How would negative mass behave in a gravitational field?
Answer: exactly the same way positive mass would.
That’s unusual, because if you try to push a negative mass, it behaves in surprising ways. While a positive mass will accelerate in the direction you push it in, in accordance with Newton’s second law,
where F is directed toward the central mass M. For a negative mass, this force would be in the opposite direction — the central mass M would repel the negative mass m.
Since (1) the gravitational force on a negative mass m is directed away from the (positive) central mass M, and (2) the negative mass accelerates in the direction opposite to the applied force, these two strange effects cancel each other out. A negative mass in a gravitational field will behave in exactly the same way as a regular mass.
It seems that this is just another interesting consequence of the equality of inertial and gravitational mass. Doing some searching around, it seems that Hermann Bondi was the first to write about it, in the context of General Relativity — Bondi plays around with various scenarios involving positive and negative masses, and find some surprising results.
Of course, the evidence for negative-mass matter remains zilch. But it seems to be an interesting theoretical plaything!
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What what happens to the positive mass in the presence of a negative mass? Why from the looks of it, the acceleration will be in the same direction as the negative mass’s. While this isn’t mathematically a problem for the conservation of momentum, since the momentum of a negative mass moving in some direction is in the opposite direction, it does seem exceedingly strange. I mean, the very thought of sitting two things next to each other and having them accelerate off, forever going faster, is a bit strange, no? Classically, there’s no conservation of energy problem because the kinetic energy of a negative mass object is negative. It would seem to be a problem for special relativity, though, because the energy takes the form: sqrt( (m * c^2 )^2 + ( p * c )^2 ), and that’s positive whether or not the mass is positive.
I’m less familiar with the situation in GR, but I would imagine that the unbounded increase in the energy can be compensated for by the release of gravitational waves that adjust space-time’s curvature appropriately.
Interesting. Notice (by the total energy equation you wrote down) that invariant rest-energy (i.e., vanishing 3-velocity p) is mc^2, which is indeed negative when m is negative. So I don’t think negative mass would be a problem for energy conservation in SR.
In GR things get trickier, because it’s harder to say what you mean by energy — and hence by energy conservation. But I don’t see any prima facie problem with negative mass for the existing conservation laws. Nice idea though, I’ll have to think about it.