As far as I understand, if it exists, it must be far away from the "positive" matter because of repelling force, so it explains why there is no observations of such matter.

For one, negative mass would still be attracted to positive mass, but the positive mass would be repelled. This would lead to the negative mass "following" the positive mass.

Why is this? The force *is* a repulsive one. But we also have the fact that $\vec F=m\vec a$. Since one of the bodies has negative mass, it will be *attracted*.

If there was a body of negative mass less massive (talking about absolute values here) than our positive massed-universe, it would move much faster and eventually "catch up" with ours. If it had the same absolute value of mass, both would keep accelerating and it would never catch up. If it was larger, it would be left behind eventually.

The lack of any large quantities of negative mass in our universe excludes the case of a body of negative mass having caught up with us. The lack of any acceleration of the universe signifies that there isn't any large body of negative mass with absolute value less than or equal to the mass of the universe. So, if there is any large body of negative mass, it has larger mass than our universe and it is separated from our universe.

"all structures that exist mathematically exist also physically" -Max Tegmark.

This is a part of his MAthematical Universe *hypothesis*. Note that it's just a hypothesis, it isn't backed by much concrete evidence yet.

Despite being completely inconsistent with a common-sense approach and the expected behavior of "normal" matter, negative mass is completely mathematically consistent and introduces no violation of conservation of momentum or energy.

This is true--it lets one create energy out of thin air by introducing a body with negative mass to the system, but this doesn't conflict the principle of conservation of energy as the body has it's own energy become more negative.

Such matter would violate one or more energy conditions and show some strange properties, stemming from the ambiguity as to whether attraction should refer to force or the oppositely oriented acceleration for negative mass.

I'm not too sure about the energy conditions, I'm not strong in General Relativity. But the second half is just about some strangeness attached to it and some confusion regarding terminology (relating to what I explained near the top of this answer).

Finally, vacuum fluctuations (which exist) can have a net negative energy density, so these have "negative mass", in a way. But these aren't easy to harness with current technology.

## Best Answer

You're right, there's some confusion due to the way this paper has been publicized.

## Negative effective mass

The 'negative mass' referred to in the paper is effective mass. The idea is that while every fundamental constituent of a physical system has a known, nonnegative mass, the effective degrees of freedom of the system may behave as if they have a different mass.

This isn't a new idea; it pops up in a lot of contexts:

## Effective vs. fundamental mass

Is a negative effective mass "really" a negative mass? On a fundamental level, we think of mass as the thing that goes into $E = mc^2$; alternatively it's the mass of an object that determines how it couples to the gravitational field. If you're thinking of mass this way, then no, none of the examples I listed above have negative mass, nor does the paper.

But if you're in the business of atomic physics or condensed matter physics, it doesn't matter, because relativity is totally irrelevant to your experiments. The energies are low enough that the speed of light might as well be infinite, and the excitations you're studying really do have a preferred reference frame (the lab frame). If you're a fish that never leaves the water, it makes perfect sense to call an air bubble 'negative mass', even if people outside the water disagree.

## Does negative mass fall down?

You also asked whether an object with negative mass falls up or down. The equivalence principle tells us that gravity is indistinguishable from uniform acceleration. That means that positive and negative masses have to behave the exact same way under gravity, so negative mass falls down.

The common confusion here probably comes from the fact that an air bubble in water (with its negative effective mass) appears to fall up. This isn't actually true. If you drop a container of water containing an air bubble, the entire thing will accelerate downward uniformly, and the bubble will be stationary in the water, as required by the equivalence principle. You can see this explicitly in this video from the ISS (timestamp 1:05).

If you hold a container of water on Earth, the air bubbles will accelerate upward, but this isn't due to gravity. Gravity is pulling both the air and water down, but your hand is pushing the water up, and the water in turn pushes the air bubbles up.

The excitations in the BEC, which also have negative effective mass, are fully analogous. If you drop the BEC, they'll fall down. If you hold the BEC still, they might as well 'fall up', but this is just due to interactions within the material, not to gravity itself.