Annus MirabilisEinstein in 1905
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light quanta

The equation explained

∫0∞ϱνdν=RN8πL3T∫0∞ν2dν=∞.\int_0^\infty \varrho_\nu d \nu = \frac{R}{N} \frac{8 \pi}{L^3} T \int_0^\infty \nu^2 d \nu = \infty.
In wordsThe radiation's energy per unit volume and unit frequency, added up over every frequency from zero upward, equals the gas constant over the number of molecules in a gram-equivalent, times eight π over the speed of light cubed, times the temperature, times the frequency squared added up over every frequency; and that is infinite.OverviewAdd up the density over all frequencies and the radiation's total energy comes out infinite: the wider the range of resonator frequencies, the more energy the space holds, without limit. On this picture ether and matter can never settle into a definite share of the energy.Full explanationThis is the paper's second objection to the equilibrium density. The radiation's energy per unit volume is the density ϱν\varrho_\nu added up over all frequencies. With the density of the preceding line, everything except ν2\nu^2 is the same at every frequency and comes out in front, leaving ∫0∞ν2dν\int_0^\infty \nu^2 d \nu. Up to any highest frequency that integral is a third of the highest frequency cubed, so it grows without bound as the range widens: allowing frequencies ten times higher multiplies the energy a thousandfold. Over all frequencies the total is infinite. The paper concludes that in this picture 'there can be no question of a definite distribution of energy between ether and matter'.Every step∫0∞ϱνdν\int_0^\infty \varrho_\nu d \nuϱνdν\varrho_\nu d \nu is the energy per unit volume between ν\nu and ν+dν\nu + d \nu. Adding these pieces over every frequency gives the radiation's whole energy per unit volume.∫0∞RN8πν2L3Tdν\int_0^\infty \frac{R}{N} \frac{8 \pi \nu^2}{L^3} T d \nuPut in the density that the equilibrium condition gave (the preceding display).RN8πL3T∫0∞ν2dν\frac{R}{N} \frac{8 \pi}{L^3} T \int_0^\infty \nu^2 d \nuRR, NN, LL and TT are the same at every frequency, so they come out of the integral; only ν2\nu^2 stays inside.13ν3\frac{1}{3} \nu^3Up to a highest frequency ν\nu, the integral of the frequency squared is a third of ν3\nu^3, since ν3\nu^3 grows by 3ν2dν3 \nu^2 d \nu over each small range. It grows without limit as the highest frequency is raised.RN8πL3T∫0∞ν2dν=∞\frac{R}{N} \frac{8 \pi}{L^3} T \int_0^\infty \nu^2 d \nu = \inftyWith no highest frequency the integral has no finite value, so neither has the energy: the printed conclusion.Historian's marginEinstein gives this divergence no name. The name it later carried in textbooks, the 'ultraviolet catastrophe', is Ehrenfest's, from 1911 (P. Ehrenfest, Annalen der Physik 36, 91-118). The paper's own words are that the relation lacks agreement with experience and allows no definite distribution of energy between ether and matter.Also

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