Assuming you don't mean 'effective' mass, but actual inertial mass, which, by the equivalence principle, is identical to the mass which 'generates' the gravitational field, and to the mass which responds to an external gravitational field ( by momentum conservation ), then H Bondi, W B Bonnor an R L Forward ( one of my favorite sci-fi authors ) proposed the phenomenon of 'Runaway Motion'
An excerpt from the Wiki article
https://en.wikipedia.org/wiki/Negative_mass
follows, and you might want to give the whole Wiki article a good read...
"Although no particles are known to have negative mass, physicists (primarily Hermann Bondi in 1957,[6] William B. Bonnor in 1964 and 1989,[12][13] then Robert L. Forward[14]) have been able to describe some of the anticipated properties such particles may have. Assuming that all three concepts of mass are equivalent according to the equivalence principle, the gravitational interactions between masses of arbitrary sign can be explored, based on the Newtonian approximation of the Einstein field equations. The interaction laws are then:
In yellow, the "preposterous" runaway motion of positive and negative masses described by Bondi and Bonnor.
Positive mass attracts both other positive masses and negative masses.
Negative mass repels both other negative masses and positive masses.
For two positive masses, nothing changes and there is a gravitational pull on each other causing an attraction. Two negative masses would repel because of their negative inertial masses. For different signs however, there is a push that repels the positive mass from the negative mass, and a pull that attracts the negative mass towards the positive one at the same time.
Hence Bondi pointed out that two objects of equal and opposite mass would produce a constant acceleration of the system towards the positive-mass object,[6] an effect called "runaway motion" by Bonnor who disregarded its physical existence."
The article goes on to state that R L Forward showed that no conservation laws are violated by this effect, however, quite a few unphysical situations ( mentioned in the article ) result.