Problem in understanding Length Contraction Phenomena

Question

I am in a great confusion for the past one week in deriving and understanding length contraction.

The thing is, in every text books we see that the rod is placed in the prime frame and one is able to measure its length without any difficulty. The prime frame wrt to unprime frame is moving with some velocity say v.

The problem is, in prime frame, we say that at same time we measure the length of the rod, so ${t_2'}$=${t_1'}$, and that is perfectly alright with me.

But when we use Lorentz transformation, to find the length of the rod in prime frame, we see that, time measured, when one measure the length of the rod at both the ends are not same, and no one cares to think about that point and uses the mathematics and arrives at that result.

Second thing is a bit of mathematics,

$x_2'-x_1'=l_o$ is in prime frame, is what me measure the proper length of the rod at time $t_2'=t_1'$ and the prime frame is moving with velocity v, wrt un-prime frame in right direction.

Observer in un-prime frame measure the length,

$x_2-x_1=\gamma(x_2'-x_1'-vt_2'+vt_1')=L$.

This gives us, $L=\gamma l_o$, which is the opposite that one gets.

But, but, if one starts or uses the Lorentz Transformation from the equation,

$x_2'-x_1'=\gamma(x_2-x_1-vt_2+vt_1)$, and assumes $t_2=t_1$, one gets the actual formula.

Point to be noted is that, in $1_{st}$ case, $t_2\ne t_1$, but in above case it is same.

My overall discussion of the point is, why in deriving the length contraction formula, why we don't already assume that whether in prime or un-prime frame, the times measured to measure the length of the rod be same?

Also i referred to a YouTube video by Dr. Don Lincoln, the link is https://www.youtube.com/watch?v=-Poz_95_0RA&t=430s , to understand the length contraction. He also raises a points and tells the readers to note that time are not equal for the moving observer, when he measures the rod, which should be. And at timestamp 6:52, he uses the Lorentz equation, but yet again didn't mentioned or said anything about the times in un-prime frame being zero, in the first case., and uses the result to arrive at the length contraction formula.

Edit-Need your valuable comments/discussion to this problem of mine, as to where I am going wrong in anyway and every single minor doubt raised by me would be very much appreciable, as these small doubts would make my understand a way more stronger and would have a good hand in this Special Relativity subject.

Answer 1

Length contraction is a result of the relativity of simultaneity.

If you want to measure the length of a moving object, such as a train, you have to measure the position of the leading edge and the trailing edge at the same time. If you measure the position of the leading edge of the train first, and the trailing edge of the train a little later, you will get the wrong result, because the trailing edge of the train will have moved forward in the interval between the two measurements, so the train will seem shorter.

That is what causes length contraction. Suppose I am on a platform and I measure both ends of a train at the same time in my frame of reference. If you are on the train, my two measurements occur at two different times in your frame of reference, my measurement of the front of the train occurring before my measurement of the rear. As a result, you will say that I allowed the train to move forward between my two measurements, so in your frame I have mistakenly measured the train as being too short. I, on the other hand, believe I have made the measurements at the same time, so to me the shorter length is a valid result.