How does Modified Newtonian Dynamics (MOND) violate Newton's laws?

Question

I have read some articles on MOND which indicate that a modification of gravity violate Newton's third law, but I don't know how exactly that works. I do know that if Newtonian acceleration is smaller than a critical acceleration $a_0$ the particle undergoes MOND acceleration but can't link it to the violation of Newton's third law. Any explanation is appreciated.

Answer 1

Let's look at F = ma. This is not true for each force individually, for sure - each particle experiences many forces but only one acceleration. It's really Σ(all forces)F = ma. Even if we say just the force of gravity, we're looking at Σ(other objects)Fg + Σ(other forces)F = ma.

So let's think about applying MOND. We have a particle that's being accelerated very weakly, a < a0, at time T0. Let's even allow that ΣFg/m < a0 and set aside the issue of what's going on with any other forces.

So you begin picking up this extra MOND force, that makes 'gravity act stronger'. In order to balance forces, the other particles this object is attracted to must experience an equal and opposite force. But which ones? Suppose MOND acts by scaling them ALL up and making this object just be pulled harder by gravity, period. Fair enough, I suppose.

Now, if a < a0, most of the gravity the object feels is going to be its self-gravitation (suppose the object is a rogue planet), so it's going to feel heavier to itself. Hrm. Um. That doesn't seem right. So let's suppose there's some long-range exclusionary thing going on that keeps it from kicking in with short-ranged gravitational forces. Now we've already complicated the gravitational field from 'hey, a field' to 'some horrible mess'. But it gets worse.

In order to maintain energy+momentum conservation, we have to get the extra energy+momentum that this object has just picked up from somewhere. The only candidate is the gravitational field, since the other particles aren't right there to negotiate how much MOND boost they should give this particle. Forces take time to propagate. In order for the other particles to know how much force to experience, they would all have to know if this particle was going to need a MOND boost or not. Before it did. That's garbage.

So we need to go through the field. Now we have to have the gravitational field carrying lots and lots of momentum in ways it never has before. It's not at all clear where this momentum would go, how it would ever interact with anything else, etc.

There IS a theory that handles this, called Tensor-Vector-Scalar gravity (TeVeS). Basic MOND doesn't do this. It just sets an imbalanced force law and ignores the issue.

TeVeS has its own issues, but momentum conservation and violating Newton's 3rd law isn't one of them.