Made a bunch of changes to this page today. I'm not a scientist nor a mathematician, just a Game-Master. Which means if someone more credentialed than myself feels I've made any mistakes on the page, they should edit in cold blood. In particular, I'm worried my ignorance of the implications of negative energy might screw up some of my math somewhere. Short of that oversight, though, the basic equations seem sound to me.

What follows is sort of a rough and dirty mathematical proof of the concept that exotic matter and normal matter can and do have the crazy interactions described in the "Negative Inertial Mass" section of the page. This represents my understanding of how this concept of negative mass interacts with physics and math. If you're not sure where I'm drawing my conclusions from, it's in this stuff somewhere:

Here's the equation that is the backbone of Newton's Law of Universal Gravitation:

(1)That tells us the force attracting two objects is equal to the mass of the two objects multiplied by each other, divided by the square of the distance between them. To know what affect this force has, we need to run it through another equation:

(2)That tells us the acceleration each object experiences is that Force divided by the objects mass.

To keep the math really simple, let's ignore G and r ^{2} in the first equation, basically treating them each as if their value is "1". We're not looking for precision values, just rough estimates, and you can't get much easier than multiplying or dividing by 1. With that in mind, we'll also assume the normal matter object has a mass of 1, and the negative mass object has a mass of -1. Running all those simple numbers through the equation (F = 1 * (1 * -1) / 1^{2}), we find that F = -1. If they were both normal objects, F would equal positive 1.

Continuing the simplified math, the Acceleration of the normal matter object is -1. (A = -1/1) This means gravity is pushing the normal matter object away from the exotic one. We don't really care how fast something moves, just what direction it's going in, and -1 means it's the opposite direction of what we normally expect gravity to do.

Meanwhile, the Acceleration of the exotic matter object is positive 1. (A = -1/-1) Gravity is pulling the exotic object towards the normal one.

The net result is that they move at the same speed and never get further or closer apart, so they continue to exert this force until disrupted by some other outside influence. That perpetual application of force causes them to incrementally accelerate forever - or at least until they hit the limits set by relativity and lightspeed.

This of course assumes the absolute value of the masses is equal: +1 and -1. If they were instead unequal, you'd end up with one object moving much faster than the other. +1 and -2 would result in the normal object being hurled away (or they might form some sort of weird orbiting trajectory around their combined center of mass maybe?), whereas +2 and -1 would result in a collision. I think. Physics isn't really my area of expertise, hence the simplified math of +1 and -1, which I could reasonably figure out.

Which is the place I think my lack of understanding of negative energy might possibly be making my overestimate the cool-factor or the implications. How long would it take something to accelerate anywhere near C? I have no fuzzy clue, but it's probably a really long time.

The equations for impacts and pushing are similar, again relying on A = F/m. Instead of the impact slowing things down (pushing in the opposite direction they were going) it speeds them up (by accelerating them in the same direction of their initial velocity). That little minus-sign has some big implications.

Natter with negative gravitational mass but positive inertial mass effectively uses the absolute value of it's mass for just the calculation of A=F/m. The other equation (with the G and the r^{2}) uses the negative value for m. This explains why that form of Natter attracts and repels somewhat differently than the exotic matter with negative values for both gravitational and inertial mass.

Extra details like air resistance would make that math all get rather more complicated… but on the plus side, that complexity may allow a bit more handwaving and artistic license for the GM or storyteller. You might be able to justify shockwaves, magnetism-like effects, and other surprises as needed for the plot, and as dictated by just how hard or squishy you want the science to be. Have fun with it.