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Section eighteen Special and general principle of
relativity. The basal principle, which was

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the pivot of all our previous considerations, was the special principle of relativity,

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i e. The principle of the
physical relativity of all uniform motion. Let

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us once more analyze its meaning carefully. It was at all times clear that,

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from the point of view of the
idea it conveys to us, every

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motion must be considered only as a
relative motion. Returning to the illustration we

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have frequently used of the embankment and
the railway carriage, we can express the

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fact of the motion here taking place
in the following two forms, both of

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which are equally justifiable. A.
The carriage is in motion relative to the

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embankment. B. The embankment is
in motion relative to the carriage. In

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A the embankment n b the carriage
serves as the body of reference in our

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statement of the motion taking place.
If it is simply a question of detecting

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or of describing the motion involved,
it is in principle immaterial to what reference

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body we refer the motion. As
already mentioned, this is self evident,

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but it must not be confused with
the much more comprehensive statement called the principle

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of relativity, which we have taken
as the basis of our investigations. The

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principle we have made use of not
only maintains that we may equally well choose

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the carriage or the embankment as our
reference body for the description of any event,

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for this too is self evident.
Our principle rather asserts what follows.

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If we formulate the general laws of
nature as they are obtained from experience,

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by making use of a the embankment
as reference body, b the railway carriage

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as reference body, then these general
laws of nature e g. The laws

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of mechanics or the law of the
propagation of light in vacuo have exactly the

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same form in both cases. This
can also be expressed as follows for the

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physical description of natural processes. Neither
of the reference bodies k k prime is

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unique literally specially marked out as compared
with the other. Unlike the first,

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this latter statement need not of necessity
hold a priori. It is not contained

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in the conceptions of motion and reference
body, and derivable from them. Only

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experience can decide as to its correctness
or incorrectness. Up to the present However,

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we have by no means maintained the
equivalence of all bodies of reference K

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in connection with the formulation of natural
laws. Our course was more on the

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following lines. In the first place, we started out from the assumption that

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there exists a reference body K,
whose condition of motion is such that the

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Galilean law holds with respect to it. A particle left to itself and sufficiently

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far removed from all other parts,
moves uniformly in a straight line with reference

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decay Galileyan reference body. The laws
of nature were to be as simple as

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possible, but in addition to K, all bodies of reference K prime should

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be given preference in this sense,
and they should be exactly equivalent to K

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for the formulation of natural laws,
provided that they are in a state of

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uniform, rectilinear and non rotary motion
with respect decay. All these bodies of

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reference are to be regarded as Galilean
reference bodies. The validity of the principle

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of relativity was assumed only for these
reference bodies, but not for others,

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e g. Those possessing motion of
a different kind. In this sense,

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we speak of the special principle of
relativity or special theory of relativity. In

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contrast to this, we wish to
understand, by the general principle of relativity

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the following statement, all bodies of
reference, K, K, prime,

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etc. Are equivalent for the description
of natural phenomena or formulation of the general

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laws of nature, whatever may be
their state of motion. But before proceeding

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farther, it ought to be pointed
out that this formulation must be replaced later

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by a more abstract one, for
reasons which will become evident at a later

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stage. Since the introduction of the
special principle of relativity has been justified,

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every intellect which strives after generalization must
feel the temptation to venture the step towards

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the general principle of relativity. But
a simple and apparently quite reliable consideration seems

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to suggest that for the present,
at any rate, there is little hope

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of success in such an attempt.
Let us imagine ourselves transferred to our old

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friend the railway carriage, which is
traveling at a uniform rate. As long

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as it is moving uniformly, the
occupant of the carriage is not sensible of

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its motion, and it is for
this reason that he can, without reluctance

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interpret the facts of the case as
indicating that the carriage is at rest,

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but the embankment in motion. Moreover, according to the special principle of relativity,

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this interpretation is quite justified also from
a physical point of view. If

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the motion of the carriage is now
changed into a non uniform motion, as

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for instance, by a powerful application
of the breaks, then the occupant of

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the carriage experiences a correspondingly powerful jerk
forwards. The retarded motion is manifested in

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the mechanical behavior of bodies. Relative
to the person in the railway carriage,

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the mechanical behavior is different from that
of the case previously considered, and for

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this reason it would appear to be
impossible that the same mechanical laws hold relatively

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to the non uniformly moving carriage as
hold with reference to the carriage when at

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rest or in uniform motion. At
all events, it is clear that the

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Galileyan law does not hold with respect
to the non uniform moving carriage. Because

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of this, we feel compelled at
the present juncture to grant a kind of

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absolute physical reality to non uniform motion
in opposition to the general principle of relativity.

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But in what follows we shall soon
see that this conclusion cannot be maintained

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Section nineteen the gravitational field. If
we pick up a stone and then let

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it go, why does it fall
to the ground. The usual answer to

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this question is because it is attracted
by the Earth. Modern physics formulates the

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answer rather differently, for the following
reason. As a result of the more

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careful study of electromagnetic phenomena, we
have come to regard action at a distance

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as a process impossible without the intervention
of some intermediary medium. If, for

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instance, a magnet attracts a piece
of iron, we cannot be content to

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regard this as meaning that the magnet
acts directly on the iron through the intermediate

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empty space. But we are constrained
to imagine, after the manner of Faraday,

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that the magnet always calls into being
something physically real in the space around

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it, that something being what we
call a magnetic field. In its turn,

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this magnetic field operates on the piece
of iron, so that the latter

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strives to move towards the magnet.
We shall not discuss here the justification for

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this incidental conception, which is indeed
a somewhat arbitrary one. We shall only

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mention that With its aid, electromagnetic
phenomena can be theoretically represented much more satisfactorily

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than without it, and this applies
particularly to the transmission of electromagnetic waves.

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The effects of gravitation also are regarded
in an analogous manner. The action of

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the Earth on the stone takes place
indirectly. The Earth produces in its surroundings

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a gravitational field, which acts on
the stone and produces its motion of fall.

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As we know from experience, the
intensity of the action on a body

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diminishes according to a quite definite law
as we proceed farther and farther away from

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the Earth from our point of view. This means the law governing the properties

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of the gravitational field in space must
be a perfectly definite one in order correctly

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to represent the diminution of gravitational action
with the distance from operative bodies. It

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is something like this. The body
e g. The Earth, produces a

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field in its immediate neighborhood directly.
The intensity and direction of the field at

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points farther removed from the body are
thence determined by the law which governs the

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properties and space of the gravitational fields
themselves. In contrast, to electric and

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magnetic fields, the gravitational field exhibits
a most remarkable property which is of fundamental

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importance for what follows. Bodies which
are moving under the sole influence of a

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gravitational field receive an acceleration which does
not in the least depend either on the

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material or on the physical state of
the body. For instance, a piece

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of lead and a piece of wood
fall in exactly the same manner and a

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gravitational field in vacuo when they start
off from rest or with the same initial

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velocity. This law, which holds
most accurately, can be expressed in a

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different form in the light of the
following consideration. According to Newton's law of

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motion, we have force equals inertial
mass times acceleration, where the inertial mass

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is a characteristic constant of the accelerated
body. If now gravitation is the cause

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of the acceleration, we then have
force equals gravitational mass times intensity of the

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gravitational field, where the gravitational mass
is likewise a characteristic constant for the body.

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From these two relations, follows acceleration
equals the fraction gravitational mass over inertial

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mass times intensity of the gravitational field. If now, as we find from

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experience, the acceleration is to be
independent of the nature and the condition of

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the body, and always the same
for a given gravitational field, then the

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ratio of the gravitational to the inertial
mass must likewise be the same for all

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bodies. By a suitable choice of
units, we can thus make this ratio

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equal to unity. We then have
the following law, The gravitational mass of

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a body is equal to its inertial
mass. It is true that this important

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law had hitherto been recorded in mechanics, but it had not been interpreted.

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A satisfactory interpretation can be obtained only
if we recognize the following fact. The

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same quality of a body manifests itself
according to the circumstances as inertia or as

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weight literally heaviness. In the following
section we shall show to what extent this

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is actually the case, and how
this question is connected with the general postulate

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of relativity. Section twenty the equality
of inertial and gravitational mass. As an

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argument for the general postulate of relativity. We imagine a large portion of empty

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space, so far removed from stars
and other appreciable masses that we have before

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us approximately the conditions required by the
fundamental law of Galiley. It is then

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possible to choose a Galileyan reference body
for this part of space world, relative

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to which points at rest remain at
rest, and points in motion continue permanently

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in uniform rectilinear motion as reference body. Let us imagine a spacious chest resembling

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a room with an observer inside,
who is whipped with apparatus. Gravitation naturally

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does not exist for this observer.
He must fasten himself with strings to the

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floor, otherwise the slightest impact against
the floor will cause him to rise slowly

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toward the ceiling of the room.
To the middle of the lid of the

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chest is fixed externally a hook with
rope attached, And now a being,

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what kind of a being is immaterial
to us, begins pulling at this with

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a constant force. The chest,
together with the observer, then begin to

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move upwards with a uniformly accelerated motion. In course of time, their velocity

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will reach unheard of values, provided
that we are viewing all this from another

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reference body, which is not being
pulled with a rope. But how does

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the man in the chest regard the
process, the acceleration of the chest will

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be transmitted to him by the reaction
of the floor of the chest. He

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must therefore take up this pressure by
means of his legs, if he does

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not wish to be laid out full
length on the floor, or he is

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then standing in the chest in exactly
the same way as anyone stands in a

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room of a house on our earth. If he release a body which he

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previously had in his hand, the
acceleration of the chest will no longer be

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transmitted to this body, and for
this reason the body will approach the floor

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of the chest with an accelerated relative
motion. The observer will further convince himself

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that the acceleration of the body towards
the floor of the chest is always of

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the same magnitude, whatever kind of
body he may happen to use for the

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experiment, Relying on his knowledge of
the gravitational field as it was discussed in

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the preceding section, The man in
the chest will thus come to the conclusion

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that he and the chest are in
a gravitational field which is constant with regard

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to time. Of course, he
will be puzzled for a moment as to

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why the chest does not fall in
this gravitational field. Just then, however,

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he discovers the hook in the middle
of the lid of the chest and

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the rope which is attached to it, and he consequently comes to the conclusion

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that the chest is suspended at rest
in the gravitational field. Ought we to

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smile at the man and say that
he errs in his conclusion. I do

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not believe we ought to. If
we wish to remain consistent, we must

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rather admit that his mode of grasping
the situation violates neither reason nor known mechanical

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laws, even though it is being
accelerated with respect to the Galilean space.

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First considered, we can nevertheless regard
the chest as being at rest. We

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have thus good grounds for extending the
principle of relativity to include bodies of reference

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which are accelerated with respect to each
other, and as a result we have

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gained a powerful argument for a generalized
postulate of relativity. We must note carefully

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that the possibility of this mode of
interpretation rests on the fundamental property of the

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gravitational field of giving all bodies the
same acceleration, or what comes to the

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same thing. On the law of
the equality of inertial and gravitational mass.

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If this natural law did not exist, the man in the accelerated chest would

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not be able to interpret the behavior
of the bodies around him on the supposition

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of a gravitational field, and he
would not be justified on the grounds of

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experience in supposing his reference body to
be at rest. Suppose that the man

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in the chest fixes a rope to
the inner side of the lid, and

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that he attaches a body to the
free end of the rope. The result

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of this will be to stretch the
rope so that it will hang vertically downwards.

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If we ask for an opinion of
the cause of tension in the rope,

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the man in the chest will say, the suspended body experiences a downward

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force in the gravitational field, and
this is neutralized by the tension of the

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rope. What determines the magnitude of
the tension of the rope is the gravitational

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mass of the suspended body. On
the other hand, an observer who is

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poised freely in space will interpret the
condition of things. Thus, the rope

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must perforce take part in the accelerated
motion of the chest, and it transmits

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this motion to the body attached to
it. The tension of the rope is

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just large enough to effect the acceleration
of the body. That which determines the

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magnitude of the tension of the rope
is the inertial mass of the body.

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Guided by this example, we see
that our extension of the principle of relativity

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implies the necessity of the law of
the equality of inertial and gravitational mass.

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Thus we have obtained a physical interpretation
of this law. From our consideration of

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the accelerated chest, we see that
a general theory of relativity must yield important

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results on the laws of gravitation.
In point of fact, the systematic pursuit

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of the general idea of relativity has
supplied the laws satisfied by the gravitational field.

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Before proceeding farther, however, I
must warn the reader against a misconception

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suggested by these considerations. A gravitational
field exists for the man in the chest,

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despite the fact that there was no
such field for the coordinate system first

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chosen. Now we might easily suppose
that the existence of a gravitational field is

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always only an apparent one. We
might also think that regardless of the kind

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of gravitational field which may be present, we could always choose another reference body

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such that no gravitational field exists with
reference to it. This is by no

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means true for all gravitational fields,
but only for those of quite special form.

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It is, for instance, impossible
to choose a body of reference such

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that, as judged from it,
the gravitational field of the Earth in its

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entirety vanishes. We can now appreciate
why that argument is not convincing, which

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we brought forward against the general principle
of relativity at the end of section eighteen.

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It is certainly true that the observer
in the railway carriage experiences a jerk

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forwards as a result of the application
of the break, and that he recognizes

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in this the non uniformity of motion
or retardation of the carriage. But he

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is compelled by nobody to refer this
jerk to a real acceleration or retardation of

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the carriage. He might also interpret
his experience thus my body of reference,

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the carriage remains permanently at rest with
reference to it. However, there exists,

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during the period of application of the
breaks a gravitational field which is directed

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forwards, and which is variable with
respect to time. Under the influence of

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this field, the embankment, together
with the Earth, moves non uniformly in

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such a manner that their original velocity
in the backwards direction is continuously reduced.

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End of Section twenty