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

[yuiop]

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?

The expression that all clocks tick at at a rate of one second per second relates to a comparison of one ideal clock to another ideal clock that are adjacent to each other and stationary with respect to each other. When clocks are moving relative to each other then they can experience different tick rates relative to each other and accumulate different times when reunited. For example consider a clock (A) that is transported from Earth to Mars by an inertial rocket. Another clock (B) leaves Earth at exactly the same time (Earth time) as clock A, but takes the scenic route to Mars via Jupiter and arrives at Mars at exactly the same time (Earth time) as clock B. Clock B takes the longer route so of necessity the rocket that transports clock B has to travel faster than the rocket that transports clock A. If clocks A and B were synchronised just before they left Earth, then clock B would show less elapsed time than clock A when they both arrive at Mars. Do you agree that clock B must have been ticking slower than clock A?

The elapsed (proper) times recorded by both clock A and B are in turn less than the time recorded by the clocks at rest with respect to Earth and Mars. The latter time interval is a coordinate time interval as is a calculated time rather than a time measured by a single clock that is present at all events. The times measured by clocks A and B are proper time intervals as they are measured by single clocks present at both events. Note that clock B is a non inertial clock, but it nevertheless records a proper time interval that all other observers can agree on. The proper time interval is not necessarily the same as the invariant time interval which only applies to the inertially moving clock and this interval is always the longest proper time recorded by a single clock present at both events. Also note that while people here state that coordinate clocks record proper time, that the coordinate time interval between the two events can be longer than the invariant interval. The important concept here is that while individual "coordinate clocks" that are at rest in given reference frame record proper time just like any other ideal clock, coordinate time intervals are calculated using multiple clocks or radar measurements and are not proper time intervals.

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

Except in some weird coordinate systems, clocks are assumed to run naturally and are not built to run at different rates in different reference frames. Any difference in clock rates is completely natural. For example the "clock" could be a lump of radioactive material which records time by measuring how much un-decayed material is left.

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.

I think it is clear that most people here are not happy to use the terms "proper clock" or "coordinate clock". Perhaps for the sake of this thread we should stick to terms like "proper time interval", "invariant time interval" and "coordinate time interval", that hopefully most of can live with.
As for the expression "moving clocks run slow", I think it would be better to expand that to "in a given inertial reference frame, moving clocks run slower than stationary clocks".

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

Hi Dale, it seems that you now accept there is such a thing as a proper clock and it is defined as a single inertial clock that is present at both events, but I would agree is it a non standard term.

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

Yes, this a non standard term and I guess they mean a stationary synchronised clock in a given inertial reference frame. However, being non standard its use is open to interpretation. I think a better term is "coordinate time interval". I think you would agree that while all clocks record proper time, that the coordinate time interval between two events is observer dependent and can be longer than the invariant time interval. Without being an expert on terminology, I think we need a term that conveys the measurement of a time interval between two events that is not a proper time interval, even though all clocks measure proper time.

 

[Dale]

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?

My intro text was Serway.

Yes, SR can easily handle a single moving clock.

I would find it using the proper time formula that I posted earlier. As I have repeated multiple times that is the formula for any clock undergoing any motion in any frame.

 

[PhilDSP]

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

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

Pauli's textbook on SR and Minkowski's EM (plus the basics of GR) is pretty decent IMO. Though it is oriented towards serious physics or engineering students. It's inexpensive and was written in the 1920's so it doesn't have newer, more abstract, more exotic developments in the field.

 

[harrylin]

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?[..]

Without following that discussion, I saw a later post by you from which it appears that it's still not clear to you. Maybe useful if someone else gives a try?

- A and B are different "stationary" points (xA, yA, zA) and (xB, yB, zB).

- the straight constant speed trajectory AB is one line of the polygon.

- the straight same constant speed trajectory BC is another line of the polygon. The Lorentz transformation relating "time" in a co-moving frame along AB is identical to that along BC: x is by definition the direction of motion. And obviously from point B the clock has to continue its counting from where it was the moment before - that's just common sense.

- A polygon implies in practice a high acceleration during a very short time. Einstein was writing for physicists who know that such quick changes of direction have only a small effect on common clocks compared to the clock count over long straight trajectories. So, if that's an issue for you: yes obviously he neglected that as well as the other common things that are commonly neglected in physics as they are usually small as well as clock dependent, and instead he did an "ideal" clock calculation. The transformation relationships also have zero memory effect.
Thus he compared an implicit classical calculation according to which "ideal" clocks are unaffected by such a trajectory with the prediction of the new theory.

- What you didn't ask, is why he wrote "If we assume that the result proved for a polygonal line is also valid for a continuously curved line"; but that implicitly answered one of your questions. For in that case we cannot assume that we may neglect the effect of acceleration because of the very short duration, since there is a constant acceleration over the whole trajectory. However, clocks can be made to be insensitive to acceleration according to classical mechanics and SR predicts no effect from acceleration itself, so he merely assumed no effect from that. Note that SR ignores the effect from gravitation and if he suspected a possible effect from that then he overlooked the non-round shape of the earth, but that's another topic.

- The Lorentz transformations impose corrections to classical laws; and exactly there, in that passage, Einstein concluded what consists of a law - the law of "time dilation" or "clock retardation". That law should be valid for clocks anywhere - for example at the equator and at the pole. You can also relate it back to transformations if you remember that all ideal stationary clocks are supposed to remain in synch, so that according to SR a clock at the pole will remain in synch with a clock at the equator which does not participate in the rotation of the earth. And you can then compare that clock with the one that is rotating with the earth.

Does that help?

 

[ghwellsjr]

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?

I use the term "coordinate clock" all the time. I never realized that it doesn't have a formal definition but it seems to me that everyone would understand that it is referring to the clocks Einstein described at the end of section 1 of his 1905 paper introducing Special Relativity:

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”

The term was in use on this forum before I signed on in Sept 2010, for example this one from Feb 2010:

Proper time is just the amount of time elapsed on a physical clock. If a clock is moving inertially, then the proper time between two events on its worldline is the same as the coordinate time between those events in the clock's own rest frame (remember, coordinate time in an inertial frame is defined in terms of the readings on a set of clocks at rest in that frame, so if the clock whose proper time you're interested is also at rest in some frame then it'll be at rest right next to one of these coordinate clocks, so naturally both keep time with one another). Likewise, if a clock is moving inertially, then in a frame where the clock is moving at velocity v, if the coordinate time between two events on its worldline is t then the proper time the clock experiences between those events is t*squareroot(1 - v²/c²), that's the physical meaning of the time dilation equation. But again, proper time is more general than either of these descriptions since you can talk about proper time for a non-inertial clock too.

I don't recall if this is where I picked up the use of the term but I just thought it was one of those things that was in such wide general use that everyone would know what it meant even if it wasn't specifically defined somewhere.

 

[Dale]

I use the terms coordinate time and proper time, but I don't think that i have ever used the terms coordinate clock or proper clock. I don't like either of those terms. A clock is an object which marks proper time along its worldline. If an object does that then it is a clock, if not then it isn't. I don't see the need to add additional qualifiers, nor the benefit of doing so.

Also, you get into problems in coordinate systems where the time coordinate is non-uniform. E.g. you might have a coordinate system where 1 s of coordinate time was equal to 3 s of proper time for a clock at rest in the coordinate system. So would a "coordinate clock" mark 1 s or 3 s in such a case? The distinction doesn't add any clarity.

 

[ghwellsjr]

I use the terms coordinate time and proper time, but I don't think that i have ever used the terms coordinate clock or proper clock. I don't like either of those terms. A clock is an object which marks proper time along its worldline. If an object does that then it is a clock, if not then it isn't. I don't see the need to add additional qualifiers, nor the benefit of doing so.

I agree with regard to the term "proper clock", as I told JM back in post #112:

"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.

But since Einstein repeatedly used the term "stationary clock" and I often used the term "stationary coordinate clock" in this thread to mean exactly the same thing that Einstein meant, I don't see the problem with sometimes using "coordinate clock" in the same context. We are talking about the inertial clocks that have been previously synchronized to establish an inertial coordinate system, as Einstein said:

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

Also, you get into problems in coordinate systems where the time coordinate is non-uniform. E.g. you might have a coordinate system where 1 s of coordinate time was equal to 3 s of proper time for a clock at rest in the coordinate system. So would a "coordinate clock" mark 1 s or 3 s in such a case? The distinction doesn't add any clarity.

I have no idea what you are talking about here but I don't think it can be related to what Einstein said with regard to inertial coordinate systems in Special Relativity, which is what this thread is about.

 

[Dale]

I don't think it can be related to what Einstein said with regard to inertial coordinate systems in Special Relativity, which is what this thread is about.

You are correct. It is just that, for pedagogical reasons, I do not like to introduce non-standard terminology, particularly when it provides little or no present benefit and may cause future problems.

 

[ghwellsjr]

What would you call the clocks in the latticework described by Kip Thorne on page 3 of his upcoming http://www.pma.caltech.edu/Courses/ph136/yr2011/1102.2.K.pdf?

 

[JM]

My intro text was Serway.

Yes, SR can easily handle a single moving clock.

I would find it using the proper time formula that I posted earlier. As I have repeated multiple times that is the formula for any clock undergoing any motion in any frame.

Serway is one I haven't seen.

My answer is t' = [1/m + v²/c² - uv/c<sup>2t.
Whats yours?

JM</sup>

 

[JM]

The expression that all clocks tick at at a rate of one second per second relates to a comparison of one ideal clock to another ideal clock that are adjacent to each other and stationary with respect to each other.

HI yuiop, thanks for checking in. My latest thoughts on tick rate are in Post 140. What do you think of them?
JM

 

[JM]

Pauli's textbook on SR and Minkowski's EM (plus the basics of GR) is pretty decent IMO. Though it is oriented towards serious physics or engineering students. It's inexpensive and was written in the 1920's so it doesn't have newer, more abstract, more exotic developments in the field.

What are the titles and publisners ( or other source) for Pauli and Mink?
JM

 

[JM]

Without following that discussion, I saw a later post by you from which it appears that it's still not clear to you. Maybe useful if someone else gives a try?

- A and B are different "stationary" points (xA, yA, zA) and (xB, yB, zB).

- the straight constant speed trajectory AB is one line of the polygon.

Thanks, harrylin. I follow your ideas. My need is for the math that connects the time t'' of moving rotated frame with the time t of the original stationary frame K. Assume the points A, (0,0,0 ) and B, ( 1,1,0) wrt K. I accept that the relation between the frames with their respective 'x' axes aligned with these points is the same as the relation between the original K and K' axes. But a clock moving at v along the line between A and B moves at only vcos45 wrt K, for example. So what is the math that connects t'' with t?
JM

 

[JM]

I have no idea what you are talking about here but I don't think it can be related to what Einstein said with regard to inertial coordinate systems in Special Relativity, which is what this thread is about.

Right on George!

 

[Dale]

What would you call the clocks in the latticework described by Kip Thorne on page 3 of his upcoming http://www.pma.caltech.edu/Courses/ph136/yr2011/1102.2.K.pdf?

I would call them "clocks".

 

[Dale]

My answer is t' = [1/m + v²/c² - uv/c^{2t.
Whats yours?}

^{What is the question and what are m, u, and v?}

 

[harrylin]

Thanks, harrylin. I follow your ideas. My need is for the math that connects the time t'' of moving rotated frame with the time t of the original stationary frame K. Assume the points A, (0,0,0 ) and B, ( 1,1,0) wrt K. I accept that the relation between the frames with their respective 'x' axes aligned with these points is the same as the relation between the original K and K' axes. But a clock moving at v along the line between A and B moves at only vcos45 wrt K, for example. So what is the math that connects t'' with t?
JM

It looks as if my next remark didn't reach:
"
- the straight same constant speed trajectory BC is another line of the polygon. The Lorentz transformation relating "time" in a co-moving frame along AB is identical to that along BC: x is by definition the direction of motion. And obviously from point B the clock has to continue its counting from where it was the moment before - that's just common sense.
"
I'll try again. For the first leg, the X-axis of K and K' is by definition chosen along the line AB. It is you who draws the lines and defines the frames for the calculation. Thus you give A and B the same Y and Z coordinate (in this case you can keep them both 0), and v along x is simply v. That's how the Lorentz transformations are defined. And how the math between the polygon lines is connected I explained next. So, I'm afraid that you could not follow me.
To elaborate: you choose for the calculation for BC new reference frames with X and X' oriented along BC.

 

[ghwellsjr]

What would you call the clocks in the latticework described by Kip Thorne on page 3 of his upcoming http://www.pma.caltech.edu/Courses/ph136/yr2011/1102.2.K.pdf?

I would call them "clocks".

Then what would you do to JesseM's quote to make it satisfactory to you?

 

[JM]

What is the question and what are m, u, and v?

m is the coefficient in the LT often referred to as gamma. v is the speed of the frame moving in the x direction of the stationary frame K. u is the speed of a single clock moving in the + x direction of K. The question is 'what is the time on the single moving clock'?

 

[JM]

It looks as if my next remark didn't reach:

I think you didn't understand my question.
Section 4 of 1905 envisions a single clock moving along a polygon path wrt a stationary frame K. The clock starts at a point of K and returns to the same point of K. What you have described is the time of the clock wrt the polygon path. What you have not described is the time of the clock wrt the original frame K. This is the time that is required in order to make a valid comparison with the K time at the end of the path.

 

[GAsahi]

I think you didn't understand my question.
Section 4 of 1905 envisions a single clock moving along a polygon path wrt a stationary frame K. The clock starts at a point of K and returns to the same point of K. What you have described is the time of the clock wrt the polygon path. What you have not described is the time of the clock wrt the original frame K. This is the time that is required in order to make a valid comparison with the K time at the end of the path.

If the time on the clock at rest wrt the polygon is $T$ and the speed of the second observer is $v$, then the clock for the observer moving wrt the polygon will show $t=T \sqrt{1-(v/c)^2}$ when the observers are reunited so they can compare clocks.
Does this answer your question?

 

[Dale]

m is the coefficient in the LT often referred to as gamma. v is the speed of the frame moving in the x direction of the stationary frame K. u is the speed of a single clock moving in the + x direction of K. The question is 'what is the time on the single moving clock'?

For approximately the 100th time I refer you to the formula I posted back in post 36. The time displayed on the clock is:

$$\tau = \int \sqrt{1-v(t)^2/c^2} dt$$
So in frame K
$$\tau = t \sqrt{1-u^2/c^2}+C_1$$
And in the other frame
$$\tau = t \sqrt{1-\frac{(u+v)^2}{c^2(1+uv/c^2)^2}}+C_2$$