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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