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

Superphysics Note: This is a simplified version. The equations will be fixed in the future. We just want to know what Einstein is really trying to say.

September 27, 1905

My principle of relativity states that any changes to two 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.

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 coordinates (ξ, η, ζ) moving in uniform parallel translation with respect to the system (x, y, z), and having its origin of co-ordinates 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

…

where c denotes the speed of light.

Let there be a stationary body in the system (x, y, z). 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 1/2 L.

• This is 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).

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

E0 = E1 + 
L + 1
L,
H0 = H1 +

L1 −
vc
cosφ
p
1 − v
2/c2
+
1
2
L
1 + v
c
cosφ
p
1 − v
2/c2
= H1 +
L
p
1 − v
2/c2.

By subtraction we obtain from these equations H0 − E0 − (H1 − E1) = L (1 p

1 − v
2/c2
− 1
)

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

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 (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, which 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,

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

K0 − K1 = L (
1
p
1 − v
2/c2
− 1
)

The kinetic energy of the body with respect to (ξ, η, ζ) diminishes as a result of the emission of light. The amount of diminution 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:

K0 − K1 =
1
2
L
c
2
v
2

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/c2.

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 L/9 × 1020, the energy being measured in ergs, and the mass in grammes.

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.

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