# Application of the principle of causality to classical electrodynamics

I am reading Griffiths' Introduction to Electrodynamics in which he shows that the retarded and advanced potentials, e.g. the retarded scalar potential

$$V(\mathbf{r},t) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\mathbf{r}',t_r)}{\lvert \mathbf{r} - \mathbf{r}' \rvert} \mathrm{d}\tau', \quad t_r \equiv t - \frac{\lvert \mathbf{r} - \mathbf{r}' \rvert}{c},$$

are solutions to the inhomogeneous wave equation and satisfy the Lorenz condition. He then rejects the advanced potentials by invoking the principle of causality, stating that it is not unreasonable to believe that electromagnetic influences propagate forward and not backward in time.

I can only imagine that the reasonableness of this belief is rooted in our naive experience of the world as time-asymmetric. However, the time asymmetry that we experience on a daily basis are (at least usually) not due to a fundamental asymmetry in the fundamental laws of physics, like Maxwell's equations (insofar as a classical theory can be fundamental, but to my knowledge QED is also time symmetric), but to the emergent second law of thermodynamics. It is not clear to me that we can use our intuition about macroscopic phenomena to reason about microscopic phenomena, and even if we could, the second law is only probabilistic and would not allow us to reject forward propagation of electromagnetic influences outright and declare them impossible, only conclude that they are unlikely.

Furthermore, I assume that experiments have verified the retarded and not the advanced potentials? If so, then why invoke the principle of causation at all? And more importantly, would this not show a time asymmetry in the laws of classical electrodynamics? I suppose the principle could still considered external to the theory (like the homogeneity and isotropy of space), but at least the application to this particular case would have been shown experimentally to be justified.

So what exactly is it that allows us to apply the principle of causality?

• Is this different to physics.stackexchange.com/questions/365023/… ? May 17 '18 at 16:05
• I had initially thought so, but now I'm not so sure. I figured that that question asked specifically about waves, whereas my question was more general, but since all (nice?) solutions to Maxwell's equations in vacuum have to obey the wave equation, perhaps not. Even so, I think the answers given are not entirely satisfactory. The one by higgsss is illuminating, but the connection to the principle of causality is not entirely clear. May 17 '18 at 16:38
• Of potential interest are the Wheeler-Feynman time symmetric theory of electromagnetism, in such articles as "Classical electrodynamics in terms of direct interparticle action". They explored the idea and consequences of allowing advanced potentials May 18 '18 at 13:08

I think you are confusing the concept of causality with that of reversibility: the emergent second law of thermodynamics states that certain events can't happen in reverse, while the principle of causality implies that the cause of an event can't happen before that event. Another way of stating it is saying that information can't travel faster than light, i.e. if event A is the cause of event B, A must fall in the light-cone of B in the Minkowsky space. Reversible processes don't violate causality principle; the fact that they are reversible implies that they are still solution of the equations of temporal evolution even under time reversal transformation, not that this solution is dependent on later time. For what concerns the advanced potential, I once read that there are models that include it (e.g. Wheeler-Feynman absorber theory, but I don'tknow much about it) but it should still respect the principle of causality; that's an important check to see if a theory is valid.

• Sure, if A causes B, then A must fall in the past light cone of B. But under time reversal, doesn't A fall in the future light cone of B, violating causality? And that still leaves the question of how to justify the principle of causality in the first place (and my thermodynamics digression was just a sketch of how I imagine people do that). Also, it seems very intuitive that, if some reversible process depends on past time, then the same process, under time reversal, depends on future time, but apparently not. I'm having some trouble understanding that part of your answer. May 17 '18 at 16:53

I do not believe that Maxwell's equations are time symmetric in the following sense: When you restrict yourself to finite size matter and sources then any wave propagation will eventually cause scattering that will involve diffraction around the finite sized bodies. There is no way to reverse these diffracted orders completely with finite size scatterers for they, too, will induce diffraction somewhere, and so on. There will result some "leakage" somewhere that you will not be able to reverse and that minimum leakage related to the geometry of the scatterers/source configuration is a kind of irreversibility in a thermodynamic sense.

Furthermore, I assume that experiments have verified the retarded and not the advanced potentials?

You can't verify the one and not the other. This is because of how they come about.

Take the simplest case: AC voltage on a big straight wire creates a dipole radiation going “out.” Surely, you say, these are the delayed (or “retarded”) waves with the delayed potentials and not the advanced waves with the advanced potentials. And I agree. “QED,” you say, “experiment proves it!”

Now, wait just a second, experiment didn’t prove anything because you did not give both sides a fair shake! This is why I loved the MythBusters so much, they wouldn't stop at “what did our first experiment say?” but usually tried to go on to “so if we wanted to actually make this happen, what would it take?”°

Well, you can regard the outgoing wave as a big sum of plane waves, this is the Fourier transform. And a similar sum of plane waves with the opposite phase could be added to it in order to perfectly cancel out the outgoing wave. But to not have a source in space, you ultimately need these waves to come in from -∞, and you realize that we're talking about a boundary condition. I needed causality to impose the boundary condition that before the antenna was oscillating, like way before at t=-∞, because the Maxwell equations do not require that we start from a zero field, but that is what we usually assume that we did. I do we assume this? You could say causality as well