Retarded time

The retarded time is the time at which a signal was produced, as opposed to the time at which it is received. For an electromagnetic wave, which travels at the speed of light, it is

t_{r}\equiv t- \frac{\mathfrak{r}}{c}

where $t$ is the time of arrival and $\mathfrak{r}$ is the distance between the source and the receiver, under the assumption that it travels in a straight line.

It is also possibile to define the reverse, the advanced time, which only has a different sign:

t_{a}\equiv t+ \frac{\mathfrak{r}}{c}

This, however, is of little use in physics as it is fundamentally incompatible with the world we live for it breaks causality, the principle by which cause must precede effect. Where the advanced time to be used for anything (such as electrodynamic potentials), we'd have to accept the idea that what happens now is dependent not on what was, but on what will be¹. It makes for a wonderful concept in sci-fi (and do play Outer Wilds!), but scarcely so in physics.

For what it's worth, advanced time is actually perfectly valid in electrodynamics, since the d'Alembertian (and by extension the wave equation) is dependent on $t^{2}$ and is therefore time-reversal invariant. The choice not to allow it is entirely artificial and based off the fact that no such violation of causality has ever been encountered. For all purposes, electromagnetic signal propagate only forward in time, never backwards. ↩

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