- #1

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I have seen quite a lot of demonstrations of time dilation and length contraction that used standard Minkowski diagrams WITHOUT any scales on the axes at all. If I understand them correctly they seem to directly compare lengths, which would imply (I think) that the scaling on the ##ct/x## and ##ct'/x'## axes is the same.

But it is not. If in a standard Minkowski diagram the unit length on the axes of ##ct## and ##x## is given as ##U##, the unit length on the axes of ##ct'## and ##x'## is

$$U'=U\cdot \sqrt{\frac{1+\beta^2}{1-\beta^2}} .$$

I can see how Minkowski diagrams without scales can be used to talk about simultaneity or the temporal order of events in different inertial systems, but can they illustrate time dilation and length contraction where the actual length of intervals in time or space need to be compared? I doubt it because in the derivation of the angle between the ##ct## and ##ct'## axes one looks at all events with ##x'=0## and uses the Lorentz transform

$$0=x'=\gamma\cdot (x-vt)\Rightarrow \tan\alpha=\frac{x}{ct}=\frac{v}{c}$$

and analogously for the angle between the ##x## and ##x'## axes, which turns out to be ##\alpha## as well. The Lorentz factor ##\gamma## drops out in the derivation of the angle, so why should it turn up again in a graphical derivation of time dilation or length contraction?

Am I right about this and is it merely a coincidence that those demonstrations explain time dilation and length contraction qualitatively correct?

Second question:

Is there a way to construct ##U'## geometrically in a Minkowski diagram with given angle? I know there are symmetric Minkowski (Loedel) diagrams that cirumvent this problem, but they are far less common than the standard diagrams where the ##ct/x## system is at rest.