Is a finite lightspeed necessary for cause and effect to exist?

Question

In this question I asked in somewhat unclear (unclear enough to be closed) fashion about the relationship between an infinite speed of light and the existence of cause and effect.

In thís question, I like to clarify it and make it more articulate.

Consider a non-relativistic, non-quantum mechanical Newtonian Universe, in which the speed of light is infinite. Now the e.m. field is the vehicle by which a cause is transmitted to create an effect somewhere else.

If the speed of this vehicle, the speed of light, becomes infinite then what happens to cause and effect? All particles affect each other at the same absolute time in an absolute space in this Universe (this is, of course not our real Universe, but it just serves to investigate the Nature of cause and effect; besides, in this question a similar kind of Universe is used).

Now when all particles affect each other simultaneously this means they will also affect themselves simultaneously and these in return backfire simultaneously to all other particles which again simultaneously backfire to all particles etc. So a particle, if it moves, has an instantaneously effect on its own movement by the instantaneous backfiring of all other particles, and this holds for all particles. Aren't the particles "frozen" somehow (this is an intuitive feeling on which I can't put my finger exactly)? Can the particles move at all if this is the case and cause and effect can't be separated? If not then a finite speed of light is necessary.

This question differs from the possible duplicate because I ask the question of what will happen in this Universe (which is non-quantum mechanical, contrary to the possible duplicate) when cause and effect are non-separable and all particles interact simultaneously. What effect will this have on the movements (if they are able to move at all) on the particles?

Let's boil down the problem to two charged elementary particles present in this imaginary Universe. What will happen? It seems to me that because of the non-separability of cause and effect, a problem arises because the particles don't "know" what to do. Both particles can't "decide" if they are the cause or the effect, ending up in no movement at all.

So the crux is not being there a distinction between cause and effect.

Answer 1

It would certainly be a different universe, but I don't see that it would create a contradiction that would prevent it from being logically consistent.

The vector sum of all the charges in the universe acting on one charge, would be the force on that charge. It's that way now except that they're trailing off in time. Since they would be instant, if you knew the exact state of everything at one time then you would know as much as could be known, while to do that in our world you'd have to know the entire history of the universe to get that far.

Acceleration would work the same. a=F/m Even though the force is instant, it still takes time for things to move. So the parts of Maxwell's equations that depend on position and velocity would be the same.

The parts that depend on acceleration would be different. Because with everything instantaneously accelerating everything else, the numbers would come out different. One charge's acceleration instantly affects the amount and direction of force it puts onto everything else. If their acceleration is different that instantly affects the amount and direction of force they put on the first charge.

That would make it harder for us to compute all that. It probably doesn't mean things blow up in an infinite feedback. Nature would find its balance. At each instant every charge would be getting the force from everything else that gets it to contribute precisely the right force onto everything else. It would just be tedious for us to do the math.

Answer 2

An infinite speed of light would mean that the universe would be somewhat similar to a rigid body in classical mechanics.

This would not necessarily prevent any dynamic from happening: you could for instance have separate components free of moving with some cause-effect relation. For instance if such a universe was made of spherical disconnected balls (like infinite-light-speed black holes) these could even bounce when coming in contact with each other (they could also merge, I would say). Such objects will not radiate light and will not communicate at infinite speed though. It is difficult to say if the "rigid" components allow for any dynamics.

Forgetting now about the possibility of having horizons (i.e. disconnected "rigid" components), since a past light cone covers half of the (connected) spacetime you have that the set of events that influence any event has space measure equal to t of the whole component (for instance the observable universe, which could not be infinite..). So basically everything that happens at a certain instant influences everything that happens at another instant in a fixed frame. This could still allow for some kind of physics, as we still would be able to receive information from all the points in the past light cone of any event we observe.

Note that however such physics would not even be the same for different observers since having infinite propagation speed breaks the requirement of covariance under a change of frame (see this nice article https://aapt.scitation.org/doi/10.1119/1.10490 ).

By breaking the possibility of changing reference frame without altering the physics, you have that past, present and future would still be defined even if particles propagate at infinite speed. This would look like a non local theory in each of the "rigid" components. Let me stress that you cannot even be sure that the laws of nature seen from two different points would be the same.

I hope this gives you some thought-material.

P.S. It is difficult to say if the different components can really exist. You would need to recreate something that looks like a black hole but you are in a theory in which general relativity is heavily violated (you don't even have lorentz covariance..). Still the changes of frames (and coordinates) should be realized in some (bad baad) way, so one could even be able to deal with those..