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 thicksource 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

- I(ν) = 2.h.ν
^{3}/(exp(h.ν/k/T) &minusl1)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

dilutedintogrey-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,

- T[observed]/T[emitted] = ν[observed]/ν[emitted] for a given photon.
[Note that the redshifts can be

Dopplerin origin,cosmologicalin origin,gravitationalin origin, or some inseparable mixture. All that matters is the fact that the parallel-transport law for a photon's 4-momentum,∇= 0, guarantees that the redshift ν[observed]/ν[emitted] is independent of frequency emitted.]_{p}p

Now, it has just been given (in the text preceding the exercises) that
N(ν) = I(ν)/ν^{3} 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.

Written by Eddy.