The term is also casually attached to any material which is difficult to produce (such as metallic hydrogen or a Bose-Einstein condensate) or which exhibits unusual properties (such as fullerenes or nanotubes), even though these materials have been created and are relatively well understood. It can also refer to material composed of some form of exotic atom.
Ever since Newton first formulated his theory of gravity, there have been at least three conceptually distinct quantities called mass: inertial mass, "active" gravitational mass (that is, the source of the gravitational field), and "passive" gravitational mass (that is, the amount of force produced in response to gravity). The Einstein equivalence principle postulates that inertial mass must equal passive gravitational mass; while the law of conservation of momentum requires that active and passive gravitational mass must be identical. All experimental evidence to date has found these are indeed always the same. In considering hypothetical particles with negative mass, it is important to consider which of these concepts of mass are negative; however, in most analysis of negative mass, it is assumed that the equivalence principle and conservation of momentum continue to apply.
In 1957, Hermann Bondi suggested in a paper in Reviews of Modern Physics that mass might be negative as well as positive . He pointed out that this does not entail a logical contradiction, as long as all three forms of mass are all negative, but that the assumption of negative mass involves some counter-intuitive form of motion.
From Newton's second law:
Thus it can be seen that an object with negative inertial mass would be expected to accelerate in the opposite direction to that in which it was pushed, which is arguably a strange concept.
If one were to treat inertial mass , passive gravitational mass , and active gravitational mass distinctly, then Newton's law of universal gravitation would take the form
Thus objects with negative gravitational mass (both passive and active), but with positive inertial mass, would be expected to be repelled by positive active masses, and attracted to negative active masses. If all such negative matter were like this, then gravity would work similarly to the electric force except that like masses would attract and unlike masses would repel.
Although no particles are known to have negative mass, physicists (primarily 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 would produce a system where negative masses are attracted to positive masses, yet positive masses are repelled away from negative masses. As well, negative masses would produce an attractive force on one another, but would be repelled because of their negative inertial masses.
For a negative value of with positive value of , is negative (repulsive). At first glance it would appear that a negative mass would accelerate away from a positive mass, but because such an object would also possess negative inertial mass it would accelerate in the opposite direction from . Furthermore, it can be shown that if both masses are of equal but opposite mass, Bondi pointed out then the combined system of positive and negative particles will accelerate indefinitely without any additional input into the system.
This behavior is bizarre in that it is completely inconsistent with our 'normal universe' commonsense expected behavior from working with positive masses. Yet it is completely mathematically consistent and introduces no apparent contradictions when physics analysis is performed on the behaviours.
First impressions may be that this arrangement violates conservation of momentum and/or energy, but in fact if the masses are equal in magnitude, one being of positive value and the other negative, then the momentum of the system is zero if they both travel together and accelerate together, no matter what speed:
...and an equivalent equation can be calculated for Ke:
Forward extended Bondi's analysis to additional cases, and showed that even if the two masses m(-) and m(+) are not the same, the equations remain still consistent.
Some of the behaviours this seems to introduce are bizarre, such as a comingled positive matter gas and negative matter gas having 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, for electrical forces, like charges would attract each other (in distinction to positive-mass particles, where like particles repel.) In effect, this means that for negative mass particles, gravitational and electrostatic forces would be switched.
Forward has proposed a design for spacecraft propulsion using negative mass that requires no energy input and no reaction mass to achieve arbitrarily high acceleration, though of course a major obstacle to the construction of such a spacecraft is the fact that negative mass remains purely hypothetical. See diametric drive.
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. If equal and opposite types of matter are involved, no energy would be left over. However, it is easy to show that some momentum would be left over (none is left over when they move in the same direction, as described above, but they have to move in opposite directions to be able to meet and mutually nullify). This can in turn explain why equal quantities of ordinary and negative matter don't spontaneously appear out of nowhere (the opposite of nullification): Momentum would not be conserved by that event, either.
In general relativity, exotic matter is generalized to refer to any region of space in which for some observers the mass density is measured to be negative. This can occur due to negative mass, or could be a region of space in which the stress component of the Einstein stress energy tensor is larger in magnitude than the mass density. All of these are violations of one or another variant of the positive energy condition of Einstein's General Theory of Relativity; however, the positive energy condition is not a required condition for the mathematical consistency of the theory. (Various versions of the positive energy condition, weak energy condition, dominant energy condition, etc., are discussed in mathematical detail by Visser.)
Morris, Thorne and Yurtsever pointed out that the quantum mechanics of the Casimir effect can be used to produce a locally mass-negative region of space-time. In this article, and subsequent work by others, they showed that negative matter could be used to stabilize a wormhole. Cramer et al. argue that such wormholes might have been created in the early universe, stabilized by negative-mass loops of cosmic string. Stephen Hawking has proved that negative energy is a necessary condition for the creation of a closed timelike curve by manipulation of gravitational fields within a finite region of space; this proves, for example, that a finite Tipler cylinder cannot be used as a time machine.
If the rest mass is imaginary, then the denominator must be imaginary (if one is to avoid a complex value for energy); therefore the quantity under the square root must be negative, which can only happen if v is greater than c. The theory of tachyons, as worked out by Feinberg, is straightforward in one dimension, but is difficult to analyze in three dimensions. As noted by Benford et al., among others, the special theory of relativity implies that tachyons, if they existed, could be used to communicate backwards in time. . (see Tachyonic antitelephone article). Since time travel is considered to be non-physical, tachyons are believed by physicists to either not exist, or else to be incapable of interacting with normal matter.
In quantum field theory imaginary mass would induce tachyon condensation
Bubble chamber experiments are often cited as evidence that antiparticles have a positive inertial mass equivalent to their normal counterparts, but a reversed electric charge. In these experiments, the chamber is subjected to a constant magnetic field which causes charged particles to travel in helical paths. The radius and direction of these paths correspond to the ratio of electric charge to inertial mass. Particle/antiparticle pairs are observed to travel in helices with opposite directions, but identical radii. Certainly, this observation implies that their ratios differ only in sign, but it does not make clear whether it is charge or inertial mass which is negative.
However, particle/antiparticle pairs are observed to electrically attract one another, often as the prelude to annihilation. This behavior implies that both have positive inertial mass and opposite charges. If the reverse were true, and antiparticles had negative inertial mass and the same charge, then the normal particle with positive inertial mass would still be repelled by its anti-particle. Also, the equation E=mc² describes both the creation and annihilation of matter/anti-matter particle-pairs in terms of ordinary (positive) energy. If anti-particles had negative mass, then negative energy needs to be part of the equation, and it isn't. Every ordinary particle that is involved always has exactly half the creation/annihilation mass/energy that is plugged into the equation, and therefore the anti-particles must be associated with the other half, and thereby must have positive mass/energy (and consequently would fall downward in Earth's gravitational field).