Is this simple derivation of length contraction and time dilation corr

[ash64449]

well,i found this article and i find it simple to understand. But is this article totally correct?
Here is the link:http://m.sparknotes.com/physics/specialrelativity/kinematics/section2.rhtml

 

[ghwellsjr]

I don't like the article. I don't like articles that say things like, "Unfortunately, the most confusing part is yet to come", as if it was already confusing but it's getting worse. Whenever you see a writeup that admits it is confusing, it's more than likely that the author is confused. And he is confused in the same way that a great many writers are confused. He is confused between what he sees and what the observers in his explanation see. He states that the observers see the Time Dilation of the other ones clock which is not true. We can see it because we are able to see every event in the scenario simultaneously but the observers in the scenario have to wait for the light from the events to propagate to them.

So how does this bear out in the article? He talks about one observer seeing the other one flying past. Well if that is going to happen, then the first one will see the second ones clock ticking faster than his own while approaching and then suddenly switch to ticking slower than his own while departing. This is Relativistic Doppler which is what the observers see when looking at each others clock. They never see Time Dilation.

He attributes the reciprocal Time Dilation to the Relativity Principle (Einstein's first postulate) which is not true. the Relativistic Doppler can be attributed to the Relativity Principle. At least he correctly points out that the Time Dilation is frame dependent but he shouldn't be saying that each observer sees things differently depending on whether they are at rest in their own frame (How could they not be? That's the accepted definition of "their own frame".)

It's so easy to explain Time Dilation and what observers see that I have no tolerance for explanations that seem to relish confusing their readers.

 

[ash64449]

george,i somewhere in his article saw that he used old velocity addition for the "light" too.. Isn't that wrong? I think he used that in proving length contraction...

 

[ash64449]

george,then what is 'time dilation' according to you?

 

[ghwellsjr]

george,i somewhere in his article saw that he used old velocity addition for the "light" too.. Isn't that wrong? I think he used that in proving length contraction...

I actually didn't read the length contraction part but I don't think there is anything wrong with regard to velocity addition. He's showing things from the point of view of the frame which is the correct way to do it but explains it from the point of view of the observers which is not something they can view.

 

[ghwellsjr]

george,then what is 'time dilation' according to you?

Time Dilation is the ratio of the passage of Coordinate Time to Proper Time for a clock moving in an Inertial Reference Frame (IRF). It is therefore frame dependent and not visible to any observer.

I recently gave an explanation on a thread called Time Dilation with this post (#10):

The best way I know to understand Time Dilation is to look at the geometry of the spacetime diagrams for a couple of Inertial Reference Frames (IRFs) depicting the same situation and using the Lorentz Transformation process to get from one IRF to the other. The situation we will consider is a clock that is stationary in the first IRF. The spacetime diagram is simply a plot of the position (or distance from the spatial origin) of the clock along the horizontal axis versus time along the vertical axis. Here is the first diagram:

Pretty boring, isn't it? It shows that as time progresses from 0 seconds to 10 seconds, the clock stays at the spatial origin with the coordinate at 0. I also show each second of time as a blue dot.

Now we use the Lorentz Transformation to see how an IRF moving to the left at 60% of the speed of light would depict this same situation:

In this IRF, the clock is moving to the right at 0.6c. You can see that because at the Coordinate Time of 10 seconds, the blue line is at the Coordinate Distance of 6 light-seconds. You can also see that what took 10 seconds in the first IRF takes 12.5 seconds in this IRF.

Things take longer when they are moving in an IRF. And this is the simplest explanation of Time Dilation that I know of based on Special Relativity. Does it seem simple to you? If not, please let me know where you need some more explanation.

Since the OP didn't respond to my question if it was simple, maybe you could.

 

[ash64449]

I actually didn't read the length contraction part but I don't think there is anything wrong with regard to velocity addition. He's showing things from the point of view of the frame which is the correct way to do it but explains it from the point of view of the observers which is not something they can view.

the part that i think is wrong is that he used c+v and c-v which he shouldn't use.. As it violate second postulate of relativity..
And do you know how to calculate relativistic doppler?? I think it will be very usefull to me if you can explain in simpler manner. It is a request from me...

 

[ghwellsjr]

the part that i think is wrong is that he used c+v and c-v which he shouldn't use.. As it violate second postulate of relativity..

Perhaps a couple spacetime diagrams will help to illustrate his explanation of Length Contraction. First, the IRF for the rest frame of the train:

The observer is shown in blue and her mirror is shown a measured six feet away in red. I'm showing just one cycle of a light pulse traveling from her to the mirror and back.

Now, the IRF for the ground rest frame:

Can you see that while the light is moving away from the observer on the train, it is going at c-v relative to the observer and after it reflects off the mirror it is going c+v?

 

[ash64449]

Perhaps a couple spacetime diagrams will help to illustrate his explanation of Length Contraction. First, the IRF for the rest frame of the train:

The observer is shown in blue and her mirror is shown a measured six feet away in red. I'm showing just one cycle of a light pulse traveling from her to the mirror and back.

Now, the IRF for the ground rest frame:

Can you see that while the light is moving away from the observer on the train, it is going at c-v relative to the observer and after it reflects off the mirror it is going c+v?

Yes,then why speed of light "appears" constant?

 

[ash64449]

George,why i said it violates 2nd postulate is that light is not treated like that way.. i.e.light moves at the velocity c+v relative to train in the direction of motion and light moves c-v in the direction opposite to that of train...

This is a part of what Einstein said:

"""""""""Of course we must refer the process of the propagation of light (and indeed every other
process) to a rigid reference-body (co-ordinate system). As such a system let us again
choose our embankment. We shall imagine the air above it to have been removed. If a ray
of light be sent along the embankment, we see from the above that the tip of the ray will be
transmitted with the velocity c relative to the embankment. Now let us suppose that our
railway carriage is again traveling along the railway lines with the velocity v, and that its
direction is the same as that of the ray of light, but its velocity of course much less. Let us
inquire about the velocity of propagation of the ray of light relative to the carriage. It is
obvious that we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity w of
the man relative to the embankment is here replaced by the velocity of light relative to the
embankment. w is the required velocity of light with respect to the carriage, and we have
w = c-v.
The velocity of propagation ot a ray of light relative to the carriage thus comes cut
smaller than c.
But this result comes into conflict with the principle of relativity set forth in Section V."""""""""""

light appears moving in less velocity is not allowed..

 

[ghwellsjr]

Yes,then why speed of light "appears" constant?

It depends on what you are asking.

If you are asking why the speed of light measures to be a constant (a result of Einstein's first postulate, the Principle of Relativity), then the above two diagrams illustrate that. In both cases the observer sends out a light pulse at his time of 4 nanoseconds and detects the reflection from the mirror that he measured to be six feet away at his time of 16 nanoseconds. So the light, as far as he is concerned traveled 12 feet (round-trip) in 12 nanoseconds which is 1 foot per nanosecond. When we measure the speed of light, it is always a round-trip measurement and we cannot tell if the two trips are the same as in the first IRF or different as in the second IRF.

But if you are asking about the stipulation that light propagates at c in any IRF (Einstein's second postulate) then that is something that does not appear to us. We cannot observe the propagation of light. We can't know what it is unless we define it in some way and that's what the second postulate does for us. You can see that in any IRF that I draw. The light signals always travel along 45 degree diagonals which in this case is 1 foot per nanosecond. We can see it in our diagrams but the observers in our diagrams cannot tell whether we are using the first IRF or the second IRF, both of which have the light traveling at c but they can't tell if the light takes the same amount of time to get to the mirror as it takes for the reflection to get back to them.

 

[ghwellsjr]

George,why i said it violates 2nd postulate is that light is not treated like that way.. i.e.light moves at the velocity c+v relative to train in the direction of motion and light moves c-v in the direction opposite to that of train...

This is a part of what Einstein said:

"""""""""Of course we must refer the process of the propagation of light (and indeed every other
process) to a rigid reference-body (co-ordinate system). As such a system let us again
choose our embankment. We shall imagine the air above it to have been removed. If a ray
of light be sent along the embankment, we see from the above that the tip of the ray will be
transmitted with the velocity c relative to the embankment. Now let us suppose that our
railway carriage is again traveling along the railway lines with the velocity v, and that its
direction is the same as that of the ray of light, but its velocity of course much less. Let us
inquire about the velocity of propagation of the ray of light relative to the carriage. It is
obvious that we can here apply the consideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage. The velocity w of
the man relative to the embankment is here replaced by the velocity of light relative to the
embankment. w is the required velocity of light with respect to the carriage, and we have
w = c-v.
The velocity of propagation ot a ray of light relative to the carriage thus comes cut
smaller than c.
But this result comes into conflict with the principle of relativity set forth in Section V."""""""""""

light appears moving in less velocity is not allowed..

You need to read to the end of the chapter and then subsequent chapters.

 

[ash64449]

It depends on what you are asking.

If you are asking why the speed of light measures to be a constant (a result of Einstein's first postulate, the Principle of Relativity), then the above two diagrams illustrate that. In both cases the observer sends out a light pulse at his time of 4 nanoseconds and detects the reflection from the mirror that he measured to be six feet away at his time of 16 nanoseconds. So the light, as far as he is concerned traveled 12 feet (round-trip) in 12 nanoseconds which is 1 foot per nanosecond. When we measure the speed of light, it is always a round-trip measurement and we cannot tell if the two trips are the same as in the first IRF or different as in the second IRF.

But if you are asking about the stipulation that light propagates at c in any IRF (Einstein's second postulate) then that is something that does not appear to us We cannot observe the propagation of light. We can't know what it is unless we define it in some way and that's what the second postulate does for us. You can see that in any IRF that I draw. The light signals always travel along 45 degree diagonals which in this case is 1 foot per nanosecond. We can see it in our diagrams but the observers in our diagrams cannot tell whether we are using the first IRF or the second IRF, both of which have the light traveling at c but they can't tell if the light takes the same amount of time to get to the mirror as it takes for the reflection to get back to them.

So in reality light travels in different speeds but we aren't able to detect them because of time dilation and length contraction..right?? Now,i find that it is similar to absolute frame of reference...

I feel like there is absolute frame of reference due to this...

 

[ash64449]

You need to read to the end of the chapter and then subsequent chapters.

Well,i have already read it 2 to 3 times and i think light traveling at different velocities is the reason for simultaneous events being non-simultaneous and vice-versa..

 

[ash64449]

And Einstein gave relationship..Time and Relativity of simultaneity.. They are closely linked to each other... Since Simultaneity changes in different frames(base on relative motion) because of light,light traveling at different velocities in different frames is the reason for time dilation and length contraction..

 

[ash64449]

It depends on what you are asking.

If you are asking why the speed of light measures to be a constant (a result of Einstein's first postulate, the Principle of Relativity), then the above two diagrams illustrate that. In both cases the observer sends out a light pulse at his time of 4 nanoseconds and detects the reflection from the mirror that he measured to be six feet away at his time of 16 nanoseconds. So the light, as far as he is concerned traveled 12 feet (round-trip) in 12 nanoseconds which is 1 foot per nanosecond.

What about half-way?? here we considered two-ways,that is why light speed is measured constant.what about half-way? How light appears constant?

 

[ghwellsjr]

And do you know how to calculate relativistic doppler?? I think it will be very usefull to me if you can explain in simpler manner. It is a request from me...

The formula for Relativistic Doppler is:

√((1+β)/(1-β))

where β is the approaching speed as the ratio v/c.

For a departing speed it is simply the inverse of the above.

So in the second IRF shown for Time Dilation, β = 0.6 so:

√((1+β)/(1-β)) = √((1+0.6)/(1-0.6)) = √(1.6/0.4) = √4 = 2

This means that while approaching at 0.6c, each observer will see the other ones clock ticking twice as fast as their own and while departing at that speed, they will each see the other ones clock ticking at half the rate of their own.

Unlike Time Dilation for which each clock/observer/object is independent of all the others and has a Time Dilation dependent only on their individual speeds in the IRF, Relativistic Doppler is an effect between two inertial clocks/observers/objects. So I have added a stationary red observer in the second IRF from above. First I show the light signals sent each nanosecond from the traveling blue observer:

Note how in the bottom of the diagram, while the blue observer is approaching the red observer, the red observer sees two ticks of blue's clock for every one of his own. Then after they cross paths while they are departing, red sees blue's clock ticking at one-half the rate of his own.

Now what does blue see of red's clock?

Same thing--even though we are only using one IRF in which red is at rest.

And please note that the Time Dilation factor of 1.25 is not observable by either observer, only the Relativsitc Doppler factors are.

 

[ash64449]

My Guess of relativistic Doppler was absolutely right! I understood from your comment.. Thank you!

 

[ghwellsjr]

So in reality light travels in different speeds but we aren't able to detect them because of time dilation and length contraction..right?? Now,i find that it is similar to absolute frame of reference...

I feel like there is absolute frame of reference due to this...

When we define light to travel in an IRF at c in all directions, it is identical to what scientists believed was the case for a single unknown IRF.

Well,i have already read it 2 to 3 times and i think light traveling at different velocities is the reason for simultaneous events being non-simultaneous and vice-versa..

Maybe it would be helpful for you to read the first few sections of Einstein's first paper on SR. It's much shorter and gets right to the point.

And Einstein gave relationship..Time and Relativity of simultaneity.. They are closely linked to each other... Since Simultaneity changes in different frames(base on relative motion) because of light,light traveling at different velocities in different frames is the reason for time dilation and length contraction..

I hope I'm understanding what you mean by this-that comparing one IRF to another, light is traveling at different speeds relative to the observers/objects/clocks, correct? But you should only think of one IRF at a time and say that light travels at c in that IRF. To get to another IRF, you use the Lorentz Transformation to convert all the coordinates of the events in the first IRF to the second IRF and then you don't mix any coordinates between them.

What about half-way?? here we considered two-ways,that is why light speed is measured constant.what about half-way? How light appears constant?

I think by half-way, you mean what I mean by one-way. Each "half" of the round-trip is a one-way trip so the two one-way trips add up to the round-trip.

 

[ash64449]

I hope I'm understanding what you mean by this-that comparing one IRF to another, light is traveling at different speeds relative to the observers/objects/clocks, correct?


.

WOW! exactly! isn't that correct?

 

[ash64449]

I think by half-way, you mean what I mean by one-way. Each "half" of the round-trip is a one-way trip so the two one-way trips add up to the round-trip.

Well,Your are quite understanding what i am saying!

Well,yes,then what is the answer? Will light be measured same if we consider only one-way trip?

 

[ash64449]

when we define light to travel in an irf at c in all directions, it is identical to what scientists believed was the case for a single unknown irf.
.

what?

 

[ghwellsjr]

WOW! exactly! isn't that correct?

I'm just saying that it's not a good idea to compare the coordinates in one frame with the coordinates in another frame. Each IRF, by itself, is sufficient to describe and handle everything that is going on in a scenario.

Well,Your are quite understanding what i am saying!

Well,yes,then what is the answer? Will light be measured same if we consider only one-way trip?

We can't measure the one-way speed of light. That's the whole point of Einstein's second postulate. In any IRF we want to consider, we define each half of the round-trip measurement of the speed of light to take the same amount of time. That allows us to synchronize remote clocks in a consistent way and then we can "measure" the one-way speed of light but then it can't help but to come out to be c so we are not really measuring it.

what?

If there were an ether in which light propagated at c like the early scientists believed, then any IRF we chose will have that same property that light propagates at c in it. I hope I didn't confuse you on this point. The problem with believing in an ether is that there is no way to determine which IRF it is at rest in and so it is pointless to hang on to that concept.

 

[SammyS]

well,i found this article and i find it simple to understand. But is this article totally correct?
Here is the link:http://m.sparknotes.com/physics/specialrelativity/kinematics/section2.rhtml

By the way, the link appears to be to an online publishing site. A citation for this material is:

SparkNotes Editors. “SparkNote on Introduction to Special Relativity.” SparkNotes LLC. n.d.. http://www.sparknotes.com/physics/specialrelativity/intro/ (accessed April 13, 2013).​

The name of the author is not given in any place on that site, as far as I could determine.


Regarding the correctness of the article:

I found a problem fairly early on. The development of time dilation looks reasonable up to and including the time ratio, t_B/t_A . That's the ratio of the time for one time interval of the "clock" as measured by a "stationary" observer, observer B, who measures the time as t_B and one time interval of the "clock" as measured by an observer on a moving train, observer A, who measures the time as t_A.


How does O_B, Observer B, actually determine t_B ?

The author of the article has O_A wave every time that the clock completes a cycle. Presumably, the train (along with O_A) is moving at a speed, v, which is a fairly large fraction of the speed light with respect to O_B. The author seems to imply that if O_B uses the time, Δt, that he observes as being the time between "seeing" one wave and the next wave --- that this Δt can be used as t_B. However, that doesn't work.

Let's suppose that O_B stands very close to the train track --- so we don't complicate matters via Pythagoras. The train moves a distance, v∙t_B, between successive waves by O_A . In order for O_B to determine t_B by observing the waves by O_A, O_B needs to consider the extra time it takes light to reach him from one wave to the next. (I presume O_B is using sight to observe O_A waving.)

As the train and O_A approach O_B, suppose O_A waves when the O_A is distance, d₁ from O_B, as measured in O_B's FoR (Frame of Reference). The next time O_A waves, the train will be v∙t_B closer to O_B than she was for the previous wave, so it takes light a time of (v∙t_B)/c less to reach O_B than light from the previous wave.

The time, Δt, between O_B's observation of successive waves of O_A is longer than t_B by an amount (v∙t_B)/c .

Therefore , t_B = Δt - (v∙t_B)/c .


When teaching relativity, I liked to use the word "infer" rather than the word "observe" when discussing time-dilation / length-contraction, as in: The stationary observer, O_B infers that the moving clock ticks at a slower rate than an identical clock which is stationary.


By the way: I do realize that there are several other ways for observer, O_B, obtain t_B from observer, O_A . The results will be equivalent.

 

[Dale]

I just want to remind the participants to be careful in discussing LET. It is a legitimate interpretation of SR, but it is not the only legitimate interpretation. Please avoid any assertions or claims that it uniquely represents reality.