Translation:On the Theory of Radiation in Moving Bodies. Correction

On the Theory of Radiation in Moving Bodies. Correction  (1905)
by Friedrich Hasenöhrl, translated from German by Wikisource

On the Theory of Radiation in Moving Bodies. Correction.

By Fritz Hasenöhrl.

In a paper of same title published a short time ago [1], I have stated the concept of an apparent mass of cavity radiation, and as its value I have given the quantity[2]

${\displaystyle {\frac {8}{3}}{\frac {h\epsilon _{0}}{c^{2}}}}$

where ${\displaystyle h\epsilon _{0}}$ is the amount of radiation energy contained in the stationary cavity, and ${\displaystyle c}$ is the speed of light. Namely, this value was only valid when quantities of order ${\displaystyle \beta ^{4}}$ were neglected.

Now, M. Abraham was so kind to report to me by letter a new method for the calculation of this mass, yet which gives a different result.

Here, I state the simple method of Abraham with his permission, while I am using the notation of my cited paper. The total relative radiation in the cavity is given by[3]

${\displaystyle 2\pi i\sin \psi \ d\psi =2\pi i_{0}{\frac {c}{c'\cos \alpha }}\sin \psi \ d\psi }$

The absolute radiation that corresponds to it, is:

 1) ${\displaystyle 2\pi i_{0}\left({\frac {c}{c'}}\right)^{4}\sin \varphi \ d\varphi }$.[4]

Now, according to Abraham, the density of the electromagnetic momentum is equal to the absolute radiation divided by ${\displaystyle c^{2}}$.[5] Thus if we calculate the total electromagnetic momentum which coincides with the direction of the system and which is contained in the cavity, we have to calculate expression (1) with ${\displaystyle cos\varphi }$, then to integrate with respect to ${\displaystyle \varphi }$ from O to ${\displaystyle \pi }$, and then to multiply the result with the volume of cavity ${\displaystyle h}$. If we additionally substitute for ${\displaystyle c'}$ its value[6], then the momentum becomes

${\displaystyle {\begin{array}{rl}G&={\frac {2\pi i_{0}}{c^{2}}}h{\overset {\pi }{\underset {0}{\int }}}{\frac {\sin \varphi \cos \varphi \ d\varphi }{(1+\beta ^{2}-2\beta \cos \varphi )^{2}}}\\\\&={\frac {\epsilon _{0}h}{c}}\left({\frac {1}{2\beta }}{\frac {1+\beta ^{2}}{(1-\beta ^{2})^{2}}}-{\frac {1}{4\beta ^{2}}}\log {\frac {1+\beta }{1-\beta }}\right).\end{array}}.}$

Now, the longitudinal electromagnetic mass is given by ${\displaystyle {\tfrac {1}{c}}{\tfrac {dG}{d\beta }}}$;[7] thus it becomes equal to (when higher terms are neglected):

${\displaystyle {\frac {4}{3}}{\frac {h\epsilon _{0}}{c^{2}}}.}$

This is half of the value given by me.

After it was sought in vain after a principal difference, I found that this difference stems from a calculation error, unfortunately committed by me in my paper. At p. 362, line 6 from above,

not ${\displaystyle {\frac {2\beta _{1}}{c^{2}(1-\beta _{1}^{2})^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\dots }$ shall be stated, but ${\displaystyle {\frac {4\beta _{1}}{c^{2}(1-\beta _{1}^{2})^{2}}}{\overset {\pi /2}{\underset {0}{\int }}}\dots }$,

therefore the heat absorbed by the walls of the cavity when the system is accelerated, is:

${\displaystyle Q=h\epsilon _{0}\left(\varkappa _{1}-2\delta \beta {\frac {\partial \varkappa _{1}}{\partial \beta }}\right);}$

however, since furthermore the walls have given off the heat ${\displaystyle h\epsilon _{0}\varkappa _{1}}$, we can say that the walls of the cavity (when accelerated by ${\displaystyle \delta _{\tau }}$) have altogether given off the heat

${\displaystyle 2h\epsilon _{0}\delta \beta {\frac {\partial \varkappa }{\partial \beta }}=2h\epsilon _{0}\delta \varkappa }$

Therefore, the amount ${\displaystyle (2\varkappa -1)}$ stems from the total radiating energy in the moving cavity:

${\displaystyle h\epsilon _{0}(1-\beta ^{2})^{-2}\qquad (=h\epsilon _{0}(\varkappa +\tau ))}$

thus

${\displaystyle h\epsilon _{0}\left({\frac {\beta ^{2}}{1-\beta ^{2}}}+{\frac {1}{2\beta }}\log {\frac {1+\beta }{1-\beta }}\right)=\varkappa 'h\epsilon _{0}}$

from the heat supply of the walls, while the amount

 ${\displaystyle h\epsilon _{0}((1-\beta ^{2})^{-2}-\varkappa ')=h\epsilon _{0}\left({\frac {1-\beta ^{2}+\beta ^{4}}{(1-\beta ^{2})^{2}}}-{\frac {1}{2\beta }}\log {\frac {1+\beta }{1-\beta }}\right)=\tau 'h\epsilon _{0}}$

is gained from the work.

This result is now in full agreement with the one of Abraham. Because the work which was spent to bring the system to a certain velocity, can be calculated from momentum by

${\displaystyle \int w\ dt\ {\frac {dG}{dt}}={\overset {\beta }{\underset {0}{\int }}}wd\beta {\frac {dG}{d\beta }};}$

if one inserts for ${\displaystyle G}$ its value, then the integration indeed provides the value ${\displaystyle h\epsilon _{0}\tau '}$.

If one neglects magnitudes beginning with order ${\displaystyle \beta ^{4}}$, then it is

${\displaystyle {\begin{array}{lr}\varkappa '=&1+{\frac {4}{3}}\beta ^{2},\\\\\tau '=&{\frac {2}{3}}\beta ^{2}.\end{array}}.}$

(These values are related to quasi-stationary, reversible velocity changes; twice of the work must be spent at sudden accelerations of the system. In the latter case, one obtains ${\displaystyle {\tfrac {8}{3}}{\tfrac {\epsilon _{0}}{c^{2}}}}$ for the apparent mass; the relevant calculation executed by me in an earlier work[8], is free of the mentioned calculation error. Though the concept of an apparent mass is probably to be confined to quasi-stationary motions.)

The following thermodynamic considerations remain unchanged in principle; one only has to understand by ${\displaystyle \varkappa }$ the corrected value ${\displaystyle \varkappa '}$. However, now the contraction hypothesis of Lorentz and Fitzgerald doesn't suffice anymore as the solution of the resulting contradiction with the second thermodynamic theorem, so one would still have to add the hypothesis, that the true emission capability of a black body explicitly depends on the motion, namely by the factor ${\displaystyle 1-{\tfrac {2}{3}}\beta ^{2}}$, a hypothesis whose possibility I already considered in the mentioned paper.

If it is generally assumed, that the dimensions of matter (in the direction of motion) have to be multiplied by ${\displaystyle \lambda }$, and the true emission capability with ${\displaystyle \sigma }$, then it must be

${\displaystyle \lambda ^{-3/4}\cdot \sigma ^{-1}=1+{\frac {4}{3}}\beta ^{2}}$

Vienna, January 1905.