I did a little scouting around as to what "negative matter" might be. There is such stuff as antimatter, which is a lot like ordinary matter, but has electric charge (and some other interesting things) switched about it. Antimatter has positive mass and energy, however. We’ve known about antimatter for most of the last century, and use it routinely in particle physics experiments. You can find out more about antimatter by searching the answers on this site.
Negative matter appears to be some people’s idea of "stuff which has negative mass". One consequence that is commonly reported in discussion boards on the web is that the gravitational force between negative matter and matter may be repulsive -- kind of like the analogy to positive and negative electrical charges which may electrostatically attract or repel.
Life is different for gravitation, since you get gravitational effects around collections of matter or energy. Either one may create a gravitational field. In fact, gravity really cannot tell the difference between a box containing matter and one containing other forms of energy, such as lots of photons or just a small amount of matter heated very hot.
So the addition of negative matter to a collection of ordinary matter must cancel it out -- what’s left should have no gravitational field if the amounts balance. This means that the negative matter would have negative energy. This causes lots of trouble already.
Physical systems in nature prefer to settle down to their lowest energy states, usually dissipating excess energy in friction, heat, radiating it away as light, etc. Unfortunately, no matter what the state the universe is in, it can be made to have lower energy by creating some negative matter. Even if the rules of making matter and negative matter are like those of making matter and antimatter together in pairs, there still would be no barrier to spontaneous creation of arbitrary amounts of the stuff all by itself. We’d say that "the vacuum is unstable". Maybe this conflicts more with observation than whether such a theory can be consistently put together which includes negative matter.
Not that this doesn’t mean such things can’t be speculated about. One recent observation by the astrophysicists is that the expansion of the universe may be speeding up somehow. A large number of strange hypotheses have been advanced to explain this, while the astronomers go and repeat their measurements and devise more sensitive ways to measure the universe’s expansion. One of the strange ideas to explain an accelerating universe is that there is some kind of energy everywhere which doesn’t interact much with ordinary matter except gravitationally so we haven’t noticed it yet. In some models, this energy can even be negative. I approach this kind of model with skepticism, and rely on the scientific method to put it to the test. Careful tests have to be devised that can convince people that the original effect (i.e., the observation of an accelerating expansion of the universe) is real and not some ordinary but badly understood feature of the observations that is throwing us off course. Other tests must be performed to show that one hypothesis and not another is responsible for our observations. So far, observations have not shown a need to introduce negative matter into our models, but who knows? We can certainly learn more as time goes by.
(published on 10/22/2007)