There is, as I see it, no 'set distances'. You need to use the concepts of clocks and rulers to make them 'connect' for two different frames. And there I'm still wondering what they really are:) Distance is such an 'normal' everyday concept to us, we expect them to be the same from day to day, and they are too at least inside our own frame.

This made me smile.

I had a look for links and all I can find (that look helpful) are these two...

http://wapedia.mobi/en/Minkowski_diagram [nofollow]http://www.einsteins-theory-of-relativity-4engineers.com/loedel-diagrams.html [nofollow]Both of them dive in quite quickly.

The way I think about the diagrams is like this (not why they work, only how to interpret)...

A normal space vs time diagram, which you know from school looks like this, except I have swapped x axis (distance) and time axis (t) over, because this is what Loedel diagrams do.

[BTW, I've put events on all these diagrams but they are not related, they are just examples of how to read coordinates from the axis.]

[diagram=575_0]

Note how there is a grid.

The horizontal lines represent things which all happen at the same time (from the origin). The vertical lines represent things which all happen at the same distance (from the origin). In relativity an observer makes all measurements relative to themselves i.e. they never move. So their 'world line' is the vertical 't' axis at x=0.

The units on the axis are defined in terms of C. Any unit can be used for distance, but the time unit used will then be "the time it takes light to travel one distance unit (in a vacuum)". Thus a light beam is always at 45 degrees on the diagram (to move one distance unit along, it takes one time unit up).

At low relative speeds two observers can use the same diagram. But at high relative speeds the effects of relativity can be modelled by 'compressing' the observers grid. It is just as if you had a trellis (like for plants) and squeeze it.

So for one observer we get a grid like this.

[diagram=576_0]

[Note how due to our choice of units for the time and distance axis, C is still at 45 degrees. The grey lines represent light signals which pass through the origin.]

And the other observer gets this.

[For some reason the events dots don't show on this diagram, just imagine

]

[diagram=577_0]

Note that (apart from drawing accuracy) the two grids are mirror images of each other, both up-down and left-right. The only thing that doesn't reflect is the +/- on the axis.

So the Loedel diagram works by making a reference event where both observers were present and synchronised clocks, and puts this event at the origin. Then the two grids can then be overlaid. Note, I am not explaining why this works, just how to do it.

The resulting grids are not easy to draw on one diagram and it would just look a mess anyway. What usually happens is that only the axis are drawn, the events marked, and the lines from events to the axis to read off times and distances.

Note that for events remote to an observer, they see its time as being along their time grid line, and the distance as being along their distance grid lines.

[diagram=578_0]

So using this diagram as an example, there are two events marked and the construction lines to show what times and distances the observers measure.

Mr Green, from along his time line measures that there was some time before an event occurred, and an even longer time before the next one. However, Mr Red measures that he waited for some time in between what Mr Green thinks, and then both events happened at almost the same time, he even thinks they happened in reverse order.

Now, neither observer can define absolute positions where any event occurs, only the distance from themselves. But by measuring how far away from them each event is, they can determine the distance between events. By looking at the total distance between events on each observer's 'x' axis it can be seen that Mr Red think there was less distance between events than Mr Green, but not a huge difference compared to the time discrepancy.

There are good reasons why these diagrams work [to do with s^2 = (Ct)^2 - x^2] but I can't remember the full explanation any more. Suffice to say that if you try out any of the common 2 party scenarios, such as the OP, or why observers each think the other has a slow clock and a short ruler, or how causality is preserved even though observers can disagree on the order of events, then they always seem to give the correct explanation.

As a true exercise in relativity, you should do the following, which is actually quite tricky but worth the effort.

Place a straight edge (like a piece of paper, or a ruler) along one observers 'x' axis.

Mark the position of the 't' axis (x=0) on the straight edge. This mark represents the observer.

Slide the edge up the diagram, always keeping it parallel to that observers 'x' axis and always keeping the mark on that observers 't' axis (x=0). This takes a bit of practice.

As you move it up the page you can see what the observer (the mark) thinks is happening.

Events hit the straight edge in the time order he measures and at the distance he measures. It is his world view.

You can now do the same thing for the other observer using their axis and see their version of events.

Once you have done this enough times you can kind of do it in your head.

Note that for either observer, light beams either move toward them or away from them at velocity C. For example, the grey diagonals shown could represent light beams that coincided at the origin, along with our observers. Both observers measure that these beams move away from themselves in opposite directions, at velocity C.

It goes without saying that these diagrams only show one space dimension but for the armchair physicist that seems to be enough.

Have fun!