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I'm currently taking "Understanding Einstein" Coursera course before reading Smith's "Introduction to Special Relativity", and there's something I probably miss regarding relativistic length contraction.

Let's use an example from the course - Bob flies over Alice with the speed v on a platform, and both measure the length of Bob's platform.

If each one of them uses their own clocks to perform the measurement they should get the same result, right? Their clocks are at rest in their frames of reference and time dilation will have no effect. Only when each one of them compares the result obtained using his/her own clock with result obtained using another one's clock in his/her frame of reference, they will find the difference.

For Bob, for example, Alice's clock will "tick" slower, so if he measures the length of his platform using his clock and using Alice's' clock, lengths will be different. But the instructor said that even in the first case (each one used only own clocks for the experiment) they will record a different result. What do I miss here?

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  • $\begingroup$ Why would you measure the length of a platform with a clock? $\endgroup$
    – Physiker
    Jun 4, 2021 at 18:55
  • $\begingroup$ The length is measured using time and known speed. Alice notices when the leading edge of Bob's platform passes over her and writes down time, then she does the same when rear edge of Bob's platform passes over her. Bob uses his clocks to write down times when leading and rear edges of his platform pass over Alice. $\endgroup$
    – Alx Mx
    Jun 4, 2021 at 19:00
  • $\begingroup$ @AlxMx It will be helpful if you include the above comment in the question to explain what it means to use the clocks to measure length $\endgroup$ Jun 5, 2021 at 5:59

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The rule is that a length that is stationary in one reference frame will always be shorter in a frame moving relative to it.

The discrepancy arises from the relativity of simultaneity.

In the frame in which the length is moving, the observers believe that they are pinning down each end of the object simultaneously- however, from the perspective of the stationary frame, those observers are measuring the position of the front end of the object before they measure the position of the rear end, which results in a shorter measurement.

If you are in any doubt about that, imagine noting the position of two ends of a moving ship, say. If you note the position of the front and rear of the ship at the same time, then the difference will give you the true length of the ship. However, if you note the position of the front, and then a few seconds later note the position of the rear, the rear will have moved on during the intervening period while your were waiting, so that it will be ahead of the position it would have been in when you pinned down the position of the front. As a result, you will calculate a shorter length for the ship.

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Let us consider the rest length of the platform to be $l_o$ and their relative velocity to be $v$.

The length of the platform with respect to Alice will be contracted because for her, $$l=l_o/\gamma$$

where $\gamma=\sqrt{1/(1-v^2/c^2)}$

Thus, the time that she measures for the carriage to pass is-

$$t=l_o/(\gamma \cdot v)$$

What is the time measured by Bob? $l/v$

Therefore, Alice measured less time than Bob since $v\leq c$

Where is the confusion?

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No, Even when they use their own clocks, the result is different. Length contraction means that a moving object (wrt to observer) seems smaller or contracted to the observer compared to its proper length in the rest frame of the object. This can be seen also in the definition on Length Contraction.

It has nothing to do with exchange of information between the two observers Alice and Bob. Even if they observe in their own time, the length is different. There are some quantities that they need to agree upon which are called scalar quantities and length is not one of them. Length is a relative quantity. If let's say, Alice and Bob had to find the length that Bob measures, I mean if Alice calculates not the length that she observes but the length she thinks Bob measured, then they will get the same result.

Length contraction is one of the results of the fact that the speed of light is absolute.

You can also try to understand it this way:

t=0

There is a moving train and three observers in total. Observer 1 and 2 are on the train while Observer 3 is on a stationary platform. The train is moving with velocity v wrt to Obs 3. Obs 1 is at the front of the train while Observer 2 is at the rear. When Obs 1 crosses Obs 3, they all synchronize their clocks to t=0 (for Obs 3: rest frame) and t'=0 (for Obs 1 & 2: moving frame).

t=t'

When Obs 2 crosses Obs 3 they note down the times again. Now, $t_3=t$ and $t_1=t_2=t'$. Now, the length of the train for Obs 3 is given by speed*time which is: $$l=v \times t$$

For Obs 1 and 2: Obs 3 is going with a velocity of v in the opposite direction. Hence, for them, the length of train is: $$l'=v \times t'$$

Since, $t \neq t'$ and magnitude(v) is same for both the frames: $l \neq l'$ needs to be satisfied. Note: that these lengths are measured by the observers in their own time frame and is irrespective of each other's measurements. Hence, length contraction is independent of 'if they compare their observations'. Even if they do not compare their observations or take observations in different frames, length contract does exist.

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