Negative mass is very diferent from antimatter. Antimatter is opposite to normal matter in charge and quantum numbers (such as baryon number, etc.), but still has positive mass.
Negative mass reacts oppositely to both gravity and intertia. Oddly, that means that negative mass still falls down in a gravitational field: The gravitational force is opposite, but negative mass responds negatively to force (a=F/m, where both F and m are negative). So negative mass particles repel each other gravitationally, but are attracted to positive mass objects.
Trying my best to ignore my intuition, which is heavily biased toward "all mass is positive":
A negative mass would fall down in a gravitational field (generated by a positive mass) -- it would experience a force directed away from the positive mass, and it would respond to that force by moving toward the positive mass.
However, the negative mass would repel the positive mass gravitationally -- effectively, exert a force directed away from itself -- correct?
It seems to me that if you had two equal but opposite masses in freefall, the negative mass would accelerate toward the positive, the positive would accelerate away from the negative, and the negative mass would chase the positive mass off the edge of the universe at constant acceleration.
It also seems like two negative masses would repel each other (exert a force directed away from each other), but respond to that repulsion by accelerating toward each other.
What am I missing?