# What is the logical status of locality in classical physics?

By "classical" I mean not quantum, so classical mechanics, electrodynamics and relativity. Perhaps a different label is more appropriate.

First of all, on Wikipedia it is claimed that "Local realism is a feature of classical mechanics, and of classical electrodynamics", so I assume also of SR. However, I can't quite square that with my understanding of Newtonian mechanics. As far as I know, Newton postulated five things: Absolute time, absolute space and his three laws of motion (very roughly put). Add the gravitational law, and you get accelerations that are caused by forces acting instantaneously at any distance. Is this wrong? If not, then how can Newtonian mechanics be local? I might also be misunderstanding the principle of locality or local realism.

Secondly, in classical electrodynamics. For instance, Griffiths claims that "electromagnetic 'news' travels at the speed of light" (Introduction to Electrodynamics, 10.2.1). First of all, this is consistent with locality, since there is a finite speed at which the information is transferred. But it seems a stronger claim than that, since the news are claimed to travel at a particular speed.

I am not clear on the exact postulates of classical electrodynamics (apart from Maxwell's equations and the Lorentz force law), so I have some trouble seeing exactly what the justification for this claim is. Griffiths has previously shown that electromagnetic waves propagate with the speed of light (at least after taking into account the experimental discovery that $\mu_0 \epsilon_0 = 1/c^2$), and I can sort of see how that might be relevant when discussing retarded potentials. But lacking a rigorous definition of "news" or "information", it is not clear to me if this is enough to conclude that information also propagates at this speed.

Griffiths even considers advanced potentials, but claims that they are non-physical since they violate the principle of causality. But as usual he is very vague and calls it the "most sacred tenet" of physics. Do we then, in classical electrodynamics, postulate time-asymmetry? Stepping outside of this theory for a moment, I was under the impression that the asymmetry of time (at least up to a CP transformation, which ) is a purely macroscopic phenomenon due to the second law of thermodynamics, which itself results from the boundary conditions of the universe, i.e. a very low entropy at the Big Bang. It makes sense then that, inside a supposedly self-contained theory such as classical ED, we would take it as a postulate, since we cannot appeal to theoretical results outside of the theory. I find this satisfactory enough, but I would like it to be stated explicitly. (Also, in the light of the above, Laplace's ideas of conservation of information, and in particular Hume's critique of causation, I find it a bit unsatisfactory to brush all discussion aside and simply take the principle as self-evident. There might be something I'm missing.)

Thirdly, special relativity. I find relativistic dynamics very confusing, but to my knowledge relativistic kinematics only contain Einstein's two postulates. And the dynamical theory of SR doesn't seem to postulate much else, only define relativistic generalisations of classical observables such as energy and momentum which reduce to their classical counterparts in the non-relativistic limit. It takes on further experimental results, such as the conservation of four-momentum, but I don't know that that is relevant. I might be wrong, of course. Thus it is again unclear to me exactly what the status of locality is in SR, and why we should assume that the speed of light puts an upper limit on the speed at which we can transfer information.

Finally, since I assume that this is a topic much too large to be explicated here, any literature recommendations would be appreciated as well.

• The same wikipedia article that you referenced also says that "In the 17th century Newton's law of universal gravitation was formulated in terms of 'action at a distance', thereby violating the principle of locality." – David Hammen Apr 28 '18 at 16:32
• Right, I missed that (obviously). Then surely the claim that I cited above is false? – Danny Hansen Apr 28 '18 at 18:03

Locality is the inability of a part of a physical system to influence another part of the system, which is far away. The rejection of locality would imply to accept the instantaneous action-at-distance, which is denied by SR (special relativity) setting an upper speed limit. So far SR has always been confirmed by experimental evidences.
Note: As for entanglement in quantum physics, it is a probabilistic relation, not influence. However I know that there are different interpretations.

• But does SR actually set an upper speed limit? As in, do Einstein's postulates imply that there exists a speed such that no particle can travel at higher speeds? To my knowledge, while tachyons have never been observed, they are consistent with SR, and so presumably one needs more than Einstein's postulates to make such a claim. I suppose they would violate the principle of causality, but as I said above, that's a further claim which I would just like someone to explicitly postulate. – Danny Hansen Apr 28 '18 at 18:12
• Another thing: Even if SR sets an upper speed limit on particles, how does that translate into an upper speed limit for electromagnetic waves? And I mean without quantising the EM-field, purely within classical field theory. – Danny Hansen Apr 28 '18 at 18:21
• SR formally sets the speed of light as an upper speed limit for bradyons (massive particles) and as a lower speed limit for tachyons (hypothetical imaginary mass particles). However, according to SR, a faster-than-light particle could be used to communicate backwards in time, thus violating causality, a fundamental principle in physics. Moreover, so far, no experimental evidence of tachyons exists. The SR speed limit was set to the speed of light as a consequence of observations. Light travels at a constant speed independently of the inertial reference frame. – Michele Grosso Apr 29 '18 at 16:04
• I think that makes sense. If I understand you correctly, we take the principle of causality as a fundamental postulate and reject theories that break it. So that establishes why, within the theory, information cannot be transferred faster than the speed of light, but why, as Griffiths writes, does electromagnetic "news" travel at exactly that speed? – Danny Hansen Apr 29 '18 at 17:58
• The first principle of SR, that is the equivalence of any inertial reference frame, allows to set a speed limit, but not its magnitude. Experimentally light fits the meaning of that limit, being the speed of light constant in any frame. That is why of the second principle. I am not aware of other explanations. Could be just regarded as an accident of nature. – Michele Grosso Apr 30 '18 at 16:01