Section eighteen Special and general principle of relativity. The basal principle, which was
the pivot of all our previous considerations, was the special principle of relativity,
i e. The principle of the physical relativity of all uniform motion. Let
us once more analyze its meaning carefully. It was at all times clear that,
from the point of view of the idea it conveys to us, every
motion must be considered only as a relative motion. Returning to the illustration we
have frequently used of the embankment and the railway carriage, we can express the
fact of the motion here taking place in the following two forms, both of
which are equally justifiable. A. The carriage is in motion relative to the
embankment. B. The embankment is in motion relative to the carriage. In
A the embankment n b the carriage serves as the body of reference in our
statement of the motion taking place. If it is simply a question of detecting
or of describing the motion involved, it is in principle immaterial to what reference
body we refer the motion. As already mentioned, this is self evident,
but it must not be confused with the much more comprehensive statement called the principle
of relativity, which we have taken as the basis of our investigations. The
principle we have made use of not only maintains that we may equally well choose
the carriage or the embankment as our reference body for the description of any event,
for this too is self evident. Our principle rather asserts what follows.
If we formulate the general laws of nature as they are obtained from experience,
by making use of a the embankment as reference body, b the railway carriage
as reference body, then these general laws of nature e g. The laws
of mechanics or the law of the propagation of light in vacuo have exactly the
same form in both cases. This can also be expressed as follows for the
physical description of natural processes. Neither of the reference bodies k k prime is
unique literally specially marked out as compared with the other. Unlike the first,
this latter statement need not of necessity hold a priori. It is not contained
in the conceptions of motion and reference body, and derivable from them. Only
experience can decide as to its correctness or incorrectness. Up to the present However,
we have by no means maintained the equivalence of all bodies of reference K
in connection with the formulation of natural laws. Our course was more on the
following lines. In the first place, we started out from the assumption that
there exists a reference body K, whose condition of motion is such that the
Galilean law holds with respect to it. A particle left to itself and sufficiently
far removed from all other parts, moves uniformly in a straight line with reference
decay Galileyan reference body. The laws of nature were to be as simple as
possible, but in addition to K, all bodies of reference K prime should
be given preference in this sense, and they should be exactly equivalent to K
for the formulation of natural laws, provided that they are in a state of
uniform, rectilinear and non rotary motion with respect decay. All these bodies of
reference are to be regarded as Galilean reference bodies. The validity of the principle
of relativity was assumed only for these reference bodies, but not for others,
e g. Those possessing motion of a different kind. In this sense,
we speak of the special principle of relativity or special theory of relativity. In
contrast to this, we wish to understand, by the general principle of relativity
the following statement, all bodies of reference, K, K, prime,
etc. Are equivalent for the description of natural phenomena or formulation of the general
laws of nature, whatever may be their state of motion. But before proceeding
farther, it ought to be pointed out that this formulation must be replaced later
by a more abstract one, for reasons which will become evident at a later
stage. Since the introduction of the special principle of relativity has been justified,
every intellect which strives after generalization must feel the temptation to venture the step towards
the general principle of relativity. But a simple and apparently quite reliable consideration seems
to suggest that for the present, at any rate, there is little hope
of success in such an attempt. Let us imagine ourselves transferred to our old
friend the railway carriage, which is traveling at a uniform rate. As long
as it is moving uniformly, the occupant of the carriage is not sensible of
its motion, and it is for this reason that he can, without reluctance
interpret the facts of the case as indicating that the carriage is at rest,
but the embankment in motion. Moreover, according to the special principle of relativity,
this interpretation is quite justified also from a physical point of view. If
the motion of the carriage is now changed into a non uniform motion, as
for instance, by a powerful application of the breaks, then the occupant of
the carriage experiences a correspondingly powerful jerk forwards. The retarded motion is manifested in
the mechanical behavior of bodies. Relative to the person in the railway carriage,
the mechanical behavior is different from that of the case previously considered, and for
this reason it would appear to be impossible that the same mechanical laws hold relatively
to the non uniformly moving carriage as hold with reference to the carriage when at
rest or in uniform motion. At all events, it is clear that the
Galileyan law does not hold with respect to the non uniform moving carriage. Because
of this, we feel compelled at the present juncture to grant a kind of
absolute physical reality to non uniform motion in opposition to the general principle of relativity.
But in what follows we shall soon see that this conclusion cannot be maintained
Section nineteen the gravitational field. If we pick up a stone and then let
it go, why does it fall to the ground. The usual answer to
this question is because it is attracted by the Earth. Modern physics formulates the
answer rather differently, for the following reason. As a result of the more
careful study of electromagnetic phenomena, we have come to regard action at a distance
as a process impossible without the intervention of some intermediary medium. If, for
instance, a magnet attracts a piece of iron, we cannot be content to
regard this as meaning that the magnet acts directly on the iron through the intermediate
empty space. But we are constrained to imagine, after the manner of Faraday,
that the magnet always calls into being something physically real in the space around
it, that something being what we call a magnetic field. In its turn,
this magnetic field operates on the piece of iron, so that the latter
strives to move towards the magnet. We shall not discuss here the justification for
this incidental conception, which is indeed a somewhat arbitrary one. We shall only
mention that With its aid, electromagnetic phenomena can be theoretically represented much more satisfactorily
than without it, and this applies particularly to the transmission of electromagnetic waves.
The effects of gravitation also are regarded in an analogous manner. The action of
the Earth on the stone takes place indirectly. The Earth produces in its surroundings
a gravitational field, which acts on the stone and produces its motion of fall.
As we know from experience, the intensity of the action on a body
diminishes according to a quite definite law as we proceed farther and farther away from
the Earth from our point of view. This means the law governing the properties
of the gravitational field in space must be a perfectly definite one in order correctly
to represent the diminution of gravitational action with the distance from operative bodies. It
is something like this. The body e g. The Earth, produces a
field in its immediate neighborhood directly. The intensity and direction of the field at
points farther removed from the body are thence determined by the law which governs the
properties and space of the gravitational fields themselves. In contrast, to electric and
magnetic fields, the gravitational field exhibits a most remarkable property which is of fundamental
importance for what follows. Bodies which are moving under the sole influence of a
gravitational field receive an acceleration which does not in the least depend either on the
material or on the physical state of the body. For instance, a piece
of lead and a piece of wood fall in exactly the same manner and a
gravitational field in vacuo when they start off from rest or with the same initial
velocity. This law, which holds most accurately, can be expressed in a
different form in the light of the following consideration. According to Newton's law of
motion, we have force equals inertial mass times acceleration, where the inertial mass
is a characteristic constant of the accelerated body. If now gravitation is the cause
of the acceleration, we then have force equals gravitational mass times intensity of the
gravitational field, where the gravitational mass is likewise a characteristic constant for the body.
From these two relations, follows acceleration equals the fraction gravitational mass over inertial
mass times intensity of the gravitational field. If now, as we find from
experience, the acceleration is to be independent of the nature and the condition of
the body, and always the same for a given gravitational field, then the
ratio of the gravitational to the inertial mass must likewise be the same for all
bodies. By a suitable choice of units, we can thus make this ratio
equal to unity. We then have the following law, The gravitational mass of
a body is equal to its inertial mass. It is true that this important
law had hitherto been recorded in mechanics, but it had not been interpreted.
A satisfactory interpretation can be obtained only if we recognize the following fact. The
same quality of a body manifests itself according to the circumstances as inertia or as
weight literally heaviness. In the following section we shall show to what extent this
is actually the case, and how this question is connected with the general postulate
of relativity. Section twenty the equality of inertial and gravitational mass. As an
argument for the general postulate of relativity. We imagine a large portion of empty
space, so far removed from stars and other appreciable masses that we have before
us approximately the conditions required by the fundamental law of Galiley. It is then
possible to choose a Galileyan reference body for this part of space world, relative
to which points at rest remain at rest, and points in motion continue permanently
in uniform rectilinear motion as reference body. Let us imagine a spacious chest resembling
a room with an observer inside, who is whipped with apparatus. Gravitation naturally
does not exist for this observer. He must fasten himself with strings to the
floor, otherwise the slightest impact against the floor will cause him to rise slowly
toward the ceiling of the room. To the middle of the lid of the
chest is fixed externally a hook with rope attached, And now a being,
what kind of a being is immaterial to us, begins pulling at this with
a constant force. The chest, together with the observer, then begin to
move upwards with a uniformly accelerated motion. In course of time, their velocity
will reach unheard of values, provided that we are viewing all this from another
reference body, which is not being pulled with a rope. But how does
the man in the chest regard the process, the acceleration of the chest will
be transmitted to him by the reaction of the floor of the chest. He
must therefore take up this pressure by means of his legs, if he does
not wish to be laid out full length on the floor, or he is
then standing in the chest in exactly the same way as anyone stands in a
room of a house on our earth. If he release a body which he
previously had in his hand, the acceleration of the chest will no longer be
transmitted to this body, and for this reason the body will approach the floor
of the chest with an accelerated relative motion. The observer will further convince himself
that the acceleration of the body towards the floor of the chest is always of
the same magnitude, whatever kind of body he may happen to use for the
experiment, Relying on his knowledge of the gravitational field as it was discussed in
the preceding section, The man in the chest will thus come to the conclusion
that he and the chest are in a gravitational field which is constant with regard
to time. Of course, he will be puzzled for a moment as to
why the chest does not fall in this gravitational field. Just then, however,
he discovers the hook in the middle of the lid of the chest and
the rope which is attached to it, and he consequently comes to the conclusion
that the chest is suspended at rest in the gravitational field. Ought we to
smile at the man and say that he errs in his conclusion. I do
not believe we ought to. If we wish to remain consistent, we must
rather admit that his mode of grasping the situation violates neither reason nor known mechanical
laws, even though it is being accelerated with respect to the Galilean space.
First considered, we can nevertheless regard the chest as being at rest. We
have thus good grounds for extending the principle of relativity to include bodies of reference
which are accelerated with respect to each other, and as a result we have
gained a powerful argument for a generalized postulate of relativity. We must note carefully
that the possibility of this mode of interpretation rests on the fundamental property of the
gravitational field of giving all bodies the same acceleration, or what comes to the
same thing. On the law of the equality of inertial and gravitational mass.
If this natural law did not exist, the man in the accelerated chest would
not be able to interpret the behavior of the bodies around him on the supposition
of a gravitational field, and he would not be justified on the grounds of
experience in supposing his reference body to be at rest. Suppose that the man
in the chest fixes a rope to the inner side of the lid, and
that he attaches a body to the free end of the rope. The result
of this will be to stretch the rope so that it will hang vertically downwards.
If we ask for an opinion of the cause of tension in the rope,
the man in the chest will say, the suspended body experiences a downward
force in the gravitational field, and this is neutralized by the tension of the
rope. What determines the magnitude of the tension of the rope is the gravitational
mass of the suspended body. On the other hand, an observer who is
poised freely in space will interpret the condition of things. Thus, the rope
must perforce take part in the accelerated motion of the chest, and it transmits
this motion to the body attached to it. The tension of the rope is
just large enough to effect the acceleration of the body. That which determines the
magnitude of the tension of the rope is the inertial mass of the body.
Guided by this example, we see that our extension of the principle of relativity
implies the necessity of the law of the equality of inertial and gravitational mass.
Thus we have obtained a physical interpretation of this law. From our consideration of
the accelerated chest, we see that a general theory of relativity must yield important
results on the laws of gravitation. In point of fact, the systematic pursuit
of the general idea of relativity has supplied the laws satisfied by the gravitational field.
Before proceeding farther, however, I must warn the reader against a misconception
suggested by these considerations. A gravitational field exists for the man in the chest,
despite the fact that there was no such field for the coordinate system first
chosen. Now we might easily suppose that the existence of a gravitational field is
always only an apparent one. We might also think that regardless of the kind
of gravitational field which may be present, we could always choose another reference body
such that no gravitational field exists with reference to it. This is by no
means true for all gravitational fields, but only for those of quite special form.
It is, for instance, impossible to choose a body of reference such
that, as judged from it, the gravitational field of the Earth in its
entirety vanishes. We can now appreciate why that argument is not convincing, which
we brought forward against the general principle of relativity at the end of section eighteen.
It is certainly true that the observer in the railway carriage experiences a jerk
forwards as a result of the application of the break, and that he recognizes
in this the non uniformity of motion or retardation of the carriage. But he
is compelled by nobody to refer this jerk to a real acceleration or retardation of
the carriage. He might also interpret his experience thus my body of reference,
the carriage remains permanently at rest with reference to it. However, there exists,
during the period of application of the breaks a gravitational field which is directed
forwards, and which is variable with respect to time. Under the influence of
this field, the embankment, together with the Earth, moves non uniformly in
such a manner that their original velocity in the backwards direction is continuously reduced.
End of Section twenty