## Mass in relativity::Mass

### ::concepts

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Mass in relativity

### Special relativity

{{#invoke:main|main}} In special relativity, there are two kinds of mass: rest mass (invariant mass),<ref group="note">It is possible to make a slight distinction between "rest mass" and "invariant mass". For a system of two or more particles, none of the particles are required be at rest with respect to the observer for the system as a whole to be at rest with respect to the observer. To avoid this confusion, some sources will use "rest mass" only for individual particles, and "invariant mass" for systems.</ref> and relativistic mass (which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object. Relativistic mass is the total quantity of energy in a body or system divided by c2. The two are related by the following equation:

$m_\mathrm{relative}=\gamma (m_\mathrm{rest})\!$

where $\gamma$ is the Lorentz factor:

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$

The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's frame of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy E and the magnitude of its momentum p by the relativistic energy-momentum equation:

$(m_\mathrm{rest})c^2=\sqrt{E_\mathrm{total}^2-(|\mathbf{p}|c)^2}.\!$

So long as the system is closed with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Particles of matter may be converted to types of energy (e.g. light, kinetic energy, the potential energy in magnetic, electric and other fields) but this does not affect the amount of mass. Although things like heat may not be matter, all types of energy still continue to exhibit mass.<ref group="note">For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another.</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=book }}</ref> Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy appear without the other.

Both rest and relativistic mass can be expressed as an energy by applying the well-known relationship E = mc2, yielding rest energy and "relativistic energy" (total system energy) respectively:

$E_\mathrm{rest}=(m_\mathrm{rest})c^2\!$
$E_\mathrm{total}=(m_\mathrm{relative})c^2\!$

The "relativistic" mass and energy concepts are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse among physicists.<ref>{{#invoke:citation/CS1|citation|CitationClass=arxiv}}</ref> There is disagreement over whether the concept remains useful pedagogically.<ref name="okun">{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref><ref>{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref><ref>{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref>

In bound systems, the binding energy must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. Mass is not conserved in this process because the system is not closed during the binding process. For example, the binding energy of atomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leaving nuclides which have less mass than the free particles (nucleons) of which they are composed.

### General relativity

{{#invoke:main|main}} In general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass. At the core of this assertion is Albert Einstein's idea that the gravitational force as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudo-force experienced by an observer in a non-inertial (i.e. accelerated) frame of reference.

However, it turns out that it is impossible to find an objective general definition for the concept of invariant mass in general relativity. At the core of the problem is the non-linearity of the Einstein field equations, making it impossible to write the gravitational field energy as part of the stress–energy tensor in a way that is invariant for all observers. For a given observer, this can be achieved by the stress–energy–momentum pseudotensor.<ref>{{#invoke:citation/CS1|citation |CitationClass=book }}</ref>

Mass sections
Intro   Units of mass    Definitions of mass    Pre-Newtonian concepts    Newtonian mass    Atomic mass    Mass in relativity    Mass in quantum physics    See also    Notes    References    External links

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