Clearly, the blackbody spectrum is derived using quantum considerations; however It should be entirely feasible that given the blackbody distribution and considering isotropic radiation, one can calculate the classical stress energy tensor (or actually more useful for me the Faraday tensor).

Strange thing is, I can't figure out how to begin the calculation since the Fields are zero statistically at some given time. Any idea of where to start?

In cosmology it's pretty typical to consider radiation as a fluid with energy density and pressure, I was just curious is there was something a bit more quantitative.


The energy-momentum tensor of the black-body radiation is the same as that of a perfect fluid, i.e.

\begin{equation}\tag{1} T^{\mu\nu} = (\rho_0 + p_0)\frac{dx^\mu}{ds}\frac{dx^{\nu}}{ds} + g^{\mu\nu}p_0, \end{equation}

with \begin{equation}\tag{2} \rho_0 = 3p_0. \end{equation} where $\rho_0$ is the proper macroscopic density of the radiation at the point of interest, $p_0$ its proper pressure, and the velocities $dx^\mu/ds$ correspond to the macroscopic motion of the radiation.

The applicability of the energy-momentum tensor of a perfect fluid to the black-body radiation is justified by the fact that in relativistic mechanics any system whose local properties can be specified by the two scalars $\rho_0$ and $p_0$ has such an energy-momentum tensor. And this is surely the case for the black-body radiation.

A more rigorous derivation is based on the electromagnetic nature of the black-body radiation, using the covariant form of the Maxwell’s equations.

A very good treatment of this you can find in the Tolman and Ehrenfest paper:

  Tolman R.C., Ehrenfest P., *Phys. Rev.*, **35**, 1791, (1930) 
  • $\begingroup$ The important point that is missing from this answer is that for a photon gas, $p_0=(1/3)\rho_0$. You might want to edit your answer to say this. $\endgroup$ – user4552 Nov 30 '18 at 20:40
  • $\begingroup$ @Ben Crowell Absolutely right ! The black- body radiation is essentially a special perfect fluid with $\rho_0 = 3p_0$ , where the subscript 0 means the quantities are measured in the local rest frame. Thanx $\endgroup$ – kderakhshani Nov 30 '18 at 23:43
  • $\begingroup$ Thank you very much for the reference. I was aware (as stated in the question) of the fluid treatment, I'll take a look at Tolman and Ehrenfest's paper. It's really the Faraday Tensor I'm interested in but I suppose we could find the most general allowed form going backwards from the fluid tensor (with zero trace of course as Crowell mentioned) $\endgroup$ – R. Rankin Dec 1 '18 at 3:55

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