B Clarification on Length Contraction

[NoahsArk]

I appreciate all the help.

A spacetime diagram below suggest the following interpretation:
It describes how the Earth frame
compares its x-axis tickmarks with the rocket frame's x'-axis tickmarks.
The correspondence follows the Earth frame lines.
Is that how you interpret it?

Yes. The blue rectangles are the rocket's space units and the red diamonds are Earth's space units right?

That is the spacetime interval between the explosion event and the event where your x′=−7.5x′=−7.5x′ = - 7.5 line of simultaneity (the dotted line) crosses the worldline of the S′ origin (the red line). So you should find that √(Δx)2−(Δt)2=12.5

I will have to give that some thought. This looks like the invariant interval in reverse. I remember reading that you use this instead when you have two events that are separated by space by not by time in a certain frame. The invariant interval for time gives us proper time, so does this version of the invariant inverval give proper distance?

Sorry my latex botches things in quotes.

 

[PAllen]

I will have to give that some thought. This looks like the invariant interval in reverse. I remember reading that you use this instead when you have two events that are separated by space by not by time in a certain frame. The invariant interval for time gives us proper time, so does this version of the invariant inverval give proper distance?

Sorry my latex botches things in quotes.

Yes, the invariant interval for spacelike separated events is proper distance i.e. distance in a frame where the events are simultaneous.

 

[robphy]

The blue rectangles are the rocket's space units and the red diamonds are Earth's space units right?

Yes, the red diamonds are for the earth/lab frame.
The blue diamonds are for the rocket frame.

"diamonds" or "parallelograms" [or "boxes"] are accurate descriptions.

However, "rectangles" or "squares" are not...
because (despite Euclidean appearances), the edges are not "Minkowski-perpendicular" to each other.
More generally, one set of lightlike lines (the 45-degree lines) are not "Minkowski-perpendicular" to the other set of lightlike lines (at minus-45-degrees). Their Minkowski-dot-product is not zero.
The diagonals, however, are Minkowski-perpendicular.
(A lot of the geometry of Minkowski spacetime is encoded in those diamonds.)