Retarded time

Retarded and advanced times

Retarded time t_r or t′ is calculated with a "speed-distance-time" calculation for EM (Electro-Magnetic) fields.

If the EM field is radiated at position vector r′ (within the source charge distribution), and an observer at position r measures the EM field at time t, the time delay for the field to travel from the charge distribution to the observer is |r − r′|/c. Subtracting this delay from the observer's time t then gives the time when the field began to propagate, i.e. the retarded time t′.

The retarded time is: ${\displaystyle t^{\prime }=t-\frac{|\mathbf{r}-({\mathbf{r}}^{\prime })|}{c}}$

(which can be rearranged to ${\displaystyle c=|\mathbf{r}-{\mathbf{r}}^{\prime }|/(t-t^{\prime })}$, showing how the positions and times of source and observer are causally linked).

A related concept is the advanced time t_a, which takes the same mathematical form as above, but with a “+” instead of a “−”:

${\displaystyle t_a=t+\frac{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{c}}$

This is the time when the EM field originating at ${\displaystyle \left(\mathbf{r},t\right)}$ reaches a position ${\displaystyle {\mathbf{r}}^{\prime }}$by traveling over a distance ${\displaystyle |\mathbf{r}-{\mathbf{r}}^{\prime }|}$. Corresponding to retarded and advanced times are retarded and advanced potentials.