In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum. It can be written as the following equation:

${\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,}$

(1)

This equation holds for a body or system, such as one or more particles, with total energy E, invariant mass m₀, and momentum of magnitude p; the constant c is the speed of light. It assumes the special relativity case of flat spacetime^[1][2][3] and that the particles are free. Total energy is the sum of rest energy and kinetic energy, while invariant mass is mass measured in a center-of-momentum frame.

For bodies or systems with zero momentum, it simplifies to the mass–energy equation ${\displaystyle E=m_{0}{\textrm {c}}^{2}}$, where total energy in this case is equal to rest energy (also written as E₀).

The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy–momentum relation.

Connection to E = mc²

The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: E = mc² relates total energy E to the (total) relativistic mass m (alternatively denoted m_rel or m_tot ), while E₀ = m₀c² relates rest energy E₀ to (invariant) rest mass m₀.

Unlike either of those equations, the energy–momentum equation (1) relates the total energy to the rest mass m₀. All three equations hold true simultaneously.

Special cases

1. If the body is a massless particle (m₀ = 0), then (1) reduces to E = pc. For photons, this is the relation, discovered in 19th century classical electromagnetism, between radiant momentum (causing radiation pressure) and radiant energy.
2. If the body's speed v is much less than c, then (1) reduces to E = 1/2m₀v² + m₀c²; that is, the body's total energy is simply its classical kinetic energy (1/2m₀v²) plus its rest energy.
3. If the body is at rest (v = 0), i.e. in its center-of-momentum frame (p = 0), we have E = E₀ and m = m₀; thus the energy–momentum relation and both forms of the mass–energy relation (mentioned above) all become the same.

A more general form of relation (1) holds for general relativity.

The invariant mass (or rest mass) is an invariant for all frames of reference (hence the name), not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime (see below). However the total energy of the particle E and its relativistic momentum p are frame-dependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures E and p, while the other frame measures E′ and p′, where E′ ≠ E and p′ ≠ p, unless there is no relative motion between observers, in which case each observer measures the same energy and momenta. Although we still have, in flat spacetime:

${\displaystyle {E'}^{2}-\left(p'c\right)^{2}=\left(m_{0}c^{2}\right)^{2}\,.}$

The quantities E, p, E′, p′ are all related by a Lorentz transformation. The relation allows one to sidestep Lorentz transformations when determining only the magnitudes of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to;

${\displaystyle {E}^{2}-\left(pc\right)^{2}={E'}^{2}-\left(p'c\right)^{2}=\left(m_{0}c^{2}\right)^{2}\,.}$

Since m₀ does not change from frame to frame, the energy–momentum relation is used in relativistic mechanics and particle physics calculations, as energy and momentum are given in a particle's rest frame (that is, E′ and p′ as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. E and p as determined by particle physicists in a lab, and not moving with the particles).

In relativistic quantum mechanics, it is the basis for constructing relativistic wave equations, since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is Lorentz invariant. In relativistic quantum field theory, it is applicable to all particles and fields.^[4]

Origins and derivation of the equation

The energy–momentum relation goes back to Max Planck's article^[5] published in 1906. It was used by Walter Gordon in 1926 and then by Paul Dirac in 1928 under the form ${\textstyle E={\sqrt {c^{2}p^{2}+(m_{0}c^{2})^{2}}}+V}$, where V is the amount of potential energy.^[6][7]

The equation can be derived in a number of ways, two of the simplest include:

1. From the relativistic dynamics of a massive particle,
2. By evaluating the norm of the four-momentum of the system. This method applies to both massive and massless particles, and can be extended to multi-particle systems with relatively little effort (see § Many-particle systems below).

Heuristic approach for massive particles

For a massive object moving at three-velocity u = (u_x, u_y, u_z) with magnitude |u| = u in the lab frame:^[1]

${\displaystyle E=\gamma _{(\mathbf {u} )}m_{0}c^{2}}$

is the total energy of the moving object in the lab frame,

${\displaystyle \mathbf {p} =\gamma _{(\mathbf {u} )}m_{0}\mathbf {u} }$

is the three dimensional relativistic momentum of the object in the lab frame with magnitude |p| = p. The relativistic energy E and momentum p include the Lorentz factor defined by:

${\displaystyle \gamma _{(\mathbf {u} )}={\frac {1}{\sqrt {1-{\frac {\mathbf {u} \cdot \mathbf {u} }{c^{2}}}}}}={\frac {1}{\sqrt {1-\left({\frac {u}{c}}\right)^{2}}}}}$

Some authors use relativistic mass defined by:

${\displaystyle m=\gamma _{(\mathbf {u} )}m_{0}}$

although rest mass m₀ has a more fundamental significance, and will be used primarily over relativistic mass m in this article.

Squaring the 3-momentum gives:

${\displaystyle p^{2}=\mathbf {p} \cdot \mathbf {p} ={\frac {m_{0}^{2}\mathbf {u} \cdot \mathbf {u} }{1-{\frac {\mathbf {u} \cdot \mathbf {u} }{c^{2}}}}}={\frac {m_{0}^{2}u^{2}}{1-\left({\frac {u}{c}}\right)^{2}}}}$

then solving for u² and substituting into the Lorentz factor one obtains its alternative form in terms of 3-momentum and mass, rather than 3-velocity:

${\displaystyle \gamma ={\sqrt {1+\left({\frac {p}{m_{0}c}}\right)^{2}}}}$

Inserting this form of the Lorentz factor into the energy equation gives:

${\displaystyle E=m_0{c}^2\sqrt{1+}}$Zdroj:https://en.wikipedia.org?pojem=Relativistic_energy
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