Mass

Mass is a property of physical objects that, roughly speaking, measures the amount of matter they contain. It is a central concept of classical mechanics and related subjects.

Relativistic relation among mass, energy and momentum

Special relativity is a necessary extension of classical physics. In particular, special relativity succeeds where classical mechanics fails badly in describing objects moving at speeds close to the speed of light.

Related Topics:
Special relativity - Classical physics - Speed of light

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In relativistic mechanics, the mass (m) of a free particle is related to its energy (E) and momentum (p) by the equation

Related Topics:
Energy - Momentum

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:rac{E^2}{c^2} = m^2 c^2 + p^2.

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where c is the speed of light. This is sometimes referred to as the mass-energy-momentum relation.

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The first thing to notice about this equation is that it can cope with massless objects (m = 0), for which it reduces to

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:E = pc ,

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In classical mechanics, massless objects are an ill-defined concept, since applying any force to one would produce, via Newton's second law, an infinite acceleration - a nonsensical result. In relativistic mechanics, they are objects that are always travelling at the speed of light; an example being light itself, in the form of photons. The above equation says that the energy carried by a massless object is directly proportional to its momentum.

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Let us now consider objects with non-zero mass. For these, the quantity m has a simple physical meaning: it is the inertial mass of the object as measured in its rest frame, the frame of reference in which its velocity is zero. (Note: massless objects do not possess a rest frame; they are moving at the speed of light in any frame of reference.) The way we would measure m is exactly the same as in classical mechanics, which we described above: bouncing it off a reference object and measuring the accelerations. As long as the velocity of each object remains much smaller than the speed of light during this procedure, relativistic corrections to classical mechanics will be utterly negligible.

Related Topics:
Rest frame - Frame of reference - Speed of light

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In the rest frame, the velocity is zero, and thus so is the momentum p. The mass-energy-momentum relation thus reduces to

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:E = mc^2 ,

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which states that the energy of an object as measured in its rest frame - its "rest energy" - is equal to its mass times the square of the speed of light.

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Some books follow this up by stating that "mass and energy are equivalent", but this is somewhat misleading. The mass of an object, as we have defined it, is a quantity intrinsic to the object, and independent of our current frame of reference. The energy E, on the other hand, varies with the frame of reference; if the frame is moving at a high velocity relative to the object, E will be very large, simply because the object has a lot of kinetic energy in that frame. Thus, E = mc2 is not a "good" relativistic statement; it is true only in the rest frame of the object.

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Some authors define a quantity known as the relativistic mass, which is basically the quantity E/c2. This makes the "equivalence" of "mass" and energy true by definition, though neither quantity is frame-independent! "Relativistic mass" was used in many early writings on relativity, and it is still used in books for laymen as well as introductory physics classes. However, the concept is downplayed or discouraged by many physicists nowadays, for reasons explained in the article on relativistic mass. Following the modern usage, whenever we refer to "mass" in this article we always mean the rest mass, unless otherwise identified.

Related Topics:
Relativistic mass

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Having defined the mass of an object, let us look at how it behaves when not at rest. We can arrange the mass-energy-momentum relation in the following way:

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:E = mc^2 sqrt{1 + left( {p over mc} ight)^2}

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When the momentum p is much smaller than mc, we can Taylor expand the square root, with the result

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:E = mc^2 + {p^2 over 2m} + cdots

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The leading term, which is the largest, is of course the rest energy. The object always has this minimum amount of energy, regardless of its momentum. The second term is the classical expression for the kinetic energy of the particle, and the higher-order terms are basically relativistic corrections for the kinetic energy.

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Under normal circumstances, the rest energy of an object is inaccessible, in the sense that it cannot be used to do mechanical work. When the object hits something, it can do work by transferring its momentum, and thus its kinetic energy, to whatever it hit. However, the rest energy depends only on the mass of the object, which does not change during collisions, so it cannot be transferred along with the kinetic energy.

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On the other hand, it is possible to access the rest energy using processes that split or combine particles. The reason is that mass, as we have defined it, is not conserved during such processes. The simplest example is the process of electron-positron annihilation, in which an electron and a positron annihilate each other to produce a pair of photons: the electron and positron both have non-zero mass, but the photons are massless. Other examples include nuclear fusion and nuclear fission. Metabolism, fire and other exothermic chemical processes also convert mass to energy, however the mass change from these is negligible.

Related Topics:
Electron-positron annihilation - Electron - Positron - Nuclear fusion - Nuclear fission - Metabolism - Fire - Exothermic

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Energy, unlike mass, is always conserved in special relativity, so, roughly speaking, what is happening in these reactions is that the rest energy of the reactants is being transformed into the kinetic energy of the reaction products. The fact that rest energy can be liberated in this way is one of the most important predictions of special relativity.

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