Annus MirabilisEinstein in 1905
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mass energy

The equation explained

l∗=l1−vVcos⁡φ1−(vV)2l^* = l \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left(\frac{v}{V}\right)^2}}
In wordsThe energy of the light measured in the moving system is its energy in the system the light was described in first, times one less the ratio of the speed of the moving system to the speed of light times the cosine of the angle between the light's direction of travel and the x-axis, all of it divided by the square root of one less the square of that same ratio.OverviewThe same light carries different amounts of energy for two observers in relative motion, and how much depends on the direction the light travels. This is the one result the paper takes over from the electrodynamics paper, and everything after it is built on this line.Full explanationA system of plane light waves has energy ll referred to the coordinate system (x,y,z)(x, y, z), and its ray direction, the wave normal, makes the angle φ\varphi with the xx-axis. Now introduce a second system (ξ,η,ζ)(\xi, \eta, \zeta) in uniform parallel translation, whose origin moves with the speed vv along the xx-axis. The same quantity of light, measured in (ξ,η,ζ)(\xi, \eta, \zeta), has the energy l∗l^* this display gives, where VV is the speed of light. Two things vary. The square root in the denominator does not depend on direction and makes the measured energy larger. The factor 1−vVcos⁡φ1 - \frac{v}{V} \cos \varphi does depend on direction: light sent forward along the motion counts for less in the moving system, light sent backward for more. Einstein does not derive this here. He states that he derived it in section 8 of the electrodynamics paper, and says of it only: we make use of this result in what follows.Every stepl∗=l1−vVcos⁡φ1−(vV)2l^* = l \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left(\frac{v}{V}\right)^2}}The result as the paper prints it on page 639, taken over from section 8 of the electrodynamics paper. The steps below do not derive it; they read what it says in the three directions the argument will use.l∗=l1−vV1−(vV)2=l1−vV1+vVl^* = l \frac{1 - \frac{v}{V}}{\sqrt{1 - \left(\frac{v}{V}\right)^2}} = l \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}}Light sent along the xx-axis, in the direction the second system travels, has cos⁡φ=1\cos \varphi = 1. The square root factors as 1−vV1+vV\sqrt{1 - \frac{v}{V}} \sqrt{1 + \frac{v}{V}}, one of which cancels the numerator. The moving observer measures less energy than ll.l∗=l1+vV1−vVl^* = l \sqrt{\frac{1 + \frac{v}{V}}{1 - \frac{v}{V}}}Light sent the opposite way has cos⁡φ=−1\cos \varphi = -1, which turns the numerator into 1+vV1 + \frac{v}{V} and reverses the ratio. The moving observer measures more energy than ll. The paper uses this case and the one above together, since the body emits equally in two opposite directions.l∗=l1−(vV)2l^* = \frac{l}{\sqrt{1 - \left(\frac{v}{V}\right)^2}}Light sent across the motion, at right angles to the xx-axis in the system (x,y,z)(x, y, z), has cos⁡φ=0\cos \varphi = 0, and only the square root is left. Whatever the direction, the measured energy of a fixed quantity of light differs between the two systems.Historian's marginThe paper carries a footnote at this point, 'A. Einstein, Ann. d. Phys. 17. p. 891. 1905', which is the electrodynamics paper of the same year, and the text says the result was derived there in section 8. A second footnote adds that the principle of the constancy of the speed of light used there is contained in Maxwell's equations. VV is the letter the 1905 papers print for the speed of light; a reader today writes cc. The quantity transformed is the energy of a light complex, and the paper's whole argument needs nothing else from relativity.Also

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