# If negative mass should exist, how would conservation of momentum work?

Imagine that a particle with negative mass has been discovered. It is known to obey both the equivalence principle and Newton's second law. This means that the particle is a subject to repulsive gravitational force from ordinary matter but also accelerates towards any repulsive force (resulting in the infamous runaway motion).

Let's also say that the particle is able to interact with ordinary matter in the process of "nullification" (particless of oposite energy get close enough to each other and disappear). The problem is that this interaction would violate the conservation of momentum every time the two nullifying particles didn't have exactly identical velocities, which is practically never the case. And from what I understand, conservation of momentum is pretty important to all of our current theories.

My question is: Is there a mechanism that would allow for nullification but also leave the conservation of momentum unviolated and wouldn't break GR and QM? (Energy conditions may be broken.)

Originally I wanted to post this question to Worldbuilding but I decided it's too technical and generally better suited for this site, even if it's quite hypothetical.

Edit:

Nullification (as it is defined in the last paragraph of Wikipedia's Runaway motion) is the interaction where particless of equal mass of opposite signs “cancel each other out”, completely disappearing from existence.

However, at least in classic physics, the probability that two particles with exactly opposite energy and momentum will find each other in finite time is zero. Therefore nullification would either never happen, or it would break some of the fundamental symmetries.

The answer should either show that this premise is wrong (eg. “in QM the uncertainty of momentum is large enough to have the particles match exactly with a reasonable probability”), propose an interaction that can cause the energy and momentum of the two particles to align by interactions with the outside world, or show that no such interaction is possible and therefore nullification cannot happen.

• The conservation of momentum is not violated. Take two equal but opposite masses at rest and calculate the total momentum after they accelerate for some time: they would have equal but opposite momenta which would result in the same total momentum, i.e., zero. Of course, one of them would have opposite direction of velocity with respect to its momentum. – Oktay Doğangün Jan 4 at 22:56
• More on negative mass. – Qmechanic Jan 4 at 23:26
• @OktayDoğangün Momentum is conserved if you only consider the two particles, however if you add a third particle, say with a positive mass that hits the negative mass particle and they nullify a problem emerges. For simplicity, imagine the 3rd particle had extremely high velocity so it hits the negative mass particle before the forces have enough time to significantly change the velocities. In the reference frame of the 2 initial particles the total p in the beginning is only the p of the 3rd particle, after the nullification it is only the p of the original positive mass particle. – K. Kirilov Jan 5 at 1:35
• @OktayDoğangün The particles would have to have the same direction of velocity, not opposite. $0 = p = p_1 + p_2 = v_1 m + v_2 (-m) \implies v_1 = v_2$. This is exactly what I say in the question. My problem with this is that you never find particles with the exact same velocity (at least in classical physics and GR, I don't know about QM) and therefore nullification would either never occur, or would violate the conservation of momentum. – m93a Jan 5 at 6:26
• @OktayDoğangün You should post answers as answers, not as comments. – Aaron Stevens Jan 5 at 8:17