Do the Principles of Special Relativity Require Synchronized Clocks in K and k?

[JM]

In reading over the record of this thread, I'm glad that so many thought the subject of enough interest to contribute. But the comments are so varied that I don't know what to think. It has also been suggested that I restate my views. So let's start with the following example.

If we use one ruler to measure the base of a right isosceles triangle and the other identical ruler to measure the hypotenuse then we will find that the measurements differ by a factor of $\sqrt{2}$..

We can picture how to lay out the triangle, measure the sides, and calculate the ratio. The question is: Doesn't the result sqrt2 imply that each ruler must start at the crossing of the base and the hypotenuse, and that each ruler is marked off with equal spacing of the dimension marks, i.e. the length of one inch on one ruler must equal the length of one inch on the other ruler?

 

[Dale]

We can picture how to lay out the triangle, measure the sides, and calculate the ratio. The question is: Doesn't the result sqrt2 imply that each ruler must start at the crossing of the base and the hypotenuse,

I don't know why that would be implied. You could just as easily measure two opposite sides of a quadrilateral.

and that each ruler is marked off with equal spacing of the dimension marks, i.e. the length of one inch on one ruler must equal the length of one inch on the other ruler?

This is the key question. In what sense is "each ruler marked off with equal spacing"? What does it mean that "one inch on one ruler must equal the length of one inch on the other"? How would you experimentally determine that?

 

[JM]

I don't know why that would be implied. You could just as easily measure two opposite sides of a quadrilateral.

What are you talking about? Are you measuring a triangle or not?

Yes, this is the key question. I assume you understand the concept of 'equal'. In the case of the triangle, the rulers can just be brought together and the scales compared. In your post 48 you mentioned 'co-moving reference standards'. In any case, the spacings must be equal (or correctable, as from inches to cm) or there is no way that a comparison of the readings can be meaningful. What do you mean when you say that the rulers are ' identical'?

 

[Dale]

the rulers can just be brought together and the scales compared

Exactly. Same with clocks.

 

[JM]

Exactly. Same with clocks.

So, considering clocks: If the clocks are identical, aren't the intervals marked off on the two clocks the same, i.e. the time interval between one second marks on one clock is the same as the time interval between one second marks on the other. Isn't this the same as saying that the two clocks advance in time at the same rate, i.e. when one clock advances one second, the other clock does too?

 

[JM]

Haven't published, my knowledge comes from majoring in physics as an undergrad and doing my own reading on various physics-related subjects since then.

So, from your reading in SR, what books/papers/publications would you recommend for the basic principles? I'm thinking of things like Einsteins 1905 paper that emphasizes the physics, rather than things like Minkowskis that emphasize the mathematics.

 

[JesseM]

So, considering clocks: If the clocks are identical, aren't the intervals marked off on the two clocks the same, i.e. the time interval between one second marks on one clock is the same as the time interval between one second marks on the other.

Can you give a physical definition of the word "same"? Normally if physicists were asking if two clocks are ticking at the same rate, they'd be asking if they both had the same rate relative to some spacetime coordinate system, so if one clock took a time interval of $$\Delta t$$ to make one tick and the other clock also took the same time interval of $$\Delta t$$ to make one tick, they'd both be ticking at the same rate relative to that frame's time coordinate.

We could use a similar sort of definition for rulers--the intervals marked off by rulers are "the same" relative to the x-coordinate of some xy coordinate grid if the difference in x-coordinate $$\Delta x$$ between two markings one one ruler is the same as difference in x-coordinate $$\Delta x$$ between two markings of the other ruler. You can see from this definition that even if the intervals are "the same" for two rulers when they're parallel, if you orient them at different angles the intervals may no longer be the same. The reason is that the total distance between markings is given by the pythagorean theorem $$\sqrt{\Delta x^2 + \Delta y^2}$$, and the distance between markings is the same for both rulers, so if one ruler's orientation is closer to the orientation of the y-axis than its $$\Delta y$$ will be larger and its $$\Delta x$$ will be smaller.

Similarly for clocks, if the clocks are moving inertially and we're using an inertial coordinate system, then the proper time between a pair of successive ticks $$\sqrt{\Delta t^2 - \Delta x^2}$$ (where $$\Delta x$$ is the total spatial distance between the location of two successive ticks, in units where c=1) will be "the same" for both clocks. But if one clock's worldline has an orientation that's closer to the orientation of the t-axis (i.e. its velocity is lower in this coordinate system), then its $$\Delta t$$ and $$\Delta x$$ will be smaller, while the $$\Delta t$$ and $$\Delta x$$ for successive ticks of the other clock will be larger, in such a way that both clocks have the same value for $$\sqrt{\Delta t^2 - \Delta x^2}$$

 

[Dale]

the rulers can just be brought together and the scales compared

Exactly. Same with clocks.

So, considering clocks: If the clocks are identical, aren't the intervals marked off on the two clocks the same, i.e. the time interval between one second marks on one clock is the same as the time interval between one second marks on the other. Isn't this the same as saying that the two clocks advance in time at the same rate, i.e. when one clock advances one second, the other clock does too?

Yes, provided the clocks are together during the measurement, just like the rulers mentioned above. When you compare two rulers to determine if they are identical you obviously don't put them together at one point and then have one lay at an angle to the other, you put them together parallel to each other along their length and compare their scales.

Similarly with clocks. Your repeated errors stem from from treating clocks differently than you would treat rulers.

 

[JM]

Yes, provided the clocks are together during the measurement, just like the rulers mentioned above. When you compare two rulers to determine if they are identical you obviously don't put them together at one point and then have one lay at an angle to the other, you put them together parallel to each other along their length and compare their scales.

Similarly with clocks. Your repeated errors stem from from treating clocks differently than you would treat rulers.

It does seem that we are not making much progress. I sense that your answerw are based on the full theory of SR. I am looking at a more basic level, specifically the principle that comparison of two measurements requires that both be made using the same units of measure.
Does this principle apply to SR? If observers K, at rest, and k, in motion, measure the duration of an event, must their units be the same? If not, what does it mean to say that T=t?
Considering the triangle example, the result sqrt2 requires that the base and hypotenuse must be measured using the same units. If we find that the rulers are identical when parallel, then they are considered identical when applied to the triangle.
Clocks are different from rulers, position doesn't advance of its own accord as time does. Suppose we have two clocks at rest at the same position and find that they 'tick' at the same rate. What I have read of SR doesn't make clear what happens when one begins to move. Do the two clocks continue to 'tick' at the same rate as at rest ( as the rulers do, analagously speaking) so the measurements can be compared validly? Or does something happen to one clock to make it tick at a different rate, if so what happens,and how can a valid comparison be made with each clock ticking at a different rate?

 

[Dale]

It does seem that we are not making much progress.

I agree, this is pointless. If you are going to ignore the answers that are provided then why bother asking the questions at all?

I sense that your answerw are based on the full theory of SR. I am looking at a more basic level, specifically the principle that comparison of two measurements requires that both be made using the same units of measure.
Does this principle apply to SR? If observers K, at rest, and k, in motion, measure the duration of an event, must their units be the same? If not, what does it mean to say that T=t?
Considering the triangle example, the result sqrt2 requires that the base and hypotenuse must be measured using the same units. If we find that the rulers are identical when parallel, then they are considered identical when applied to the triangle.

I have already answered these questions at least twice. See above.

Clocks are different from rulers, position doesn't advance of its own accord as time does.

It is clear that you just don't get the geometric approach to relativity.

Suppose we have two clocks at rest at the same position and find that they 'tick' at the same rate. What I have read of SR doesn't make clear what happens when one begins to move. Do the two clocks continue to 'tick' at the same rate as at rest ( as the rulers do, analagously speaking) so the measurements can be compared validly? Or does something happen to one clock to make it tick at a different rate, if so what happens,and how can a valid comparison be made with each clock ticking at a different rate?

It is simply not true that SR is not clear on the subject of what happens when a clock moves. That is what the whole theory describes in great detail.

If you believe your statement here then it is clear that you do not understand what SR says. I am clearly not getting through to you, so I would suggest that a different approach for learning may be more successful. I would recommend buying a textbook on SR and working the problems. Possibly together with watching e.g. Leonard Susskind's video lectures on the subject.

 

[matheinste]

Does this make any sense?

Take two identical ideal clocks side by side at rest in an inertial frame of reference. They tick at the same rate. Accelerate both identically and declerate both identically for a time as measured by the clocks themselves and with an acceleration measured by accelerometers accompanying each clock, so that they are eventually both at rest in inertial frames which are moving relative to each other.

Pick a third inertial reference frame relative to which the inertial frames in which the clocks are at rest are moving in opposite directions with equal speeds. Observed fom this third frame the two clocks will have the same rate as each other and so their "units" will be the same as viewed form this frame. In this sense the clocks can be said to be ticking at the same rate.

If the original clocks were not identical before being set in motion, and if their accelerations were not identiacal, it would still be possible to choose a third frame observed from which the two clocks would tick at the same rate as each other, but in this case their speeds relative to this frame would not be equal.

Matheinste.

 

[JM]

I would recommend buying a textbook on SR and working the problems. Possibly together with watching e.g. Leonard Susskind's video lectures on the subject.

Is there a particular textbook that you can recommend? Where are the Susskind lectures available?

 

[George Jones]

So, from your reading in SR, what books/papers/publications would you recommend for the basic principles?

It's hard to beat A Traveler's Guide to Spacetime: An Introduction to the Theory of Special Relativity by Thoma A. Moore,

https://www.amazon.com/dp/0070430276/?tag=pfamazon01-20.

 

[Dale]

Is there a particular textbook that you can recommend? Where are the Susskind lectures available?

The Susskind lectures are on YouTube from Stanford.

 

[JM]

It's hard to beat A Traveler's Guide to Spacetime: An Introduction to the Theory of Special Relativity by Thoma A. Moore,
.

Thanks for the suggestion, George, I'll give it a look. Its one I haven't seen.
I wonder that DaleSpam and JesseM declined to suggest some book that would describe their viewpoints.

 

[JesseM]

Thanks for the suggestion, George, I'll give it a look. Its one I haven't seen.
I wonder that DaleSpam and JesseM declined to suggest some book that would describe their viewpoints.

It's a bit pricey but the best one I've seen is https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20 by Taylor and Wheeler. For an online source I'd recommend this wikibook on SR, especially the first three sections after the introduction.

 

[Dale]

I wonder that DaleSpam and JesseM declined to suggest some book that would describe their viewpoints.

No need to wonder. I don't like my textbook for relativity and I won't recommend one I don't own.

 

[JM]

Summation:
The properties of clocks are listed in the first post. The third item is clarified in post 4 to emphasize that T and t are not meant to be equal. It was pointed out that in SR the term ‘synchronized’ refers to the Einstein convention using exchange of light signals. So, different terms are used later to describe the conclusion given in post 1.
The properties are affirmed in posts 27, 29, and 33. In addition post 33 includes examples of rulers measuring a triangle and clocks measuring spacetime lines. Further discussion of the clocks and rulers and their respective scale markings is given in posts 71, 72, 73, 74, 75, and78. This discussion leads to the following result. If two rulers are laid side by side and parallel and the scales are found to be identical, then the scales are taken to continue to be identical when the rulers are moved into position to measure the base and hypotenuse of the triangle. Thus, one inch along the base is equal to one inch along the hypotenuse. Similarly with clocks. If two clocks are brought together and found to count the time at the same rate, then they are taken to continue to agree when they are moved to time the two spacetime lines. Thus, one second along the ‘vertical’ line is equal to one second along the ‘diagonal’ line.
The picture that arises from this is that of two observers each assigning coordinates to events, while both using the same units of measure of space and time. In particular, posts 32 and 33, “If the units are the same then the clocks measuring T, t must advance at the same rate.” So, if the clocks are advancing at the same rate, then neither one can be ‘fast’ or ‘slow’ compared to the other. And the idea that ‘moving clocks run slow’ is not related to the properties of the clocks. In post 59, Al68 affirms this result and suggests Einsteins original 1905 paper for an explanation of the ‘time dilation’ equation t=T/gamma. The web location of the paper is given in post 59.
The equation is derived in par.4, page 10, the paragraph beginning ‘ Further, we imagine….”. It’s a short discussion and worth reading. Just a few comments. 1. It is mentioned more than once that the moving clock runs slow. Doesn’t it seem likely that the idea was picked up from here and repeated in the later literature? 2. However, after pointing out that t can be less than T, maybe it would be appropriate to draw attention to the Lorentz transforms. They are presented on page 9 just above par.4. If c becomes unlimited large then beta goes to 1 and the second term of the time equation drops out. The resulting equations are the Galilean transforms, the old technology before SR, where time is the same for both observers. The difference between the Galilean and the Lorentz is the terms necessary to satisfy Einsteins Light Postulate that light speed must be the same for both observers. Therefore the difference between t and T appears to be due to the Light Postulate.
Thank you all for your comments.

 

[JM]

It's hard to beat A Traveler's Guide to Spacetime: An Introduction to the Theory of Special Relativity by Thoma A. Moore,
.

George, I have reviewed the books by Moore, Taylor and Wheeler, and French for ideas related to the subject of this thread. The properties of clocks have not been called out explicitly as in post 1 and 88. But the way equations are developed and used indicates that the properties of clocks are implicit in their analysis, i.e. they appear to support the ideas given in post 88. Do you, or anyone else, have information to the contrary or supporting?

I wonder why there has been no comment on my post 88. Do you all agree, disagree, don't understand , or ?

 

[JM]

No need to wonder. I don't like my textbook for relativity and I won't recommend one I don't own.

I don't like any of the books or papers that I've seen on relativity, either. They are incompletely explained and raise more question than answers. The topic of this thread is an example. If you would like to say what text you are looking at and what you don't like, maybe we could compare notes.

 

[Taissir]

With curved and flat spacetimes in mind, does the photon (light) in it self produces any kind of curvature in flat spacetime. and when we speak of vacuum do we mean one that is devoid of light and matter too, and does such vacuum exist in the universe.?