Four-momentum

A massive particle has four-momentumFor a massless particle, $P^{\mu }=(E/c,\mathbf{p})$ with $E=c|\mathbf{p}|$ and $P^2=0$.

For future-directed particles with total four-momentum $P$ and metric signature $(-+++)$, the positive square root of $-P_{\mu }{P}^{\mu }$ is their energy in the frame where total spatial momentum vanishes. Its invariant definition remains valid at a collision in spacetime.

The mass shell is the locus of four-momenta satisfying the relativistic dispersion relation for a fixed mass $m$. With mostly-minus Minkowski metric, $p^2=m^2$; with mostly-plus Minkowski metric, $p^2=-m^2$. The positive-energy sheet represents particle four-momenta. For $m=0$ this is the light cone. The condition of being on shell refers more generally to satisfying the field equations.

In an isolated relativistic interaction, the sum of incoming four-momenta equals the sum of outgoing four-momenta.

When an electron of Lorentz factor $\gamma >1$ annihilates with a stationary equal-mass positron, transverse four-momentum conservation makes the photons' transverse components opposite. Eliminating their positive frequencies gives the displayed relation for noncollinear photon angles to the incoming electron. The exactly collinear forward/backward configuration is physically allowed but makes the quotient $0/0$; its undivided cross-multiplied identity remains valid. The angle relation thus requires nondegeneracy or an appropriate limit, rather than an assertion that every allowed angle pair defines the quotient.

When two massive particles coalesce without additional outgoing radiation, four-momentum conservation makes the final four-momentum their sum. For collinear incident velocities, its rest mass is the positive square root of the displayed invariant, and its final velocity equals the center-of-momentum frame velocity $c^2{p}_{\mathrm{total}}/E_{\mathrm{total}}$. Relative kinetic energy can become internal rest energy, so the final rest mass is generally larger than the sum of the incident rest masses.

The rest energy of a particle of invariant mass $m$ is $m{c}^2$, the energy measured in its rest frame.

The center-of-momentum frame is an inertial frame in which the total spatial momentum of a system vanishes. For a two-particle initial state, the momenta are equal and opposite and the total invariant energy is $\sqrt{s}$.

The centre-of-momentum energy is the total energy measured in the centre-of-momentum frame. For a scattering process with total four-momentum $P$, it is the Lorentz-invariant quantity $\sqrt{P^2}=\sqrt{s}$ in the mostly-minus convention.

For a particle of rest mass $m$,

Rest mass is the invariant mass $m$ in the relativistic energy-momentum relation. It equals the energy measured in the particle's rest frame divided by $c^2$.

In an isolated relativistic collision, total four-momentum is conserved; its time component states conservation of total relativistic energy.

For $\mathbf{E}=(0,0,E)$, $\mathbf{B}=0$, and initial momentum $(p_0,0,0)$,With the initial position at the origin,

For three-momentum $p=\gamma mv$, force and acceleration satisfy

Under a constant force $F$ with $p(0)=0$, momentum is $p=Ft$ andThe speed approaches $c$ without reaching it at finite time.

For a rest-frame decay $1\to 2+3$,The decay is allowed exactly when $m_1\ge m_2+m_3$.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.

The Källén function controls two-body relativistic kinematics. In the rest frame of total invariant mass $\sqrt{s}$, the magnitude of each final momentum is $\sqrt{\kappa (s,m_1^2,m_2^2)}/(2\sqrt{s})$. It also determines the invariant flux factor in a two-particle collision.

For two-to-two scattering $A+B\to C+D$,They satisfy $s+t+u=m_A^2+m_B^2+m_C^2+m_D^2$ in units $c=1$.

With a specified ordering of external legs, this crossed channel carries the difference between an incoming and an outgoing energy-momentum vector. A one-particle exchange pole requires that difference to be on the exchanged particle's mass shell. Ordered amplitudes in one spatial dimension can label the crossed invariant using the difference of the two rapidity-parameterized external momenta; the external-leg convention fixes which crossed invariant is named $t$.

The direct channel combines the two incoming four-momentum vectors. A one-particle pole in this channel has the intermediate particle's squared mass equal to $s$. Imaginary rapidity differences can satisfy this mass relation for a bound state even though there is no pole at real physical scattering rapidity.