Retarded potential

In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past.

The fields propagate at the speed of light c, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution (the point of cause) to another point in space (where the effect is measured), see figure below.

[1] The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and

[2] Solving these gives the retarded potentials below (all in SI units).

From φ(r, t) and A(r, t), the fields E(r, t) and B(r, t) can be calculated using the definitions of the potentials: and this leads to Jefimenko's equations.

operators of the fields are zero, and Maxwell's equations reduce to where ∇2 is the Laplacian, which take the form of Poisson's equation in four components (one for φ and three for A), and the solutions are: These also follow directly from the retarded potentials.

In the Coulomb gauge, Maxwell's equations are[5] although the solutions contrast the above, since A is a retarded potential yet φ changes instantly, given by: This presents an advantage and a disadvantage of the Coulomb gauge - φ is easily calculable from the charge distribution ρ but A is not so easily calculable from the current distribution j.

However, provided we require that the potentials vanish at infinity, they can be expressed neatly in terms of fields: The retarded potential in linearized general relativity is closely analogous to the electromagnetic case.

plays the role of the four-vector potential, the harmonic gauge

replaces the electromagnetic Lorenz gauge, the field equations are

In gravitation, there are application examples for calculating deviations in orbits of satellites[7], moons[8] or planets.

[9] The anomalies in the rotation curves of more than one hundred spiral galaxys of different types could also be explained.

The data of the “SPARC (Spitzer Photometry and Accurate Rotation Curves) Galaxy collection”, which were recorded with the Spitzer Space Telescope, were used for this purpose.

In this way, neither the assumption of dark matter nor a modification of general relativity is required to explain the observations.

[10] On even larger scales, the retarded gravitational potentials result in effects such as an accelerated expansion, which leads to an isotropic, but not homogeneous universe with an outer shell of dark matter with an increased mass density as well as a strong gravitational redshift of distant astronomical objects.

[11] The potential of charge with uniform speed on a straight line has inversion in a point that is in the recent position.