English translation · The unsectioned argument · explanatory reading

Translation

Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?

The results of an electrodynamic investigation recently published by me in these Annalen^[1)] lead to a very interesting conclusion, which is to be derived here.

There I took as a basis the Maxwell-Hertz equations for empty space, together with Maxwell’s expression for the electromagnetic energy of space, and in addition the principle:

The laws according to which the states of physical systems change are independent of which of two coordinate systems, in uniform motion of parallel translation relative to each other, these changes of state are referred to (principle of relativity).

Relying on these foundations^[2)] I derived, among other things, the following result (l. c. § 8):

Let a system of plane light waves, referred to the coordinate system $(x, y, z)$ , possess the energy $l$ ; let the ray direction (wave normal) make the angle $\varphi$ with the $x$ -axis of the system. If one introduces a new coordinate system $(\xi, \eta, \zeta)$ , in uniform parallel translation relative to the system $(x, y, z)$ , whose origin moves with the velocity $v$ along the $x$ -axis, then the quantity of light in question — measured in the system $(\xi, \eta, \zeta)$ — possesses the energy: $l^* = l \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left(\frac{v}{V}\right)^2}},$ Energy of one light pulse, moving frameEnergy of one light pulse, rest frameSigned observer speedSpeed of lightPropagation angle (stationary system)Explain this equation where $V$ denotes the velocity of light. We shall make use of this result in what follows.

Now let there be in the system $(x, y, z)$ a body at rest whose energy — referred to the system $(x, y, z)$ — is $E_0$ . Relative to the system $(\xi, \eta, \zeta)$ , moving as above with the velocity $v$ , let the energy of the body be $H_0$ .

Let this body emit, in a direction making the angle $\varphi$ with the $x$ -axis, plane light waves of energy $L/2$ (measured relative to $(x, y, z)$ ), and at the same time an equal quantity of light in the opposite direction. In this process the body remains at rest with respect to the system $(x, y, z)$ . The energy principle must hold for this process, and indeed (by the principle of relativity) with respect to both coordinate systems. If we call $E_1$ and $H_1$ the energy of the body after the emission of light, measured relative to the system $(x, y, z)$ and $(\xi, \eta, \zeta)$ respectively, then, using the relation given above, we obtain: $E_0 = E_1 + \left[ \frac{L}{2} + \frac{L}{2} \right],$ Body energy before emission, rest descriptionBody energy after emission, rest descriptionEmitted energy in the body's rest frameExplain this equation $\begin{aligned} H_0 &= H_1 + \left[ \frac{L}{2} \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left(\frac{v}{V}\right)^2}} + \frac{L}{2} \frac{1 + \frac{v}{V} \cos \varphi}{\sqrt{1 - \left(\frac{v}{V}\right)^2}} \right] \\ &= H_1 + \frac{L}{\sqrt{1 - \left(\frac{v}{V}\right)^2}}. \end{aligned}$ Body energy before emission, moving descriptionBody energy after emission, moving descriptionEmitted energy in the body's rest frameSigned observer speedSpeed of lightEmission angleExplain this equation By subtraction one obtains from these equations: $(H_0 - E_0) - (H_1 - E_1) = L \left\{ \frac{1}{\sqrt{1 - \left(\frac{v}{V}\right)^2}} - 1 \right\}.$ Body energy before emission, moving descriptionBody energy before emission, rest descriptionBody energy after emission, moving descriptionBody energy after emission, rest descriptionEmitted energy in the body's rest frameSigned observer speedSpeed of lightExplain this equation The two differences of the form $H - E$ occurring in this expression have simple physical meanings. $H$ and $E$ are energy values of the same body, referred to two coordinate systems moving relative to each other, the body being at rest in one of the two systems (system $(x, y, z)$ ). It is therefore clear that the difference $H - E$ can differ from the kinetic energy $K$ of the body with respect to the other system (system $(\xi, \eta, \zeta)$ ) only by an additive constant $C$ , which depends on the choice of the arbitrary additive constants of the energies $H$ and $E$ . We can therefore set: $\begin{aligned} H_0 - E_0 &= K_0 + C, \\ H_1 - E_1 &= K_1 + C, \end{aligned}$ Body energy before emission, moving descriptionBody energy before emission, rest descriptionEnergy of motion before emissionUnchanged additive energy offsetBody energy after emission, moving descriptionBody energy after emission, rest descriptionEnergy of motion after emissionExplain this equation since $C$ does not change during the emission of light. We therefore obtain: $K_0 - K_1 = L \left\{ \frac{1}{\sqrt{1 - \left(\frac{v}{V}\right)^2}} - 1 \right\}.$ Energy of motion before emissionEnergy of motion after emissionEmitted energy in the body's rest frameSigned observer speedSpeed of lightExplain this equation The kinetic energy of the body with respect to $(\xi, \eta, \zeta)$ decreases as a result of the emission of light, and indeed by an amount that is independent of the properties of the body. The difference $K_0 - K_1$ , moreover, depends on the velocity in the same way as the kinetic energy of the electron (l. c. § 10).

Neglecting quantities of the fourth and higher orders, we can set: $K_0 - K_1 = \frac{L}{V^2} \frac{v^2}{2}.$ Energy of motion before emissionEnergy of motion after emissionEmitted energy in the body's rest frameSpeed of lightSigned observer speedExplain this equation From this equation it follows immediately:

If a body gives off the energy $L$ in the form of radiation, its mass diminishes by $L/V^2$ . Here it is evidently immaterial that the energy withdrawn from the body turns into energy of radiation in particular, so that we are led to the more general conclusion:

The mass of a body is a measure of its energy content; if the energy changes by $L$ , the mass changes in the same sense by $L/9 \cdot 10^{20}$ , when the energy is measured in ergs and the mass in grams.

It is not ruled out that with bodies whose energy content is variable to a high degree (e.g. with the radium salts) a test of the theory will succeed.

If the theory corresponds to the facts, radiation transfers inertia between the emitting and absorbing bodies.

Bern, September 1905.

(Received 27 September 1905.)

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Does the Inertia of a Body Depend on Its Energy Content?