special relativity

The equation explained

\frac{E'}{E} = \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .

In wordsFor an observer moving along the direction of the light, the energy of the light complex in the moving system divided by its energy in the stationary system is the square root of this quotient: one minus the speed over the velocity of light, divided by one plus it.OverviewThe energy of a light complex for an observer moving along the direction of the light: it changes by exactly the factor by which the frequency changes.Full explanationWith

\varphi = 0

the general formula simplifies as the Doppler formula did in § 7. The paper then remarks that the energy and the frequency of a light complex change with the observer's state of motion according to the same law, and leaves the remark there. At

v = 0.6 \, V

both are halved.Every step

\frac{E'}{E} = \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}

The energy ratio for any angle, the previous display.

\frac{E'}{E} = \frac{1 - \frac{v}{V}}{\sqrt{\left( 1 - \frac{v}{V} \right) \left( 1 + \frac{v}{V} \right)}}

Put

\varphi = 0

and factor the difference of squares under the root.

\frac{E'}{E} = \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}}

Cancel one square root of

1 - \frac{v}{V}

. This is the factor of the Doppler formula for the same direction (§ 7, p. 912).AlsoOverviewFull explanationEvery step

Read it in the paper