21. In What Respects are the Foundations of Classical Mechanics and of the Special Theory of Relativity Unsatisfactory?22. A Few Inferences

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Episode Transcript

Speaker 1 (00:01):
This is LibriVox recording. All LibriVox recordings are in the
public domain. For more information or to volunteer, please visit
LibriVox dot org. Recorded by Annie Coleman www dot Annie
Cooleman dot com. Relativity the Special and General Theory by

(00:23):
Albert Einstein, Continuing Part two, sections twenty one through twenty three.
Section twenty one in what respects are the foundations of
classical mechanics and of the special theory of relativity unsatisfactory?

(00:45):
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

(01:06):
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

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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

(01:52):
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

(02:13):
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
am surprised at this, even if I have never seen

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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

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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

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to 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 attribute the different behavior of bodies considered with respect
to the reference systems K and K prime begin footnote.

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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

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and attempted to invalidate it, but without success. But E.
Mock 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

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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

(04:52):
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 i e.

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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
since a gravitational field exists with respect to this new
body of reference k prime, our consideration also teaches us

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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 Galilee, is executing an accelerated and
in general curvelin in yar motion with respect to the

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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

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essentially new. However, we obtain a new result of fundamental
importance when we carry out the analogous consideration 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

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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.

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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 rays 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

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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 neighbourhood of the Sun,
and are thus capable of observation during a total eclipse

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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,

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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

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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

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two fundamental assumptions in the special theory of relativity, and
to which we have all 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

(09:19):
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.

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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,

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the laws of electrostatics were looked upon as the laws
of electricity. At the present time. We know that electric
field 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

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saying that for this reason electrostatics is overthrown by the
field equations of Maxwell in 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

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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

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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

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us consider this for a moment. We are acquainted with
spacetime domains which behave approximately in a Galilean fashion under
suitable choice of reference body i e. Domains in which
gravitational fields are absent. If we now refer such a
domain to a reference body k prime possessing any kind

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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,

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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

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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

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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 purposely refrained from speaking about the physical interpretation of

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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 relativity, 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

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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

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then a Galilean 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

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our ideas, we shall imagine K prime to be in
the form of a plane circular disc which rotates uniformly
in its own plane about its center. An observer who
is sitting eccentrically on the disc k prime is sensible
of a force which acts outward in a radial direction,
and which would be interpreted as an effect of inertia

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centrifugal force by an observer who was at rest with
respect to the original reference body K. But the observer
on the disc may regard his disc as a res
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

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bodies which are at rest relative to the disc 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

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disc and increases proportionally to the distance from the center
as we proceed outwards 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,

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a law which not only explains the motion of the
stars correctly, but also the field of force experience 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,

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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 disc, so that they are at rest relative to it.

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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 disc is in
motion relative to K in consequence of the rotation. According

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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
disc i e. 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

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center of the circular disc. 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

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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,

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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 disc tangentially to the edge of the disc, then,
as judge from the Galilean system, the length of this

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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, the measuring rod will not experience
a shortening in length, as judge 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

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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, et cetera, but a larger number
begin footnote. Throughout this consideration we have to use the

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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,

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of course, for a disk which is at rest with
respect to k, this operation would yield pi exactly. This
proves that the propositions of Euclidean geometry cannot hold exactly
on the rotating disc, nor in general in a gravitational field,
at least if we attribute the length i to the

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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 disc by means of the
method used in discussing the special theory, And as long

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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

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to be able to apply the postulate of general relativity exactly.
I shall prepare the reader for this in the following
paragraphs end of sections twenty one to twenty three. Read
by Annie Coleman in Saint Louis, Missouri, on August thirteen,
two thousand six,