AN EXPERIMENTAL ANALOGY
Forms of Physical Laws.--Aids to Understanding.--Popular
Descriptions.--Optical Signals.--Simultaneity.--Experiments in Similes.
"I WISH to ask you. Professor, to help me over a difficulty and to treat
me as the spokesman of a great number who are similarly troubled. In
most accounts of your theory of relativity, there is a dearth of
definite, concrete, illustrative examples on which we can fix our minds
whenever the theorem is to be applied generally without limitation. Let
me express this more precisely: Your simplified picture of the structure
of the universe is achieved in the theory of relativity by emancipating
all observations from fixed co-ordinate systems, and by proclaiming the
equivalence of all systems of reference. One of your earliest theorems
states that physical laws describing how the states of physical systems
alter, remain the same, no matter to which of two co-ordinate systems
these states are referred, provided that the co-ordinate systems are
moving rectilinearly and uniformly relatively to one another. This
theorem entails the following statement. If we--erroneously--adopt a
non-relativistic view, we shall come to the conclusion that physical
laws depend on the particular system of reference chosen, and will thus
assume a different form for each different system. At this point we
experience a desire to hear definite examples. What varying forms may a
certain given physical law, known under a definite form, assume, and how
can we use this law to show that it must adapt itself to the postulate
of relativity?"
Einstein explained that such examples cannot be given in special cases,
but only in very general terms. If we were to suggest the elliptic
orbits of the planets (at which I had hinted in my remarks), we should
fall into error, for the law of elliptic orbits is no such law. For,
from another point of view, the elliptic paths of the planets might be
drawn out into wavy lines, or into spirals, and they would remain
ellipses only as long as the lines of motion are referred to the central
attracting body. But the constancy of the velocity of light is such a
law, as also is the law of inertia, according to which a body that is
left to itself moves uniformly in a straight line.
I confessed to him that this limitation to a few very general laws would
be a painful matter for many an enthusiast of average attainments, who
has great difficulty in distinguishing the laws that are _generally_
valid from those that hold only within circumscribed limits. But if this
were not so, we should have to alter our conception of what is conveyed
by a popular exposition. For it is called _popular_, not because it now
and then uses the patronizing words "dear reader," but because it
anticipates the questions and doubts of the man of average sense, and
examines them, proving some to be unjustified and others to be
reasonable or unreasonable, as the case may be. "Then there is a further
matter that troubles me," I continued. "Let us suppose an ordinary
reader of such a popular account to get a first insight into the new
conception of Time. He is glad to feel the ideas dawning in him, and, to
get a more lasting view of the idea, he repeats the arguments through
which he has just threaded his way, and, in doing so, again encounters
the phrase 'uniform motion.' At the first reading he imagined that he
understood the expression quite well, but the second time he pauses and
considers. For now that he knows how much depends on it, he is anxious
to find out the exact meaning of a 'uniform motion.' He looks for a
definition, and if he cannot find one in the book he is perusing, he
endeavours to reason it out for himself. With good luck he arrives at
the usual statement: a body moves with 'uniform motion' if it traverses
equal distances in equal intervals of time. But equal intervals of time
are clearly those during which a body in uniform motion traverses equal
distances. In other words, he explains A by means of B, and B by means
of A, so that he has involved himself in a vicious circle from which he
cannot escape. This is his hour of need, due to the difficulty of
'time.'
"He hopes that further study will remove this obstacle. He meets with
the conception of 'simultaneity,' which is defined for him anew, and is
disclosed as being 'relative.' He manœuvres further towards the
fundamental theorem that every body of reference has its own particular
time.
"His popular booklet makes this clear to him by quoting the example of a
flying-machine, or, better still, a railway train that is rushing along
an embankment at a very great speed, and that carries a passenger. Two
strokes of lightning I and II are to take place at two widely distant
points on the embankment. The question is then: When are these two
flashes of lightning to be considered 'simultaneous'? What conditions
must be fulfilled to ensure this? It is found--incontrovertibly--that
the light-rays starting out from the two strokes of lightning must meet
at the mid-point of the embankment.
"It now follows from a short chain of argument that the observer in the
train will see flash II earlier than flash I, if they reach the
observer, who is at rest, at the same moment. That is, two events that
are simultaneous with respect to the embankment are not simultaneous for
a moving system (such as a train or a flying-machine); the converse is,
of course, also true.
"Here, again, the eager layman encounters difficulty, for he asks
himself: Why should the two events be characterized or defined by
lightning-flashes in particular? If acoustic signals were used instead,
nothing would be altered in the fundamental determination, for the sound
rays (sound-waves) would likewise meet at the mid-point of the line
joining the sources of disturbance. What is the reason that the
relativity of time arises only when phenomena are regarded optically,
and that rays of light play the deciding part in all later developments?
"And this particular query is followed by one which is more general: Why
does the popular pamphlet not read this question in my mind? I know that
the author of it is more skilled in these matters than I, but just this
superiority should help him to divine what is passing in my mind when I
make efforts to follow his reasoning."
Einstein had listened to me patiently, and then he explained to me at
considerable length why in this case optical signals cannot be replaced
by sound signals: light is the only mode of motion that shows itself to
be entirely independent of the carrier of the motion, of the
transmitting medium. Thus the constancy of velocity is assumed in the
above argument, and as this constancy is an exclusive property of light,
every other method must be discarded as unallowable for investigating
the conception "simultaneity." Furthermore, he showed me how, on the
basis of relativity, starting from the embankment-experiment, we may
arrive at a perfectly consistent representation of the conception of
Time. He certainly did this by applying subtle physical arguments that
exceed the scope of the present book.[7] He added, in substance, that it
was futile and impossible to discuss in detail all the conceivable
objections that might arise in the mind of one reading a popular work of
this kind: it was a futile undertaking, because the true purpose was
defeated, inasmuch as a clear development of the fundamental thought
would be almost impossible under the cross-fire of so many random
questions.
[Footnote 7: In these arguments, arrangements of synchronous clocks
occur, which are fixed into the co-ordinate systems, the positions of
their hands being compared with one another. The "time" of an event is
then defined as the position of the hands of a clock immediately
adjacent to the scene of the event.]
Thus, in this matter, Einstein takes the same stand as Schopenhauer in
the preface of his chief work, in which he says: "To understand this
work no better way can be advised than to read it twice (at least),
inasmuch as the beginning assumes the end, almost as much as the end
assumes the beginning; the smallest part cannot be understood if the
whole has not already been understood." Whoever accepts and follows this
advice will find that the intermediate objections will gradually balance
and cancel one another, and that it is not necessary that they should
interrupt the steady and consistent line of development.
The position would be different if a disciple of the new theory should
resolve to dispense with strictly scientific reasoning altogether, and
should wish to meet the wishes of his readers or hearers by discarding
accuracy entirely. Such a programme seems quite feasible.
"This would be merely following the sketchy method of a magazine,"
Einstein remarked, "but you do not seriously think that it would lead to
anything?"
"It would not be a true explanation, which is reserved for technical
productions. But I can imagine that it would not be unprofitable to help
one who is entirely ignorant on these questions by using makeshifts, in
the form of allegories or analogies, which will serve as supports if he
should take fright during the course of his earlier studies. These
shocks are bound to occur, as, for instance, when he learns that a
moving rigid rod undergoes contraction in the direction of motion."
"But this is _proved_ to him!"
"Nevertheless, he does not easily accept it. For the general reader will
say to himself: 'A superhuman effort is imposed on my mind. A rigid rod
is the most constant of all things, and never before has one been
compelled to regard something that is constant as variable.'"
"If he does not grasp it, no analogy will teach him."
"But perhaps it is possible. The analogy is to show him that the effort
is not _superhuman_, and that thinking Man has already had occasion to
become familiar with such transformations from constant to variable
factors."
"I am afraid your analogy will prove a failure."
"From the scientific point of view this is probably true, inasmuch as
all comparisons are imperfect, but the analogy may yet be of service as
a last resort. For example, I should say to my general reader: 'Picture
to yourself a savant of the Middle Ages who reflects on the constitution
of animals and plants. One fact seems to him to be irrevocably true,
namely, that the species are unchangeable! A palm tree is a palm tree, a
horse is a horse, a worm a worm, and what is once a reptile remains a
reptile. A species in itself denotes something absolutely _invariant_.'"
"The expression is wrong when taken in this connexion; you mean
_invariable_."
"A little inaccuracy more or less does not affect the analogy. For the
sake of my picture I should like to retain the conception-couple,
_variable_ and _invariant_. Well, then, the species give our savant the
impression of invariance, as in the view that was held by Linné and
Cuvier. This view necessarily has its counterpart in his thought. He
argues that every species has its own original root, and that, in this
sense, there is very extensive variation. The fundamental roots are
extremely manifold; Nature has produced innumerable variations in her
individual acts of creation. But now the _Theory of Descent_ of Lamarck,
Goethe, Oken, Geoffroy St. Hilaire, enters the field and produces a
complete inversion of these two elements; the two parts of the earlier
point of view change places. Our savant has to revise his whole world of
thought. Now all organisms are to be traced back to a single original
root: the latter, which was variable before, becomes an invariable
unicellular primitive organism, but the apparently unchangeable species
now becomes variable, in the widest possible sense. And even if this
savant should exclaim: 'How am I to reconcile myself to this view?' his
descendants later find no difficulty in accepting the idea that the
organic roots are uniform, and that it is the species that are subject
to all manner of variation as a compensating feature."
Einstein expressed himself very little pleased with this attempt at an
analogy, and found that it was so far fetched that it could not be
considered admissible.
"Then I must ask your permission to continue my attempt; perhaps
something useful may yet result from it. I now picture to myself a human
being who lived in classical times and who, following Ovid and the great
majority of his contemporaries, regards the earth as a disc. On this
disc, each inhabitant of the earth has his own particular position, for
the disc has a centre with reference to which the position of a person
can be specified if his distance and his angular displacement from a
given initial radius is specified. Thus, there is a variation of
position if various persons are considered. On the other hand, the
_Above_ and the _Below_ is absolutely invariable for all persons, for
the lines running between _Above_ and _Below_ are all parallel for them,
since they all have uniformly the same disc under their feet and the
same heaven above their heads. Ovid would therefore have refused to
entertain for a moment the suggestion that _Above-Below_ is a variable.
But his distant descendants accepted the view that the earth is
spherical and that there are antipodes as self-evident, and they found
not the slightest difficulty in considering the line _Above-Below_ to
vary with their own position, making all possible angles with an initial
line extending to direct oppositeness. Referred to the centre of the
sphere, all people have now an 'invariant' position, whereas, in
compensation, the _Above-Below_ is subject to every conceivable
variation. And now I again address myself to the average reader, and say
that the meaning of these analogies is that every doctrine that leads to
a great _uniformity_ converts what was formerly invariant into a
variable quantity, and vice versa. The theory of relativity makes all
considerations about the physical world independent of all co-ordinate
systems; it establishes completely invariable uniformity, removed from
all changes due to varying points of view. Hence what was previously
invariable--such as a rigid measuring-rod--will now become variable. It
is not surprising that this requires a new method of thought, a revision
of our mode of reasoning, for the above analogies show that these
radical adjustments are characteristically necessary in the case of
comprehensive theories, and that such theories are able to overcome
apparently firmly established ideas. The parallels that I drew above
will at least inspire the average reader with a certain confidence, for
they show him how results of reasoning that were once considered
incredible were regarded as self-evident by later generations."
I have already emphasized sufficiently that Einstein regards as
inadequate these auxiliary pictures that have presented themselves to
me. Yet in the course of the conversation I gained the impression that
his judgment grew somewhat milder, and that, with certain reservations,
he was disposed to let them pass as tolerably useful helps--and they are
not intended to be more than this. I think, therefore, that I am not
acting counter to his wishes in citing these allegorical examples here,
particularly as they arose in the course of our talks.
Since then, I have had many opportunities of testing these examples on
certain persons, and may mention that they proved quite useful.
Analogies of this kind may offer a friendly help in moments when the
uninitiated feel themselves in peril, and encounter a difficulty which
they imagine to be insurmountable. They do not remove the difficulty,
but they impart a certain power of expansion to the intellect and
encourage a continuation of effort, which would probably otherwise be
relaxed at the first sign of something which is imagined to be
inconceivable. There is thus no room in textbooks for such helps, but
they may justifiably find a place in a book that departs from the
methodical route, and hopes to discover in by-ways things that are
suggestive and instructive.