Does the Inertia of a Body Depend on Its Energy-content?

September 27, 1905

My principle of relativity states that any changes to 2 systems of coordinates that are in uniform motion parallel to each other are governed by laws which do not depend on either of them.

I combined the following to answer the question whether inertia depends on energy-content:

• the Maxwell-Hertz equations for empty space
• the Maxwellian expression for the electromagnetic energy of space
• my principle of relativity

Let:

• a system of plane waves of light, referred to the system of co-ordinates (x, y, z), possess the energy l.
• the direction of the ray (the wave-normal) make an angle φ with the axis of x of the system.

We introduce a new system of coordinates (ξ, η, ζ) moving in uniform parallel translation with respect to the system (x, y, z).

The origin of its co-ordinates are in motion along the axis of x with the velocity v.

Then this quantity of light in the system (ξ, η, ζ) possesses the energy l

…

where c denotes the speed of light.

Let there be a stationary body in the system (x, y, z).

• Its energy is E0.
• Its energy relative to the system (ξ, η, ζ) moving as above with the velocity v, is H0.

This body send out plane waves of light of energy 1/2 L.

• These waves are in a direction making an angle φ with the axis of x
• This is measured relatively:
  • to (x, y, z), and
  • simultaneously to an equal quantity of light in the opposite direction.*

The body remains at rest with respect to the system (x, y, z).

Energy must apply to this process and to both systems of co-ordinates via Relativity.

After the emission of light E1 or H1 respectively, we measure the energy of the body relatively to the system (x, y, z)`` or (ξ, η, ζ)` respectively. We obtain:

…

By subtraction we obtain from these equations:

…

The two differences of the form H − E in this expression have simple physical significations.

H and E are energy values of the same body referred to 2 systems of co-ordinates which are in motion relative to each other.

The body is at rest in one of the two systems (system (x, y, z)).

Thus, 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.

This C depends on the choice of the arbitrary additive constants of the energies H and E.

Thus we may place:

H0 − E0 = K0 + C,
H1 − E1 = K1 + C,

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.

The amount of reduction is independent of the properties of the body.

Moreover, the difference K0 − K1, like the kinetic energy of the electron (§ 10), depends on the velocity.

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 as radiation, its mass reduces by L/c2.

The fact that the energy withdrawn from the body becomes energy of radiation makes no difference.

This leads us 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 L/9 × 1020, the energy being measured in ergs, and the mass in grams*.

This theory may be tested with bodies whose energy-content is very variable (e.g. with radium salts).

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