Retarded potentials

Retarded potentials are general forms of the electric potential and magnetic vector potential in terms of the retarded time

V(r,t)=14πε0∫ρ(r′,tr)r dτ′,A(r,t)=μ04π∫J(r′,tr)r dτ′V(\mathbf{r},t)=\frac{1}{4\pi \varepsilon_{0}}\int \frac{\rho(\mathbf{r}',t_{r})}{\mathfrak{r}} \ d\tau',\qquad \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int \frac{\mathbf{J}(\mathbf{r}',t_{r})}{\mathfrak{r}} \ d\tau'

These are necessary when the electric charge and electric current distributions change in time in order to preserve causality.

Derivation

In the Lorenz gauge, Maxwell's equations for potential read

□2V=−ρε0,□2A=−μ0J(1)\square ^{2}V=- \frac{\rho}{\varepsilon_{0}},\qquad\square ^{2}\mathbf{A}=-\mu_{0}\mathbf{J}\tag{1}

These reduce to

∇2V=−ρε0,∇2A=−μ0J\nabla^{2}V=- \frac{\rho}{\varepsilon_{0}},\qquad\nabla^{2}\mathbf{A}=-\mu_{0}\mathbf{J}

in the static case, where the solutions are

V(r)=14πε0∫Vρ(r′)r dτ′,A(r)=μ04π∫VJ(r′)r dτ′V(\mathbf{r})=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}} \frac{\rho(\mathbf{r}')}{\mathfrak{r}} \ d\tau',\qquad \mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int_{\mathcal{V}} \frac{\mathbf{J}(\mathbf{r}')}{\mathfrak{r}} \ d\tau'

In the dynamic case, we must consider the change in time, which is to say the propagation of the fields. This occurs at the speed of light cc, which means that the potential that an object feels from an object a distance r\mathfrak{r} away is not due to the potentials right now, but to the potentials when they were emitted at some time in the past. This time is right now minus the time it took for the potential to travel to its destination at light speed, called the retarded time:

tr≡t−rct_{r}\equiv t- \frac{\mathfrak{r}}{c}

The potentials then naturally generalize to

V(r,t)=14πε0∫Vρ(r′,tr)r dτ′,A(r,t)=μ04π∫VJ(r′,tr)r dτ′\boxed{V(\mathbf{r},t)=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}} \frac{\rho(\mathbf{r}',t_{r})}{\mathfrak{r}} \ d\tau',\qquad \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int_{\mathcal{V}} \frac{\mathbf{J}(\mathbf{r}',t_{r})}{\mathfrak{r}} \ d\tau' }

To prove that this heuristic logic is valid, we must check that it satisfies the Maxwell's equations (1)(1), which means that we need the Laplacian of VV and A\mathbf{A} and their second time derivative. Let's start from the Gradient of VV. We need to take derivatives in space. Notice how VV depends on space not just in r=∣r′−r∣\mathfrak{r}=\lvert \mathbf{r}'-\mathbf{r} \rvert at the denominator, but also implicitly due to tr∝rt_{r}\propto \mathfrak{r} inside of ρ\rho. As such

∇V=14πε0∫V[∇ρr+ρ∇(1r)]dτ′\nabla V=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\left[ \frac{\nabla \rho}{\mathfrak{r}} + \rho \nabla\left( \frac{1}{\mathfrak{r}} \right) \right]d\tau'

Separately, we get

∇ρ=ρ˙∇tr=−1cρ˙∇r=−1cρ˙r^\nabla \rho=\dot{\rho}\nabla t_{r}=- \frac{1}{c}\dot{\rho}\nabla \mathfrak{r}=- \frac{1}{c}\dot{\rho}\hat{\boldsymbol{\mathfrak{r}}}

since ∇r=r^\nabla \mathfrak{r}=\hat{\boldsymbol{\mathfrak{r}}} and using dot notation for time derivatives. Now, we know from electrostatics that ∇(1/r)=−r^/r2\nabla(1/\mathfrak{r})=-\hat{\boldsymbol{\mathfrak{r}}}/\mathfrak{r}^{2}, so

∇V=14πε0∫V[−ρ˙cr^r−ρr^r2]dτ′\nabla V=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\left[ - \frac{\dot{\rho}}{c} \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}}-\rho \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}} \right]d\tau'

Take the Divergence to obtain the Laplacian, recalling that the divergence of a product between a Scalar aa and a Vector field F\mathbf{F} is ∇⋅(aF)=a(∇⋅F)+F⋅(∇a)\nabla\cdot(a\mathbf{F})=a(\nabla\cdot \mathbf{F})+\mathbf{F}\cdot (\nabla a):

∇2V=14πε0∫V[−ρ˙c∇⋅(r^r)−1c(∇ρ˙)⋅r^r]−[ρ∇⋅(r^r2)+(∇ρ)⋅r^r2]dτ′\nabla ^{2}V=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\left[-\frac{\dot{\rho}}{c}\nabla\cdot\left( \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}} \right)-\frac{1}{c}(\nabla \dot{\rho})\cdot\frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}}\right]- \left[\rho \nabla\cdot\left( \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}} \right)+(\nabla \rho)\cdot \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}} \right]d\tau'

Notice that

∇ρ˙=−1cρ¨∇r=−1cρ¨r^,∇(r^r2)=4πδ3(r)\nabla \dot{\rho}=- \frac{1}{c}\ddot{\rho}\nabla \mathfrak{r}=- \frac{1}{c}\ddot{\rho}\hat{\boldsymbol{\mathfrak{r}}},\qquad \nabla\left( \frac{\hat{\boldsymbol{\mathfrak{r}}}}{\mathfrak{r}^{2}} \right)=4\pi \delta^{3}(\boldsymbol{\mathfrak{r}})

and we already know the other two gradients, so

∇2V=14πε0∫V[1c2ρ¨r−4πρδ3(r)]dτ′=1c2∂2V∂t2−1ε0ρ(r,t)\nabla ^{2}V=\frac{1}{4\pi \varepsilon_{0}}\int_{\mathcal{V}}\left[ \frac{1}{c^{2}}\frac{\ddot{\rho}}{\mathfrak{r}}-4\pi \rho \delta^{3}(\mathfrak{\mathbf{r}}) \right]d\tau'=\frac{1}{c^{2}}\frac{ \partial^{2} V }{ \partial t^{2} }- \frac{1}{\varepsilon_{0}}\rho(\mathbf{r},t)

or

□2V=−ρ(r,t)ε0\square ^{2}V=- \frac{\rho(\mathbf{r},t)}{\varepsilon_{0}}

which confirms that the retarded electric potential does in fact satisfy the potential wave equation. An analogous process also proves that the retarded vector potential satisfies (1)(1).

You can rerun this method again to prove that it also applies just as well to the advanced time ta≡t+r/ct_{a}\equiv t+ \mathfrak{r}/c and the associated advanced potentials (same equations, just with tat_{a} instead of trt_{r}). This is a curious and rather absurd result; think about it for a moment. The retarded time is necessary to keep the speed of light into account. A signal, even one moving at the speed of light, takes time to arrive to destination. This extra time is r/c\mathfrak{r}/c of course, which we remove to go back into the past when the signal was created... but if we add it, we go forward into the future when... the signal will be created? Which implies that technically the potential we feel now may also be due to a signal that has not even been created yet. This is obvious witchcraft and it flagrantly violates causality, the principle that cause must precede effect, but it's not the math that proves it wrong, we can only say that is wrong because the theory has no problem allowing it. This is a property known as time-reversal symmetry, the fact that a theory does not change if you invert the time coordinate (you set tt to −t-t). Time-reversal asymmetry is introduce into electrodynamic theory right here, completely artificially, by arbitrarily choosing that advanced potentials are invalid choices. The choice is driven not by mathematical proof, but by overwhelming empirical evidence in favor of it.