*Reference: *Einstein’s 1920 Book

*This paper presents Part 1, Chapter
15 from the book **RELATIVITY:
THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this
book by Henry Holt and Company, New York (1920).*

*The paragraphs of the original
material (in black) are accompanied by brief comments (in color) based on the present
understanding. Feedback on these comments is appreciated.*

*The heading below is linked to
the original materials.*

.

## General Results of the Theory

*NOTE: These
comments are temporary as more work is required. Better results are expected if
velocity is replaced by acceleration, kinetic energy is replaced by force, and
potential energy and mass are replaced by inertia.*

It is clear from our
previous considerations that the (special) theory of relativity has grown out
of electrodynamics and optics. In these fields it has not appreciably altered
the predictions of theory, but it has considerably simplified the theoretical
structure, *i.e. *the derivation of laws, and —what is incomparably more
important —it has considerably reduced the number of independent hypotheses
forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz
theory so plausible, that the latter would have been generally accepted by physicists
even if experiment had decided less unequivocally in its favour.

*The theory of
relativity has grown out of the effort to align the laws of electrodynamics
with the laws of mechanics. This goes back to Einstein’s paper on light quanta,
from which it may be inferred that radiation is a substance that quantizes into
matter. The laws of electrodynamics apply to radiation, whereas, the laws of
mechanics apply to matter.*

*The special theory
of relativity has succeeded in explaining the observed motions better than
classical mechanic, by somehow accounting for inertia. Classical mechanics assumes
inertia to be uniform because velocity is uniform. This is not so because
velocity can be different while uniform, and so can inertia. Not accounting of
inertia in classical mechanics is causing the error that is being corrected by
special theory of relativity. *

Classical mechanics
required to be modified before it could come into line with the demands of the
special theory of relativity. For the main part, however, this modification
affects only the laws for rapid motions, in which the velocities of matter ** v**are not very
small as compared with the velocity of light. We have experience of such rapid
motions only in the case of electrons and ions; for other motions the
variations from the laws of classical mechanics are too small to make themselves
evident in practice. We shall not consider the motion of stars until we come to
speak of the general theory of relativity. In accordance with the theory of
relativity the kinetic energy of a material point of mass

*m*is no longer given by the well-known expression

*mv*^{2}*/2,*

but by the expression

*mc ^{2} / *

*√(*

**1 – v**^{2}/c^{2}

*)*

*.**Velocity is different
for different reference-bodies, and it can be chosen arbitrarily. This makes
kinetic energy dependent on the reference-body too. This arbitrary dependency goes
away when acceleration is chosen in place of velocity; force is chosen in place of kinetic energy,
and inertia is chosen in place of potential energy and mass.*

This expression
approaches infinity as the velocity ** v**approaches the velocity of light

**. The velocity must therefore always remain less than**

*c***, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain**

*c**Each velocity seems to represent a certain “potential force”, or inertia that is in balance. This is the case with both v and c. The value of c is very high because the inertia in balance is very small.*

When ** v^{2}/c^{2}**
is small compared with unity, the third of these terms is always small in
comparison with the second, which last is alone considered in classical mechanics.
The first term

*mc*

^{2}does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-mass depends on the velocity. We shall speak of its essential significance later.

*The first term (mc ^{2})
is more representative of the inertia of the body. The second term seems to be
more representative of force in some manner.*

The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about, and what meaning is to be attached to it.

*Faraday had
combined the two conservation laws of mass and energy into one single law of the
conservation of force. In Faraday’s framework mass corresponds to inertia.
Active energy corresponds to force.*

The principle of
relativity requires that the law of the conservation of energy should hold not only
with reference to a co-ordinate system ** K**,
but also with respect to every co-ordinate system

**which is in a state of uniform motion of translation relative to**

*K’***, or, briefly, relative to every “Galileian” system of co-ordinates. In contrast to classical mechanics, the Lorentz transformation is the deciding factor in the transition from one such system to another.**

*K**Lorenz
transformations somehow account for the difference in inertia, when velocities
are different.*

By means of
comparatively simple considerations we are led to draw the following conclusion
from these premises, in conjunction with the fundamental equations of the
electrodynamics of Maxwell: A body moving with the velocity ** v**, which absorbs

^{1}an amount of energy

*E*

_{0}*in the form of radiation without suffering an alteration in velocity in the process, has, as a consequence, its energy increased by an amount*

*E _{0} / *

*√(*

**1 – v**^{2}/c^{2}

*)*

*.*^{1} ** E_{0}
**is the energy taken up, as judged from a co-ordinate system moving with the
body.

*This is more like
an increase in inertia with the absorption of radiation force.*

In consideration of the expression given above for the kinetic energy of the body, the required energy of the body comes out to be

Thus the body has the same energy as a body of mass **( m + E**

_{0}**/**

*c*

^{2}

*)**moving with the velocity*

**. Hence we can say: If a body takes up an amount of energy**

*v*

*E***, then its inertial mass increases by an amount**

_{0}

*E*_{0}**/**

*c***; the inertial mass of a body is not a constant, but varies according to the change in the energy of the body. The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy. Writing the expression for the energy in the form**

^{2}we see that the term ** mc^{2}**, which has hitherto attracted our attention, is
nothing else than the energy possessed by the body

^{2}before it absorbed the energy

**.**

*E*_{0}^{2} As
judged from a co-ordinate system moving with the body.

*The special theory
of relativity interprets mass as the configuration of substance formed from an equivalent
amount of energy. In the relationship E _{0} = m_{0}c^{2 },
however, c cannot be taken to its limiting value of infinity. A finite solution
must exists when c is taken to infinity.*

A direct comparison
of this relation with experiment is not possible at the present time, owing to the
fact that the changes in energy ** E_{0}**to which we can
subject a system are not large enough to make themselves perceptible as a
change in the inertial mass of the system.

*E*_{0}**/**

*c***is too small in comparison with the mass**

^{2}*m*, which was present before the alteration of the energy. It is owing to this circumstance that classical mechanics was able to establish successfully the conservation of mass as a law of independent validity.

Let me add a final
remark of a fundamental nature. The success of the Faraday-Maxwell interpretation
of electromagnetic action at a distance resulted in physicists becoming
convinced that there are no such things as instantaneous actions at a distance
(not involving an intermediary medium) of the type of Newton’s law of gravitation.
According to the theory of relativity, action at a distance with the velocity
of light always takes the place of instantaneous action at a distance or of
action at a distance with an infinite velocity of transmission. This is
connected with the fact that the velocity *c *plays a fundamental rôle in
this theory. In Part II we shall see in what way this result becomes modified
in the general theory of relativity.

.