 Solution

Choose the source motion along $+x$, write $\beta =v/c$, and let primes denote its instantaneous rest frame. A photon with rest-frame energy $E^{\prime }$ and direction ${\theta }^{\prime }$ has $p_x^{\prime }=E^{\prime }\cos {\theta }^{\prime }/c$ and transverse momentum $E^{\prime }\sin {\theta }^{\prime }/c$. A Lorentz boost to the laboratory givesSince $|\mathbf{p}|=E/c$, the relativistic aberration relation and its inverse areThe mapping is monotone. The boundary of the rest-frame forward hemisphere, $\cos {\theta }^{\prime }=0$, therefore maps to $\cos \theta =\beta$. An isotropic photon count assigns exactly half the photons to that hemisphere, provingThe photon count inside a relativistic beaming cone concerns an emitted ensemble and counts photons, not their total energy or the arrival-time-weighted power at a detector. The laboratory photon energies depend on direction, so “half the photons” need not mean half the radiated energy. At large $\gamma$ the angular scale is ${\theta }_{1/2}\simeq 1/\gamma$.