- #1

Leepappas

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- TL;DR Summary
- Simple case of two rulers moving past one another in constant relative motion both in inertial reference frames.

In the case of two rulers of equal rest length moving past one another ibex posted the minkowski diagram and I have a question about the one diagram that follows the following comment by him:

Note that, in the primed frame, "State 2" is after "State 3"! This is the relativity of simultaneity striking - the definitions of the states are anchored to spacelike separated events, and the ordering of spacelike separated events is frame-dependent.

Again we can transform this last diagram into the primed frame:

Now I wrote the formula for the equation of the line that goes from State two to State One in the primed frame and I got an interesting result that I want to share. Ibex can back me up on this.

The lower thick red line intersects the A' axis at

## \frac {L_0}{v}##

And it's slope is found from the lorentz transformation. For the unprimed frame we have

## t = \frac{t'}{\gamma}+\frac{vx}{c^2}##

So the slope is ##\frac{v}{c^2}##

So for the primed frame the slope of a thick red line is

## \frac {-v}{c^2}##

So the equation of the lower thick red line is

## t' = \frac{-vx'}{c^2} + \frac{L_0}{v}## (1)

Where x' is the distance from the B' axis to the lower thick red line. When ##t'=\frac {L}{v}## focus on the triple intersection point where the top thick red line intersects the B prime axis and also meets the thin Blue line representing state three.

Now I'm focused on the time in the prime frame corresponding to state three. For that state we have

##t' =\frac{L}{v} ## and ##x'=L##

Now I checked and equation (1) can be solved explicitly for v. My question is what is v?

Thank you

Here's the attachment of his diagram

Note that, in the primed frame, "State 2" is after "State 3"! This is the relativity of simultaneity striking - the definitions of the states are anchored to spacelike separated events, and the ordering of spacelike separated events is frame-dependent.

Again we can transform this last diagram into the primed frame:

Now I wrote the formula for the equation of the line that goes from State two to State One in the primed frame and I got an interesting result that I want to share. Ibex can back me up on this.

The lower thick red line intersects the A' axis at

## \frac {L_0}{v}##

And it's slope is found from the lorentz transformation. For the unprimed frame we have

## t = \frac{t'}{\gamma}+\frac{vx}{c^2}##

So the slope is ##\frac{v}{c^2}##

So for the primed frame the slope of a thick red line is

## \frac {-v}{c^2}##

So the equation of the lower thick red line is

## t' = \frac{-vx'}{c^2} + \frac{L_0}{v}## (1)

Where x' is the distance from the B' axis to the lower thick red line. When ##t'=\frac {L}{v}## focus on the triple intersection point where the top thick red line intersects the B prime axis and also meets the thin Blue line representing state three.

Now I'm focused on the time in the prime frame corresponding to state three. For that state we have

##t' =\frac{L}{v} ## and ##x'=L##

Now I checked and equation (1) can be solved explicitly for v. My question is what is v?

Thank you

Here's the attachment of his diagram

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