Shifted thermal radiation

The cosmological microwave background radiation is the most perfect fit yet seen to the theoretical form of black-body radiation as studied in the 1800s and explained by Planck and Einstein at the start of the next century. One of its more interesting features is that it's been massively red-shifted, by the universe's expansion, since it was emitted (about 13.7 Gyr ago); but its spectrum is still that of black-body radiation, despite the red-shifting, albeit at a greatly reduced temperature. One of the exercises (22.17, p588) in Misner, Thorne and Wheeler addresses the fact that red-shifting one black-body spectrum yields another:

An optically thick source of black-body radiation (e.g., the surface of a start, or the hot matter filling the universe shortly after the big bank) emits photons isotropically with a specific intensity, as seen by an observer at rest near the source, given (Planck Radiation law) by

Here T is the temperature of the source. Show that any observer, in any local Lorenz frame, anywhere in the universe, who examines this radiation as it flows past him, will also see a black-body spectrum. Show, further, that if he calculates a temperature by measuring the specific intensity I(ν) at any one frequency, and if he calculates a temperature from the shape of the spectrum, those temperatures will agree. (Radiation remains black body rather than being diluted into grey-body.) Finally, show that the temperature he measures is red-shifted by precisely the same factor as the frequency of any given photon is redshifted,

[Note that the redshifts can be Doppler in origin, cosmological in origin, gravitational in origin, or some inseparable mixture. All that matters is the fact that the parallel-transport law for a photon's 4-momentum, ∇p p = 0, guarantees that the redshift ν[observed]/ν[emitted] is independent of frequency emitted.]

Now, it has just been given (in the text preceding the exercises) that N(ν) = I(ν)/ν3 is invariant, from observer to observer and, along the world-line of any given photon, from event to event. The Planck Radiation law gives N(ν).(exp(h.ν/k/T) −1) = 2.h and the rule for red-shifting is that all of the light has its frequency changed by the same factor. (That is, if the Fourier transform of the spectrum starts out as F, the red-shifting it by a factor r yields (: F(r.ν) ←ν :) as the Fourier transform of the shifted spectrum.)

Our observer gets to record I(ν) at any frequency ν and can, from it, infer N(ν); if the radiation has been red-shifted by factor r since its emission, this is equal to the originally emitted value of N(ν/r), which is 2.h/(exp(h.ν/k/T/r) −1), which is exactly the value of N(ν) that would arise from a local source at temperature r.T. This being, plainly, true at all frequencies we obtain the results claimed.

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