Part II: The General Theory of Relativity18. Special and General Principle of Relativity19. The Gravitational Field20. The Equality of Inert

None

Episode Transcript

Speaker 1 (00:01):
This is a LibriVox recording. All LibriVox recordings are in
the public domain. For more information or to volunteer, please
visit LibriVox dot org. Relativity The Special and General Theory
by Albert Einstein, recorded by Laurie Ann Walden, Part two,
The General Theory of Relativity, Sections eighteen through twenty. Section

(00:28):
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

(00:50):
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.

(01:11):
The carriage is in motion relative to the embankment. B
the embankment is in motion relative to the carriage in
A the embankment in 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

(01:35):
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

(01:56):
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

(02:17):
is 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

(02:37):
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 arrivable from them. Only experience can decide as to
its correctness or incorrectness. Up to the present, however, we

(03:01):
have by no means maintained the equivalents 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.

(03:23):
A particle left to itself and sufficiently far removed from
all other particles, moves uniformly in a straight line with
reference to K 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

(03:45):
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 to K. 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.

(04:08):
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, et cetera. Are

(04:29):
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

(04:51):
principle of relativity has been justified, every intellect which strives
after generalization must feel the temptation to venture the step
towards the su 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

(05:14):
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,

(05:35):
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 brakes, then the occupant of the carriage experiences
a correspondingly powerful jerk forwards. The retarded motion is manifested

(05:58):
in the mechanical behaveavior 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

(06:19):
in uniform motion at all events, it is clear that
the Galilean law does not hold with respect to the
non uniformly 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

(06:42):
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,

(07:05):
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

(07:26):
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

(07:48):
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

(08:09):
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

(08:30):
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

(08:51):
distance from operative bodies. It is something like this. The
body e g. The Earth produces a field in its
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 in space
of the gravitational fields themselves. In contrast to electric and

(09:16):
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

(09:39):
piece of wood fall in exactly the same manner in
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,

(10:05):
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

(10:30):
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

(10:53):
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,

(11:16):
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

(11:37):
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

(12:00):
we have before us approximately the conditions required by the
fundamental law of galilee. It is then possible to choose
a Galilean 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

(12:21):
reference body, Let us imagine a spacious chest resembling a
room with an observer inside who is equipped 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

(12:43):
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 the uniformly accelerated motion.
In course of time, their velocity will reach unheard of values,

(13:06):
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. He is then standing in the

(13:29):
chest in exactly the same way as any one 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.

(13:50):
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

(14:10):
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

(14:31):
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

(14:53):
it is being accelerated with respect to the Galilean space.
First considered, we can nevertheless regard the chess 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

(15:15):
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

(15:36):
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

(15:57):
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

(16:18):
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

(16:40):
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

(17:01):
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

(17:23):
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

(17:44):
field for the co ordinate 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

(18:07):
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

(18:31):
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

(18:53):
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 brakes a gravitational
field which is directed forwards, and which is variable with

(19:15):
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