Unlike electric charge, as far as we know all mass is positive, and positive masses attract one another. Could negative mass exist, and if so, what would be the ramifications?

Antigravity between a positive mass and a hypothetical negative mass has received a fair bit of attention in both physics and science fiction over the years. For example, in 1901's "First Men in the Moon" author H. G. Wells imagines a substance he calls "cavorite" which creates a negative force of gravity and thus acts as a gravity shield. In Newton's theory of gravity, negative mass would effectively appear as his same equation but with the gravitational constant G replaced by -G. But Einstein's version of gravity, general relativity, is not so kind, and does not seem to consistently allow anti-gravity.

In a new paper in Physics of the Dark Universe, Shin'ichi Nojiri from Japan and S.D. Odintsovc from Spain dig deeper into the possibility of negative mass objects (NMOs) and conclude that the idea may not be as exotic as is thought. Using theoretical tools, they show that negative mass "does not always lead to any inconsistency."

Where negative mass could arise
Mass comes from a particle's interaction with the Higgs field, and most of a particle's mass is actually binding energy (remember Einstein: m=E/c²) between its constituents. Protons, 1,836 times more massive than electrons, are composed of three quarks and gluons bound together. The quark masses are only about 9% of the proton's mass, according to lattice gauge theories of quantum chromodynamics. The rest comes from the field energy of gluons that mediate the dynamics inside the proton...