Basic Information
It is allowable within the laws of physics for there to be matter with negative mass, which would have interesting properties. It is worth distinguishing between inertial mass (which affects how matter responds to forces exerted on it) and gravitational mass (how it affects and is affected by gravity) in this context.
For story purposes it is usually convenient to assume that even though a material's gravitational mass is negative, its inertial mass is still positive (or equivalently, that inertial laws turn out to actually be based on the absolute value of mass). This technically violates current science, but this can be handwaved since current scientists have never encountered any matter with negative mass. Negative-mass objects of this kind attract each other normally but repel positive-mass objects and vice versa. On regular planets, negative-mass objects fall upwards at the same rate positive-mass objects fall down.
Effectively, negative mass (let's call it "natter") would mean that mass is working like electric charge - there are positive and negative charges, but a positive and negative ones repel and two positive or two negative ones attract. That would mean natter can form stars, planets and life, just like matter, anti-matter or even anti-natter.
If you use negative mass in a science-fiction story, you may want to use the term "exotic matter" instead and only go into the details when someone asks. Exotic matter sounds more "hard science"-y.
Sources
Game and Story Use
- If available in large quantities, negative mass can allow for all kinds of otherwise preposterous engineering projects. You can make anything float if you add enough negative mass counterweights and fine-tune using lifting gas and ballast.
- An alloy of negative-mass and regular metals can be effectively weightless (or have whatever weight you choose) while offering the same protection and structural integrity as ordinary metal. This makes it a very useful material for air- and spacecraft.
- It'll be useful for a lot of other things too.
- Storing negative mass in such alloys makes the logistics easier as you don't have to worry about it falling off your planet.
- Negative-mass material can be a highly fought-over resource.
- When inertial mass is negative too, you get a highly bizarre substance that's impossible to use for mundane purposes but might do any of the following:
- Serve as the key ingredient in the diametric drive, which allows for arbitrary acceleration without consuming anything.
- Stabilize a wormhole.
- Be used to create a Tipler Cylinder.
- Generally serve as phlebotinum for the settings dimension-bending supertech.

