In a non-relativistic compressible fluid, the turbulent energy spectra are well-understood and appear to follow the Kolmogorov hypothesis. It would also appear that relativistic turbulence also follows the expected -5/3 slope in the inertial range of the spectra.
However, one point (of many) that I am unsure about, is how one could actually compute either a frequency spectrum or a wavenumber spectrum in a relativistic turbulent flow. What does a frequency or a wavenumber mean when the Lorentz factor leads to time dilation and length contraction?
Do frequencies appear smaller (time dilated) depending on the velocity of a particular eddy? Do wavenumbers appear larger depending on the velocity? It would seem that if there are "eddies" with velocity $u_1 \ll c$ and a wavenumber $\kappa_1$ and other "eddies" with velocity $u_2$ where $u_1 \ll u_2 < c$ with a wavenumber $\kappa_2$, it is possible that $\kappa_1 = \kappa_2$ with the Lorentz factor but $\kappa_1 \neq \kappa_2$ without it.
It looks like it must not change because studies show that turbulence still follows the $k^{-5/3}$ slope in the inertial range and if the Lorentz factor lead to shifts in the frequency or wave number, it seems extremely unlikely that the slope would match traditional turbulence. But on the other hand, relativistic speeds change both time and lengths, so I don't know how that doesn't influence the frequency or wavenumber depending on the velocity.
If I had my giant hotwire probe in space as some relativistic grid turbulence passed by, and my probe measures the velocity every 1 second, what am I really measuring in the fluid when the fluid has a different "clock"? And how do I use that signal to compute a frequency spectrum?
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