Annus MirabilisEinstein in 1905
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mass energy

The equation explained

E0=E1+[L2+L2],E_0 = E_1 + \left[ \frac{L}{2} + \frac{L}{2} \right],
In wordsThe body's energy before it emits, measured in the system it rests in, equals its energy after plus the two halves of the energy carried off by the light.OverviewThe body sends out two equal amounts of light in opposite directions and stays at rest. In the system where it rests, the books balance in the plainest way: the energy it loses is the energy the light carries off.Full explanationA body rests in the system (x,y,z)(x, y, z) and has the energy E0E_0 there. It now emits plane light waves of energy L2\frac{L}{2} in a direction making the angle φ\varphi with the xx-axis, and at the same time an equal quantity of light in the opposite direction. Because the two emissions are equal and opposite, the body stays at rest in (x,y,z)(x, y, z). Call E1E_1 its energy after the emission, in the same system. The principle of energy conservation then reads as this display: what the body had is what it has left plus what the two light complexes carry away. The bracket holds the two emissions separately rather than their sum LL, and that is deliberate: in the moving system of the next display they no longer have equal energies, and each must be transformed on its own.Every stepE0E_0The energy of the body before the emission, referred to the system (x,y,z)(x, y, z) in which it rests. The paper assigns no absolute value to it, and never needs one.L2+L2\frac{L}{2} + \frac{L}{2}The two light complexes, each of energy L2\frac{L}{2} measured in (x,y,z)(x, y, z), one at the angle φ\varphi to the xx-axis and one in the opposite direction. LL is the name for what they carry off together.E0=E1+[L2+L2]E_0 = E_1 + \left[ \frac{L}{2} + \frac{L}{2} \right]The energy principle applied in (x,y,z)(x, y, z): the energy before equals the energy left in the body plus the energy of the light.E0−E1=LE_0 - E_1 = LThe two halves add to LL, so in the body's own system the drop in its energy is exactly the energy emitted. On its own this says nothing about mass; the second system is what makes the argument.Historian's marginThe paper states that the body remains at rest in (x,y,z)(x, y, z) during the emission, which is why an unchanged body energy may be compared before and after. This holds because the two light complexes are equal and opposite. A single pulse would set the body moving and the comparison would not be between two states of rest.Also

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