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Relativistic energy and momentum

There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy, one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.

The four-momentum of an object is straightforward, identical in form to the classical momentum, but replacing 3-vectors with 4-vectors:

The energy and momentum of an object with invariant mass m 0 {\displaystyle m_{0}}, moving with velocity v {\displaystyle \mathbf {v} } with respect to a given frame of reference, are respectively given by

The factor γ {\displaystyle \gamma } comes from the definition of the four-velocity described above. The appearance of γ {\displaystyle \gamma } may be stated in an alternative way, which will be explained in the next section.

The kinetic energy, K {\displaystyle K}, is defined as

and the speed as a function of kinetic energy is given by

The spatial momentum may be written as p = m v {\displaystyle \mathbf {p} =m\mathbf {v} }, preserving the form from Newtonian mechanics with relativistic mass substituted for Newtonian mass. 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 prefer to use the rest mass and account for γ {\displaystyle \gamma } explicitly through the 4-velocity or coordinate time.

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

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

dividing the definition of energy by γ {\displaystyle \gamma } and squaring,

and substituting:

This is the relativistic energy–momentum relation.

While the energy E {\displaystyle E} and the momentum p {\displaystyle \mathbf {p} } depend on the frame of reference in which they are measured, the quantity E 2 − (p c) 2 {\displaystyle E^{2}-(pc)^{2}} is invariant. Its value is − c 2 {\displaystyle -c^{2}} times the squared magnitude of the 4-momentum vector.

The invariant mass of a system may be written as

Due to kinetic energy and binding energy, this quantity is different from the sum of the rest masses of the particles of which the system is composed. 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 or momentum with its surroundings, its rest mass will not change and can be calculated with the same result in any reference frame.