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I don't quite get the following: Every year I shoot a laser beam into space, say into the x-direction of my frame of reference. Then the world-lines of the photons are specified by $x_{\mathrm{Photons}}=ct-n$, where $n=1,2,3,...$ represents the year.

For me this looks like all the beams start in the past. Intuitively I would have said it should be $x_{\mathrm{Photons}}=ct+n$. Where is my error in reasoning?

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Your diagram looks like:

Spacetime diagram

The worldlines of the light have the equation:

$$ x = ct - ncT $$

where $T$ is the time you wait before launching each beam (1 year) and $n$ has the values $0$, $1$, $2$, etc. However it's not obvious why this implies the beams should have started in the past. The equation for the lines does continue to negative values of $x$, but that continuation doesn't have any physical meaning because you created the light beam at $x=0$.

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