How does special relativity account for the time on a single moving clock?

[ghwellsjr]

By the way, JM, I just noticed that Taylor and Wheeler have a similar explanation to the one in wikipedia if you back up to page 156. There in Figure 5-12, they show two worldlines going between two events labeled O and B. The straight vertical worldline is the one for what they call a "proper clock" because it is constant velocity--no acceleration--and it has the "maximal lapse of proper time", 10, in this case. By contrast, they say a clock carried along the kinked worldline OQB has a proper time of 6, and then they say of the proper clock, "the direct worldline displays maximum proper time".

Then in the next paragraph, they contrast two different comparisons of time between two events. The first is what they call map time, frame time, latticework time, but what everyone else calls co-ordinate time and they make the point that different frames will generate different times but the least amount of time is the frame in which the two events are at the same location. This would be the case in which a "proper clock" is not moving. In other frames the "proper clock" is moving and runs slower than the co-ordinate time difference for the two events. So it is in this sense that "moving proper clocks run slow". They then proceed to the second contrast and repeat the statement that the "proper clock" with the straight worldline "registers maximal passage of proper time" meaning it runs the fastest not slower like a clock that accelerates, meaning that it is not a "proper clock".

 

[JM]

He's not concerned about the actual time displayed on the clock but how its rate of ticking compares to the rate of ticking of the clocks in the stationary system. You are looking at the actual times on the clocks. What you need to do is what I showed you in my previous post which is to compare two events in both frames where the the clock is stationary in the moving frame and moving in the stationary frame.

George- I see what you are doing in Post 80, you are following a clock as it moves wrt the stationary frame by specifying its two values of time t' for the same value of x', and working backwards to find the corresponding values of x and t. The only quibble I might make is that transforming from x',t' to x,t usually uses the + sign instead of the -sign. The transformed values are different but the 'deltas' are the same and the 'slow clock' formula is confirmed. Also, the two points in the stationary frame follow the relation Δx = 0.8 Δt, as Einstein assumed.
I'm fine with 'slow clocks' now.
JM

 

[JM]

Surely every clock is a single clock, moving along its worldline between events ?

If one takes the moving clocks to be moving in a straight line parallel to the stationary x axis, and the coordinates x,t to represent (perhaps isolated) events of interest (such as two lightning bolts hitting a train track), then there seems to be the possibility that no clock will be present at any two events. This is the picture that Einsteins 1905 paper suggests to me.
I gather from the discussion that there are additional ideas for clocks moving in arbitrary directions, accelerating , etc . Can you suggest a reference describing such theories?
JM

 

[JM]

By the way, JM, I just noticed that Taylor and Wheeler have a similar explanation to the one in wikipedia if you back up to page 156. There in Figure 5-12, they show two worldlines going between two events labeled O and B. The straight vertical

George-
(I've shortened the quote only to save space)
Thanks for the ideas. 'Proper clocks' is evidently not a simple subject.
My efforts to date have been on understanding Einsteins 1905 paper. I feel comfortable with most of it ( there are a few questions about rod shortening, how the time t' is made to appear on the moving clocks, and the theory behind the linking of frames moving in different directions).
I'm looking for references for the theory that 'everyone else uses'. You have mentioned Wikipedia and Taylor/Wheeler. Are these the introductory authorities, or is there something better? I have looked at Taylor but find it difficult because of manner of presentation and the many 'off the wall ideas'.
JM

 

[ghwellsjr]

George- I see what you are doing in Post 80, you are following a clock as it moves wrt the stationary frame by specifying its two values of time t' for the same value of x', and working backwards to find the corresponding values of x and t. The only quibble I might make is that transforming from x',t' to x,t usually uses the + sign instead of the -sign. The transformed values are different but the 'deltas' are the same and the 'slow clock' formula is confirmed. Also, the two points in the stationary frame follow the relation Δx = 0.8 Δt, as Einstein assumed.
I'm fine with 'slow clocks' now.
JM

Yes, well since deltas never care about the sign of the difference, it can be done either way. I'm glad you scrutinized my post enough to notice the difference. And I'm glad you are fine with slow clocks now.

 

[ghwellsjr]

If one takes the moving clocks to be moving in a straight line parallel to the stationary x axis, and the coordinates x,t to represent (perhaps isolated) events of interest (such as two lightning bolts hitting a train track), then there seems to be the possibility that no clock will be present at any two events. This is the picture that Einsteins 1905 paper suggests to me.
I gather from the discussion that there are additional ideas for clocks moving in arbitrary directions, accelerating , etc . Can you suggest a reference describing such theories?
JM

Moving clocks do not have to be moving just along the x-axis nor do they have to be moving at a constant velocity. That's merely the way Einstein developed the equation to show the tick rate of a moving clock compared to the tick rate of the stationary co-ordinate clocks. In his 1905 paper, after he derives the formula, he immediately moves on to a clock that is not moving in a straight line along the x-axis but rather is moving in a circular path so that it returns to a stationary clock and he determines that the moving clock will have accumulated less time on it than the stationary clock during the same time interval.

This is an example of what Taylor and Wheeler discuss on page 156 where the two events in question are when Einstein's two clocks start out together and when they end up together. The stationary clock is following a straight line through spacetime and qualifies as what they call a "proper clock" since its velocity is constant (actually zero) and it is present at both events. The moving clock is constantly accelerating even though its speed is constant it's velocity is not. So it is not a "proper clock". It takes a curved line through spacetime and so its accumulated proper time is less than the accumulated proper time on the stationary "proper clock".

As Taylor and Wheeler point out on page 11, if two events have a spacelike spacetime interval, then it won't be possible for a single clock to traverse between the two events at a constant speed because that speed would have to be greater than the speed of light. But it has nothing to do with any axes.

 

[ghwellsjr]

George-
(I've shortened the quote only to save space)
Thanks for the ideas. 'Proper clocks' is evidently not a simple subject.
My efforts to date have been on understanding Einsteins 1905 paper. I feel comfortable with most of it ( there are a few questions about rod shortening, how the time t' is made to appear on the moving clocks, and the theory behind the linking of frames moving in different directions).
I'm looking for references for the theory that 'everyone else uses'. You have mentioned Wikipedia and Taylor/Wheeler. Are these the introductory authorities, or is there something better? I have looked at Taylor but find it difficult because of manner of presentation and the many 'off the wall ideas'.
JM

"Proper clocks" is a very simple subject once you know what Taylor and Wheeler mean by them. This term was brought up by you and you brought up the Taylor and Wheeler reference. I don't like their approach nor their confusing, non-standard, "proper clocks" term. I would advise that you just forget about them as a bad experience. Einstein's 1905 paper develops everything you need to know about Special Relativity although some of his other writings are also helpful, such as his 1920 book.

 

[Mentz114]

If one takes the moving clocks to be moving in a straight line parallel to the stationary x axis, and the coordinates x,t to represent (perhaps isolated) events of interest (such as two lightning bolts hitting a train track), then there seems to be the possibility that no clock will be present at any two events. This is the picture that Einsteins 1905 paper suggests to me.

I'm sorry, JM, I find that paragraph incomprehensible. Whether or not a clock is present at an event is irrelevant to the 1905 train scenario.

I gather from the discussion that there are additional ideas for clocks moving in arbitrary directions, accelerating , etc .
JM

All clocks are the same whatever path they move on. They record the proper time along their worldlines. Every worldline may have its own proper time.

Can you suggest a reference describing such theories?

I don't know what you mean by 'theories'. Maybe start with finding out about worldlines and the proper time

dτ² = c²dt² - dx² - dy² - dz²

 

[Dale]

OK, now I see what's going on. I did a search and found this reference to Taylor and Wheeler's Spacetime Physics where they mention a proper clock on page 160:

http://books.google.com/books?id=PDA8YcvMc_QC&pg=PA160&dq#v=onepage&q&f=false

However, they define a proper clock on page 10 which is not available online [at least it wasn't yesterday, today it is?] so I checked the book out of the library and what they mean by a proper clock is one that travels between two events at a constant speed (without regard to any frame). In other words, it is measuring the frame invariant spacetime interval but this can only work for timelike intervals.

Hi ghwellsjr, thanks for this information, I was unaware of this definition. So I must correct my previous statements, there is a such thing as a proper clock. A proper clock is not the same thing as proper time.

All clocks (proper or not) measure proper time according to the formula I gave above. However, for a proper clock the proper time formula simplifies even further. The proper time formula is more general than any proper clock formulas.

 

[PAllen]

dτ² = c²dt² - dx² - dy² - dz²

Atypical mixing of units here. If you use c^2 dt^2, you usually call the invariant differential ds^2. If you use dt^2 on the rhh, the dτ^2.

 

[Mentz114]

Atypical mixing of units here. If you use c^2 dt^2, you usually call the invariant differential ds^2. If you use dt^2 on the rhs, the dτ^2.

Whoops. I should have said proper interval.

 

[JM]

"Proper clocks" is a very simple subject once you know what Taylor and Wheeler mean by them. This term was brought up by you and you brought up the Taylor and Wheeler reference. I don't like their approach nor their confusing, non-standard, "proper clocks" term. I would advise that you just forget about them as a bad experience. Einstein's 1905 paper develops everything you need to know about Special Relativity although some of his other writings are also helpful, such as his 1920 book.

George- Re the 1905 paper, what is the theory that supports the use of linked/accelerating frames? Doesn't 1905 restrict to inertial frames? I sense that many have divorced the time on the moving clock from its roots in the stationary frame and the 'events' that happen there, when 'viewing from the stationary frame'. The early posts in this thread explain my thoughts on this. I don't see this in 1905.
Re the 1920 book, what's the name and publisher?
JM

 

[Dale]

what is the theory that supports the use of linked/accelerating frames?

Special relativity.

Doesn't 1905 restrict to inertial frames?

Yes, but accelerating frames are obtained simply by a coordinate transform from an inertial frame in SR. You don't need to change theories to GR until you want to add gravity.

Do you now understand that moving clocks always tick slow in an inertial frame?

 

[ghwellsjr]

"Proper clocks" is a very simple subject once you know what Taylor and Wheeler mean by them. This term was brought up by you and you brought up the Taylor and Wheeler reference. I don't like their approach nor their confusing, non-standard, "proper clocks" term. I would advise that you just forget about them as a bad experience. Einstein's 1905 paper develops everything you need to know about Special Relativity although some of his other writings are also helpful, such as his 1920 book.

George- Re the 1905 paper, what is the theory that supports the use of linked/accelerating frames?

I don't know what you mean by a linked frame and I see no advantage or need for an accelerating frame when any inertial frame will do everything that needs to be done and so much more simply. So I'm not the one to ask about other types of frames but I see DaleSpam has provided an answer. It's a good bet to trust what he says.

Doesn't 1905 restrict to inertial frames?

Yes, and so do I.

I sense that many have divorced the time on the moving clock from its roots in the stationary frame and the 'events' that happen there, when 'viewing from the stationary frame'.

I'm sorry, I can't figure out what you mean here. What are the moving clock's roots in the stationary frame and what 'events' are you talking about?

The early posts in this thread explain my thoughts on this.

What posts would those be? I thought we resolved that the confusion was over Taylor and Wheeler's restrictive definition of a 'proper clock' and you were fine (post #107) with the fact that any moving clock, inertial or not, will tick more slowly than the co-ordinate clocks in the frame in which it is moving

I don't see this in 1905.

Using two frames, Eisntein showed the derivation for the equation to determine the tick rate of a moving clock as a function of its speed in an inertial frame and then he proceeds to show how two clocks, one stationary and one accelerating in a circle, both with respect to a single frame, will have accumulated different times on them every time they are colocated. In contrast, the 'proper clock' in this scenario is the stationary one because it is inertial between the two events of successive meetings of the two clocks. It's tick rate is not slowed down but is identical to the tick rate of the co-ordinate clocks in the single inertial frame. The other clock is the moving one and its tick rate is slowed down as it travels in a circle and each time it meets up with the stationary clock, it has accumulated less time on it. In other word, the proper time on the stationary 'proper clock' has advanced more than the proper time on the moving clock between each of the events when they meet.

Re the 1920 book, what's the name and publisher?
JM

The link to the book was provided by harrylin in post #18 and quoted by you in post #23 so I thought you had taken a look at it.

 

[JM]

All clocks are the same whatever path they move on. They record the proper time along their worldlines. Every worldline may have its own proper time.

So, what is your definition of proper time?

I don't know what you mean by 'theories'. Maybe start with finding out about worldlines and the proper time

OK, so where do I find out about these things?
JM

 

[JM]

I know this is an early post, but it seems relavent now.

I don't know why you would claim that. Isn't t the time according to clocks in the moving frame?

Clocks do only what they are told. The theory says that the time of the moving frame is given by c t' = gamma(c t - v x / c ). This means that the moving clock is instructed ( or built ) to accept t, x, and v/c as inputs and to display the result as t'. Thus the moving clock has no initiative of its own to decide what time to display, but must display what the stationary frame tells it to.
JM

 

[Dale]

Clocks do only what they are told.

Clocks measure proper time. If something doesn't measure proper time then it isn't a clock.

Thus the moving clock has no initiative of its own to decide what time to display, but must display what the stationary frame tells it to.

Proper time doesn't need any reference to a mythical "stationary" frame.

 

[JM]

I don't know what you mean by a linked frame and I see no advantage or need for an accelerating frame when any inertial frame will do everything that needs to be done and so much more simply. So I'm not the one to ask about other types of frames but I see DaleSpam has provided an answer. It's a good bet to trust what he says.

Refer to 1905 section 4: "It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide."
It is not apparent to me. If it is to you,can you explain it to me?
Where are the points A and B in terms of x,y,z,t, and where is the polygonal line? The theory of section 3 refers to clocks moving parallel to x, so how to make a polygon out of that? The picture that sentence suggests to me is a series of stationary frames, each one aligned along one segment of the polygonal line, with an accompanying moving frame. The change of direction from one segment to another implies an acceleration of the clock. I don't see anything in section 3 about that. If one clock is on the equator and the other is at the pole then their positions will never coincide. So what is the explanation?

I'm sorry, I can't figure out what you mean here. What are the moving clock's roots in the stationary frame and what 'events' are you talking about?

See the posts on page one of this thread, and the one just above.

I thought we resolved that the confusion was over Taylor and Wheeler's restrictive definition of a 'proper clock' and you were fine (post #107) with the fact that any moving clock, inertial or not, will tick more slowly than the co-ordinate clocks in the frame in which it is moving

The confusion was about the meaning of the phrase 'moving clocks run slow'. It was cleared up with the qualifier that proper clocks run slow, not arbitrary coordinate clocks. We didn't get to what a correct definition of a proper clock is. I am fine with inertial clocks being slow, but not non-inertial ones. As I noted above, I don't see how the inertial analysis of section 3 applies to non-inertial clocks.

The link to the book was provided by harrylin in post #18 and quoted by you in post #23 so I thought you had taken a look at it.

I have that book, and I have read it. I don't recall anything about clocks moving in various directions, or all clocks being proper. I was hoping that you could suggest a text better than Taylor.
Thanks again for your efforts.
JM

 

[ghwellsjr]

I don't know what you mean by a linked frame and I see no advantage or need for an accelerating frame when any inertial frame will do everything that needs to be done and so much more simply. So I'm not the one to ask about other types of frames but I see DaleSpam has provided an answer. It's a good bet to trust what he says.

Refer to 1905 section 4: "It is at once apparent that this result still holds good if the clock moves from A to B in any polygonal line, and also when the points A and B coincide."
It is not apparent to me. If it is to you,can you explain it to me?
Where are the points A and B in terms of x,y,z,t, and where is the polygonal line? The theory of section 3 refers to clocks moving parallel to x, so how to make a polygon out of that?

In section 3, the clocks were moving parallel to x because it is conventional in the standard configuration of the Lorentz Transformation to align the axes so that the motion is along the x-axis. It doesn't matter physically which direction the motion is in, we just assign the two co-ordinate systems so that the relative motion between them is along the x-axis. Remember, all frames are equally valid, including ones where the only difference is the orientation of their axes.

So once Einstein establishes that any clock that moves in a reference frame along the x-axis will tick at a slower rate than the co-ordinate clocks of that reference frame, he generalizes the concept to include any clock moving in any direction and he says to pick any two additional clocks, one at any point A and one at any other point B, not necessarily aligned along the x-axis, which had previously been synchronized with each other when at relative rest, and move the one at A to the position of the one at B at some relatively slow velocity v, then when it gets there, it will be slow by ½tv²/c² compared to the clock at B. (Note that this formula is approximate and only applies to a slow-moving clock.)

Then he says that we can repeat the process, moving the A clock from the first B position to another B position in any other direction and we will get the same additional difference in clock time when it gets there. We can repeat the process as many times and in as many directions as we want, even to the point where we eventually return the A clock to its original location and the same formula applies if we take the total time t for the clock to make its round trip. This is what he means by the A and B points coinciding.

The picture that sentence suggests to me is a series of stationary frames, each one aligned along one segment of the polygonal line, with an accompanying moving frame. The change of direction from one segment to another implies an acceleration of the clock. I don't see anything in section 3 about that.

You can do the analysis with multiple additional frames if you want, but it is just more complicated with no additional increase in knowledge.

If one clock is on the equator and the other is at the pole then their positions will never coincide. So what is the explanation?

Prior to space travel (or sustained air travel), this was the only way to carry out the experiment. And it still would work, neglecting any effects from gravity, even if the clocks don't ever come to the same location because we are considering just one inertial rest frame, that of the clock at the pole. But of course nowadays, we just have the moving clock take off in a spaceship (or airplane, which has been done).

Don't be confused by the oft-repeated statement that clocks have to be co-located at the start and end of the journey of one of them to compare times. All frames will show that there is a difference in accumulated times, even if they don't agree on the absolute times on the two clocks (because of simultaneity issues).

I'm sorry, I can't figure out what you mean here. What are the moving clock's roots in the stationary frame and what 'events' are you talking about.

See the posts on page one of this thread, and the one just above.

You repeated several times that man-made clocks do what we tell them to do but let's assume that they all have one thing in common, they tick once per second. Then the only issue is how many ticks have transpired between point A and point B, agreed? In this sense, we can treat them as stop watches, even if they actually display time as hours, minutes, and seconds or if they count backwards.

But the point is that no one makes a clock that is designed to tick slowly when it is traveling with respect to some rest frame--how in the world would they do that? And you overlook that fact that two identical clocks in inertial relative motion would each tick more slowly compared to its own tick rate. How do you design clocks to do that? No, it happens independently of any purposeful design, in fact if you tried to make it happen, it wouldn't be reciprocal.

I thought we resolved that the confusion was over Taylor and Wheeler's restrictive definition of a 'proper clock' and you were fine (post #107) with the fact that any moving clock, inertial or not, will tick more slowly than the co-ordinate clocks in the frame in which it is moving

The confusion was about the meaning of the phrase 'moving clocks run slow'. It was cleared up with the qualifier that proper clocks run slow, not arbitrary coordinate clocks. We didn't get to what a correct definition of a proper clock is.

It wasn't cleared up by the qualifier that only "proper clocks" run slow and we did get to the correct definition of a "proper clock". But it is not a generally acknowledged definition. It is what I would call a private definition made by Taylor and Wheeler on page 10 of their book which you pointed out. No one else talks about a "proper clock". Instead, we keep repeating that all clocks keep track of "proper time". This applies to inertial clocks and non-inertial clocks, moving clocks, stationary clocks, accelerating clocks and co-ordinate clocks. All clocks keep track of their own proper time. They don't have any choice. Of course we are talking about carefully designed clocks that aren't affected by environmental effects, such as a pendulum clock.

I am fine with inertial clocks being slow, but not non-inertial ones. As I noted above, I don't see how the inertial analysis of section 3 applies to non-inertial clocks.

Well, I hope you can see it now.

The link to the book was provided by harrylin in post #18 and quoted by you in post #23 so I thought you had taken a look at it.

I have that book, and I have read it. I don't recall anything about clocks moving in various directions, or all clocks being proper. I was hoping that you could suggest a text better than Taylor.
Thanks again for your efforts.
JM

Here is the link to Einstein's 1920 book: http://www.bartleby.com/173/.

Now if you look at the end of chapter 12, you will see this statement:

As a consequence of its motion the clock goes more slowly than when at rest.

Then if you look at chapter 23, you will see where Einstein once again discusses a clock moving in a circle with respect to a stationary clock.

 

[Dale]

The confusion was about the meaning of the phrase 'moving clocks run slow'. It was cleared up with the qualifier that proper clocks run slow, not arbitrary coordinate clocks. We didn't get to what a correct definition of a proper clock is. I am fine with inertial clocks being slow, but not non-inertial ones.

All moving clocks run slow, not just proper clocks. See the formula I posted above. It applies to all clocks, inertial or non inertial.

What is a coordinate clock? That is also a non standard term. Is it defined somewhere or are you just making things up?

 

[Mentz114]

So, what is your definition of proper time?

The accepted definition of proper time is the Lorentzian ( or proper) length of a segment of a worldline. Consider a piece of string. The length of the string is independent of its shape because we use the Euclidean definition ,
Length = √( x²+y²+z²)

But worldlines are 4-dimensional and the proper length is given by the Lorentzian length,
L = √( c²t²-x²-y²-z²) or T = √( t²-x²/c²-y²/c²-z²/c²)

with this definition of length, the strings length depends on its shape. A twisty bent piece of string is shorter than it would be if measured stretched out.

This is why the traveling twin is younger than the stay at home twin

OK, so where do I find out about these things?
JM

Read, ask and listen. You could start here

http://en.wikipedia.org/wiki/Proper_time

 

[JM]

In section 3, the clocks were moving parallel to x because it is conventional in the standard configuration of the Lorentz Transformation to align the axes so that the motion is along the x-axis. It doesn't matter physically which direction the motion is in, we just assign the two co-ordinate systems so that the relative motion between them is along the x-axis. Remember, all frames are equally valid, including ones where the only difference is the orientation of their axes.

OK.

..., he generalizes the concept to include any clock moving in any direction and he says to pick any two additional clocks, one at any point A and one at any other point B, not necessarily aligned along the x-axis, which had previously been synchronized with each other when at relative rest, and move the one at A to the position of the one at B...

Suppose that A is at (x,y,z) = (0,0,0) and B is at (x,y,z) = ( 1,1,0). Within section 3, where all clocks move parallel to x, there is no clock that passes through these two points. One option is to add an extra clock that is not at rest in either K', moving or in K, stationary. But would that clock follow the same 'slow clock' formula? Another option is to align the axes of the moving frame with the line passing through these two points. For this option the moving frame must then be re-aligned again to get the clock to the second point B. This is the option that I refer to. Do you see another option?

You repeated several times that man-made clocks do what we tell them to do but let's assume that they all have one thing in common, they tick once per second. Then the only issue is how many ticks have transpired between point A and point B, agreed? In this sense, we can treat them as stop watches, even if they actually display time as hours, minutes, and seconds or if they count backwards.

In general,OK. But let's examine the idea that they all tick once per second. The clocks of K, stationary, can be considered as reference clocks and ,for convenience, assumed to be adjusted to match the day-night cycle of the earth( on average). This is what I call everyday time. But what about the moving clocks? Are you saying that the moving clocks also are adjusted to everyday time? i.e. that one second on a moving clock is the same as one second on a statioinary clock? The transform equations seem to demand this; if t is entered in seconds then the resulting t' must be also measured in the same seconds. Also, section 3 asserts that ' the clocks are in all respects alike'. But if this is so, then all clocks are running at the same rate, so how can a moving clock be said to run slow? Could it be a matter of terminology (or translation), with the meaning actually being that an interval between two events is different as measured by moving clocks than by stationary clocks, even thouth both clocks are running at the same rate? This seems to me to be a better view, because when comparing two things it is necessary to use the same units, if the clocks are running at different rates then the measurements are meaningless.

No one else talks about a "proper clock". Instead, we keep repeating that all clocks keep track of "proper time". This applies to inertial clocks and non-inertial clocks, moving clocks, stationary clocks, accelerating clocks and co-ordinate clocks. All clocks keep track of their own proper time. They don't have any choice.

Can I infer from this that your definition of proper time is simply the time of any clock? With no need for any 'events' for the clock to mark?

Here is the link to Einstein's 1920 book: http://www.bartleby.com/173/.
Now if you look at the end of chapter 12, you will see this statement:
Then if you look at chapter 23, you will see where Einstein once again discusses a clock moving in a circle with respect to a stationary clock.

I have checked and my book has the same statements in the same places, so we're talking about the same book. Aren't the items you refer to just re-statements of the same things as presented in 1905?

I am still puzzled by your, and DaleSpams, reluctance to identify published sources for your ideas. Surely there must be some, what gives? The responses to my posts suggest that there is a line of theory that is not wholly included in Einsteins works. I have heard of world lines, maybe in French, and Taylor hints at a different viewpoint. Us old timers prefer paper books to internet, maybe because of editing and reviewing.

I can see that you have put much effort into this conversation, and I appreciate it.
JM

 

[JM]

But worldlines are 4-dimensional and the proper length is given by the Lorentzian length,
L = √( c²t²-x²-y²-z²) or T = √( t²-x²/c²-y²/c²-z²/c²)

Don't these definitions mean that a line between two points connected by a light ray has zero length? Does that make sense?

[/QUOTE]http://en.wikipedia.org/wiki/Proper_time[/QUOTE]
Thanks for the reference, I'll look into it.
JM

 

[Dale]

I am still puzzled by your, and DaleSpams, reluctance to identify published sources for your ideas.

I don't understand this comment at all. I posted the Wikipedia link on proper time back in post 30-something when I first joined this thread. Please start there, it will be the best resource for an introduction, and is entirely sufficient for this conversation.

Once you understand the Wikipedia page then you can search for "spacetime interval" or "line element" or "spacetime metric" or "Riemannian metric" for more information, but most of that will be too advanced until you have mastered the material on the Wikipedia page.

If you specifically want paper-published sources then any introductory SR textbook will include material on proper time although it may be called "spacetime interval", or "invariant interval". You have some textbooks already, just start in the index there if you don't like Wikipedia.

 

[Dale]

Don't these definitions mean that a line between two points connected by a light ray has zero length? Does that make sense?

Yes. Such lines are called "null" or "lightlike".

 

[ghwellsjr]

In section 3, the clocks were moving parallel to x because it is conventional in the standard configuration of the Lorentz Transformation to align the axes so that the motion is along the x-axis. It doesn't matter physically which direction the motion is in, we just assign the two co-ordinate systems so that the relative motion between them is along the x-axis. Remember, all frames are equally valid, including ones where the only difference is the orientation of their axes.

OK.

..., he generalizes the concept to include any clock moving in any direction and he says to pick any two additional clocks, one at any point A and one at any other point B, not necessarily aligned along the x-axis, which had previously been synchronized with each other when at relative rest, and move the one at A to the position of the one at B...

Suppose that A is at (x,y,z) = (0,0,0) and B is at (x,y,z) = ( 1,1,0). Within section 3, where all clocks move parallel to x, there is no clock that passes through these two points. One option is to add an extra clock that is not at rest in either K', moving or in K, stationary. But would that clock follow the same 'slow clock' formula?

Since Einstein was living in an era where fast space travel was not possible, he was merely simplifying the analysis by using the 'slow clock' formula but the exact formula will still work and should be used where the difference between the two answers would be significant. So this option is perfectly viable.

Another option is to align the axes of the moving frame with the line passing through these two points. For this option the moving frame must then be re-aligned again to get the clock to the second point B. This is the option that I refer to.

This is the option I put in bold above.

Do you see another option?

I don't see the need for another option, do you?

You repeated several times that man-made clocks do what we tell them to do but let's assume that they all have one thing in common, they tick once per second. Then the only issue is how many ticks have transpired between point A and point B, agreed? In this sense, we can treat them as stop watches, even if they actually display time as hours, minutes, and seconds or if they count backwards.

In general,OK. But let's examine the idea that they all tick once per second. The clocks of K, stationary, can be considered as reference clocks and ,for convenience, assumed to be adjusted to match the day-night cycle of the earth( on average). This is what I call everyday time. But what about the moving clocks? Are you saying that the moving clocks also are adjusted to everyday time? i.e. that one second on a moving clock is the same as one second on a statioinary clock? The transform equations seem to demand this; if t is entered in seconds then the resulting t' must be also measured in the same seconds. Also, section 3 asserts that ' the clocks are in all respects alike'. But if this is so, then all clocks are running at the same rate, so how can a moving clock be said to run slow? Could it be a matter of terminology (or translation), with the meaning actually being that an interval between two events is different as measured by moving clocks than by stationary clocks, even thouth both clocks are running at the same rate? This seems to me to be a better view, because when comparing two things it is necessary to use the same units, if the clocks are running at different rates then the measurements are meaningless.

There was a time, many decades ago, when the rotation of the Earth was the most stable standard for a second, but now that we can make atomic clocks that are more stable, it would be meaningless to continue with that standard and so now we use atomic clocks as a standard to define what a second means. But that presents the problem that you are asking about. Not only will moving clocks tick at different rates (as analyzed by Special Relativity) but clocks at different altitudes will also (as analyzed by General Relativity). So it is a real problem that has to be dealt with and fortunately we have very smart people who have come up with a solution to provide us with a coordinated everyday time which is called "Coordinated Universal Time". The clocks on board GPS satellites are examples of moving clocks that have to be adjusted to everyday time and our GPS devices take care of the problem so that we can all make it to our meetings at the same agreed upon time. But if you were doing physics experiments, such as measuring the speed of light, you would not use the time standard provided by GPS because you will get the wrong answer. Instead, you have to use your own atomic clock to provide you with the measurement of time.

We can't ignore the issue of moving clocks ticking at different rates and rather than saying it is all meaningless, we have agreed upon conventions to make the best sense out of the situation.

No one else talks about a "proper clock". Instead, we keep repeating that all clocks keep track of "proper time". This applies to inertial clocks and non-inertial clocks, moving clocks, stationary clocks, accelerating clocks and co-ordinate clocks. All clocks keep track of their own proper time. They don't have any choice.

Can I infer from this that your definition of proper time is simply the time of any clock? With no need for any 'events' for the clock to mark?

Yes, except it's not my definition, it was promoted by Minkowski in 1908. Einstein apparently didn't see the need to have a special term for something that applies to all clocks. But in terms of talking about the time between two arbitrary events, there is no single answer to that question because two clocks traveling in different ways between those two events can have a different answer. The term "proper clock" was coined to apply to an inertial clock that travels unaccelerated between those two events.

Here is the link to Einstein's 1920 book: http://www.bartleby.com/173/.

Now if you look at the end of chapter 12, you will see this statement:

Then if you look at chapter 23, you will see where Einstein once again discusses a clock moving in a circle with respect to a stationary clock.

I have checked and my book has the same statements in the same places, so we're talking about the same book. Aren't the items you refer to just re-statements of the same things as presented in 1905?

Yes, that was my point.

I am still puzzled by your, and DaleSpams, reluctance to identify published sources for your ideas. Surely there must be some, what gives? The responses to my posts suggest that there is a line of theory that is not wholly included in Einsteins works. I have heard of world lines, maybe in French, and Taylor hints at a different viewpoint. Us old timers prefer paper books to internet, maybe because of editing and reviewing.

I can see that you have put much effort into this conversation, and I appreciate it.
JM

I think you are referring to Minkowski's re-interpretation and re-statement of Einstein's ideas. Einstein gave passing mention of his work in chapter 17 and near the end of his 1920 book. It is basically a graphical presentation of the Lorentz Transform and served as an important graphical aid in an era in which calculators and computers and videos were not available. But it is of necessity limited to one dimension of space. Nowadays, two-dimensional animated presentations are readily available to communicate the same ideas much more effectively. I never bothered to study Minkowski's work so I don't know what would be a good reference but I'm sure there are plenty.

 

[ghwellsjr]

But worldlines are 4-dimensional and the proper length is given by the Lorentzian length,
L = √( c²t²-x²-y²-z²) or T = √( t²-x²/c²-y²/c²-z²/c²)

Don't these definitions mean that a line between two points connected by a light ray has zero length? Does that make sense?

When Mentz says "length", he means "spacetime interval" which can physically be either a spatial length or a time period, depending on the two events.

If an inertial clock can be present at the two events, then the spacetime interval is "timelike" and is the accumulated time on the clock. This is Taylor and Wheeler's definition of a "proper clock". There is an inertial frame in which the clock is at rest.

If the two events are so far apart that a physical clock could not get from the first event to the second event, then the spacetime interval is "spacelike" and is measured with an inertial ruler spanning between the two events and in a frame in which it is at rest. Taylor and Wheeler did not call this a "proper ruler" but they could have.

Since light rays don't have rest frames, the concept of a spacetime interval is meaningless for events that a light ray is present at both of them.

 

[yuiop]

Since light rays don't have rest frames, the concept of a spacetime interval is meaningless for events that a light ray is present at both of them.

I wouldn't agree that a null spacetime interval is entirely meaningless. It lies on the boundary between spacelike and timelike intervals, is called a "lightlike" interval and the events can be causally connected and is very useful for a lot of calculations.

 

[JM]

I don't understand this comment at all. I posted the Wikipedia link on proper time back in post 30-something when I first joined this thread. Please start there, it will be the best resource for an introduction, and is entirely sufficient for this conversation.

I looked at that Wike page and it looked like just a list of formulas, with no supporting theoretical foundation. Where is the foundation?

If you specifically want paper-published sources then any introductory SR textbook will include material on proper time although it may be called "spacetime interval", or "invariant interval". You have some textbooks already, just start in the index there if you don't like Wikipedia.

Yes, I have textbooks, but they don't answer the questions that I've asked in this thread, if they did I wouldn't have asked.
So, can you recommend a specific ' introductory SR textbook' , or not?
JM

 

[JM]

We can't ignore the issue of moving clocks ticking at different rates and rather than saying it is all meaningless, we have agreed upon conventions to make the best sense out of the situation.

Well, George, we seem to be out of synch again.
I still don't see the principles or math that justify the generalization from clocks moving along x to clocks moving in arbitrary directions.
At one point you seemed to say that all clocks (did you mean both moving and stationary?)tick at the same rate, ie one tick per second. I cited reasons to believe this. Then, above, you say that moving clocks tick at a different rate. So, which is it?
So you don't read Minkowski, don't like Taylor, and don't have a suggested text. Then where do you get your ideas about SR?
JM

 

[Dale]

I looked at that Wike page and it looked like just a list of formulas, with no supporting theoretical foundation. Where is the foundation?

Wikipedia always puts a list of references and external links down at the bottom. In this case, the theoretical foundation is pretty simple so there are only a couple of references. The rest is a more practical introduction, which I found very helpful.

Yes, I have textbooks, but they don't answer the questions that I've asked in this thread, if they did I wouldn't have asked.
So, can you recommend a specific ' introductory SR textbook' , or not?

I cannot recommend an introductory SR textbook, mine was terrible and I found Wikipedia much better. I would recommend starting with chapter 1 of Sean Carroll's lecture notes on GR:
http://arxiv.org/abs/gr-qc/9712019

Chapter 1 is just SR, and he introduces the spacetime interval on page 3 and makes the connection to proper time on page 26.

 

[ghwellsjr]

Well, George, we seem to be out of synch again.

Then that must be because there is a relative speed between us.

I still don't see the principles or math that justify the generalization from clocks moving along x to clocks moving in arbitrary directions.

Well, it's simple. The direction of the x-axis is arbitrary. So once we establish that a clock moving along the x-axis in our arbitrarily defined Frame of Reference ticks slower than the coordinate clocks in that Frame of Reference we certainly don't want to conclude that if it moved in some other direction, it would not also tick slower, would we? If that bothers you, then just do the arithmetic to rotate the frame and show that the same thing happens in the rotated frame.

At one point you seemed to say that all clocks (did you mean both moving and stationary?)tick at the same rate, ie one tick per second. I cited reasons to believe this. Then, above, you say that moving clocks tick at a different rate. So, which is it?

All clocks that are stationary in a given Frame of Reference tick at the same rate. All clocks that are moving in that Frame of Reference tick slower than the clocks that are stationary.

So you don't read Minkowski, don't like Taylor, and don't have a suggested text. Then where do you get your ideas about SR?
JM

From Einstein's 1905 paper and his 1902 book.

JM, I think your confusion stems from the fact that when we define a Frame of Reference, we establish a time coordinate that extends spatially infinitely in all directions so that we can say that all the clocks read zero when the clock at the origin reads zero and then when discussing the Lorentz Transform we talk about a second Frame of Reference moving with respect to the first one and having their two origins coincide so that all the clocks in that second Frame of Reference are also zero everywhere and this leads you to the conclusion that all the clocks in both frames are synchronized to each other at time zero. But this is not true. At every location there are two clocks, one for each frame and except for the single pair at their common origin which both read zero, all the other pairs of colocated clocks read different times from each other. Do you understand this?

 

[ghwellsjr]

From Einstein's 1905 paper and his 1902 book.

I meant, of course, his 1920 book, as mentioned earlier:

Einstein's 1905 paper develops everything you need to know about Special Relativity although some of his other writings are also helpful, such as his 1920 book.

 

[JM]

I cannot recommend an introductory SR textbook, mine was terrible and I found Wikipedia much better. I would recommend starting with chapter 1 of Sean Carroll's lecture notes on GR:
http://arxiv.org/abs/gr-qc/9712019

Thanks for the reference. I've downloaded it for later study.
Out of curiosity, I wonder what your intro text was. Maybe I've read it.
Do you accept a single moving clock, not at rest in either the stationary frame K or the moving frame k', as a valid element in a SR analysis? From some of your posts I would think so, but to be sure. So, suppose there is K, k' moving at v, and a single clock moving at speed u along the x axis. What is the time on the single clock and how would you find it?
JM

 

[JM]

Then that must be because there is a relative speed between us.

Good one. But did we both start at zero at post 1?

Well, it's simple. The direction of the x-axis is arbitrary. So once we establish that a clock moving along the x-axis in our arbitrarily defined Frame of Reference ticks slower than the coordinate clocks in that Frame of Reference we certainly don't want to conclude that if it moved in some other direction, it would not also tick slower, would we? If that bothers you, then just do the arithmetic to rotate the frame and show that the same thing happens in the rotated frame.

All clocks that are stationary in a given Frame of Reference tick at the same rate. All clocks that are moving in that Frame of Reference tick slower than the clocks that are stationary.

George, I think the following is appropriate, even if it doesn't directly answer your points.
Regarding the axis rotation, re. the 'twins': Suppose the stationary frame K, and another stationary frame rotated to pass through the points mentioned before, and a moving fraame aligned with the rotated frame. I can see the relation between the rotated frames being the same as the relation between K and k'. But I don't see the details of the relation between the rotated moving frame and the original frame K.

I think that the 'tick rate' needs explanation. The time transform and the slow clock formula both demand that t and t' are measured in the same units. In 1920 chapt 12, the discussion seems to support this idea. So if t and t' are in seconds, and tick rate is defined as one tick per second then the tick rate is the same for both. But ch 12 does something different. It calculates the time elapsed by t during the time between ticks of t'. Consider the slow clock for v/c = 0.8: t' = 0.6 t. If the tick rate of t' is defined as the time elapsed by t' between two ticks of t, then the moving clock ticks at a different rate than than the stationary.

Re your last point: If t = 0 then ct' = -mvx/c. So the distant stationary clock reads 0, while the distant 'moving' clock doesn't.

You asked what I meant by the 'roots of t' '. Consider section 4, 1905, where it is stated that x= v t. This statement identifies the locations of points wrt K that transform to the point x' = 0 of k'. The analysis process, though not explicitly stated, is as follows:
1.Identify the position of a clock wrt k'. For the above x' = 0.
2.Enter this value in the space transform, x' = m(x-vt). This leads to x = vt.
3. Enter this value into the 'time' transform, ct' = m(c t -vx/c). This leads to t' = t/m.
Thus the time t' depends on the values of x and t. ie the 'roots' of t' in K ( when viewing from K ). Without the values of x and t how would t' be found?

This process can be used in other cases.
1. Let t' = 0. From the 'time' transform ct = vx/c. Enter this value in the 'space' transform, to get x' = x/m, the 'space contraction' formula.
JM