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A.~Einstein, Ann. d. Phys., {\bf 17,} 891 \hfill {\large \bf 1905}\\
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{\bf Does the Inertia of a Body Depend on its Energy Content?}\\
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A.~Einstein\\
Bern\\
(Received 1905)
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The results of the previous investigation lead to a very interesting concl
usion, 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\footnote{The principle of the constancy of
the velocity of light is of course contained in Maxwell's equations.}
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 $\phi$ with the axis of $x$ of the system.
If we introduce a new system of co-ordinates $(\xi, \eta, \zeta)$
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
($\xi, \eta, \zeta$) --- possesses the energy
$$
l^{\ast} = l \cdot \frac{1 - \frac{\displaystyle v}{\displaystyle V}
\cos \phi}{\sqrt{1 - (v/V)^2}},
$$
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 $E_0$.
Let the energy of the body relative to the system $(\xi, \eta, \zeta$),
moving as above with the velocity $v$, be $H_0$.
Let this body send out, in a direction making an angle $\phi$ 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 $E_1$ or $H_1$ respectively, measured relatively to
the system ($x, y, z$) or
($\xi, \eta, \zeta$) respectively, then by employing the relation given above
we obtain
$$
E_0 = E_1 + \left( \frac{L}{2} + \frac{L}{2} \right),
$$
$$
H_0 = H_1 + \left[ \frac{L}{2} \cdot \frac{1 -
\frac{\displaystyle v}{\displaystyle V} \cos \phi}{\sqrt{1 -
(v/V)^2}} + \frac{L}{2} \cdot \frac{1 + \frac{\displaystyle v}{\displaystyle V}
\cos \phi}{\sqrt{1 - (v/V)^2}}
\right] = H_1 + \frac{L}{\sqrt{1 - (v/V)^2}}.
$$
By subtraction we obtain from these equations
$$
\left( H_0 - E_0 \right) - \left( H_1 - E_1 \right) = L \cdot \left\{
\frac{1}{\sqrt{1 - (v/V)^2}} -1 \right\}.
$$
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 ($\xi, \eta, \zeta$), 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
$$
H_0 - E_0 = K_0 + C,
$$
$$
H_1 - E_1 = K_1 + C,
$$
since $C$ does not change during the emission of light. So we have
$$
K_0 - K_1 = L \cdot \left\{ \frac{1}{\sqrt{1 - (v/V)^2}} - 1 \right\}.
$$
The kinetic energy of the body with respect to ($\xi, \eta, \zeta$)
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.
Neglecting magnitudes of fourth and higher orders we may place
$$
K_0 - K_1 = \frac{L}{V^2} \cdot \frac{v^2}{2}.
$$
From this equation it directly follows that:
If a body gives off the energy $L$ in the form of radiation, its mass
diminishes by $L/V^2$. 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
$L/9 \times 10^{20}$, 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.\\
%SB. = ENCODED 15 OCT 1998 BY NIS;
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