Relativistic beaming

Angular concentration and Doppler enhancement of radiation from a relativistically moving emitter. Frequency, photon arrival rate and apparent solid angle transform together; inferred brightness depends on viewing angle and the emission geometry.

For an unresolved, isotropically emitting rest-frame blob with power-law spectrum $F_{\nu }^{\prime }\propto {\nu }^{-\alpha }$, the observed spectral flux density at a fixed frequency is enhanced by $D^{3+\alpha }$. The factors follow from Lorentz-invariant specific intensity and the frequency shift. The bolometric or corresponding-logarithmic-bin luminosity transforms with $D^4$. A spatially steady continuous jet has a different equal-arrival-time sampling and can have exponent $2+\alpha$ instead.

The photon phase-space occupation is a Lorentz scalar. For radiation propagating through vacuum, this makes specific intensity divided by cubed frequency invariant under a change of inertial observer. With Doppler factor $D$, $I_{\nu }(\nu )=D^3{I}_{{\nu }^{\prime }}^{\prime }(\nu /D)$. This describes corresponding ray directions; it does not assert that an angularly integrated radiation field is isotropic in both frames.

An approaching jet can be enhanced and its receding counterpart suppressed by different Doppler factors. A discrete moving emitter with rest-frame power-law spectrum $F_{\nu }\propto {\nu }^{-s}$ has the usual boost ${\delta }_D^{3+s}$, whereas a steady continuous-jet convention changes the exponent. Absorption and environmental asymmetry can also affect the contrast.

For velocity $\beta c$ at angle $\theta$ toward the observer, ${\delta }_D=[\Gamma (1-\beta \cos \theta ){]}^{-1}$. Frequencies transform as $\nu ={\delta }_D{\nu }^{\prime }$ before cosmological redshift. Specific-intensity transformation follows invariance of $I_{\nu }/{\nu }^3$.