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A physical quantity determining the inertial and gravitational properties of matter. In classical mechanics inertial mass is the coefficient of proportionality between force and acceleration in Newton's second law, which is a constant for a given body (cf. Newton laws of mechanics). The gravitational mass is defined as the coefficient of proportionality in the law of universal gravitation. According to the principle of equivalence, inertial and gravitational mass are proportional to each other and, in the usual system of units, are equal. In classical physics mass is additive: the mass of a system is equal to the sum of the masses of its parts. In the theory of special relativity mass (the so-called rest mass) can be defined as the coefficient of proportionality in the formula connecting the momentum $ p $ of a body and its velocity $ v $,

$$ p = \ \frac{m v }{\sqrt {1 - v ^ {2} / c ^ {2} } } , $$

where $ c $ is the velocity of light in vacuum. Sometimes one introduces the quantity

$$ m _ { \mathop{\rm mot} } = \ \frac{m}{\sqrt {1 - v ^ {2} / c ^ {2} } } , $$

called the motion mass of the body. By this definition, in relativity theory the momentum and velocity are related by the classical formula $ p = m _ { \mathop{\rm mot} } \cdot v $, where the mass depends on the velocity. The mass of a body is related to its energy $ E $ by the relation $ E = m _ { \mathop{\rm mot} } c ^ {2} $. In relativity theory mass is not additive: the mass of a stable system is less than the sum of the masses of its parts by the quantity $ \Delta m = \Delta E / c ^ {2} $, where $ \Delta E $ is the binding energy of the system, equal to the energy produced on the formation of the system. The quantity $ \Delta m $ is called the mass defect. The rest masses of all known physical bodies are non-negative (the rest mass of a photon, for example, is equal to zero).



[a1] M. Jammer, "Concept of mass in classical and modern physics" , Harper & Row (1964)
[a2] H. Weyl, "Philosophy of mathematics and natural science" , Princeton Univ. Press (1949)
How to Cite This Entry:
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This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article