Four-momentum conservation

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

In an isolated collinear completely inelastic capture, add incoming total energies and signed momenta. Their invariant fixes the composite rest mass, including internal excitation. If it later converts all energy to two opposite axial photons, the two conservation equations immediately give the displayed photon energies. The laboratory frequency ratio is $(E+Pc)/(E-Pc)$, not generally one; equality holds in the composite rest frame.

In energy units $m=m_{\mathrm{in}}{c}^2$ and $M=M_A{c}^2$, capture of a photon of energy $e$ by a massive particle into one resonance satisfies $E-P\cos \theta =(M^2-m^2)/(2e)=2E_0$, with $P=pc$. Head-on incidence gives the largest invariant mass at fixed $E$, so the threshold solves $E+P=2E_0$. Using $E^2-P^2=m^2$ gives the title expression. It is the minimum over incident angle; above that energy the exact angle is fixed by the invariant equation, not arbitrary. The soft-photon regime $2E_0\gg m$ excludes exactly co-propagating incidence.

When $e\ll m\ll E_0$, the incoming massive-particle energy is $E\simeq 2E_0/(1-\cos \theta )$. Momentum addition then gives $\tan \varphi \simeq [e/(2E_0)]\sin \theta (1-\cos \theta )$. Differentiating the angular factor gives its maximum at $\cos \theta =-1/2$. This is an asymptotic result for both the energy-angle constraint and final deflection; corrections are controlled by $m^2/E_0^2$ and $e/E_0$.

For a photon and a massive particle with zero total momentum, their momenta are opposite and equal. Conservation of total energy fixes the same momentum magnitude before and after elastic scattering, independently of direction. A right-angle photon turn therefore makes the recoiling particle move perpendicular to the original photon direction in that frame. Transform both velocity components, or the complete four-momentum, to obtain the laboratory recoil; a transverse component is reduced by the boost Lorentz factor when the particle has zero longitudinal velocity in the zero-momentum frame.

If the total incoming four-momentum is timelike and has centre-of-momentum energy $W$, production of particles with rest masses ${\mu }_i$ requires $W\ge {\sum }_i{\mu }_i{c}^2$. In the centre-of-momentum frame, total final energy is a sum of $\sqrt{{\mu }_i^2{c}^4+c^2|{\mathbf{p}}_i{|}^2}$, each at least its rest energy. For massive products equality occurs with all at rest in that frame. A single product requires equality of the entire invariant mass, not just an inequality; a massless product cannot have nonzero energy and be at rest.

For an incident particle of rest mass $m_1$ and total energy $E_1$ hitting a stationary target of mass $m_2>0$, four-momentum conservation gives $W^2=m_1^2{c}^4+m_2^2{c}^4+2m_2{c}^2{E}_1$. Applying the relativistic collision mass threshold gives the displayed necessary condition. The incoming energy must also satisfy $E_1\ge m_1{c}^2$; other conserved quantum numbers or reaction dynamics can impose further restrictions.

For a photon of energy $h\nu$ hitting a stationary target of mass $m_2$, the incoming invariant energy satisfies $W^2=m_2^2{c}^4+2m_2{c}^{2}h\nu$. The massive multi-product threshold is the displayed value when ${\sum }_i{\mu }_i>m_2$. If the algebraic bound is nonpositive, the lower limit of allowed positive frequencies can be zero. A single final product instead fixes the frequency by invariant-mass equality, and a final massless particle can make a nominal threshold an infimum rather than an attained value.

// Target: classical-mechanics.bigb

For a photon incident on a stationary electron, producing an additional electron and positron leaves three massive particles in the final state. Four-momentum conservation gives $W^2=m_e^2{c}^4+2m_e{c}^2{E}_{\gamma }$ for the centre-of-momentum energy. Its minimum is $3m_e{c}^2$, so the incident threshold is $4m_e{c}^2$. At threshold all final particles are at rest in their centre-of-momentum frame, which moves at $4c/5$ relative to the target laboratory frame. This differs from a heavy-target threshold because the electron target's recoil cannot be neglected.

A photon hitting a resting mass $m$ has squared total energy invariant $m^2{c}^4+2m{c}^{2}E$. Producing a larger mass $M$ and an outgoing photon requires this invariant to exceed $M^2{c}^4$ if the photon has positive energy. The displayed threshold is therefore the limiting infimum as the final photon becomes soft, not an attained physical state with a zero-energy photon. Above threshold the outgoing photon energy in the centre-of-momentum frame is positive.

An incident particle of rest mass $m$ and energy $E$ striking a stationary identical target has total squared invariant energy $W^2=2m{c}^2(E+m{c}^2)$. Producing $N\ge 2$ final particles of the same rest mass requires $W\ge Nm{c}^2$, giving the threshold. Equality means no final kinetic energy in the centre-of-momentum frame. From the Lorentz factor, the corresponding incident speed is at least $cN\sqrt{N^2-4}/(N^2-2)$.

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.