Page:Zur Thermodynamik bewegter Systeme (Fortsetzung).djvu/5

Then instead of $\phi$, $\phi '$ is the angle read by a co-moving observer at a transporter, whose dimensions are contracted in the direction of motion in the ratio $1:\varkappa$. If we set

$i\ \sin \phi \ \cos \phi \ d\phi =i'\ \sin \phi '\ \cos \phi '\ d\phi ',$

then $i'$ is the true radiation intensity observed by the co-moving observer, whose measuring rods have experienced the mentioned contraction.

The quantity $i'$ must be constant, i.e. independent of angle $\phi '$, i.e. the true radiation is uniformly distributed into all directions in the contracted system; it obeys Lambert's $\cos$-law. Then two arbitrary oriented, equally moving surface elements are radiating the same amount of heat to each other. A mirror brought into a cavity doesn't change the distribution of radiation, since the ordinary reflection laws hold for the relative ray path in the contracted system. (This was shown in the most general way by H. A. Lorentz^[1] and can be directly proven in this case.)

From (20) it easily follows:

$\sin \phi '\cos \phi '\ d\phi '=\sin \phi \cos \phi \ d\phi {\frac {\varkappa ^{2}}{(1-\beta ^{2}\sin ^{2}\phi )^{2}}},$

thus it is:

$i=i'{\frac {\varkappa ^{2}}{(1-\beta ^{2}\sin ^{2}\phi )^{2}}}=i'{\frac {(1-\beta ^{2}\cos ^{2}\phi ')^{2}}{\varkappa ^{2}}}$.[2]

1. H. A. Lorentz, Versl. kon. Akad. v. Wetensch. Amsterdam, 7, p. 507 (1899) and 12, p. 886 (1904). – See also M. Abraham, Theorie der Elektrizität, II, p. 282 (1905).
2. The absolute radiation intensity can be calculated from this equation. It is equal to

   $J_{abs}=i\left({\frac {c}{c'}}\right)^{4}$

   (see. F. Hasenöhrl, Ann. d. Phys., 16, p. 589 [1905]), where

   ${\frac {c'}{c}}={\sqrt {1+\beta ^{2}-2\beta \cos \varphi }}$

   and $\varphi$ is again the direction of the absolute rays. Thus it is