Suppose a 2d space-time and two frames of reference moving with velocity $v$ relative to each other. The Lorentz transformation is:
$$ t' = \gamma\left(t - \frac{v}{c^2}x\right) $$ $$ x' = \gamma\left(x -vt\right) $$
However this seems to apply only if the origins "coincide" or "are synchronized". Citing Wikipedia for example:
At $t = t′ = 0$, the origins of both coordinate systems are the same, $(x, y, z) = (x′, y′, z′) = (0, 0, 0)$. In other words, the times and positions are coincident at this event.
So what if they are not? Suppose Alice $(t', x')$ is moving relative to Bob $(t, x)$ and Bob measures at $t = 0$ that Alice is at $x = x_0$. How do I incorporate this into the above Lorentz transformation?
Is it correct to assume that the origins will "coincide" at $t = -x_0 / v$ (as Bob will then measure Alice to be at $x = 0$) and to adjust all times by this value in order to compute $(t', x')$? That is using
$$ t \rightarrow t + \frac{x_0}{v} $$
for the Lorentz transformation?