Albert Einstein "Does the Inertia of a Body Depend upon its Energy Content?" Harlow Shapley Annalen der Physik 18 639–641 1905

# THE E=Mc^{2} EQUATION

The results of my recently published electrodynamic 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 change do not depend on which of two systems of co-ordinates, in uniform motion of parallel translation relatively to each other, these alterations of state are referred to (principle of relativity).

With these principles^{1} as my basis I deduced *inter alia* the following result:—

Let a system of plane waves of light, referred to the system of co-ordinates (*x, y, z*), possess the energy *l;* let the direction of the ray (the wave-normal) make an angle ø 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 its origin of co-ordinates moving along the axis of *x* with the velocity *v*, then this quantity of light measured in the system (*ξ, η, ζ*)—possesses the energy

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 *E*_{0}. Let the energy of the body relative to the system (*ξ, η, ζ*), moving as above with the velocity *v*, be *H*_{0}.

Let this body send out, in a direction making an angle ø with the axis of *x,* plane waves of light, of energy ½*L* (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 *E*_{1} or *H*_{1} respectively, measured relatively to the system (*x, y, z*) or (*ξ, η, ζ*) respectively, then by employing the relation given above we obtain

By subtraction we obtain from these equations

The two differences of the form *H — E* occurring in this expression nave simple physical meanings. *H* and *E* are energy values of the same body referred to two systems of co-ordinates which are in motion relatively to each other, the body being at rest in one of the two systems (namely, (*x, y, z*)). Thus it is clear that the difference *H — E* can differ from the kinetic energy *K* of the body, with respect to the other system (*ξ, η, ζ*), only by an additive constant *C,* which depends on the choice of the arbitrary additive constants of the energies *H* and *E*. Thus we may place

since *C* does not change during the emission of light. So we have

The kinetic energy of the body with respect to (*ξ, η, ζ*) diminishes as a result of the emission of light, and the amount of diminution is independent of the properties of the body. Moreover, the difference *K*_{0} — *K*_{1}, like the kinetic energy of the electron, depends on the velocity.

Fig. 1. Einstein’s original statement of the mass-energy equivalence.
Neglecting magnitudes of fourth and higher orders we may place

From this equation it directly follows that:—

*If a body gives off the energy L in the form of radiation, its mass diminishes by L/c*^{2}. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference so that we are led to the more general conclusion that:

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

the energy being measured in ergs, and the mass in grams.

It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.

^{1} The principle of the constancy of the velocity of light is of course contained in Maxwell’s equations.