  We present an exact calculation of the power spectrum of the electromagnetic
fluctuations in a relativistic equilibrium plasma described by
Maxwell-J\"uttner distribution functions. We consider the cases of wave vectors
parallel or normal to the plasma mean velocity. The relative contributions of
the subluminal and supraluminal fluctuations are evaluated. Analytical
expressions of the spatial fluctuation spectra are derived in each case. These
theoretical results are compared to particle-in-cell simulations, showing a
good reproduction of the subluminal fluctuation spectra.
