How to read spacetime (Minkowski) diagram for proper length

[techsingularity2042]

Homework Statement

I just started learning special relativity a week ago so spare me if I'm using any wrong terminology. I am not sure how to read the diagram. Can you confirm if this is the correct proper length? I will indicate it in the diagram. L0 is the proper length, and L is the contracted length.

Relevant Equations

A rod R is rest in a frame of reference S'. S' is moving relative to a reference of frame S.

 

[PeroK]

I think things are a bit mixed up. The diagram on the left looks like the rod of length $L$ in frame S'. Not frame S. But, the right-hand end of the rod's wordline should be vertical. And the diagram on the right looks more like frame S! I think you should start again as follows:

The ends of the rod in S' are at $x' = 0$ and $x' = L_0$. Their wordlines are vertical lines - i.e. parallel to the $ct'$ axis. That gives you the diagram in frame S'. This should look more like the diagram you have on the left.

In frame S, the coordinate axes $x'$ and $ct'$ are as you have shown them in the right-hand diagram. The left-hand end of the rod's world is the $ct'$ axis. And the right-hand end is the green line you've marked as R.

The difficult question is what is the length of the rod as measured in S? Note that Minkowski diagrams are not Euclidean, so things are not the lengths they appear.

Maybe the first thing is to get the diagrams right and then we can figure out how to calculate $L$?

 

[Orodruin]

Here's a snap from my SR lecture notes with the correct diagrams:

Green are the end-points of the rod, red and blue lines are the simultaneities of the respective frames. Using c = 1.

 

[techsingularity2042]

Thank you for your reply.

This was my thought process:

the axes ct' and x' are of frame S', and the axes ct and x are of frame S. I tried to the combine the two frames into one because that was how it was done on the textbook.

R is at rest in frame S', and because the frame S' is moving relative to the frame S, R is moving relative to S at the velocity S' does S.
So R should have a shallow gradient in frame S.
And because R is at rest in the frame S', ct'-axis and R should have the same gradient - as they travel at the same speed according to the observer in frame S.

Now I look at it again after checking your reply and pondering over some time, the proper length, L0 should be a line parallel to the x'-axis because the left end of the rod is at x'=0 and the right end on the starting point of line R.

(I labelled the diagrams to indicate the measured L is in which frame)

The contracted length L is a line parallel to the x-axis (as in frame S) and perpendicular to ct-axis because the measurement of both ends should occur at the same time.

I just realized that another one of you left a reply. Thank you for the explanation and the diagrams. Huge thanks to both of you.

 

[PeroK]

Both diagrams look like frame S to me. S' is the reference frame in which the rod is at rest.

 

[techsingularity2042]

Both diagrams look like frame S to me. S' is the reference frame in which the rod is at rest.

Yes, I labelled the diagrams to indicate in which frame the L was measured. I understand what you mean is that I construct fundamental axes with ct' and x' and make the worldline of the object R vertical.

But it seems like the axes of both frames should be drawn in one diagram in my course. That's why I put both the axes of frame S and frame S' in one diagram.

 

[PeroK]

Yes, I labelled the diagrams to indicate in which frame the L was measured. I understand what you mean is that I construct fundamental axes with ct' and x' and make the worldline of the object R vertical.

View attachment 355086

But it seems like the axes of both frames should be drawn in one diagram in my course. That's why I put both the axes of frame S and frame S' in one diagram.

If a diagram is labelled S, then the $x$ and $ct$ axes need to be the main ones. There's no point in having two diagrams unless the diagrams have different axes. You have essentially two copies of the same diagram.

For example. If S' was the frame of a car, then the motion of a wheel would be rotation about a fixed point. And, if S was the frame of the road, then the wheel would also have linear motion, and the motion of a point on the rim of the wheel would be a cycloid. The purpose of having two diagrams in this case would be to highlight the different motion of the wheel in the two reference frames.

 

[Orodruin]

I agree with @PeroK . When you draw a Minkowski diagram on Euclidean paper, it is invariably based on the frame whose axes are orthogonal in the Euclidean sense. You can of course draw the axes of other frames into the same diagram as well, but they will not be orthogonal in the Euclidean sense (they are of course in the Minkowski space). Drawing diagrams of the same thing based on different frames can be quite instructive.

In this case, you can mark both the measured lengths in a single diagram as well. The main thing is to realize that what we call the ”length” of something is the distance between the end-points at a given time and that what a given time means depends on the frame. After that it is just geometry using the Minkowski metric, as illustrated in #3.