

Hot, Cold and Warm Ideas in Particle Physics 
If we pick up on the idea of negative energy and mass as implied by the Dirac Equation, where does this lead in terms of Gravity?
For low speeds, we can use Newton's Law of Gravity:
F =  G m_{1} m_{2} ⁄ r^{2}
and the 2nd of Newton's Laws of Motion:
F = m_{i} a
Now consider a system in which:
Where does that get us? Well, we're already in murky waters because we've allowed for the possibility that mass may be negative as well as positive.
Surely that can't be right, otherwise we're off into the realms of antigravity, warp drive, aliens and trekkies? Well, maybe ... but at least it's a suffciently welltrodden path that Wikipedia has seen fit to provide an article: Negative Mass.
Hermann Bondi briefly considered negative mass, but didn't take long to reject it as being nonphysical.
The killer argument was that if you take a negative mass and place it next to a positive mass, then do the maths, you get to one of two places:
Mass 1 is positive and repelled by mass 2, which is negative, therefore mass 1 moves away from mass 2, since positive mass moves in the same direction in which it's pushed.
Mass 2 is negative and repelled by mass 1 which is positive. However, negative mass moves in the oppositive direction to which it's pushed, so it moves towards mass 1 after all.
This was the interpretation considered by Bondi et al, taking Newton's laws at face value and insisting that gravitational and inertial mass are the same thing. Since every largescale experiment ever done points in this direction, this is a pretty reasonable stance to take.
Essentially the 2 masses move in tandem, in the same direction, accelerating all the while, up to light speed...
This was all too much to bear, certainly for largescale everyday objects which clearly don't behave in this way. It also violates various other preconceptions about what mass and energy mean in standard physics. So it's not surprising that negative mass gets a bad press after only half a page of reasoning.
What if the underlying point is right and this scanario happens all the time at the smallest scales? What if by combining positive and negative mass at the quantum level, you get something that accelerates very quickly up to a very high speed?
The trouble with this scenario, is that it leads to a fairly boring universe. Assuming we start with an equal mix of negative and positive mass randomly distributed in a little model universe, then the positive masses initially clump together and the negative masses fly apart. However, the negatives go after the positives, systematically pairing up and zooming off until the universe is full of stuff exiting the scene rapidly in all directions.
You can do the simulation yourself if you like, using software from www.gravsim.com. Just plug in negative numbers for masses and see what happens. Boom!
But what if the combination of masses is only half the story?
Even at this level, it's possible to get confused by doing the maths differently. There are several places on the internet where you can read about negative mass, but they often take the attitude that when a mass is negative, it can only be negative in terms of its gravitational attraction, because negative inertial mass is impossible (i.e. "don't be daft, things don't move in the opposite direction to which they're pushed").
In this case, we're not taking Newton's laws at face value, we're assuming that gravitational and inertial mass are 2 different things and can take different numerical values, at least in the case where the gravitational mass is negative but the inertial mass is still positive.
Mass 1 is positive and repelled by mass 2 which is negative, so it moves away as before. Mass 2 is negative and repelled by mass 1, but since its inertial mass is positive, it moves in the same direction in which it's pushed, so it moves away from mass 1.
This was still too much to bear. Experience suggest that large scale objects attract each other due to the force of gravity. They don't repel. Gravity is always positive, that's what keeps us rooted to the floor and the Earth going round the Sun...
But what if this scenario happens all the time time at the smallest scales?
The following article discusses some evidence that electrons in semiconductors behave with negative inertial mass:
Negative mass and high speed: How electrons go their own ways
We will take all of this as a hint that we should be considering negative mass when we attempt to build a unified theory.
We will also take the hint that we should be considering the case where gravitational mass and inertial mass may vary.
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