Exotic matter
From Academic Kids

Exotic matter is a hypothetical concept of particle physics. It covers any material which violates one or more classical conditions or is not made of known baryonic particles. Such materials would possess qualities like negative mass or being repelled rather than attracted by gravity. It is used in certain speculative theories, such as on the construction of wormholes. The closest known real representative of exotic matter is a region of negative pressure density produced by the Casimir effect.
Contents 
Negative Mass
Ever since Newton first formulated his theory of gravity, there have been at least three conceptually distinct quantities called mass. However, these three, inertial mass, active gravitational mass, and passive gravitational mass have so far always been found to be equivalent. When considering hypothetical particles with negative mass, it is important to consider which of these concepts of mass are negative.
From Newton's law
 <math>F = m_ia \!\;<math>
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!
If one were to treat inertial mass m_{i}, passive gravitational mass m_{p}, and active gravitational mass m_{a} distinctly, then Newton's law of universal gravitation would take the form
 <math>m_ia=\frac{Gm_pm_a}{r^2}<math>
Thus, objects with negative gravitational mass, both passive and active, but with positive inertial mass, would be expected to be repeled by positive active masses, and attracted to negative active masses. If all such negative matter was like this, then gravity would work similarly to the electric force except that like masses would attract and unlike masses would repel.
Forward's Analysis
Although no particles are known to have negative mass, physicists (primarily 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 both positive and negative masses, yet positive masses are repeled away from negative masses.
For a negative value of m_{p} with positive value of m_{a}, F 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 F. Furthermore, it can be shown that if both masses are of equal but opposite mass, then the combined system of positive and negative particles will accelerate indefinitely without any additional input into the system.
This behaviour is bizarre, in that it's completely inconsistent with our 'normal universe' common sense expected behaviour from working with positive masses, but it is completely mathematically consistent and introduces no apparent contradictions when physics analysis is performed on the behaviours.
Naïve first impressions are that this 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:
 <math>P_{sys} = ( v \times m ) + ( v \times (m) ) \!\;<math>
 <math>P_{sys} = v \times ( m + m ) = v \times 0 = 0 \!\;<math>
...and an equivalent equation can be calculated for K_{e}:
 <math>K_{e\ sys} = \left( {1 \over 2} \times m \times v^2 \right) + \left( {1 \over 2} \times (m) \times v^2 \right)<math>
 <math>K_{e\ sys} = \left( {1 \over 2} \times v^2 \right) \times ( m + m ) = \left( {1 \over 2} \times v^2 \right) \times 0 = 0<math>
Forward also showed that if m() and m(+) are not the same magnitude of mass, the equations are still consistent.
Some of the behaviours this seems to introduce are really 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.
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.
Which way does antimatter fall?
Virtually every modern physicist suspects that antimatter has positive mass and should fall down just like normal matter. That being said, it is thought that this view has not yet been conclusively empirically observed. [1] (http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/antimatterFall.html) [2] (http://athenapositrons.web.cern.ch/ATHENApositrons/wwwathena/FAQ.html) It is difficult to directly observe gravitational forces at the particle level. At these small distances, electric forces tend to overwhelm any weak gravitational interaction. Furthermore, antiparticles must be kept separate from their normal counterparts or they will quickly annihilate. Worse still, the methods of production of antimatter typically have very energetic results unsuitable for observations. Understandably, this has made it difficult to directly measure the passive gravitational mass of antimatter. Fortunately, the ATHENA or ATRAP antimatter experiments may soon have the answers.
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 helixes 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 repeled by its antiparticle.
Supporters of the theory that antimatter has negative gravitational mass, and thus should fall up, argue that it solves two major cosmological paradoxes. The first is the apparent local lack of antimatter. Since by the theory, antimatter and matter would repel each other, forming separate matter and antimatter galaxies. These galaxies would also tend to repel one another, thereby preventing possible collisions and annihilations. This same galactic repulsion is also endorsed as a potential explanation to the observation of an accelerating universe.
The term Exotic matter is also casually attached to any material which would be difficult to produce (such as metallic hydrogen or a BoseEinstein condensate) or which exhibits unusual properties (such as fullerenes or nanotubes), even though these materials are relatively mundane in their composition. It could also refer to material composed of some form of exotic atom.
See also
General subfields within physics  
Classical mechanics  Condensed matter physics  Continuum mechanics  Electromagnetism  General relativity  Particle physics  Quantum field theory  Quantum mechanics  Solid state physics  Special relativity  Statistical mechanics  Thermodynamics 