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Photon absorption removes a photon and transfers its energy and momentum to matter. It contributes a negative photon-number collision source. It differs from Compton scattering, which redistributes a photon's energy and direction while preserving the count. The integrated source can vanish if photon emission and absorption balance.

Photon emission creates a photon, for example in bremsstrahlung or the decay of an excited material state. It produces a positive photon-number collision source when not balanced by inverse processes. The photon number current is conserved only if its integrated creation and destruction rates cancel.

For the metric in vector cosmological perturbation, an orthonormal tetrad is $E_0=a^{-1}{\partial }_{\eta }$, $E_i=a^{-1}(B_i{\partial }_{\eta }+{\partial }_i)$ to first order. A photon has $p^0=(ϵ/a^2)(1+B_i{e}^i)$ and $p^i=(ϵ/a^2)e^i$, so the covariant component is $p_0=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$. The derivative of the scale factor cancels by the null vector condition, proving the redshift formula.

In an orthonormal tetrad, $J^{\hat{0}}=\int f{d}^{3}p$ is the photon number density and $J^{\hat{i}}=\int f{e}^i{d}^{3}p$ is its number flux. Integrating the Boltzmann equation gives ${\nabla }_{\mu }{J}^{\mu }=\int C[f]d^{3}p/E$ when momentum-boundary terms vanish. Collisionless propagation and photon-number-conserving scattering give zero divergence. Photon emission, photon absorption, electron-positron annihilation in cosmology, bremsstrahlung and double Compton scattering generally do not. This is a number current, not an electric charge current: photons are neutral.

For isotropic planar photon emission in a rest frame, aberration of light maps its forward half of the angular directions into a lab cone of semi-angle $\alpha$ satisfying the displayed equation. This follows from $\cos \theta =(\cos {\theta }^{\prime }+\beta )/(1+\beta \cos {\theta }^{\prime })$ at ${\theta }^{\prime }=\pm \pi /2$. For a fast positive Lorentz boost, $\alpha \sim 1/\gamma$. The half fraction concerns photon count or directions: the relativistic Doppler effect changes the angular energy weights, so the cone need not contain half the radiant energy.

Two orthonormal transverse polarization vectors span the plane perpendicular to a nonzero momentum. Summing their dyadic products gives the spatial transverse projector. It has trace and rank two and acts as the identity on physical photon polarizations.

For a real on shell photon of null momentum $k$, a physical polarization satisfies $k\cdot ϵ=0$ and is defined modulo $ϵ\mapsto ϵ+\alpha k$. The two-dimensional quotient carries the two physical photon helicities. In a transverse gauge choose ${ϵ}^0=0$ and ${ϵ}^{\ast }\cdot ϵ=-1$. A Ward identity ensures that replacing an external polarization by its momentum gives zero in the summed physical amplitude.

For two photons emitted simultaneously a distance $d$ apart in one frame and moving in the same positive direction, the world-line intercept $x-ct$ transforms to $\gamma (1+v/c)(x-ct)$. Their equal-time separation in the boosted frame is therefore the displayed constant $d^{\prime }$. It is different from length contraction of the emitters, because the emission events are not simultaneous after the boost.

For photons in thermal equilibrium at temperature $T$, the number density per frequency interval is

A polariton is a hybrid light–matter excitation formed by strong coupling between a photon and a material excitation. For an exciton-polariton, the material component is an exciton.

For a free electromagnetic field with suitable boundary conditions, combine Coulomb gauge ${\partial }_i{A}_i=0$ with $A_0=0$. Gauss's law makes the second condition consistent in the source-free problem. The remaining field operator is spatially transverse and describes the two radiative photon polarizations. A constrained scalar potential must be restored when charges are present.

Equating radiation pressure to cold electron degeneracy pressure gives $T=[3P_e(\rho )/a_{\mathrm{rad}}{]}^{1/4}$. Because the two Electron pressure limits scale as ${\rho }^{5/3}$ and ${\rho }^{4/3}$, this boundary has logarithmic slopes $5/12$ and $1/3$, respectively. Above it, photon pressure can dominate even when the net Electrons remain degenerate; degeneracy of one component and dominance of the total pressure are different criteria. Thermal pairs modify the high-temperature pressure comparison.

The Lorentz transformation changes the photon four-momentum and therefore its frequency, producing the displayed formula for a boost along the first axis. Here the primed ray angle is measured in the emitting frame and $\beta =v/c$. The angle and frequency transforms must be used together: aberration of light changes the direction, and a photon-number angular fraction need not equal the corresponding radiant-energy fraction.

A charged scalar field coupled to an Abelian gauge field through $D_{\mu }={\partial }_{\mu }-ie{a}_{\mu }$. Under $\varphi \mapsto e^{i\alpha }\varphi$ and $a_{\mu }\mapsto a_{\mu }+e^{-1}{\partial }_{\mu }\alpha$, its gauge covariant derivative transforms with the same phase. The potential determines whether the perturbative spectrum has a massless photon or exhibits the Higgs mechanism.

A temperature versus density diagram separates classical, degenerate, nonrelativistic and relativistic stellar-matter limits. The Electron degeneracy boundary is $T\sim T_F(\rho )$, with slope $2/3$ at low density and $1/3$ at high density. Thermal relativity is controlled by $k_{B}T\sim m_e{c}^2$, while relativity of degenerate Electrons is controlled by $p_F\sim m_{e}c$. Photon pressure and thermal electron-positron pairs add further crossovers; all depend on composition and are gradual, not thermodynamic phase boundaries.

The physical Thomson scattering rate equals the inverse photon mean scattering time. Comparing it with the Hubble parameter gives a local approximate decoupling criterion. A more precise last-scattering temperature is obtained from optical depth and the visibility maximum; these conventions need not give exactly the same temperature.