Time Dilation Formula: Clarifying Confusion

[Grimble]

If A emits light in the direction of movement, then length contraction does come into play. In fact this is a typical scenario used by textbooks to derive the length contraction formula.

http://www.pa.msu.edu/courses/2000spring/PHY232/lectures/relativity/contraction.html

I'm sorry, but surely this example is wrong: it is merely a GALILEAN transformation and has nothing to do with SR!

As I work it out if t_o = 2L_o/c
and t = 2L(c² - v²)^-1/2
and t = γt_o
then surely when we solve this we have

L(1 - v²c^-2)^-1/2 = γL_o

and as γ = (1 - v²c^-2)^-1/2

we arrive at L = L_o which is hardly surprising if, as I say this is merely a Galilean transformation.

Grimble

 

[Ich]

and t = 2L(c2 - v2)-1/2

Where did you get that from?

 

[Grimble]

Where did you get that from?

Sorry my mistake the calculation is correct.

But as this uses light traveling at three different speeds c, c-v and c+v and we all know that light only travels at c, where does that leave one?

To use c+v and c-v it can only be describing a Galilean transformation, where either the speed of light inceases or the time increases.

 

[Ich]

But as this uses light traveling at three different speeds c, c-v and c+v

It doesn't. It calculates the time it takes for light moving at c to reach an object moving at v, that's where those denominators come from.
It is only your - wrong, btw. - interpretation that this somehow implies that the object sees the light moving at c+-v.

 

[Grimble]

Another link with animations:

http://webphysics.davidson.edu/physlet_resources/special_relativity/illustration4.html

Again purely Galilean transformations. With the addition of shortening the moving clock, which ticks slower although the text says it is the stationary clock that ticks slower.

The problem is that these derivations/demonstrations/proofs or whatever are only using Euclidean geometry so how can they have anything to do with SR? Merely adding a length contraction or time dilation proves nothing.

 

[collectedsoul]

Okay. Thanks Rasalhague for that excellent link to the animation.

Referring to the earlier derivation, http://www.pa.msu.edu/courses/2000sp...ntraction.html , isn't there something wrong with the formula? If I look at the final expression and substitute for c=L/t, I'm getting c=L₀/t₀gamma² So the speed of light is less in the moving frame? (btw how do I type symbols like the one for gamma?)

Besides, my original doubt was about what actually happens in the horizontal velocity scenario. If the receiver A is moving at velocity v, is both time dilation and length contraction occurring for him? I mean, in the above derivation they have just assumed the time dilation (I'm guessing from the previously derived light clock scenario where light moves in the vertical). But why? Why just assume this? Isn't it possible to derive time dilation in the second scenario?

I think part of my problem with these derivations is that they use Galilean transformations as Grimble says. I was looking for something not involving Euclidean geometry at all.

 

[JesseM]

Again purely Galilean transformations. With the addition of shortening the moving clock, which ticks slower although the text says it is the stationary clock that ticks slower.

No, the text says moving clocks tick slower as measured in the frame we label as "stationary":

Therefore it takes more clicks as measured by the stationary clock to measure a time interval of a moving clock. Observed from stationary frames, moving clocks run slower. This is called time dilation.

The problem is that these derivations/demonstrations/proofs or whatever are only using Euclidean geometry so how can they have anything to do with SR? Merely adding a length contraction or time dilation proves nothing.

The geometry of space at a given instant in any inertial frame is Euclidean, even if the geometry of spacetime is not. The spatial distance between points that are a distance dx apart along the x-axis of some inertial frame, dy along the y-axis and dz along the z-axis would just be \sqrt{dx^2 + dy^2 + dz^2} in that frame for example.

 

[Grimble]

The geometry of space at a given instant in any inertial frame is Euclidean, even if the geometry of spacetime is not. The spatial distance between points that are a distance dx apart along the x-axis of some inertial frame, dy along the y-axis and dz along the z-axis would just be \sqrt{dx^2 + dy^2 + dz^2} in that frame for example.

Yes of course, and I certainly agree with that, but one inertial frame observed from another is not, it is relativistic. Where in all these references is there one allusion to proper time, co-ordinate time or Lorentz transformations?

All I am seeing, as I say, are Galilean transformations, where the addition of an extra movement results in a longer path, and a longer time.

IF they hadn't complied with Einstein's 2nd postulate they could have complied with the 1st and kept the time the same but increased the velocity. THAT is the problem with Galilean transformations that Einstain was addressing - HOW to comply with the 1st and keep the time constant whilst AT THE SAME TIME complying with the 2nd and keep the speed of light constant. With these references they only do one at the expense of the other.

Einstein had to conceive of space and time not being absolute in order to comply with both postulates - thereby giving rise to transformations, and co-ordinate time.
(Which, not surprisingly has the same duration, in absolute terms, - unit length x number of units - as the proper time, for the same occurrence - (I would say event but that term has been appropriated already).)

Grimble

 

[Ich]

Don't you see that you are looking at very basic derivations of SR effects? There are no Galilean or Lorentzian transformations, they only rely on very general properties that you can derive from the postulates.
At the end of the process, one would find out how the transformations have to look like.
Your assertion "To use c+v and c-v it can only be describing a Galilean transformation" is plain nonsense. Just try to follow what they are saying, and try to understand.

 

[JesseM]

Yes of course, and I certainly agree with that, but one inertial frame observed from another is not, it is relativistic.

What do you mean it's not? The Euclidean formula for distance still works fine when dealing with objects that are moving at relativistic velocities in your frame. For example, if you have an object which in its own frame looks like a square 10 light-seconds on each side, and in your frame it's moving at 0.6c so the side moving parallel to the direction of motion is 8 light-seconds long in your frame while the side perpendicular to the direction of motion is 10 light-seconds long in your frame, then the distance between opposite corners of the square is just going to be given by the ordinary Euclidean formula \sqrt{8^2 + 10^2}. Relativity only plays a role in making one side shorter in your frame than it is in the object's own rest frame, but other than that nothing about the object's shape in your frame is any different than it would be if you were just dealing with an 8 by 10 rectangle in classical Newtonian physics.

Where in all these references is there one allusion to proper time, co-ordinate time or Lorentz transformations?

All I am seeing, as I say, are Galilean transformations, where the addition of an extra movement results in a longer path, and a longer time.

In relativity as in Galilean physics, velocity is defined as distance/time, so the time T for an object traveling at speed v to travel a given distance D must be given by T = D/v. And as I said, distances are given by the ordinary Euclidean formula, so if an object travels a horizontal distance D_h between points and a vertical distance D_v, it is just as true in an inertial SR frame as it is in Galilean physics that the total distance the object traveled must be \sqrt{{D_h}^2 + {D_v}^2}.

There is a uniquely relativistic aspect of the light clock thought-experiment though. Note that in Galilean physics, if the light bouncing between the two mirrors was moving at a speed c in the light clock's own rest frame, then it would not be moving at speed c in your frame where the light clock is in motion! In both Galilean physics and relativity, the total diagonal velocity of the light in your frame depends on both the vertical velocity in your frame (which according to the Galilean transformation would be c in your frame too if it was c in the light clock's rest frame and the light clock was moving horizontally in your frame) and the horizontal velocity v_h in your frame--the total velocity in Galilean physics would therefore be \sqrt{c^2 + {v_h}^2}, and you'd find that because the velocity was slightly greater, the time between ticks for the moving clock would not be slower than the time between ticks for your own light clock (assuming the light moves at c vertically for your clock too), despite the fact that the light has a longer path length to travel between ticks for the moving clock than for your clock. In relativity, on the other hand, it's a fundamental postulate that light travels at c in all inertial frames, so in the frame where the light clock is in motion the light must still travel at c along the diagonal path between mirrors.

IF they hadn't complied with Einstein's 2nd postulate they could have complied with the 1st and kept the time the same but increased the velocity. THAT is the problem with Galilean transformations that Einstain was addressing - HOW to comply with the 1st and keep the time constant whilst AT THE SAME TIME complying with the 2nd and keep the speed of light constant. With these references they only do one at the expense of the other.

I don't understand, why do you think the 1st postulate implies that the time should be constant? The 1st postulate implies that if you run the same experiment in different frames each frame will see the same result, so if you construct a light clock at rest in frame A and measures the time in frame A, it should give the same answer that you'd get if you constructed an identical light clock at rest in frame B and measured the time in frame B. But the 1st postulate does not say that if you construct a light clock at rest in frame A but in motion in frame B, and measure the time in frame B, that the result would be the same as if you construct a clock at rest in frame B and measure the time in frame B. In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical.

Einstein had to conceive of space and time not being absolute in order to comply with both postulates - thereby giving rise to transformations, and co-ordinate time.

I'm not sure what you mean by "space and time not being absolute"--certainly it's true that the distance and time between a pair of events will vary depending on what frame you use, but there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.

 

[Grimble]

Don't you see that you are looking at very basic derivations of SR effects? There are no Galilean or Lorentzian transformations, they only rely on very general properties that you can derive from the postulates.
At the end of the process, one would find out how the transformations have to look like.
Your assertion "To use c+v and c-v it can only be describing a Galilean transformation" is plain nonsense. Just try to follow what they are saying, and try to understand.

I'm sorry if I seem a little dim to you, but if you read http://www.bartleby.com/173/7.html" you should see why I am saying what I am saying.

 

[Grimble]

What do you mean it's not? The Euclidean formula for distance still works fine when dealing with objects that are moving at relativistic velocities in your frame. For example, if you have an object which in its own frame looks like a square 10 light-seconds on each side, and in your frame it's moving at 0.6c so the side moving parallel to the direction of motion is 8 light-seconds long in your frame while the side perpendicular to the direction of motion is 10 light-seconds long in your frame, then the distance between opposite corners of the square is just going to be given by the ordinary Euclidean formula \sqrt{8^2 + 10^2}. Relativity only plays a role in making one side shorter in your frame than it is in the object's own rest frame, but other than that nothing about the object's shape in your frame is any different than it would be if you were just dealing with an 8 by 10 rectangle in classical Newtonian physics.

Yes, I can see what you are saying BUT you have neglected to say that the 8 light-seconds is measured in co-ordinate seconds, not proper seconds and how do you apply Euclidean/Newtonian physics to a rectangle whose sides are measured in different units?

I don't understand, why do you think the 1st postulate implies that the time should be constant? The 1st postulate implies that if you run the same experiment in different frames each frame will see the same result, so if you construct a light clock at rest in frame A and measures the time in frame A, it should give the same answer that you'd get if you constructed an identical light clock at rest in frame B and measured the time in frame B. But the 1st postulate does not say that if you construct a light clock at rest in frame A but in motion in frame B, and measure the time in frame B, that the result would be the same as if you construct a clock at rest in frame B and measure the time in frame B. In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical.

You are quite right when you say that the two clocks should keep identical time; but consider; if the moving clock were to transmit a pulse of light each second, then allowing for the transmission time for the pulse of light, the stationary frame would still determine that the moving clock was 'ticking' at the same rate as its own clock.
It is only when the stationary frame observes the moving clock directly that the changes are seen, but will it really be 'ticking' at a different rate?

I'm not sure what you mean by "space and time not being absolute"--certainly it's true that the distance and time between a pair of events will vary depending on what frame you use, but there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.

By

space and time not being absolute

I am referring to Einstein's statement in http://www.bartleby.com/173/11.html" where he says

THE RESULTS of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows:

1. The time-interval (time) between two events is independent of the condition of motion of the body of reference.
2. The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.

Grimble

 

[Ich]

if you read this you should see why I am saying what I am saying.

I still don't see it. The problem Einstein refers to arises if you say that w=c-v is the speed of light as measured by the carriage. It isn't.
That does not mean that the equation w=c-v is evil and mustn't be used for whatever purposes. It's just that w is not the speed of light as measured by the carriage. It is the difference of the velocity of light and the velocity of the carriage as measured in the embankment system. That has nothing to do with any transformation laws, as it is a description in only one reference frame.

 

[Grimble]

I still don't see it. The problem Einstein refers to arises if you say that w=c-v is the speed of light as measured by the carriage. It isn't.
That does not mean that the equation w=c-v is evil and mustn't be used for whatever purposes. It's just that w is not the speed of light as measured by the carriage. It is the difference of the velocity of light and the velocity of the carriage as measured in the embankment system. That has nothing to do with any transformation laws, as it is a description in only one reference frame.

Taking the passage in question:

However to an observer who sees the clock pass at velocity v, the light takes more time to traverse the length of the clock when the pulse is traveling in the same direction as the clock, and it takes less time for the return trip.

Now I may be wrong, but I read that as saying: "to an observer of the moving clock, when observing the CLOCK'S FRAME OF REFERENCE, that the pulse of light moves slower one way and faster the other..."

For:

However to an observer who sees the clock pass at velocity v

- the moving clock

the light takes more time to traverse the length of the clock when the pulse is traveling in the same direction as the clock

Light takes longer - moves slower

and it takes less time for the return trip.

Light takes less time - moves faster

- whilst traveling the same distance.

Grimble

 

[Ich]

Now I may be wrong, but I read that as saying: "to an observer of the moving clock, when observing the CLOCK'S FRAME OF REFERENCE, that the pulse of light moves slower one way and faster the other..."

They didn't write what you read. There is no such thing as "observing the clock's frame of reference". You may observe the clock, recording time and date of such measurements as given by your frame of reference.
Why don't you read it as saying:
"However to an observer who sees the clock pass at velocity v, the light takes more time to traverse the length of the clock when the pulse is traveling in the same direction as the clock, and it takes less time for the return trip." ?

Apparently, you haven't understood the concept of a reference frame. For example

Light takes less time - moves faster

- whilst traveling the same distance.

(emphasis mine)
It isn't traveling the same distance.
Again: why don't you try to understand what they wrote instead of interpreting their every word? That has nothing to do with accusing you of being dim, but you definitely always skip the first step - reading and following their argument - and jump to the second: interpreting hat you have learned.

I think a reasonable first step would be if you draw a spacetime diagram of the described events; you can then easily read off distances, times, and velocities, and check with what the link is saying contrary to what you have assumed it was saying.

 

[Grimble]

It isn't traveling the same distance.

But it is, within the clock's frame of reference, and the speed of light has to be the same wherever it is observed from

 

[Grimble]

I still don't see it. The problem Einstein refers to arises if you say that w=c-v is the speed of light as measured by the carriage. It isn't.

Einstein wrote:

w is the required velocity of light with respect to the carriage, and we have
w = c - v.
The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c.

 

[Grimble]

They didn't write what you read. There is no such thing as "observing the clock's frame of reference". You may observe the clock, recording time and date of such measurements as given by your frame of reference.

Einstein wrote in http://www.bartleby.com/173/11.html"

How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity? In other words: Can we conceive of a relation between place and time of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to the embankment and relative to the train? This question leads to a quite definite positive answer, and to a perfectly definite transformation law for the space-time magnitudes of an event when changing over from one body of reference to another.

So I read this as Einstein being concerned about finding the space-time co-ordinates from one frame of reference and using them (transforming them) to find the corresponding space-time magnitudes within another frame of reference.

Why don't you read it as saying:
"However to an observer who sees the clock pass at velocity v, the light takes more time to traverse the length of the clock when the pulse is traveling in the same direction as the clock, and it takes less time for the return trip." ?

And that is exactly how Galileo or Newton would have read it!

But we are not concerned with a Galilean transformation.

Again: why don't you try to understand what they wrote instead of interpreting their every word? That has nothing to do with accusing you of being dim, but you definitely always skip the first step - reading and following their argument - and jump to the second: interpreting hat you have learned.

I'm sorry if I have given that impression; I have read and understood exactly what they are saying and I am, I suppose, guilty of trying to re-interpret it in relation to SR.

I think a reasonable first step would be if you draw a spacetime diagram of the described events; you can then easily read off distances, times, and velocities, and check with what the link is saying contrary to what you have assumed it was saying.

Please allow me to do just what you say and draw some diagrams and then check back with you with my interpretaions of them!

Grimble

ps thank you for your time!

 

[matheinste]

Hello Grimble.

As you say :-
Einstein wrote:

w is the required velocity of light with respect to the carriage, and we have
w = c - v.
The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c.


But if you follow the argument from beginning to end he explains that this relation cannot hold unless we abandon the principle of relativity, and as we know, he did not abandon it.

I have not had time to reread the translation of Einstein's work, but I recall that it is probably one of the best and simplest explantions of the theory. I think it may be unwise to put a different interpretation on what Einstein wrote, after all its his theory and subsequent events have so far proved him correct. He proposed the constancy of c in inertial frames and so how likely do you think it is that he would argue against it except to make a point of how the theory would not hold up if this proposition was untrue.

Matheinste.

 

[Grimble]

Hello again Matheinste!

Hello Grimble.

As you say :-
Einstein wrote:

w is the required velocity of light with respect to the carriage, and we have
w = c - v.
The velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c.


But if you follow the argument from beginning to end he explains that this relation cannot hold unless we abandon the principle of relativity, and as we know, he did not abandon it.

I have not had time to reread the translation of Einstein's work, but I recall that it is probably one of the best and simplest explantions of the theory. I think it may be unwise to put a different interpretation on what Einstein wrote, after all its his theory and subsequent events have so far proved him correct. He proposed the constancy of c in inertial frames and so how likely do you think it is that he would argue against it except to make a point of how the theory would not hold up if this proposition was untrue.

Matheinste.

I agree with everything you say here! c is constant and the same wherever we measure it and that quote would deny relativity!

I was quoting Einstein in response to an earlier post by Ich who stated that:

The problem Einstein refers to arises if you say that w=c-v is the speed of light as measured by the carriage. It isn't.

This is indeed the simplest and most elegant explanation of his theory and, to me at least, it makes perfect sense and agrees with everything one tries to put into space-time diagrams.
Particular favourites of mine are diagrams of Minkowski space, particularly the traditional Minkowski Diagrams elevated from flat two dimensional diagrams into 3 dimensional constructs which treatment is what I believe really shows everything clearly.

 

[JesseM]

Yes, I can see what you are saying BUT you have neglected to say that the 8 light-seconds is measured in co-ordinate seconds, not proper seconds and how do you apply Euclidean/Newtonian physics to a rectangle whose sides are measured in different units?

Both sides of the moving rectangle are measured in coordinate light-seconds. When you want to compute a distance in any coordinate system, you only use that coordinate system's measures of distance. Again, if you want to know the distance between two points in a given frame, you take the coordinate distance between them on the x-axis which we can call dx, the coordinate distance on the y-axis dy, and the coordinate distance on the z-axis dz, then use the pythagorean formula \sqrt{dx^2 + dy^2 + dz^2} to get the total distance between the points. The procedure is the same regardless of whether you are measuring the distance between objects (or parts of an object, like different corners of a square) that are at rest in this frame, or between objects that are in motion in this frame. Nowhere would the notion of "proper distance" enter into it at all.

I don't understand, why do you think the 1st postulate implies that the time should be constant? The 1st postulate implies that if you run the same experiment in different frames each frame will see the same result, so if you construct a light clock at rest in frame A and measures the time in frame A, it should give the same answer that you'd get if you constructed an identical light clock at rest in frame B and measured the time in frame B. But the 1st postulate does not say that if you construct a light clock at rest in frame A but in motion in frame B, and measure the time in frame B, that the result would be the same as if you construct a clock at rest in frame B and measure the time in frame B. In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical.

You are quite right when you say that the two clocks should keep identical time;

Identical time in their own respective rest frames. They certainly do not keep identical time if you measure both from the perspective of a single inertial frame in which they have different speeds--that's exactly what I meant when I said above "In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical." Do you think the 1st postulate implies their ticking rate should be identical even in this situation? If so, can you explain your reasoning?

if the moving clock were to transmit a pulse of light each second, then allowing for the transmission time for the pulse of light, the stationary frame would still determine that the moving clock was 'ticking' at the same rate as its own clock.

No, it definitely would not! Time dilation is what remains after you correct for transmission delays (i.e. correcting for the Doppler effect). For example, suppose the clock is moving at 0.6c, and in 2020 I see it next to a marker 10-light-years away from me (in my frame) with the clock showing a reading of 30 years, and then in 2036 I see it next to a marker 16-light-years-away from me with the clock showing a reading of 38 years. If I subtract off the light travel times (10 years for the first reading to travel 10 light-years from the clock to my eyes, and 16 years for the second reading to travel 16 light-years from the clock to my eyes), I will conclude that the clock "really" showed the first reading of 30 years in 2020-10=2010, and it "really" showed the second reading of 38 years in 2036-16=2020. So I will conclude that in the 10 years between 2010 and 2020, the clock itself only ticked forward by 8 years from 30 to 38, so it must have been slowed down by a factor of 0.8 in my frame. This is different from how much it appeared to be slowed down visually--visually it took 2036-2020=16 years to tick forward by 8 years, so it appeared to be running slow by a factor of 0.5, but this extra slowdown is just due to the Doppler effect (which is a consequence of the fact that light from different readings on the clock has different delays in reaching me since the clock's distance from me is changing).

I'm not sure what you mean by "space and time not being absolute"--certainly it's true that the distance and time between a pair of events will vary depending on what frame you use, but there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.

I am referring to Einstein's statement in http://www.bartleby.com/173/11.html" where he says

THE RESULTS of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows:

1. The time-interval (time) between two events is independent of the condition of motion of the body of reference.
2. The space-interval (distance) between two points of a rigid body is independent of the condition of motion of the body of reference.

OK, but in that quote he was saying exactly the same thing I was saying when I said "certainly it's true that the distance and time between a pair of events will vary depending on what frame you use". Do you agree that there is no conflict between this statement and my other statement immediately after? Namely:

there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.

 

[Grimble]

Both sides of the moving rectangle are measured in coordinate light-seconds.

I am afraid you have lost me here with yor reasoning, JesseM; for if the two sides parallel to the direction of motion have been transformed by the Lorentz factor and the other two sides are unchanged, how can they be in the same units? The Lorentz transformations change the size of the units as well as their quantity or are you saying all that was discussed previously in this thread is nonsense?

 

[Grimble]

Identical time in their own respective rest frames. They certainly do not keep identical time if you measure both from the perspective of a single inertial frame in which they have different speeds--that's exactly what I meant when I said above "In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical." Do you think the 1st postulate implies their ticking rate should be identical even in this situation? If so, can you explain your reasoning?

Yes, my reasoning is to follow what the 1st postulate says:

The Principle of Relativity – The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems in uniform translatory motion relative to each other.

or as he says in http://www.bartleby.com/173/5.html"

If K is a Galileian co-ordinate system, then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translation. Relative to K' the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K. 2
We advance a step farther in our generalisation when we express the tenet thus: If, relative to K, K' is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the restricted sense).

And if, natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K, then time and distance must be identical. (A muon's half life cannot be different and if time is identical distance has to be also cf. the speed of light)

That is; they will keep the same 'Proper Time' within their own frames of reference.

So in two independent Inertial Frames of Reference, identical clocks will keep identical time.

But yes, if viewed by an independent observer they will shew different times.

Grimble

 

[Grimble]

OK, but in that quote he was saying exactly the same thing I was saying when I said "certainly it's true that the distance and time between a pair of events will vary depending on what frame you use". Do you agree that there is no conflict between this statement and my other statement immediately after? Namely:

there is nothing in SR that implies we can't use the Euclidean formula for the distance between events in a single frame, or that we can't use the ordinary kinematical formula that velocity = distance/time in each frame. The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity, along with the uniquely relativistic notion that if the light is bouncing between mirrors at c in the light clock's own rest frame, it must also be bouncing between them at c in the frame where the light clock is in motion.

No, because

The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity

but it applies them across frames.

 

[Grimble]

No, it definitely would not! Time dilation is what remains after you correct for transmission delays (i.e. correcting for the Doppler effect). For example, suppose the clock is moving at 0.6c, and in 2020 I see it next to a marker 10-light-years away from me (in my frame) with the clock showing a reading of 30 years, and then in 2036 I see it next to a marker 16-light-years-away from me with the clock showing a reading of 38 years. If I subtract off the light travel times (10 years for the first reading to travel 10 light-years from the clock to my eyes, and 16 years for the second reading to travel 16 light-years from the clock to my eyes), I will conclude that the clock "really" showed the first reading of 30 years in 2020-10=2010, and it "really" showed the second reading of 38 years in 2036-16=2020. So I will conclude that in the 10 years between 2010 and 2020, the clock itself only ticked forward by 8 years from 30 to 38, so it must have been slowed down by a factor of 0.8 in my frame. This is different from how much it appeared to be slowed down visually--visually it took 2036-2020=16 years to tick forward by 8 years, so it appeared to be running slow by a factor of 0.5, but this extra slowdown is just due to the Doppler effect (which is a consequence of the fact that light from different readings on the clock has different delays in reaching me since the clock's distance from me is changing).

As I said earlier, identical clocks in Inertial frames of reference will keep identical time.
It is only when one is observed from the other that time dilation occurrs.
Time dilation is the phenomenon where the time observed from one frame is different from that observed from the other.
So the observer from the other frame will see the time transformed in unit size and number of units but the total duration in absolute terms has to be the same - the half-life of the muon cannot change, only how it is measured can, as demonstrated by the afore mentioned experiment where the half-life was extended to 65secs, 65 transformed seconds that are \frac{1}{29.4} of the laboratory seconds.

cf.

With regards to the first point the wording seems over complicated and still confuses me. Clocks just show time

Perhaps I can give examples, in my view, of faulty and correct reasoning with regard to the often used example of the muon's lifetime as an aid to illustrating time dialtion. These two methods lead to exactly the opposite outcome.

Let the lab frame be regarded as the stationary frame and the muon's frame the moving frame with repect to it. We can use the values of 2 microseconds 60 microseconds as being the figures used for the decay times of the muon measured by clocks in the muon and lab frame respectivley. Both explanations are non rigorous.

WRONG reasoning:- The muon's lifetime of 2 micoseconds its own frame is extended to 60 microseconds in the lab frame. This is an example of time dilation this shows that the number of seconds which the muon lives is dilated, made bigger, to 60 microseconds.

Now bear in mind the definition which says that a moving clock viewed from a stationary frame runs slow, and reason as follows.

CORRECT reasoning:- The muon has a lifetime as measured in its own frame, the time measured by a clock carried with it, its proper time, of 2 microseconds. This is an invariant and is the same for everyone, it cannot be changed. In the lab frame this is measured as 60 microseconds. This is an example of time dilation and shows that 2 microseconds in the muon's frame takes 60 microseconds to pass in the lab frame. So the time it takes 2 microseconds to pass in the moving frame is dilated to 60 microseconds as viewed from the stationary frame. That is, the preiod is extended.

Remember that although the lab frame measures 60 microseconds, the lab observers still agree that the muon's clock reads a proper, invariant time of 2 microseconds.

Matheinste.

 

[JesseM]

Identical time in their own respective rest frames. They certainly do not keep identical time if you measure both from the perspective of a single inertial frame in which they have different speeds--that's exactly what I meant when I said above "In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B--nothing about the 1st postulate suggests that their ticking rates should be identical." Do you think the 1st postulate implies their ticking rate should be identical even in this situation? If so, can you explain your reasoning?

And if, natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K, then time and distance must be identical. (A muon's half life cannot be different and if time is identical distance has to be also cf. the speed of light)

That is; they will keep the same 'Proper Time' within their own frames of reference.

So in two independent Inertial Frames of Reference, identical clocks will keep identical time.

But yes, if viewed by an independent observer they will shew different times."

Grimble

You bolded the second half of my sentence but then ignored the first half, taking the meaning out of context. I first said "In this case we are dealing with two clocks that have different velocities in frame B, but we are measuring both their ticking rates from the perspective of frame B"--so when I then said "nothing about the 1st postulate suggests that their ticking rates should be identical", I was clearly talking about their ticking rates in frame B, not their ticking rates in their own respective rest frames. Hopefully you'd agree that nothing about the first postulate suggests that their ticking rates should be identical in frame B, given that one is at rest in frame B and the other is not?

 

[JesseM]

The light clock thought-experiment makes use of these ordinary geometrical and kinematical rules which still apply in relativity

No, because but it applies them across frames.

How do you think the light clock thought-experiment applies these rules "across frames"? The light clock thought-experiment derives the slowed down rate of ticking of the moving light clock using only a single frame, namely the frame in which the light clock is moving--the derivation only uses velocities and distances and times which are measured in the coordinates of that frame.

Do you understand that just because I am observing a clock which is moving relative to myself, does not mean that I need to use any frames other than my own rest frame to analyze its behavior? That talking about the properties of an object which is moving relative to me (like the time on a moving clock) does not in any way imply I am comparing multiple frames, I can analyze these properties just fine using nothing but my own rest frame? A frame is just a coordinate system after all, I can perfectly well keep track of the way the position coordinate of the moving object changes with coordinate time using just the coordinates of my rest frame.

If you think any frames other than the observer's rest frame are used in analyzing the light clock, can you point out the specific step in the analysis where you think this happens? For example, do we need any frames other than the observer's frame to figure out the distance the mirrors have traveled horizontally in a given time if we know their velocity v? Do we need any frames other than the observer's frame to use the pythagorean theorem to figure out the diagonal distance the light must travel if we know the horizontal distance traveled by the mirrors (just v*t, where t is the time between the light hitting the top and bottom mirror and v is the horizontal velocity of the mirrors) and the vertical distance h between them? Do we need any frames other than the observer's frame to figure out the time T that would be required in order to ensure that the diagonal distance D = \sqrt{v^2 T^2 + h^2} will satisfy D/T = c? (making use of the second postulate which says light must move at c in every frame, including the observer's frame, along with the ordinary kinematical rule that speed = distance/time)

From all this, you can conclude that the time T between ticks of the light clock in the observer's frame must be equal to T = \frac{h}{\sqrt{c^2 - v^2}}. Only here do you have to consider another frame if you want to derive the time dilation formula from this--you have to figure out what the time t between ticks would be in the light clock's own rest frame, and obviously if h is the vertical distance between mirrors this would be t = h/c (here you do need to make an argument to show the vertical height h will be the same in both frames, that there will be no length contraction perpendicular to the direction of motion). Then if you divide T/t you get the gamma factor \frac{1}{\sqrt{1 - v^2/c^2}}. But this is just simple division, when deriving the time between ticks in each frame you can work exclusively with the coordinates of that frame and not worry about other frames.

 

[JesseM]

As I said earlier, identical clocks in Inertial frames of reference will keep identical time.
It is only when one is observed from the other that time dilation occurrs.
Time dilation is the phenomenon where the time observed from one frame is different from that observed from the other.
So the observer from the other frame will see the time transformed in unit size and number of units but the total duration in absolute terms has to be the same - the half-life of the muon cannot change, only how it is measured can, as demonstrated by the afore mentioned experiment where the half-life was extended to 65secs, 65 transformed seconds that are \frac{1}{29.4} of the laboratory seconds.

I'm not sure what you mean "the total duration in absolute terms". Certainly if you're talking about "proper time", meaning the time as measured by a clock moving along with the object (in this case the muon), then it is true that there is no disagreement between frames about the proper time between two events on the object's worldline (like the muon being created and then decaying, which you can average for many muons to derive the half-life). But what does this point about proper times have to do with the light clock derivation of the time dilation equation, an equation which deals with coordinate time in the frame where the light clock is moving, not proper time?

Also, you can see that time dilation is not just a sort of illusion created by using coordinate time rather than proper time by considering a case where two clocks depart from the same location and then later return to a common location, as in the twin paradox--in this case one clock may actually have elapsed less proper time (aged less) than the other. And you can calculate how much proper time each elapsed if you know the coordinate times t₀ and t₁ of the first and second meetings of the two clocks in some inertial frame, and you know a given clock's velocity as a function of time v(t) in that frame...then you can take the time dilation equation dT = \sqrt{1 - v^2/c^2} * dt (where dT is the proper time and dt is the coordinate time) and integrate it to find the total proper time elapsed on the clock, i.e. \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt integrated over the coordinate time t. So you can see that even if your ultimate interest is in knowing the proper time between two events on an object's worldline, the coordinate time between the events in some inertial frame, along with the time dilation equation and the object's velocity as a function of time in that frame, can be used to calculate the proper time.

 

[Ich]

Sorry, I missed a few posts of yours.

It isn't traveling the same distance.

But it is, within the clock's frame of reference, and the speed of light has to be the same wherever it is observed from

From this, and some of your comments before, it is obvious that you have no idea what a reference frame is good for, or even how "speed" is defined. You should get familiar with this basic stuff in relativity before you move on.
So I invite you to draw that spacetime diagram of the quoted situation, and post it here along with the derivation of light travel time and speed of light. You'll encounter some points where you don't know how to proceed; it would be most fruitful if we could help you exactly with these points.

 

[Grimble]

Sorry, I missed a few posts of yours.

From this, and some of your comments before, it is obvious that you have no idea what a reference frame is good for, or even how "speed" is defined. You should get familiar with this basic stuff in relativity before you move on.
So I invite you to draw that spacetime diagram of the quoted situation, and post it here along with the derivation of light travel time and speed of light. You'll encounter some points where you don't know how to proceed; it would be most fruitful if we could help you exactly with these points.

I am sorry Ich, that I seem to be getting lost about what I am doing and I will draw some diagrams that we can discuss, just give me a little time.

I have been picturing the stationary clock as being placed on Einstein's embankment and the moving clock riding on his train; am I misreading this situation?

As for speed, I take that as non-directional (for speed with direction is velocity?) and it is distance/time.

And thank you for your offer of help