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Rest mass and relativistic mass

The quantity $m=\gamma (\mathbf {v} )m_{0}$ is often called the "relativistic mass" of the object in the given frame of reference. Some authors use m for relativistic mass and m₀ for rest mass, while others simply use m for rest mass. This article uses the former convention for clarity.

Substituting relativistic mass for classical mass, the spatial momentum may be written as $\mathbf {p} =m\mathbf {v}$, preserving the form from classical mechanics. However, this substitution fails for some quantities, including force and kinetic energy. Moreover, the relativistic mass is not invariant under Lorentz transformations, while the rest mass is. For this reason, many people find it easier to use the rest mass and account for $\gamma$ explicitly through the 4-velocity or coordinate time.

Lev Okun has suggested that the concept of relativistic mass "has no rational justification today" and should no longer be taught. Other physicists, including Wolfgang Rindler and T. R. Sandin, argue that the concept is useful. See mass in special relativity for more information on this debate.

A simple relationship between energy, momentum, and velocity may be obtained from the definitions of energy and momentum by multiplying the energy by $\mathbf {v}$, multiplying the momentum by $c^{2}$, and noting that the two expressions are equal. This yields

$\mathbf {p} c^{2}=E\mathbf {v}$

$\mathbf {v}$ may then be eliminated by dividing this equation by $c$ and squaring,

$(pc)^{2}=E^{2}(v/c)^{2}$

dividing the definition of energy by $\gamma$ and squaring,

$E^{2}\left(1-(v/c)^{2}\right)=\left(m_{0}c^{2}\right)^{2}$

and substituting:

$E^{2}-(pc)^{2}=\left(m_{0}c^{2}\right)^{2}$

This is the relativistic energy–momentum relation.

While the energy E and the momentum p depend on the frame of reference in which they are measured, the quantity E² − (pc)² is invariant, and arises as −c² times the squared magnitude of the 4-momentum vector which is −(m₀c)².

The invariant mass of a system

${m_{0}}_{\text{tot}}={\frac {\sqrt {E_{\text{tot}}^{2}-(p_{\text{tot}}c)^{2}}}{c^{2}}}$

is different from the sum of the rest masses of the particles of which it is composed due to kinetic energy and binding energy. Rest mass is not a conserved quantity in special relativity unlike the situation in Newtonian physics. However, even if an object is changing internally, so long as it does not exchange energy with surroundings, then its rest mass will not change, and can be calculated with the same result in any frame of reference.

A particle whose rest mass is zero is called massless. Photons and gravitons are thought to be massless; and neutrinos are nearly so.