Deriving Length Contraction From Lorentz Transform

[Delzac]

Homework Statement

From Lorentz Transform,

$$x^{\prime} = \gamma (x - vt)$$

From textbooks and wikipedia,

$$L_0 = x'_2 - x'_1 = \gamma (x_2 - x_1 )$$

Where x_1 and x_2 = L

Thus,

$$\L_0 = \gamma L$$

Question is this:
If i take the same method and us the Inverse Lorentz transform, i seem to get a different answer, namely:

$$\L = \gamma L_0$$

Which obviously is wrong. I suspect the problem is with where the observing is that is implicitly assume when one use either Lorentz or inverse Lorentz. But, i cannot be sure nor can i resolve this problem.

Any help will be appreciated.

 

[ideasrule]

From textbooks and wikipedia,

$$L_0 = x'_2 - x'_1 = \gamma (x_2 - x_1 )$$

Note that this is only true if t1=t2. To put it another way, (x1,t1) represents the event of measuring x1 while (x2,t2) represents the event of measuring x2. In the reference frame where the stick is moving, the two measurements have to be performed at the same time. They don't have to be performed at the same time in the rest frame--no matter when you measure x1' or x2', they're always going to be the same.

Question is this:
If i take the same method and us the Inverse Lorentz transform, i seem to get a different answer, namely:

$$\L = \gamma L_0$$

Here, you're assuming that t1=t2, where both are rest frame coordinates. (Otherwise, the right-hand side would not equal L_0.) However, if t1=t2, t1' does not equal t2' because of relativity of simultaneity! Unlike in the previous case, neither x1' nor x2' remain the same as time passes, so x2'-x1' does not equal L.

 

[Delzac]

Ah, i see. Thanks, got it.

 

[Delzac]

But, then how dose one use Inverse Lorentz Transform to get length contraction formula? Since t' is not the same, so we use the lorentz transform for t'?

 

[rwisz]

I can understand your confusion and frustration with the different results obtained from using the Lorentz transform and the inverse Lorentz transform. It is important to note that both equations are valid and can be used to derive length contraction, but they represent different perspectives.

The Lorentz transform is used to calculate the coordinates of an event in one frame of reference, while the inverse Lorentz transform is used to calculate the coordinates of the same event in a different frame of reference. So, depending on which frame of reference you are using, you will get different results.

In the equation L_0 = \gamma L, the length L_0 is measured in the frame of reference in which the event takes place, while L is measured in a different frame of reference. This is why the inverse Lorentz transform is used to convert from one frame of reference to another.

In the equation L = \gamma L_0, the length L is measured in the frame of reference in which the event is observed, while L_0 is measured in the frame of reference in which the event takes place. This is why the Lorentz transform is used to convert from the frame of reference of the event to the frame of reference of the observer.

So, both equations are correct, but they represent different perspectives and should be used accordingly. I hope this helps to clarify the issue for you. If you have any further questions, please don't hesitate to ask.