Forward's Analysis
Although no particles are known to have negative mass, physicists (primarily Hermann Bondi and Robert L. Forward) have been able to describe some of the anticipated properties such particles may have. Assuming that all three concepts of mass are equivalent the gravitational interactions between masses of arbitrary sign can be explored.
For two positive masses, nothing changes and we have a pull on each other causing an attraction. Two negative masses would produce a pull on one another, but would repel because of their negative inertial masses. For different signs we get a push, that repels the positive mass but attracts the negative mass.
Bondi pointed out objects of equal and opposite mass this would produce a constant acceleration of the system towards the positive mass object. However, the total mass, momentum and energy of the system would remain 0.
This behavior is completely inconsistent with a common-sense approach and the expected behaviour of 'normal' matter; but is completely mathematically consistent and introduces no violation of conservation of momentum or energy. If the masses are equal in magnitude but opposite in sign, then the momentum of the system remains zero if they both travel together and accelerate together, no matter what their speed:
And equivalently for the kinetic energy :
Forward extended Bondi's analysis to additional cases, and showed that even if the two masses m(-) and m(+) are not the same, the conservation laws remain unbroken. This is true even when relativistic effects are considered, so long as inertial mass, not rest mass, is equal to gravitational mass.
This behaviour can produce bizarre results: for instance, a gas containing a mixture of positive and negative matter particles will have the positive matter portion increase in temperature without bound. However, the negative matter portion gains negative temperature at the same rate, again balancing out. Geoffrey A. Landis pointed out other implications of Forward's analysis, including noting that although negative mass particles would repel each other gravitationally, the electrostatic force would be attractive for like-charges and repulsive for opposite charges.
Forward used the properties of negative-mass matter to create the diametric drive, a design for spacecraft propulsion using negative mass that requires no energy input and no reaction mass to achieve arbitrarily high acceleration.
Forward also coined a term, "nullification" to describe what happens when ordinary matter and negative matter meet: they are expected to be able to "cancel-out" or "nullify" each other's existence. An interaction between equal quantities of positive and negative mass matter would release no energy, but since the only configuration of such particles which has zero momentum (both particles moving with the same velocity in the same direction) does not produce a collision, all such interactions would leave a surplus of momentum, which is classically forbidden.
Read more about this topic: Negative Mass
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