Suppose an observer $\mathcal{O}$ uses the coordinates $t$, $x$, and that another observer $\mathcal{O}'$, with coordinates $t'$, $x'$, is moving with velocity $\mathbb{v}$ in the $x$ direction relative to $\mathcal{O}$. Where do the coordinate axes for $t'$ and $x'$ go in the spacetime diagram of $\mathcal{O}$?
<p>$t'$-axis: This is the locus of events at constant $x'=0$ (and $y'=z'=0$, too, but we shall ignore them here), which is the locus of the origin of $O'$'s spatial coordinates. This is $O's$'s world line, and it looks like that shown in the figure below. (<em>A First Course In General Relativity</em>, Bernard Schutz, Second Edition, p. 6)</p>
Since an observer is at the origin of his coordinate system, and $\mathcal{O'}$ is moving with a relative velocity of $v$ in the $x$ direction relative to $O$, I am perfectly fine with the fact that the line which we call $t'$-axis in the diagram is the worldline of $\mathcal{O'}$ in the spacetime diagram drawn by $\mathcal{O}$. However, I do not at all get the point of why we say that this worldline of $\mathcal{O'}$ is the "$t'$-axis" of $\mathcal{O'}$ in the spacetime diagram drawn by $\mathcal{O}$, and why it has to be that way; "tilted". Can someone please explain the reasons to me?
