Section twenty one, in what respects are the foundations of classical mechanics and of
the special theory of relativity unsatisfactory? We have already stated several times that classical
mechanics starts out from the following law. Material particles sufficiently far removed from other
material particles continue to move uniformly in a straight line, or continue in a
state of rest. We have also repeatedly emphasized that this fundamental law can only
be valid for bodies of reference K, which possess certain unique states of motion,
and which are in uniform translational motion relative to each other. Relative to
other reference bodies K, the law is not valid both in classical mechanics and
in the special theory of relativity. We therefore differentiate between reference bodies K,
relative to which the recognized laws of nature can be said to hold, and
reference bodies K, relative to which these laws do not hold. But no
person whose mode of thought is logical can rest satisfied with this condition of things.
He asks, how does it come that certain reference bodies or their states
of motion are given priority over other reference bodies or their states of motion?
What is the reason for this preference? In order to show clearly what I
mean by this question. I shall make use of a comparison. I am
standing in front of a gas range. Standing alongside of each other. On
the range are two pans, so much alike that one may be mistaken for
the other. Both are half full of water. I notice that steam is
being emitted continuously from one pan, but not from the other. I'm surprised,
rised at this, even if I have never seen either a gas range
or a pan before. But if I now notice a luminous something of bluish
color under the first pan, but not under the other, I cease to
be astonished, even if I have never before seen a gas flame. For
I can only say that this bluish something will cause the emission of the steam,
or at least possibly it may do so. If, however, I
notice the bluish something in neither case, and if I observe that the one
continuously emits steam whilst the other does not, then I shall remain astonished and
dissatisfied until I have discovered some circumstance to which I can attribute the different behavior
of the two pans. Analogously, I seek in vain for a real something
in classical mechanics or in the special theory of relativity to which I can tribute
the different behavior of bodies considered with respect to the reference systems K and K
prime begin footnote. The objection is of importance more especially when the state of
motion of the reference body is of such a nature that it does not require
any external agency for its maintenance, for example, in the case when the
reference body is rotating uniformly. End footnote. Newton saw this objection and attempted
to invalidate it, but without success. But E. Mok recognized it most
clearly of all, and because of this objection he claimed that mechanics must be
placed on a new basis. It can only be got rid of by means
of a physics which is conformable to the general principle of relativity, since the
equations of such a theory hold for every body of reference, whatever may be
its state of motion. Section twenty two a few inferences from the general principle
of relativity. The considerations of section twenty show that the general principle of relativity
puts us in a position to derive properties of the gravitational field in a purely
theoretical manner. Let us suppose, for instance, that we know the space
time course for any natural process whatsoever as regards the manner in which it takes
place in the Galilean domain relative to a Galilean body of reference k by means
of purely theoretical operations by e simply by calculation. We are then able to
find how this known natural process appears as seen from a reference body K prime,
which is accelerated relatively to K. But the gravitational field exists with respect
to this new body of reference k prime. Our consideration also teaches us how
the gravitational field influences the process studied. For example, we learn that a
body which is in a state of uniform rectilinear motion with respect to K,
in accordance with the law of Galiley, is executing an accelerated and in general
curvilinear motion with respect to the accelerated reference body k prime chest. This acceleration
or curvature corresponds to the influence on the moving body of the gravitational field prevailing
relatively to K prime. It is known that a gravitational field influences the movement
of bodies in this way, so that our consideration supplies us with nothing essentially
new. However, we obtain a new result of fundamental importance when we carry
out the analogous consider duration for a ray of light with respect to the Galilean
reference body K. Such a ray of light is transmitted rectilinearly with the velocity
C. It can easily be shown that the path of the same ray of
light is no longer a straight line when we consider it with reference to the
accelerated chest reference body K. Prime. From this we conclude that in general,
rays of light are propagated curvilinearly in gravitational fields. In two respects.
This result is of great importance. In the first place, it can be
compared with the reality. Although a detailed examination of the question shows that the
curvature of light ray is required by the general theory of relativity, is only
exceedingly small for the gravitational fields at our disposal. In practice, its estimated
magnitude for light rays passing the Sun at grazing incidents is nevertheless one point seven
seconds of arc. This ought to manifest itself in the following way. As
seen from the Earth, certain fixed stars appear to be in the neighborhood of
the Sun and are thus capable of observation during a total eclipse of the Sun.
At such times, these stars ought to appear to be displaced outwards from
the Sun by an amount indicated above, as compared with their apparent position in
the sky when the Sun is situated at another part of the heavens. The
examination of the correctness or otherwise of this deduction is a problem of the greatest
importance, the early solution of which is to be expected of astronomers. Begin
footnote. By means of the star photographs of two expeditions equipped by a joint
committee of the Royal and Royal Astronomical Societies. The existence of the deflection of
light demanded by theory was first confirmed during the solar eclipse of twenty ninth May
nineteen nineteen. End footnote. In the second place, our result shows that,
according to the general theory of relativity, the law of the constancy of
the velocity of light in vacuo, which constitutes one of the two fundamental assumptions
in the special theory of relativity, and to which we have already frequently referred,
cannot claim any unlimited validity. A curvature of rays of light can only
take place when the velocity of propagation of light varies with position. Now,
we might think that as a consequence of this, the special theory of relativity
and with it the whole theory of relativity, would be laid in the dust.
But in reality this is not the case. We can only conclude that
the special theory of relativity cannot claim an unlimited domain of validity. Its results
hold only so long as we are able to disregard the influences of gravitational fields
on the phenomena, for example, of light. Since it has often been
contended by opponents of the theory of relativity that the special theory of relativity is
overthrown by the general theory of relativity, it is perhaps advisable to make the
facts of the case clearer by means of an appropriate comparison. Before the development
of electrodynamics, the laws of electrostatics were looked upon as the laws of electricity.
At the present time, we know that electric fields can be derived correctly
from electrostatic considerations only for the case which is never strictly realized, in which
the electrical masses are quite at rest relatively to each other and to the coordinate
system. Should we be justified in saying that for this reason, electrostatics is
overthrown by the field equations of Maxwell and electrodynamics not in the least. Electrostatics
is contained in electrodynamics as a limiting case. The laws of the latter lead
directly to those of the former. For the case in which the fields are
invariable with regard to time. No fairer destiny could be allotted to any physical
theory than that it should of itself point out the way to the introduction of
a more comprehensive theory in which it lives on as a limiting case. In
the example of the transmission of light just dealt with, we have seen that
the general theory of relativity enables us to derive theoretically the influence of a gravitational
field on the course of natural processes, the laws of which are already known
when a gravitational field is absent. But the most attractive problem to the solution
of which the general theory of relativity supplies the key, concerns the investigation of
the laws satisfied by the gravitational field itself. Let us consider this for a
moment. We are acquainted with space, time domains which behave approximately in a
Galileyan fashion under suitable choice of reference body, ie, domains in which gravitational
fields are absent. If we now refer such a domain to a reference body
k prime possessing any kind of motion, then relative to k prime, there
exists a gravitational field which is variable with respect to space and time. Begin
footnote. This follows from a generalization of the discussion in section twenty end footnote.
The character of this field will of course depend on the motion chosen for
k prime. According to the general theory of relativity, the general law of
the gravitational field must be satisfied for all gravitational fields obtainable in this way.
Even though by no means all gravitational fields can be produced in this way,
yet we may entertain the hope that the general law of gravitation will be derivable
from such gravitational fields of a special kind. This hope has been realized in
the most beautiful manner. But between the clear vision of this goal and its
actual realization it was necessary to surmount a serious difficulty. And as this lies
deep at the root of things, I dare not withhold it from the reader.
We require to extend our ideas of the space time continuum still farther Section
twenty three behavior of clocks and measuring rods on a rotating body of reference.
Hitherto, I have purposefully refrained from speaking about the physical interpretation of space and
time data in the case of this general theory of relativity. As a consequence,
I am guilty of a certain slovenliness of treatment, which, as we
know from the special theory of relativetivity, is far from being unimportant and pardonable.
It is now high time that we remedy this defect, But I would
mention at the outset that this matter lays no small claims on the patients and
on the power of abstraction of the reader. We start off again from quite
special cases which we have frequently used before. Let us consider a space time
domain in which no gravitational field exists relative to a reference body K, whose
state of motion has been suitably chosen. K is then a galiley in reference
body as regards the domain considered, and the results of the special theory of
relativity hold relative to K. Let Us suppose the same domain referred to a
second body of reference K prime, which is rotating uniformly with respect to K.
In order to fix our ideas, we shall imagine K prime to be
in the form of a plain circular disk which rotates uniformly in its own plane
about its center. An observer who is sitting eccentrically on the disk k prime
is sensible of a force which acts outward in a radial direction, and which
would be interpreted as an effect of inertia centrifugal force by an observer who was
at rest with respect to the original reference body K. But the observer on
the disk may regard his disk as a reference body which is at rest on
the basis of the general principle of relativity. He is justified in doing this.
The force acting on himself, and in fact on all other bodies which
are at rest relative to the disk, he regards as the effect of a
gravitational field. Nevertheless, the space distribution of this gravitational field is of a
kind that would not be possible on Newton's theory of gravitation. Begin footnote.
The field disappears at the center of the disk and increases proportionally to the distance
from the center as we proceed atwards and footnote. But since the observer believes
in the general theory of relativity. This does not disturb him. He is
quite in the right when he believes that a general law of gravitation can be
formulated, a law which not only explains the motion of the stars correctly,
but also the field of force experienced by himself. The observer performs experiments on
his circular disc with clocks and measuring rods. In doing so, it is
his intention to arrive at exact definitions for the significance of time and space data
with reference to the circular disc k prime, these definitions being based on his
observations, what will be his experience in the enterprise. To start with,
he places one of two identically constructed clocks at the center of the circular disc
and the other on the edge of the disk, so that they are at
rest relative to it. We now ask ourselves whether both clocks go at the
same rate from the standpoint of the non rotating Galilean reference body K. As
judged from this body, the clock at the center of the disc has no
velocity, whereas the clock at the edge of the disk is in motion relative
to K in consequence of the rotation. According to a result obtained in section
twelve. It follows that the latter clock goes at a rate permanently slower than
that of the clock at the center of the circular disk, ie as observed
from k It is obvious that the same effect would be noted by an observer,
whom we will imagine sitting alongside his clock at the center of the circular
disk. Thus on our circular disc, or, to make the case more
general, in every gravitational field, a clock will go more quickly or less
quickly according to the position in which the clock is situated at rest. For
this reason, it is not possible to obtain a reasonable definition of time with
the aid of clocks which are arranged at rest with respect to the body of
reference. A similar difficulty presents itself when we attempt to apply our earlier definition
of simultaneity in such a case. But I do not wish to go any
farther into this question. Moreover, at this stage, the definition of the
space coordinates also presents insurmountable difficulties. If the observer applies his standard measuring rod,
a rod which is short as compared with the radius of the disk tangentially
to the edge of the disk. Then, as judged from the Galilean system,
the length of this rod will be less than I, since, according
to Section twelve, moving bodies suffer a shortening in the direction of the motion.
On the other hand, end, the measuring rod will not experience a
shortening in length as judged from K, if it is applied to the disc
in the direction of the radius. If then the observer first measures the circumference
of the disc with his measuring rod, and then the diameter of the disk.
On dividing the one by the other, he will not obtain as quotient
the familiar number Pi equals three point one, four, etc. But a
larger number begin footnote. Throughout this consideration we have to use the Galilean non
rotating system K as reference body, since we may only assume the validity of
the results of the special theory of relativity relative to K. Relative to k
prime a gravitational field prevails end footnote, but a larger number, whereas,
of course, for a disc which is at rest with respect to K,
this operation would yield pi actly. This proves that the propositions of Euclidean geometry
cannot hold exactly on the rotating disk, nor in general in a gravitational field,
at least if we attribute the length I to the rod in all positions
in every orientation. Hence the idea of a straight line also loses its meaning.
We are therefore not in a position to define exactly the coordinates x,
y z relative to the disk by means of the method used in discussing the
special theory, And as long as the coordinates and times of events have not
been defined, we cannot assign an exact meaning to the natural laws in which
these occur. Thus, all our previous conclusions based on general relativity would appear
to be called in question. In reality, we must make a subtle detour
in order to be able to apply the postulate of general relativity. Actually,
I shall prepare the reader for this in the following paragraphs. End of sections
twenty one to twenty three.