Retarded potential

In the Coulomb gauge

In the Coulomb gauge, Maxwell's equations are

${\displaystyle {\mathrm{\nabla }}^2\phi =-{\displaystyle \frac{\rho }{{ϵ}_0}}}$

${\displaystyle {\mathrm{\nabla }}^2\mathbf{A}-{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{{\mathrm{\partial }}^2\mathbf{A}}{\mathrm{\partial }{t}^2}}=-{\mu }_0\mathbf{J}+{\displaystyle \frac{1}{c^2}}\mathrm{\nabla }\left({\displaystyle \frac{\mathrm{\partial }\phi }{\mathrm{\partial }t}}\right),}$

although the solutions contrast the above, since A is a retarded potential yet φ changes instantly, given by:

${\displaystyle \phi (\mathbf{r},t)={\displaystyle \frac{1}{4\pi {ϵ}_0}}\int {\displaystyle \frac{\rho ({\mathbf{r}}^{\prime },t)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}{\mathrm{d}}^3{\mathbf{r}}^{\prime }}$

${\displaystyle \mathbf{A}(\mathbf{r},t)={\displaystyle \frac{1}{4\pi {\epsilon }_0}}\mathrm{\nabla }\times \int {\mathrm{d}}^3{\mathbf{r}}^{\prime }{\int }_0^{|\mathbf{r}-{\mathbf{r}}^{\prime }|/c}\mathrm{d}{t}_r{\displaystyle \frac{t_r\mathbf{J}({\mathbf{r}}^{\prime },t-t_r)}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}}\times (\mathbf{r}-{\mathbf{r}}^{\prime }).}$

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:

${\displaystyle \phi (\mathbf{r},t)={\displaystyle \frac{1}{4\pi }}\int {\displaystyle \frac{\mathrm{\nabla }\cdot \mathbf{E}({\mathbf{r}}^{\prime },t)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}{\mathrm{d}}^3{\mathbf{r}}^{\prime }}$

${\displaystyle \mathbf{A}(\mathbf{r},t)={\displaystyle \frac{1}{4\pi }}\int {\displaystyle \frac{\mathrm{\nabla }\times \mathbf{B}({\mathbf{r}}^{\prime },t)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}}{\mathrm{d}}^3{\mathbf{r}}^{\prime }}$

In linearized gravity

The retarded potential in linearized general relativity is closely analogous to the electromagnetic case. The trace-reversed tensor ${\tilde{h}}_{\mu \nu }=h_{\mu \nu }-\frac{1}{2}{\eta }_{\mu \nu }h$ plays the role of the four-vector potential, the harmonic gauge ${\displaystyle {\tilde{h}}^{\mu \nu }{}_{,\mu }=0}$ replaces the electromagnetic Lorenz gauge, the field equations are ${\displaystyle ◻{\tilde{h}}_{\mu \nu }=-16\pi G{T}_{\mu \nu }}$, and the retarded-wave solution is $${\displaystyle {\tilde{h}}_{\mu \nu }(\mathbf{r},t)=4G\int \frac{T_{\mu \nu }({\mathbf{r}}^{\prime },t_r)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\mathrm{d}}^3{\mathbf{r}}^{\prime }.}$$ Using SI units, the expression must be divided by ${\displaystyle c^4}$, as can be confirmed by dimensional analysis.

Occurrence and application

A many-body theory which includes an average of retarded and advanced Liénard–Wiechert potentials is the Wheeler–Feynman absorber theory also known as the Wheeler–Feynman time-symmetric theory.

In gravitation, there are application examples for calculating deviations in orbits of satellites, moons or planets. The anomalies in the rotation curves of more than one hundred spiral galaxies 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.

In 2026 the German physicist Andreas Malcherek published about a new theory of gravitoelectromagnetism that builds on the basic work of Oliver Heaviside und Oleg D. Jefimenko. The new theory applies retarded gravitational potentials within Minkowski spacetime and is consistent with all the predictions of Albert Einstein's general theory of relativity, without taking into account the equivalence principle or for curved space.

Example

The potential of charge with uniform speed on a straight line has inversion in a point that is in the recent position. The potential is not changed in the direction of movement.