# Minkowski diagram of two frames in rest with respect to each other

I'm starting to study Minkowski diagrams, but I can't figure out how should the axis of a $$S'$$ system look if it is in rest with respect to the frame $$S$$. I've seen that in all diagrams the origin seems to stay in place. How is this possible for two frames in rest to respect with each other, one separated from the other 8 light-minutes to agree on the origin? Shouldn't the other frame be "to the right/left"?

Also, if I am an observer on a frame $$S''$$ moving with a speed $$u_x$$ with respect to them, how would their axis look to me? Are they axis with the same inclination but located in different places?

• If two observers are at rest w/r to each other the axes of their respective diagrams will coincide since the relative slope is $v/c$, with $v$ the relative of one w/r to the other. – ZeroTheHero Feb 21 at 13:35
• But why ? Wouldn't that mean that an event A(x^0,x^i) would have the same coordinates in both frames? That's clearly false, since the frames are spatialy separated. – IchVerloren Feb 21 at 13:40
• Minkowsky diagram are usually constructed so that origins coincide. You would need to simply shift the origin of one but keep the relative orientations the same. – ZeroTheHero Feb 21 at 13:42

## 1 Answer

Here is a spacetime diagram [drawn on rotated graph paper] of the back of the stick showing two reference frames, one for the back of the stick (through the origin event), and the other for the front of the stick. On a spacetime diagram drawn by another inertial observer who meets [the back of the stick at] event O and travels with velocity (-3/5)c with respect to the stick, the stick-ends' frames appear as shown. The worldline of this other observer isn't shown, but can easily be constructed. • I keep wondering, was there a specific reason to draw the diagrams on a rotated graph paper? – IchVerloren Mar 19 at 3:31
• The rotated graph paper makes it easier to visualize ticks along segments and to visualize Minkowski-orthogonality, especially for inertial observers in motion according to rest frame that drew the diagram. The key feature is that the areas of all “light clock diamonds” are equal and have their edges along the light cones, as required by a Lorentz boost transformation. – robphy Mar 19 at 9:58