Inertial Versus Gravitational
The earliest references to negative weight are due to the observation that metals gain weight when oxidizing in the study of phlogiston theory in the early 1700s.
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 mass that is evident from the force produced in a gravitational field). 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 be identical. All experimental evidence to date has found these are, indeed, always the same. In considering negative mass, it is important to consider which of these concepts of mass are negative; however, in most analyses 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 negative, but that the assumption of negative mass involves some counter-intuitive form of motion. For example, an object with negative inertial mass would be expected to accelerate in the opposite direction to that in which it was pushed.
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