DOES THE INERTIA OF A
BODY DEPEND
UPON ITS ENERGY-CONTENT?
By A. Einstein
September 27, 1905
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 (§ 8):--
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 with
the axis of x of the system. If we introduce a new system of
co-ordinates () 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 ()
--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 E0. Let
the energy of the body relative to the system () moving as
above with the velocity v, be H0.
Let this body send out, in a direction making an angle with
the axis of x, plane waves of light, of energy ½L 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 () 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 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 it is clear that the difference H - E can
differ from the kinetic energy K of the body, with respect
to the other system (), 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 () 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 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 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.
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