Length Contraction rearrangement

[mananvpanchal]

Hello All. Just consider once the above post. I expect a comment at least.

 

[Chestermiller]

If you continue to insist on using Minkowski diagrams to help gain an understanding of length contraction and time dilation, then you need to at least learn how to use them correctly. Until you do, you are not only wasting your own valuable time (which I'm sure you don't want to do), you're also wasting everyone else's. Two points about Minkowski diagrams:
1. The events A and A' are not separate points on the Minkowski diagram. They are the same event, and must coincide on the diagram. The same goes for events O and B.
2. Equal values of Δt and Δt' do not correspond to equal increments along the t and t' axes on the diagram. Equal values of Δx and Δx' do not correspond to equal increments along the x and x' axes on the diagram.
Most standard texts on SR provide a description of how to use Minkowski diagrams.

Chet

 

[mananvpanchal]

If you continue to insist on using Minkowski diagrams to help gain an understanding of length contraction and time dilation, then you need to at least learn how to use them correctly. Until you do, you are not only wasting your own valuable time (which I'm sure you don't want to do), you're also wasting everyone else's. Two points about Minkowski diagrams:
1. The events A and A' are not separate points on the Minkowski diagram. They are the same event, and must coincide on the diagram. The same goes for events O and B.
2. Equal values of Δt and Δt' do not correspond to equal increments along the t and t' axes on the diagram. Equal values of Δx and Δx' do not correspond to equal increments along the x and x' axes on the diagram.
Most standard texts on SR provide a description of how to use Minkowski diagrams.

Chet

Then please, help me here to understand Minkowski diagrams correctly or provide me the text. I want to learn Minkowski diagrams and time dilation and length contraction from Minkowski diagrams.

 

[Doc Al]

Then please, help me here to understand Minkowski diagrams correctly or provide me the text. I want to learn Minkowski diagrams and time dilation and length contraction from Minkowski diagrams.

What textbook are you using to teach yourself special relativity?

 

[darkhorror]

And the whole problem of discussion starts from here.
O' and B' is assumed as rest points in S' frame. But O' and B' is not at same time in S'. So we have to evolve idea to pick different time component for the points which is assumed as at rest in S' frame.
After derivation we get contracted length as OB in S frame. And we have discovered that OB is contracted length in S because S measure length at same time.

Again I don't know what I should call O'B' as. But I am sure that OB is proper length measured by S frame between rest points of S frame. And I am sure again that O'B' is length measured by S' between rest points of S frame.

That is why the title of the thread is "Length Contraction rearrangement".

You just want to know how to get the length starting from known length at rest in S, moving to calculate that length in S'?

if so you just use length * 1/$\gamma$

 

[Doc Al]

Now, we come to length contraction.
O and B is stationary points in S frame at same time but at different location. S measures OB length between these two points. O and B is rest points in S frame, so OB is proper length in S frame. S' defines that points as O' and B'. Now we want to know how much length S' measures between those points. We can get parallel line to $t$ axis and we get D point on $x$ axis. Or we can get parallel line to $t'$ axis and we get the same B point on $x$ axis. But, there is no meaning to get point B again. And I am stuck here... I don't know how to get length contraction. I don't know what I have to call O'B' as. But I know that OB is proper length measured by S frame between rest points in the same fame. And I am surely know that O'B' is not proper length between rest points in S' frame. O'B' is length measured by S' between rest points of S frame.

We cannot start derivation of length contraction by guessing O'B' as proper length between rest points in S' frame. It is not really. The whole idea to derive length contraction starting from S' frame is strange.

And the whole problem of discussion starts from here.
O' and B' is assumed as rest points in S' frame. But O' and B' is not at same time in S'. So we have to evolve idea to pick different time component for the points which is assumed as at rest in S' frame.
After derivation we get contracted length as OB in S frame. And we have discovered that OB is contracted length in S because S measure length at same time.

Again I don't know what I should call O'B' as. But I am sure that OB is proper length measured by S frame between rest points of S frame. And I am sure again that O'B' is length measured by S' between rest points of S frame.

That is why the title of the thread is "Length Contraction rearrangement".

The easiest way to illustrate length contraction using your diagram is to let O'B' represent a fixed length in S' (imagine it being a stick). If you want to see how S would measure the length of that stick you must draw the worldlines of the stick endpoints. Those are shown in your diagram as the slanted lines parallel to the t' axis. Now for S to measure the length of the stick, he must measure the positions of the endpoints of that stick at the same time. To do that, draw the horizontal line t = 0 and see where it intersects those world lines: The length of the stick measured in S will equal what you have shown as OB in your diagram.

If you want, you can do it the other way around. Let OB be a stick at rest in S. Draw the worldlines of its endpoints, which will be vertical lines in your diagram. Then see where S' will measure the ends of the stick at any given time by slicing those worldlines with a line parallel to the x' axis.

One thing to be careful about, as pointed out by Chestermiller, is that the units are not the same in S and S'. Done carefully, and properly accounting for units, you will always find that the length of a stick at rest in S' will be measured as shorter in S, and similarly, the length of a stick at rest in S will be measured as shorter in S'. That's how length contraction works.

Of course, all of this is trivially done with the Lorentz transformations.

I know you want to be able to 'derive' length contraction from the Minkowski diagrams, but realize that length contraction is already built into the diagrams by how you set up your coordinate systems. (You are implicitly using the Lorentz transforms when you draw a Minkowski diagram.)

 

[Jorrie]

Then please, help me here to understand Minkowski diagrams correctly or provide me the text. I want to learn Minkowski diagrams and time dilation and length contraction from Minkowski diagrams.

I have used the attached Minkowski spacetime diagram to explain what you have asked, so maybe it will help (and not distract) you. What you see is the orthogonal axes of the reference frame (x,ct) and the oblique axes of the relatively moving frame (x',ct'), with coordinates x,ct =1,1 indicated by the larger green bullet and x',ct'=1,1 by the larger red bullet.

Now look at where the line ct'=1 intersects the ct-axis and where the line x'=1 intersects the x-axis (smaller red bullets). Both time dilation and length contraction are demonstrated. The same happens where the line ct=1 intersects the ct'-axis and where the line x=1 intersects the x'-axis (smaller green bullets), because the effects are reciprocal.

The relative velocity used in the diagram was 0.4c.

 

[mananvpanchal]

Thanks to all of you guys. This seems that I am interpreting Minkowsky diagrams incorrectly. As I start interpreting it truly, the length contraction is no longer confusing to me.

Thanks again.