Consider a source that is emitting isotropically in its own rest frame (denoted by primes):

$ \frac{dL'}{d\Omega'}=\frac{L'}{4\pi} $

where $ d\Omega' $ is a unit solid angle and $ L' $ is the luminosity.

Consider an observer moving relativistically at a speed $ \beta c $ compared to the source and at an angle $ \theta $ above the line of sight. If the source emits two photons at a time difference $ \Delta t $ in the observer frame. The difference in their arrival times will be:$ \Delta t _{obs}=(1-\beta cos(\theta))\Delta t $.

The difference of their emission times in the source frame is:

$ \Delta t'=\frac{\Delta t}{\Gamma}=\frac{\Delta t _{obs}}{\Gamma(1-\beta cos(\theta))}=D \Delta t _{obs} $

where $ \Gamma $ is the Lorentz factor of the source and $ D $ is the Doppler factor.

Since the phase difference of an electromagnetic wave must be relativistically invariant: $ \Delta \Phi=\omega' \Delta t'=\omega_{obs} \Delta t_{obs} $, we find that:

$ D=\frac{\nu}{\nu'}=\frac{\omega_{obs}}{\omega'}=\frac{\Delta t'}{\Delta t_{obs}} $.

Now, consider a photon emitted by the source towards the observer. Its speed is:

$ \beta_x=cos(\theta) \equiv \mu $ in the observer frame and $ \beta_x'=cos(\theta') \equiv \mu' $ in the source frame. Using the relativistic transformation of velocities, we obtain:

$ \mu= \frac{\mu'+\beta}{1+\beta \mu'} $

From which it follows that: $ d\mu=D^{-2} d\mu' $. Since $ d\phi=d\phi' $ we get that: $ d\Omega=D^{-2} D\Omega' $.

Finally we can transform the luminosity from the source to the observer frame:$ L_{\nu}\equiv\frac{dL}{d\nu}=\frac{4\pi dE}{dt d\nu d\Omega}=L'_{\nu'}\frac{dE d\Omega}{dE' d\Omega'}=D^3 L'_{\nu'} $.

Notice that $ D $ is maximal for $ \theta=0 $ for which $ D\approx 2\Gamma $ (assuming $ \Gamma \gg 1 $). In addition, for $ \theta=1/\Gamma $ we get $ cos (\theta)=\beta $ and:

$ D=\frac{1}{\Gamma(1-\beta^2)}=\frac{\sqrt{1-\beta^2}}{1-\beta^2}=\frac{1}{\sqrt{1-\beta^2}}=\Gamma $.

So at this angle the luminosity drops by a factor $ 2^3=8 $ compared to that for the forward direction. However, for significantly larger angles, $ 1-\beta cos(\theta) \ll 1 $ and the Doppler factor drops rapidly leading to an even sharper drop in luminsoity.

We conclude that the emission that was isotropic in the source frame, becomes confined to a cone of opening angle $ 1/\Gamma $ around the direction of motion of the source in the observer frame. This effect is known as relativistic beaming, and may lead to various selection effects when observing relativistic sources.