Retardation for a moving point charge

Let the trajectory be z as a function of source time, and hold the observer at x and observation time t. The retarded event is the one for which emission time plus the light travel time equals observation time. The distance in that equation is from the emission position, not the charge's present position. This distinction persists even during uniform motion. Solving the implicit equation is the first task; differentiating fields or evaluating velocities comes afterward.

Define the unit vector n from the retarded charge position toward the observer, and beta as the source velocity divided by light speed at that same event. Increasing emission time by a small amount moves the source toward the observer by its line-of-sight displacement. The reception interval is therefore shorter when the source approaches. For a timelike trajectory the reception time grows monotonically with emission time, giving a unique intersection when the trajectory and observation event admit a retarded connection.

The point charge density is a three-dimensional spatial delta supported on its worldline. Writing the retarded source integral as a space-time integral lets the spatial delta be used first. What remains is a time delta whose argument depends on the moving distance. The identity for a delta of a function divides by the absolute derivative of its argument at a root. Here that derivative has magnitude kappa, which is positive for subluminal motion.

This calculation produces an inverse kappa factor in both potentials. The vector potential is the scalar potential multiplied by the retarded velocity divided by speed squared. Keeping that SI factor prevents confusion with conventions in which light speed is set to one. All quantities in the denominator and numerator belong to the same retarded event. Combining a present velocity with an emission distance does not approximate the exact formula in a controlled way unless slow changes are independently established.

Forward directions have smaller kappa, so successive source events can produce compressed reception intervals. This geometric fact later helps explain beaming, but the potentials themselves are gauge dependent. To determine observable radiation one must differentiate with the retarded-time chain rule and isolate the transverse inverse-distance fields. Uniformly moving charges have nonzero Liénard–Wiechert potentials while carrying no acceleration radiation to infinity in an inertial frame. Potential enhancement is therefore not a radiation criterion.

The vector lab can test the dot product entering kappa. Treat its horizontal axis as the fixed sightline and its magnitude as beta, and read the horizontal projection. Reversing direction changes the sign of that projection, even though speed is unchanged. This is useful geometry, not a worldline intersection calculation. The lab does not display the moving-source distance or either potential, and it cannot transform the gauge-dependent scalar potential into a direct measure of energy flux.