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Both sides of Maxwell's equations are equal to each other, so each of these equations connects quantities simultaneous in time, and as a consequence none of these equations can represent a causal relation:

\begin{align} \mathbf\nabla\cdot\mathbf{E}(\mathbf{r}, t) &= 4\pi\rho(\mathbf{r}, t) \label{Diff I}\\ \mathbf\nabla\times\mathbf{B}(\mathbf{r}, t) &= \dfrac{4\pi}{c} \mathbf{J}(\mathbf{r}, t)+\dfrac{1}{c}\dfrac{\partial\mathbf{E}(\mathbf{r}, t)}{\partial t} \label{Diff IV}\\ \mathbf\nabla\times\mathbf{E}(\mathbf{r}, t) &= -\dfrac{1}{c}\dfrac{\partial\mathbf{B}(\mathbf{r}, t)}{\partial t} \label{Diff III}\\ \mathbf\nabla\cdot\mathbf{B}(\mathbf{r}, t) &= 0 \label{Diff II} \end{align}

But the solutions of this equation (Jefimenko's equations) reflects the "causality", because of right hand sides involve "retarded" time:

\begin{equation} \mathbf{E}(\mathbf{r}, t) = \int \left[\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3}\rho(\mathbf{r}', t_r) + \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^2}\frac{1}{c}\frac{\partial \rho(\mathbf{r}', t_r)}{\partial t} - \frac{1}{|\mathbf{r}-\mathbf{r}'|}\frac{1}{c^2}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \mathrm{d}^3 \mathbf{r}', \end{equation}

\begin{equation} \mathbf{B}(\mathbf{r}, t) = -\frac{1}{с} \int \left[\frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^3} \times \mathbf{J}(\mathbf{r}', t_r) + \frac{\mathbf{r}-\mathbf{r}'}{|\mathbf{r}-\mathbf{r}'|^2} \times \frac{1}{c}\frac{\partial \mathbf{J}(\mathbf{r}', t_r)}{\partial t} \right] \mathrm{d}^3 \mathbf{r}', \end{equation}

where $\mathbf{r}'$ is a point in the charge distribution, $\mathbf{r}$ is a point in space, and $t_r = t - \frac{|\mathbf{r}-\mathbf{r}'|}{c}$ is the retarded time.

The question can be purely technical, how does the “retardation” arise in the solutions, if it was not in the original equations?

Sergio
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  • consider this: the "non-causal" version of Maxwell equations came about from the head of Maxwell, but not necessarily there is a one-to-one mapping between their solutions and observable phenomena. Jefimenko formulation is more strict in the sense that only selects the "experimentally observed" part of Maxwell solutions. Jefimenko specifically removes advanced potentials from the solutions – lurscher Jun 27 '21 at 23:44
  • Related, but for a scalar field (still Lorentz-symmetric): How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality? – Chiral Anomaly Jun 28 '21 at 00:40
  • The derivation of Jefimenko from Maxwell is concisely reviewed in this answer, even though the question is different. Equation (2) (and the preceding equation) in that derivation shows how retardation arises... but is that what you mean by "arises"? Or are you asking more fundamentally why Maxwell's equations necessarily respect causality, like the question linked in my previous comment? – Chiral Anomaly Jun 28 '21 at 01:00

1 Answers1

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You are correct that Maxwell's equations do not impose causality. In the Lorenz gauge, Maxwell's equations in terms of $A^\alpha$ is a wave equation $$\Box^2 A^\alpha = - \mu_0 J^\alpha$$

The solution to this can be found using the Green's function of the operator $\Box^2$. There are two possible Green's functions: The retarded one $$G_1 = -\frac{\delta\left(t-t' - \left|\mathbf{r}-\mathbf{r}'\right|/c\right)}{4\pi \left|\mathbf{r}-\mathbf{r}'\right|}$$ and the advanced one $$G_2 = -\frac{\delta\left(t-t' + \left|\mathbf{r}-\mathbf{r}'\right|/c\right)}{4\pi \left|\mathbf{r}-\mathbf{r}'\right|}$$

These two solutions are purely mathematical and can be derived using Fourier transforms. No physics has been involved yet. Therefore, in principle, $A^\alpha$ can be written as a linear combination of both solutions: $$A^\alpha = \frac{\mu_0}{4\pi} \int \left(aG_1 + bG_2\right) J^\alpha (\mathbf{r}',t')\text{d}^3{\mathbf{r}'} \text{d}t'$$

It is when we apply this to physical situations that $G_2$ is discarded because it violates causality.

Interestingly, in the 1940's, Feynman and Wheeler pointed out that there are certain physical situations that could involve both $G_1$ and $G_2$, known as Wheeler-Feynman absorber theory. It was originally proposed to explain the self-force of an accelerating charge. They propose that emitters have $a=b=1/2$ while absorbers have $a = 1/2$ and $b= -1/2$, such that the combined reaction on a test charge is purely retarded. A simple discussion can be found here.

Vincent Thacker
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