It is very easy to come up with the expression for an invariant time like interval.
A light flash at ground level goes up, hits a mirror and comes back to the source in a time T as seen from the inertial frame of the source. A fast moving observer passes this light clock and measures t^2=(vt)^2+T^2=x^2+T^2. So T^2=t^2-x^2.
I have never been able to come up with anything nearly as simple to give the expression X^2=x^2-t^2, where X is the proper length of a rigid body.
This bugs the daylights out of me. I guess the reason is that the above example depends on two dimensions of space, so finding a comparable derivation for the space-like interval fails because of a broken symmetry between space and time.
Is there a reasonable concise way to arrive at X^2=x^2-t^2? I want to do this without the use of Lorentz transformations.
The trick is, if you can show both T^2=t^2-x^2 and X^2=x^2-t^2, the Lorentz transformations are just two derivatives away.