Retarded time

Retarded time is the earlier instant t_r at which electromagnetic information must have left a moving source position r_s(t_r) in order to reach an observer at position r at the present time t. Because electromagnetic signals propagate at the finite speed c, "now here" is determined by "then there" across a light-travel delay |r − r_s(t_r)|/c. The defining equation t_r = t − |r − r_s(t_r)|/c is implicit: the retarded source position depends on the retarded time, which in turn depends on the source's trajectory. For a stationary source the equation collapses trivially to t_r = t − r/c; for a moving source it requires algebraic or numerical inversion.

Retarded time appears in three canonical places. First, the Liénard-Wiechert potentials V(r, t) = (q/4πε₀) · 1/[κ·R]_r and A(r, t) = (q v/4πε₀c²) · 1/[κ·R]_r are evaluated at t_r, with R the retarded source-observer separation and κ = 1 − n̂·v/c the Doppler-like geometric factor. Second, the radiation field of an oscillating dipole carries the argument cos(ω(t − r/c)) rather than cos(ωt), so the "now" field at distance r is set by "then" behaviour of the dipole at time t − r/c. Third, in radiation-reaction problems (§10.6) the self-force of a charge on itself is computed by integrating the retarded field of every element of the charge over every other element, with each pair picking up its own retarded-time shift. The causal structure of classical electromagnetism — effects never precede their causes — is built into retarded time, and the "advanced" solutions (t_a = t + r/c, which formally satisfy Maxwell's equations but violate causality) are discarded by fiat except in certain time-reversal-invariance thought experiments.