Does the Inertia of a Body Depend on its Energy Content?
[list]A. Einstein
Bern
(Received 1905)[/list:u]
The results of the previous investigation lead to a very interesting conclusion, which is here to be deduced.
I based that investigation on the Maxwell-Hertz equations for empty space, together with the Maxwellian expression for the electromagnetic energy of space, and in addition the principle that:
The laws by which the states of physical systems alter are independent
of the alternative, to which of two systems of coordinates, in uniform motion
of parallel translation relatively to each other, these alterations of state are
referred (principle of relativity).
With these principles as my basis I deduced inter alia the following result:
Let a system of plane waves of light, referred to the system of coordinates (x, y, z), possess the energy l; let the direction of the ray (the wave-normal) make an angle φ with the axis of x of the system. If we introduce a new system of co-ordinates (ξ, η, ζ) moving in uniform parallel translation with respect to the system (x, y, z), and having its origin of coordinates in motion along the axis of x with the velocity v, then this quantity of light-measured in the system (ξ, η, ζ) — possesses the energy
$$l^* = l·\frac{(1-v/Vcosφ)}{\sqrt{1−(v/V)^2}},$$
where c denotes the velocity of light. We shall make use of this result in what follows.
Let there be a stationary body in the system (x, y, z), and let its energy referred to the system (x, y, z) — be E0. Let the energy of the body relative to the system (ξ, η, ζ), moving as above with the velocity v, be H0. Let this body send out, in a direction making an angle φ with the axis of x, plane waves of light, of energy L/2 measured relatively to (x, y, z), and simultaneously an equal quantity of light in the opposite direction. Meanwhile the body remains at rest with respect to the system (x, y, z). The principle of energy must apply to this process, and in fact (by the principle of relativity) with respect to both systems of co-ordinates. If we call the energy of the body after the emission of light E1 or H1 respectively, measured relatively to the system (x, y, z) or (ξ, η, ζ) respectively, then by employing the relation given above we obtain…