pick any two events and in every reference frame there will always be a Coordinate Time associated with them (and a Coordinate Length) but you cannot say that there will always be a Proper Time (or a Proper Length) associated with them. And you can always say that wherever there is a Proper Time (or a Proper Length), it is never associated with any reference frame.
The thread took a roundabout path but your description cleared it up for me. The proper time or length represents the physical content and the reference frame is used to give a description if one is possible.