# Speed of electromagnetic interactions

We know that electromagnetic waves fly with the speed of light, but my question not about waves. Consider a very strong electromagnet that creates a substantial field 3 meters away. Then we send a proton accelerated to near the speed of light to fly by. The proton interacts with the field of the magnet and is deflected.

The field is generated by the electrons in the wire of the electromagnet. The interaction of the proton with the field is a quantum exchange of energy, momentum, etc. between the proton and these electrons. Whether we label these exchanges as "mediating virtual photons" or not is not really important here. My question is about the speed of these exchanges.

What are the current views on the timing of these exchanges? Are they instantaneous or limited by the speed of light? If instantaneous, would they not violate casualty by transferring information and energy faster than light?

If the exchanges happen at the speed of light, a number of problems arise. First, the exchanges must be directional. Say, if an electron in the wire emits "a virtual photon" toward the proton, then the proton would be 3 meters away from that position by the time "the virtual photon" arrives. So the electrons would have to aim at the future position of the proton to hit it. This makes no sense and probably is one of the reasons why the "virtual particle" model is not in favor. Secondly, a virtual photon would have to exist for a nanosecond that would severily limit its energy per the uncertainty principle.

Can someone please clarify the actual physics behind the electromagnetic interactions from the timing standpoint? By physics here I mean physical observables, something we can measure. Using quantum fields of mathematical probabilities if fine as long as they are linked to observable values.

• Re: feel free to replace the electromagnet there with a static charge But charge and magnetic dipole are entirely different things. In Coulomb gauge we have equation $\Delta \phi= - 4\pi \rho$ without any time derivatives, which means instantaneous propagation of $\phi$. But there is no causality violation since charge is a conserved quantity and there would be the same charge at any moment of time (no information is transferred superluminally). Magnetic dipole is not a conserved quantity, you could turn it on and off, so information about it would propagate at a speed of light… Feb 22 '20 at 17:41
• … Incidentally, the no-hair theorems of black hole physics is a good indicator for the speed of propagation: the fields carrying information on mass (grav. monopole), angular momentum (gravimagnetic dipole) and charge propagate instantaneously, so they have no trouble escaping the black hole. Information on magnetic dipole moment however propagates at the speed of light and so black hole has no independent magnetic dipole moment (it does have induced by frame dragging magnetic dipole proportional to charge and spin). This take on no-hair thm., IIRC, belongs to J. Bekenstein. Feb 22 '20 at 18:08
• So the recoil of the static charge is delayed by the speed of light relative to the deflection. And if we change the frame to make the other charge static, the situation reverses per relativity of simultaneity. Got it, thanks! An interesting insight on black holes. I get it perfectly well and it is quite elegant. Intuitively it also should be equivalent outside to all matter and charges located at the horizon. Feb 22 '20 at 18:34
• Here is a relevant paper on the subject: arxiv.org/abs/gr-qc/9909087 . Though the title speaks about “speed of gravity” it does have a section on EM as a warm-up. Feb 22 '20 at 19:25
• @A.V.S. Thanks! I had a question based on this very paper :) Also with no good answer: physics.stackexchange.com/questions/492870/… Feb 22 '20 at 21:23

If the field is initially off, then it has to be turned on at least $d/c$ seconds before the proton arrives, where $d$ is the distance from the electromagnet to the proton's path through the field and $c$ is the speed of light. When the electromagnet is turned on, the field it creates will be established at a point after a delay due to the speed of light.