Question about Two Clocks in this Relativity scenario

[ultramarinus_regis_8]

TL;DR Summary

Chapter 1.8 of Schutz: Time Dilation

I have been enjoying learning relativity from the book "A First Course on General Relativity". I came across a somewhat confusing experiment mentioned in Chapter 1.8, which I have attached here:

Namely, I am referring to how you might measure $t=0$ as "the difference in readings of two clocks at rest on $\mathcal{O}$." I can't understand how such an experiment would be done, for I would think that the difference between two clocks, one at the origin and one at the spatial position of event B with worldlines parallel to $t$, is zero, if both are allowed to run until event B.

Should I instead understand this as the difference between the readings of two clocks, one which is positioned at t = 0, and another which is positioned at $t = B$?

But, the "proper time" is also the difference between times of two clocks. It's the difference between "two clocks, one at the origin, and one which passes through event B".

I'm not sure what I am missing from the statements made in the passages I have enclosed. Thanks very much for your help.

 

[PeterDonis]

how you might measure $t=0$ as "the difference in readings of two clocks at rest on $\mathcal{O}$."

It doesn't say $t$ is the difference in readings of two clocks. It says $\Delta t$ is. They're not the same thing.

I can't understand how such an experiment would be done, for I would think that the difference between two clocks, one at the origin and one at the spatial position of event B with worldlines parallel to $t$, is zero, if both are allowed to run until event B.

The difference $\Delta t$ is the difference between the reading of the clock at the origin, when the moving clock passes it, and the reading of the clock at $\mathcal{B}$, when the clock passes it. So you're only looking at the clock at $\mathcal{B}$ at event B. You're not looking at the clock at the origin at the same time in frame $\mathcal{O}$ as event B.

Should I instead understand this as the difference between the readings of two clocks, one which is positioned at t = 0, and another which is positioned at $t = B$?

You can't position a clock at a time. What determines the events at which you take the readings of the two clocks at rest, is when the moving clock passes them, i.e., at the intersections of the worldlines.

the "proper time" is also the difference between times of two clocks

No, it isn't. It's the difference in readings of a single clock, the moving one, between the same two events--the event where its worldline crosses the worldline of the origin clock, and the event where its worldline crosses the worldline of the $\mathcal{B}$ clock.

 

[ultramarinus_regis_8]

I see, thanks so much @PeterDonis! I was confused on several points, such as the difference between events and readings of clocks. I think I misunderstood what Schutz was saying in this case. Maybe what is meant, more verbosely, is that "the difference in readings of two clocks in frame $\mathcal{O}$, one which passes through the origin and one which passes through the spatial components of event $\mathcal{B}$, the first of which is measured at $t = 0$, and the second of which is measured at the occurence of event $\mathcal{B}$, is equal to $\Delta t$."

Is this more accurate (and more wordy, I presume)?

 

[PeterDonis]

"the difference in readings of two clocks in frame $\mathcal{O}$, one which passes through the origin and one which passes through the spatial components of event $\mathcal{B}$, the first of which is measured at $t = 0$, and the second of which is measured at the occurence of event $\mathcal{B}$, is equal to $\Delta t$."

Yes, this is what Schutz was describing.

 

[PeroK]

TL;DR Summary: Chapter 1.8 of Schutz: Time Dilation

I have been enjoying learning relativity from the book "A First Course on General Relativity". I came across a somewhat confusing experiment mentioned in Chapter 1.8, which I have attached here:

View attachment 347251

View attachment 347252

Namely, I am referring to how you might measure $t=0$ as "the difference in readings of two clocks at rest on $\mathcal{O}$." I can't understand how such an experiment would be done, for I would think that the difference between two clocks, one at the origin and one at the spatial position of event B with worldlines parallel to $t$, is zero, if both are allowed to run until event B.

Should I instead understand this as the difference between the readings of two clocks, one which is positioned at t = 0, and another which is positioned at $t = B$?

But, the "proper time" is also the difference between times of two clocks. It's the difference between "two clocks, one at the origin, and one which passes through event B".

I'm not sure what I am missing from the statements made in the passages I have enclosed. Thanks very much for your help.

The point to note that is that $\Delta \bar t$ is the unambiguous proper time measured on a clock between two events (points) on its worldline. Whereas, $\Delta t$ is the coordinate time (between those two events) measured in some inertial reference frame in which the first clock is not at rest. How you measure the coordinate time is, of course, a matter that requires some thought. (In his original 1905 paper, Einstein goes into this in considerable detail!)

You could have two clocks at rest at the points in space where events squiggly A and squiggly B take place, and have a simultaneity convention for those two clocks. This is the usual theoretical assumption, using the default Einstein simultaneity convention.

Or, you could have a single clock at the origin and a light signal sent to the origin when event squiggly B takes place. When the signal arrives at the origin, you have to subtract the light signal travel time to get the coordinate time of event squiggly B. This is, of course, equivalent to the Einstein simultaneity convention. This is the usual practical situation observing cosmological events - as we cannot go out into space and place synchronised clocks wherever we need them. We can only make direct observations of distant observations here on Earth.

You may find learning Special Relativity from a GR textbook quite tough. If you get bogged down in Schutz, you could take a look at Morin's book, the first chapter of which is available free online:

https://davidmorin.physics.fas.harvard.edu/books/special-relativity/

 

[ultramarinus_regis_8]

The point to note that is that $\Delta \bar t$ is the unambiguous proper time measured on a clock between two events (points) on its worldline. Whereas, $\Delta t$ is the coordinate time (between those two events) measured in some inertial reference frame in which the first clock is not at rest. How you measure the coordinate time is, of course, a matter that requires some thought. (In his original 1905 paper, Einstein goes into this in considerable detail!)

You could have two clocks at rest at the points in space where events squiggly A and squiggly B take place, and have a simultaneity convention for those two clocks. This is the usual theoretical assumption, using the default Einstein simultaneity convention.

Or, you could have a single clock at the origin and a light signal sent to the origin when event squiggly B takes place. When the signal arrives at the origin, you have to subtract the light signal travel time to get the coordinate time of event squiggly B. This is, of course, equivalent to the Einstein simultaneity convention. This is the usual practical situation observing cosmological events - as we cannot go out into space and place synchronised clocks wherever we need them. We can only make direct observations of distant observations here on Earth.

You may find learning Special Relativity from a GR textbook quite tough. If you get bogged down in Schutz, you could take a look at Morin's book, the first chapter of which is available free online:

https://davidmorin.physics.fas.harvard.edu/books/special-relativity/

Thanks for the advice, @PeroK! The simultaneity convention you described is interesting and I didn't see it touched upon in the Schutz book, but I should probably take a look at the Einstein 1905 paper you mentioned.

Indeed, I found the explanations in Schutz quite terse, probably because it's a GR text. I'll take a look at the Morin book you recommended, and perhaps other SR texts.

Thanks to all for your help!

 

[Ibix]

Indeed, I found the explanations in Schutz quite terse, probably because it's a GR text. I'll take a look at the Morin book you recommended, and perhaps other SR texts.

One GR text I have (Ben Crowell's, I think) comments that GR texts are uniformly terrible for learning SR. They really assume you already know SR and are only recapping it in heavy-machinery terms before they start generalising that machinery into curved spacetime.

Taylor and Wheeler's Spacetime Physics is worth a look as well as Morin - it's available free for download via Taylor's website.

 

[PeroK]

Indeed, I found the explanations in Schutz quite terse, probably because it's a GR text. I'll take a look at the Morin book you recommended, and perhaps other SR texts.

The sections on SR in a GR textbook may be essentially intended as revision (and to familiarise the student with the author's style). Schutz may expect you already to be familiar with SR.

 

[ultramarinus_regis_8]

One GR text I have (Ben Crowell's, I think) comments that GR texts are uniformly terrible for learning SR. They really assume you already know SR and are only recapping it in heavy-machinery terms before they start generalising that machinery into curved spacetime.

Taylor and Wheeler's Spacetime Physics is worth a look as well as Morin - it's available free for download via Taylor's website.

Thanks, @Ibix and @PeroK, I appreciate the help. I'll add Ben Crowell's text to the list of a couple of books I will look at, and thanks for pointing me towards the right path–I'll come back to the GR texts once I have a better grasp of the fundamentals of SR. Thanks also everyone else in the thread for your helpful comments and support.

 

[PeterDonis]

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[PeterDonis]

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