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First, it may be important to identify what is meant by a negative mass. In particular, it is necessary to distinguish negative mass from anti-matter. Anti-matter is matter made up of atomic particles that have opposite electrical charges from those of ordinary matter. Anti-matter, as we know it, does not have negative mass. Anti-matter has been shown to exist in experiments, but there is no evidence that any anti-particles have negative mass. In fact, the idea that anti-particles, such as the positron, are regular particles, such as the electron, going backwards in time is probably no more than a convenient fantasy, since both positrons and electrons have positive mass. This can be demonstrated by the physics of the mutual annihilation of a positron and an electron, which creates energy in the form of photons, that is devolved by E = mc^2, to the sum of the masses of the two particles. In such collisions, charge is annihilated, but mass-energy is not.