How to find the time and space axes of an arbitary intertial reference frame?
The best approach I have ever read is in Schutz’s book.
Suppose we have a frame O which we are sitting in. Another frame O’ is moving with velocity \(v\).
The time axis is the \(x=0\) points. We simply find out the line of object moving with velocity \(v\).
The space axis is an equal time line. The invariant motion that is assumed in special relative is the motion of light. So we use light to find out this axis. We know light is always travelling with speed 1, which means it is always travelling with 45 degrees of angle in spacetime diagram, no matter what frame we are in. We fine equal time distance on t axis of frame O and t’ axis of frame O’, light emitted from \(t'=-t'_0\) reflected on the \(t'=0\) point will be back to \(x'=0\) point but at time \(t'=t'_0\). So we draw 45 degree lines from \((-t'_0,0)\) and \((t'_0,0)\) and let the two light beams intersect. The intersection point is a point on the space axis.