Perspective on Relativity and Length Contraction

[Histspec]

To me, time dilation is the name of the phenomenon: a particular clock runs at a different rate than reference clocks; for inertial frames in SR, this is more specifically a moving clock runs slow compared to stationary reference clocks. You can mathematically describe this phenomenon in multiple ways. The most common in SR is a a factor saying how many times slower the observed clock is = seconds of coordinate time per second of clock time. However, I was interested in comparing the rates the other way: seconds elapsed on observed clock compared to seconds measured by reference clocks. I was careful to define my terms, so I don't see what the problem is.

[addendum: which is right, "price to earnings ratio" or "earnings to price ratio"? The former is more common, both are used, and both describe the same underlying thing.]

Let's write it down that way:
$$\begin{align} (1) & & T_{0} & =T'\cdot\gamma\\ (2) & & T' & =T_{0}/\gamma\\ \end{align}$$

$T_0$ indicates proper time of a single clock in motion, $T$ indicates coordinate time of two synchronized clocks at rest in S'. $T_0$ is dilated with respect to coordinate time $T$.

By the relativity principle, this is symmetrically also true for a single clock at rest in S', which is compared with two synchronized clocks at rest in S:
$$\begin{align} (3) & & T'_{0} & =T\cdot\gamma\\ (4) & & T & =T'_{0}/\gamma \end{align}$$

 

[PAllen]

Let's write it down that way:
$$\begin{align} (1) & & T_{0} & =T'\cdot\gamma\\ (2) & & T' & =T_{0}/\gamma\\ \end{align}$$

$T_0$ indicates proper time of a single clock in motion, $T$ indicates coordinate time of two synchronized clocks at rest in S'. $T_0$ is dilated with respect to coordinate time $T$.

By the relativity principle, this is symmetrically also true for a single clock at rest in S', which is compared with two synchronized clocks at rest in S:
$$\begin{align} (3) & & T'_{0} & =T\cdot\gamma\\ (4) & & T & =T'_{0}/\gamma \end{align}$$

Hmm. It looks to me like you have these (consistently) backwards if I understand your convention. If T0 shows 1 second, T' > 1 is expected (for difference of two clocks at rest in s'). You have T' < 1.

 

[nitsuj]

So the precise definition of Time Dilation is the ratio of the Coordinate Time to the Proper Time of a clock moving according to the coordinate frame.

A perfect way to define time dilation; & differential aging is the comparative of elapsed proper times.

 

[PAllen]

A perfect way to define time dilation; & differential aging is the comparative of elapsed proper times.

differential aging is the comparative of elapsed proper times [for two different spacetime paths between some pair of events].

 

[Histspec]

Hmm. It looks to me like you have these (consistently) backwards if I understand your convention. If T0 shows 1 second, T' > 1 is expected (for difference of two clocks at rest in s'). You have T' < 1.

Sorry, I mismatched the symbols. It is

$$\begin{align} (1) & & T_{0} & =T'/\gamma\\ (2) & & T' & =T_{0}\cdot\gamma\\ \end{align}$$

$T_0$ indicates proper time of a single clock in motion, $T$ indicates coordinate time of two synchronized clocks at rest in S'. $T_0$ is dilated with respect to coordinate time $T$.

By the relativity principle, this is symmetrically also true for a single clock at rest in S', which is compared with two synchronized clocks at rest in S:
$$\begin{align} (3) & & T'_{0} & =T/\gamma\\ (4) & & T & =T'_{0}\cdot\gamma \end{align}$$

 

[Histspec]

In this way, it also becomes clear why length contraction of proper length L_0 is reciprocal to time dilation of proper time T_0:
$$\begin{align} T_{0}= & T/\gamma\\ L_{0}= & L\cdot\gamma \end{align}$$
The notation is similar to the one used by Max Born, "The theory of relativity", 1962.

 

[nitsuj]

differential aging is the comparative of elapsed proper times [for two different spacetime paths between some pair of events].

That's implicit with proper time and different measures of proper times. Paths is a great word to bring into it, one of those paths is shorter then the other, perhaps contracted (length) depending on the perspective (muon). This is my point why I find it odd length contraction is difficult for some to accept as being "proven" or whatever because it hasn't been "directly" observed, as if time-dilation has in some more "direct" sense. all because of differential aging being thought of as a consequence of time dilation, but not length contraction. It's solid proof of both.

 

[PAllen]

That's implicit with proper time and different measures of proper times. Paths is a great word to bring into it, one of those paths is shorter then the other, perhaps contracted depending on the perspective (muon). This is my point why I find it odd length contraction is difficult for some to accept as being "proven" or whatever because it hasn't been "directly" observed, as if time-dilation has in some more "direct" sense. all because of differential aging being thought of as a consequence of time dilation, but not length contraction. It's solid proof of both.

Except that atmospheric muons reaching the ground provides no evidence for differential aging because it is not an example of it. A muon example that is a differential aging scenario is a muon accelerator ring. Here, you can talk about comparing the time measured on one clock on the rim, with an (imaginary) clock on a muon over one circuit. The two clocks are compared at two events where they are both colocated. This comparison is invariant and would be explained as time dilation in any coordinates (thought different coordinates would disagree about interim rates of the two clocks). Differential aging never has anything to do with length contraction (because all clock comparisons are done at co-location, and no distances are measured).

In the atmosphere muon case, there are no two clocks you can directly compare. Instead, at minimum, you have a clock in the atmosphere at rest relative to the ground, and synchronized with it, another clock on the ground, and the muon clock. This shows time dilation in the Earth frame. In the muon frame, what this shows is that the atmosphere clock co-moving with the ground is out of synch with the ground clock. Thus this tells the muon nothing about why it hits the ground so fast. Instead, for the muon, all of why it reaches the gound is because the ground is close when (per the muon) the muon is created.

This is why time dilation is described as coordinate or frame dependent and is related to length contraction by the fact that what is explained by time dilation in one frame is due to length contraction in another frame. Differential aging is not frame dependent, because all frames explain it as due to different clock rates, and all agree on the amount of difference at the end, and which clock elapsed less time.

 

[nitsuj]

Acceleration doesn't confirm or deny anything, but maybe just forces a particular conclusion if you place lots of emphasis on a clock reading.

postulate...all muons exist for the same amount of time.

Differential aging is frame invariant only because of proximity, comparison of measured proper times.

You don't need an imaginary clock on the accelerator ring muon...the muon is a "clock".

and all agree on the amount of difference at the end, and which clock elapsed less time.

Really? it's because the traveler traveled less length . Looks like allot of emphasis put on the cumulative counting of time to suggest "Ah it was REALLY because the clock was ticking slower", it was equally because it REALLY traveled a shorter length (proper length). unfortunately rulers cannot show the history of measured length quite like a clock does for simple "this current reading less that current reading", that doesn't give anymore physical significance to time dilation over length contraction; in turn does not support time dilation over length contraction as an explanation for differential aging.

 

[PAllen]

Acceleration doesn't confirm or deny anything, but maybe just forces a particular conclusion if you place lots of emphasis on a clock reading.

Acceleration is relevant only to the extent that it enables two spacetime paths between the same events. In SR, this is possible only if one or both world lines have proper acceleration.

postulate...all muons exist for the same amount of time.

Differential aging is frame invariant only because of proximity, comparison of measured proper times.

Yes, this is the key point distinguishing it from time dilation. No auxiliary clocks are needed to measure it, no distances are involved in any interpretation of it.

You don't need an imaginary clock on the accelerator ring muon...the muon is a "clock".

Of course. I just added a little more description.

and all agree on the amount of difference at the end, and which clock elapsed less time.

Really? it's because the traveler traveled less length .

Each considers itself not to have traveled at all, and the other to have traveled. Yet both agree on which clock measures less time at the end, and how much less. This for the atmosphere muon case this is not true at all. The muon thinks the Earth clock has elapsed e.g. only 1 picosecond between the time the muon was born and when the ground clock reaches the muon. Meanwhile, the muon clock has elapsed e.g. 1 nanosecond (a thousand times more than the Earth clock) between birth of muon and hitting ground. Thus, the Earth and muon have opposite conclusions about which clock ran faster and elapsed more time. This is the fundamental distinction between time dilation versus differential aging. I don't get why you are having so much trouble seeing the distinction.

Looks like allot of emphasis put on the cumulative counting of time to suggest "Ah it was REALLY because the clock was ticking slower", it was equally because it REALLY traveled a shorter length (proper length). unfortunately rulers cannot show the history of measured length quite like a clock does for simple "this current reading less that current reading", that doesn't give anymore physical significance to time dilation over length contraction; in turn does not support time dilation over length contraction as an explanation for differential aging.

As explained above, there is no relevance of distance to a twin scenario. In the classic one using uniform acceleration for one of the twins, and Fermi-Normal or Rindler coordinates for this twin, each twin considers the other to have traveled the same distance!

 

[nitsuj]

Thus, the Earth and muon have opposite conclusions about which clock ran faster and elapsed more time. This is the fundamental distinction between time dilation versus differential aging. I don't get why you are having so much trouble seeing the distinction.

I just don't see the muon example as symmetric through out the entire "experiment"; the muon hits the ground, and takes a break checking out this new shared frame with Earth, looks back and says "Wow! I traveled a great Distance through space time. Pretty cool I was able to make it so far through spacetime! Well, maybe I actually didn't travel any great length for any great period of time. Both my measures of time/length were retarded to the specific point I calculated c to the same value as I do in this new Earth frame. Also the spacetime interval, or distance through spacetime, seems to be the same too!"

 

[ghwellsjr]

To me, time dilation is the name of the phenomenon: a particular clock runs at a different rate than reference clocks; for inertial frames in SR, this is more specifically a moving clock runs slow compared to stationary reference clocks. You can mathematically describe this phenomenon in multiple ways. The most common in SR is a a factor saying how many times slower the observed clock is = seconds of coordinate time per second of clock time. However, I was interested in comparing the rates the other way: seconds elapsed on observed clock compared to seconds measured by reference clocks. I was careful to define my terms, so I don't see what the problem is.

The problem is that some people think that both length and time change in the same way for an observer in motion and that is why they continue to measure the speed of light as c. Here are two examples that both came up today:

Length contraction and time dilation are nothing but an explanation for this phenomenon, is what I have known.

Your instrument will calculate speed of light, by using distance of the source and time taken by light to reach it. Since both the values decrease while in motion, when you will calculate the speed, it will turn out to be 'c'.

I just don't see the muon example as symmetric through out the entire "experiment"; the muon hits the ground, and takes a break checking out this new shared frame with Earth, looks back and says "Wow! I traveled a great Distance through space time. Pretty cool I was able to make it so far through spacetime! Well, maybe I actually didn't travel any great length for any great period of time. Both my measures of time/length were retarded to the specific point I calculated c to the same value as I do in this new Earth frame. Also the spacetime interval, or distance through spacetime, seems to be the same too!"

This happens quite often and I think if we emphasized that one is smaller and the other is larger, we can't explain the constant speed of light by simply saying the division comes out the same.

[addendum: which is right, "price to earnings ratio" or "earnings to price ratio"? The former is more common, both are used, and both describe the same underlying thing.]

But applying the value for the "earnings to price ratio" to the "price to earnings ratio" is not right. If you want to use a value less than 1, you should either say that it is the reciprocal of Time Dilation or do what DrGreg said no one does:

"Time" and "rate" are reciprocals of each other, so "rate contraction" means the same as "time dilation". But no-one ever uses the phrase "rate contraction".

 

[PAllen]

I just don't see the muon example as symmetric through out the entire "experiment"; the muon hits the ground, and takes a break checking out this new shared frame with Earth, looks back and says "Wow! I traveled a great Distance through space time. Pretty cool I was able to make it so far through spacetime! Well, maybe I actually didn't travel any great length for any great period of time. Both my measures of time/length were retarded to the specific point I calculated c to the same value as I do in this new Earth frame. Also the spacetime interval, or distance through spacetime, seems to be the same too!"

If the muon 'stops' and 'survives', we are talking about the muon changing its motion, and adopting a new frame at a certain point, corresponding to its changed motion, and using it to analyze its past when its motion was different. This has no bearing on the analysis in the prior frame. Each frame offers a complete, correct analysis of why the muon reaches the ground. If the muon adopts the Earth frame, it is it tautological that its measures now agree with the Earth frame (distance). [Again, in the twin scenario also, all frames offer complete, correct analysis. However, in this case, distances are not part of the explanation in any frame. As I explained previously, in one of the classic twin scenarios (uniform acceleration of one twin), both twins consider the other to have traveled the same distance; both also expect and find that the twin experiencing proper acceleration aged less.]

As for spacetime interval, what interval do you mean? The spacetime interval between muon creation and hitting ground is a timelike interval that is e.g. 1 nanosecond, in all frames. For distance, you have two completely different spacelike intervals involved:

- The one between the event of muon creation and the event on the ground's world line that a ground frame considers simultaneous with muon creation. This measures many kilometers in all frames, being and invariant spacelike interval between two specific events.

- The one between the event of muon creation and the event on ground's world line that the muon travel frame considers simultaneous with the creation event. The creation event is the same event as the prior case. The other event here is a completely different event on the ground's world line. This spacelike spacetime interval is e.g. 10s of meters. The spacetime interval between these two events is also invariant.

 

[nitsuj]

This happens quite often and I think if we emphasized that one is smaller and the other is larger, we can't explain the constant speed of light by simply saying the division comes out the same.

What do you mean "one is smaller and the other is larger"? If I get what you are saying, RoS explains the calculated value of c being invariant. To say that different length is dependent on defining what is simultaneous.

 

[nitsuj]

If the muon 'stops' and 'survives'...


As for spacetime interval, what interval do you mean? The spacetime interval between muon creation and hitting ground is a timelike interval that is e.g. 1 nanosecond, in all frames. For distance, you have two completely different spacelike intervals involved:

- The one between the event of muon creation and the event on the ground's world line that a ground frame considers simultaneous with muon creation. This measures many kilometers in all frames, being and invariant spacelike interval between two specific events.

- The one between the event of muon creation and the event on ground's world line that the muon travel frame considers simultaneous with the creation event. The creation event is the same event as the prior case. The other event here is a completely different event on the ground's world line. This spacelike spacetime interval is e.g. 10s of meters. The spacetime interval between these two events is also invariant.

I can't follow the last two points, The separation between muon creation and landing is time like. I'm not sure why you suggest the muon MUST survive. It "lands", and they have a "fixed" lifespan. Yes, there must be acceleration in with my idealized muon landing, but even with that, the "fixed" lifespan is a pretty solid comparative for a measure of proper time in both frames.

 

[PAllen]

I can't follow the last two points, The separation between muon creation and landing is time like. I'm not sure why you suggest the muon MUST survive. It "lands", and they have a "fixed" lifespan. Yes, there must be acceleration in with my idealized muon landing, but even with that, the "fixed" lifespan is a pretty solid comparative for a measure of proper time in both frames.

You said "the muon hits the ground, and takes a break checking out this new shared frame with Earth, looks back and says...". This is vague, I gave the the interpretation that makes sense to me: the muon stopped. If the muon is considered not to have stopped, then, just because it has reached the ground does not mean it shares a frame with the Earth in normal usage. Normal usage is that the frame of an object is short hand for the frame in which an object is at rest. If the muon doesn't change motion to match the earth, it doesn't share a frame in this sense. In any other sense, I have no idea what you could possibly mean.

I said, so obviously agree, that the timelike interval from muon creation to muon destruction is invariant - same in all frames, and is e.g. 1 nanosecond in all frames. I didn't know what spacetime interval you meant since you did not define it, so I threw out two additional intervals of interest that happen to be spacelike. If you are uninterested in these intervals, fine.

Do you agree that in the muon rest frame:

- the muon ages 1 nanosecond

- the Earth clock will run slow, e.g. elapse much less than 1 nanosecond (< 1 picosecond) between the event simultaneous with muon creation, in this frame, and when the Earth clock reaches the muon. That is, the Earth ages < 1 picosecond in the one nanosecond life of the muon.

- The Earth will have traveled e.g only .3 meters in this frame during the time between creation and destruction of the muon. Thus there is no mystery why the Earth reaches the muon in 1 nanosecond - it has only .3 meters to cover.

Do you agree that in the Earth rest frame:

- the muon ages 1 nanosecond

- the time between creation and destruction is 10 microseconds. That is the Earth ages 10 microseconds between creation and destruction of muon (per this frame).

- the distance traveled by the muon between creation and destruction is e.g. 3 km. The muon reaches the ground because it only ages 1 nanosecond in the 10 microseconds it takes to cover this distance.

In contrast, if twin A is inertial and twin B passes A at some high relative speed, but is uniformly accelerating such that they will meet up with A again later:

- A and B agree on the distance traveled by the other (using the coordinates considered most physical for B).

- A expects and finds that B will age less. B expects and finds that B will age less.

Let's focus on which of these statements you disagree with. If you agree with them all, then we only disagree on how to describe the facts, but not on the facts themselves.

 

[nitsuj]

You said "the muon hits the ground, and takes a break checking out this new shared frame with Earth, looks back and says...". This is vague, I gave the the interpretation that makes sense to me: the muon stopped. If the muon is considered not to have stopped, then, just because it has reached the ground does not mean it shares a frame with the Earth in normal usage. Normal usage is that the frame of an object is short hand for the frame in which an object is at rest. If the muon doesn't change motion to match the earth, it doesn't share a frame in this sense. In any other sense, I have no idea what you could possibly mean.

I said, so obviously agree, that the timelike interval from muon creation to muon destruction is invariant - same in all frames, and is e.g. 1 nanosecond in all frames. I didn't know what spacetime interval you meant since you did not define it, so I threw out two additional intervals of interest that happen to be spacelike. If you are uninterested in these intervals, fine.

Do you agree that in the muon rest frame:

- the muon ages 1 nanosecond

- the Earth clock will run slow, e.g. elapse much less than 1 nanosecond (< 1 picosecond) between the event simultaneous with muon creation, in this frame, and when the Earth clock reaches the muon. That is, the Earth ages < 1 picosecond in the one nanosecond life of the muon.

- The Earth will have traveled e.g only .3 meters in this frame during the time between creation and destruction of the muon. Thus there is no mystery why the Earth reaches the muon in 1 nanosecond - it has only .3 meters to cover.

Do you agree that in the Earth rest frame:

- the muon ages 1 nanosecond

- the time between creation and destruction is 10 microseconds. That is the Earth ages 10 microseconds between creation and destruction fo muon (per this frame).

- the distance traveled by the muon between creation and destruction is e.g. 3 km. The muon reaches the ground because it only ages 1 nanosecond in the 10 microseconds it takes to cover this distance.

In contrast, if twin A is inertial and twin B passes A at some high relative speed, but is uniformly accelerating such that they will meet up with A again later:

- A and B agree on the distance traveled by the other (using the coordinates considered most physical for B).

- A expects and finds that B will age less. B expects and finds that B will age less.

Let's focus on which of these statements you disagree with. If you agree with them all, then we only disagree on how to describe the facts, but not on the facts themselves.

It's the describing of the facts. Even in "scientific literature", I find allot of emphasis is placed on time and it's measurements. Like how you and everything I read say differential aging is due to time dilation. Why not equally length contraction? Because it highlights RoS? (I think that was the same issue with the spacelike interval you mentioned) Maybe because clocks give cumulative sequential readings dependent on it's history. Or maybe some blatantly obvious reason I can't see.

 

[Dale]

It's the describing of the facts.

That is pretty vague. Is there something specific about the description that you don't like, or do you just think that facts shouldn't be described at all, or what?

 

[PAllen]

It's the describing of the facts. Even in "scientific literature", I find allot of emphasis is placed on time and it's measurements. Like how you and everything I read say differential aging is due to time dilation. Why not equally length contraction? Because it highlights RoS? (I think that was the same issue with the spacelike interval you mentioned) Maybe because clocks give cumulative sequential readings dependent on it's history. Or maybe some blatantly obvious reason I can't see.

Can you explain how length contraction is relevant to the twin scenario? In the variant I described, each concludes that the the other has traveled the same distance.

 

[nitsuj]

That is pretty vague. Is there something specific about the description that you don't like, or do you just think that facts shouldn't be described at all, or what?

I went on to explain why, Even in "scientific literature", I find allot of emphasis is placed on time and it's measurements. The context is differential aging, or the muon example.

 

[PAllen]

I went on to explain why, Even in "scientific literature", I find allot of emphasis is placed on time and it's measurements. The context is differential aging, or the muon example.

And these are different situations and you seem very resistant to see the difference. The case of muon's reaching the ground is not differential aging.

 

[nitsuj]

And these are different situations and you seem very resistant to see the difference. The case of muon's reaching the ground is not differential aging.

You're being strictly technical with the use of frames/ comparative coordinates. Is the difference I'm not seeing physical or just about the post analysis?

 

[Dale]

I went on to explain why, Even in "scientific literature", I find allot of emphasis is placed on time and it's measurements. The context is differential aging, or the muon example.

But he didn't do that here. In the description that you objected to he used length just as much as time. He carefully and consistently described both the length and the time in both frames.

So that explanation didn't make sense. In fact, to me it seemed like an unrelated commentary on the scientific literature. I didn't realize you intended it to apply to his comments and now that I understand that was your intention I still don't see the applicability.

 

[PAllen]

You're being strictly technical with the use of frames/ comparative coordinates. Is the difference I'm not seeing physical or just about the post analysis?

I'ts obviously physical. In Twin (differential aging) you have two clocks (or equivalent) that are co-located at two different events. No interpretation is needed to compare them. No matter what frame is used, there is complete agreement about which twin aged more and by how much. [Also, length contraction cannot possibly be relevant because a common twin situation has both twins agreeing on the distance the other traveled - that is each thinks the other twin traveled the same distance, e.g. 1 ly.]

Muon: All you know is that muon reached ground before decaying. You can say the muon aged only a little (this part is invariant - the muon didn't decay). But you can't say (invariantly) that the Earth aged more. In one frame (muon travel frame), the Earth aged much less than the muon between creation and destruction of the muon. The existence of any time dilation at all for the muon depends on choice of frame.

It continues to boggle me how you don't see this difference.

 

[ghwellsjr]

This happens quite often and I think if we emphasized that one is smaller and the other is larger, we can't explain the constant speed of light by simply saying the division comes out the same.

What do you mean "one is smaller and the other is larger"? If I get what you are saying, RoS explains the calculated value of c being invariant. To say that different length is dependent on defining what is simultaneous.

I was reacting to your statement from post #186 where you said:

Both my measures of time/length were retarded to the specific point I calculated c to the same value as I do in this new Earth frame.

I was talking about the Length Contraction factor being less than 1 for a moving observer and the Time Dilation factor being greater than 1 which might dissuade people from jumping to the conclusion that the measurement of the speed of light continues to be c if they think both factors are "retarded", as you put it, or both less than one by the same amount.

To illustrate, let's say that we are measuring the time it takes for light to traverse 10 feet to a mirror and 10 feet back for a total distance of 20 feet. Our timer will read 20 nsecs and we will conclude that the speed of light is 20 feet per 20 nsec or 1 foot per nsec.

If we have a length contracted ruler, say to 50%, then we will think that the distance to the mirror is 20 feet and we will calculate the speed of light to be 40 feet per 20 nsec or 2 feet per nsec.

Instead, if we have a clock that runs 50% slow, then instead of measuring the time interval as 20 nsec, we will say it is 10 nsec and we will calculate the speed of light to be 20 feet per 10 nsec or 2 feet per nsec.

Now if we have both theses problems at the same time, we will calculate the speed of light to be 40 feet per 10 nsec or 4 feet per nsec.

So the the two factors being smaller don't cancel out and don't result in the measured speed of light being the same as before.

So I have to ask you, what did you mean by:

Both my measures of time/length were retarded to the specific point I calculated c to the same value as I do in this new Earth frame.

Also, think about this: the Length Contraction only occurs along the direction of motion. If we're talking about the speed of light at 90 degrees to the direction of motion, then how does your statement apply?

My point is that Length Contraction and Time Dilation are coordinate effects and are easily understood with spacetime diagrams showing the same scenario viewed from different Frames of Reference moving at different speeds. I have shown many examples of this in this thread. In these diagrams, it is obvious that a moving object takes up less distance on the drawing and its clock takes up more distance on the drawing to tick off the same amount of time.

 

[Dale]

OK, this is clearly going nowhere. And the OP is long gone. Time to close it down.