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Title: Introduction to Einstein
Author: William F. Hudgings
Editor: E. Haldeman-Julius
Release date: March 23, 2026 [eBook #78274]
Language: English
Original publication: Girard: Haldeman-Julius Company, 1923
Other information and formats: www.gutenberg.org/ebooks/78274
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*** START OF THE PROJECT GUTENBERG EBOOK INTRODUCTION TO EINSTEIN ***
TEN CENT POCKET SERIES NO. 408
Edited by E. Haldeman-Julius
Introduction to
Einstein
William F. Hudgings
Copyright, 1923, by the author
Copyright, 1923, Haldeman-Julius Company
HALDEMAN-JULIUS COMPANY
GIRARD, KANSAS
[Illustration: DR. ALBERT EINSTEIN]
_A Popular Explanation of the Einstein Theory That Approaches an
Actual Visualization of the Subject._
The preparation of this essay on the Principle of Relativity has been
undertaken after a careful study of practically all the literature
hitherto published on the subject purporting to be of a popular or
semi-popular nature. With due appreciation of the merit of many of
these works, a perusal of them has made it increasingly apparent that
the writers have not succeeded in eliminating technical phraseology
sufficiently to bring the subject down to the level of the untrained
reader, and that a _really_ popular exposition of both the Special
and the General Theory is needed.
Notwithstanding the laudable efforts of some to meet this need it
is manifest that the public continues to be in the dark and many
erroneously suppose that “only twelve men in the world” can really
comprehend this new theory of the universe. Even book reviews ofttimes
betray a surprising lack of appreciation of the fundamentals of
Einstein’s noble contribution to science.
It is the purpose of this booklet therefore, to bring within the
scope of visualization, as nearly as possible, the very essence of
relativity, and to state its principles in terms devoid of unfamiliar
phrases and illustrations. The popular mind is not satisfied with
vague statements, mathematical equations and formulæ. It calls for an
explanation in words of every day usage, and where a technical term
_must_ be employed then a definition of that term in language
simple and lucid. To meet these requirements without sacrificing
scientific accuracy has been the earnest endeavor of the author.
Acknowledgment is hereby made to Dr. S. I. Bailey and Dr. G. D.
Birkhoff of Harvard, Dr. Edward Kasner of Columbia, Prof. Victor
Schmidt of Cincinnati, and Dean J. A. Robison of the Oakland Technical
High School (Calif.), all of whom have kindly criticized the manuscript
and made valuable suggestions for improvement of the text.--W. F. H.
New York, Dec., 1922.
Introduction to Einstein and Universal Relativity
What Is Einstein’s Contribution?
Albert Einstein, orthodox Jew of German birth and ardent Zionist, is
the most discussed figure in intellectual circles in the world today.
It is not as a Jew nor as a Zionist, however, that we find him so
much in the limelight of public opinion, although as such he is a
distinguished figure and has only recently toured Europe and America
in the interest of the Zionist movement. What has lately made the name
of Einstein a byword upon the lips of scientists, philosophers and the
multitudes of more common mortals is his revolutionary treatment of
time, space and gravitation which materially affects various branches
of accepted science and thereby threatens to throw certain portions of
our school books and other scholarly works into the discard.
The ideas which led to the development of Einstein’s theory are not
new. Newton appreciated the principle of mechanical relativity and
presented it in much detail. But to Einstein belongs the credit of
completely disentangling the idea of relativity of motion from the
ancient conception of absoluteness of velocity and carrying out the
relativistic viewpoint to its logical conclusion. In so doing,
however, he found it necessary to discard or revamp many popular
theories and “laws of nature.” But consistency is a jewel wherever
found, and Einstein seems to have placed the gem into its proper
setting.
This Jewish scientist, born in 1878 of humble parentage, began
delving into the mysteries of the universe at an early age. When only
sixteen, it is said, he contributed a paper, modest in its brevity,
which was read before an assemblage of German scientists with marked
appreciation of its merit, and soon thereafter his active mind began
to traverse new and unbeaten paths of scientific thought. During the
years that followed his ideas took more definite form, and in 1905,
while occupying a chair in a Swiss university, Einstein published
his first “Principle of Relativity,” now called the “Special Theory
of Relativity,” which treats the physical laws relating to uniform,
rectilinear motion, i. e., motion in its simplest form.
Einstein’s hypothesis, unlike most theories advanced, did not originate
with any physical “discovery” or observation. In a sense it grew out
of the philosophic notion of the relativity of all knowledge. He
laid hold of the idea that everything is measured by or considered
_relative_ to something else; that our concepts of absolute
position, absolute motion and absolute time are groundless for the
reason that we have no immovable and unchangeable standard anywhere
in the universe as a starting point. While there is nothing new or
revolutionary in this fact, yet it actually produced certain scientific
“revolutions” when carried out to a consistent end.
Prior to Einstein, the principle of relativity had been recognized
and philosophized upon to a certain point and then dropped when it
ran counter to hitherto unquestioned conceptions. Einstein, on the
contrary, contended that if the principle is admitted at all it must
be admitted at every point irrespective of what it contradicts. After,
therefore, his theory was fully developed it was then subjected
to certain physical tests, with the result that it has been quite
fully confirmed. This mode of theoretical development is called _a
priori_ as against the more usual _a posteriori_ method,
wherein the theory _follows_ rather than precedes certain
particular phenomena.
It was in 1915, during the stress of the great war, that Professor
Einstein, working unconcernedly in his laboratory in Berlin, completed
and published what is now called his “General Principle of Relativity.”
Due to hostilities, the details of his developed theory did not reach
the outside world until after the armistice in 1918; but such meagre
accounts of it as did cross the German border were sufficient to awaken
Eddington and other British scientists to the extent that they hastily
equipped two research expeditions, dispatching one to the Island of
Principe off the coast of Africa and the other to Sobral, Brazil, to
test Einstein’s hypothesis about the deflection of light rays in a
gravitational field, by taking photographs of stars near the sun
during the solar eclipse of May 29, 1919, which was total in the
regions mentioned.
[Illustration: WILLIAM F. HUDGINGS]
It was the announced results of these expeditions, confirming as they
did this new theory of gravitation, that has made Albert Einstein
famous.
There are, of course, opponents of the Einstein theory in the circle of
scientists in both Europe and America, though they are comparatively
few in proportion to the number who see merit in his postulates if not,
indeed, a complete solution of gravitational and other phenomena which
have baffled the sages of every century. Sir Oliver Lodge is a leading
opponent of the Einstein theory among a certain few British scientists,
but his opposition is easily accounted for when it is remembered that
Einstein, by ignoring the ether and denying other unproven hypotheses,
has materially upset Sir Oliver’s spiritistic playground.
Einstein, to say the least, has stimulated the scientific thought
of the world to an extent unprecedented by anyone since the days of
Newton. If the Principle of Relativity is not universal, then any
theory concerning it cannot long stand; it would predict results
which actual experiment would disprove. It must be admitted, however,
that experiments to date tend to confirm the Theory of Relativity as
propounded by Einstein.
=Concepts Versus Reality=
Relativity is a term appropriately applicable to that school of
thought standing midway between mathematics, physics and astronomy,
which concerns itself with determining the relationship between
observers and their objects. Two or more observers may view a certain
phenomenon or set of phenomena and reach divergent conclusions as
to distance, position, size or other physical quantities commonly
involved. It is not for anyone to arbitrarily conclude that one
particular observation is correct and that all the others are
consequently in error.
It is the task of the relativist to consider the findings of the
various observers, note the conditions attending each, eliminate the
peculiarities of the varying viewpoints and the qualities which have
been unmindfully superimposed by each observer upon the external
object. He must then set down a general rule or law which will embody
the inter-relations of the data before him and which will also hold
good for all other observers regardless of distance, position or motion
of their frames of reference.
_A frame of reference_ is simply an arbitrary set of mutually
perpendicular lines or planes from which measurements may be made for
purposes of describing the location of points within a given area
or region. If a _two_-dimensional area (i. e., a _plane_
surface) is being dealt with, then our frame of reference would consist
merely of two _lines_ drawn perpendicular to each other. Thence,
any point within the right-angle formed by these two lines may be
described by specifying its distance from each line.
If we are to locate a given point in a _three_-dimensional region
(i.e., where length, breadth and also _height_ are involved), then
our frame of reference must consist of _planes_ instead of lines,
three in number, each perpendicular to the other two, like the floor
and two adjacent walls of a room. If any point within a room is to be
located we would first select as a frame of reference, let us say, the
east and south walls together with the floor. The location of the point
may then be described with accuracy by simply specifying its distance
from each of the two walls together with its height above the floor.
The perpendiculars of a frame of reference are called its _axes_,
and the distances from a given point to the axes are called its
_co-ordinates_. A little reflection will suffice to show that it
is quite impossible to record the exact location of anything, either in
space or in a plane, without making use of a reference frame and two or
more co-ordinates.
=Unlimited Choice of Reference Frames=
We are not limited in our selection or choice of reference frames.
They may be close by or relatively remote from the point or object
whose location we wish to describe. The axes may be physical or purely
imaginary. They may be in any direction or position whatsoever, so
long as they are mutually perpendicular and embrace the area in which
the point or object is located. They may be at rest relative to the
object and in motion relative to the observer, or vice versa, or either
at rest or in motion relative to both. If the relative velocity is
known, the co-ordinates may be determined as certainly as though both
the object and the observer, together with his reference frame, were
relatively at rest.
An observer usually prefers to select a frame of reference which
partakes of his own motion, for the sake of convenience and simplicity.
Hence for purposes of either local or astronomical observation we may
choose an imaginary axis perpendicular to the surface of the earth. But
we must admit that as the earth rotates our perpendicular rotates with
it. Consequently it is really changing its direction from moment to
moment. Such rotation, however, is not detected so long as we confine
our observations to terrestrial things, because our frame of reference
is at rest relative to the earth and to all things fixed upon the
earth; but for astronomical observations the matter becomes somewhat
complicated due to the various relative motions involved.
Astronomers often choose as a reference frame an imaginary line
reaching from the earth’s center to the center of the sun, together
with another line perpendicular thereto extending from the earth to
some other planet or point in space. But neither the earth or the sun,
or in fact any other body of matter, is at rest; nor do they move in a
perfectly straight line or plane. The axes of such a frame of reference
continuously undergo a change of direction. Indeed, a reference frame
attached to any physical body in the universe must of necessity move,
because all matter of which we have any knowledge is in motion; and a
frame of reference attached to nothing would be meaningless.
By reason of the motion of all objects and of frames of reference it
is necessary that the latter be supplemented by clocks for determining
the times of occurrences of physical events encountered in our
measurements. We may therefore say that no frame of reference, when
used for physical purposes, is complete without a set of accurately
synchronized timepieces. The element of time is an essential factor
in any calculation of physical quantities because motion is forever
involved.
Velocity (rate of motion) means distance traveled within a given
time. Only when distance and time are combined can we determine
our co-ordinates with universal accuracy. This will be more fully
understood when we come to consider the four-dimensional aspect of
the universe. But for all practical purposes the time element cannot
be divorced from the distance measurement of moving bodies even when
viewed in the ordinary three-dimensional sense.
Suppose, for sake of illustration, we are assigned the task of charting
or recording the precise course of a ball shot from a cannon. In order
to specifically state the history of its flight it will be necessary
to indicate the position of the cannon ball during each moment of
its journey. We will say this is a possibility without attempting
an explanation of the method of measurement used. When our task is
completed we will have a set of four figures for each successive
second, as follows:
(1) distance from cannon.
(2) distance from ground.
(3) distance to right or left.
(4) duration of time.
When these four sets of figures are carefully set down they may be
transmitted on paper or by telegraph and the recipient will be able to
determine therefrom the history of the fired ball with as much accuracy
as though he had been personally present and made the observations
himself. This, however, would be impossible were the time factor
omitted.
The foregoing illustration is given merely to show that time is fully
as important as distance measurement in specifying the location of
any moving body (and everything in the universe is moving). It is in
no sense intended to show how time is really a fourth dimension. This
fact will be brought out in subsequent pages; but thus far we have
followed, rather, our ordinary concepts of time as an independent
element, detached from any unit of space measurement whatsoever. Such a
conception, however, comes from a lifelong training which is based upon
a limited scope of vision.
=Knowledge Broadens Many Concepts=
Concepts do not necessarily represent the reality merely because they
are easy to believe. They may be simply following in a groove caused
by centuries of “thoughtless reasoning.” Prior to Galileo, human
concept taught that “up” and “down” were absolute directions which
never vary. It was easy to so believe, and nobody even thought to call
such concept in question. But eventually it became known that two plumb
lines which point “down” to earth do not hang parallel to each other.
They each point to a common center like the spokes of a wheel.
Now if two plumb lines happen to be about six thousand miles
(one-fourth the circumference of the earth) apart, they then hang
practically perpendicular to each other. When this erstwhile paradox
was established as a certain fact mankind were obliged to change their
former concepts of the absoluteness of the “up” and “down” directions.
Einstein now asks us to revise our concepts of the absoluteness and
independence of time and space and gives us his reasons therefor.
In the light of Einstein’s treatment of the principle of relativity it
is seen that no particular reference frame possesses any advantage over
any other for mathematical accuracy irrespective of the physical laws
involved. It would not be difficult, of course, to grasp this fact in
a mechanical sense provided units of length and of time were absolute
quantities which are unaffected by the motion of the reference frame.
But as will be subsequently seen, such units when viewed independently,
do vary constantly, and rigidity becomes in reality a meaningless term.
In the face of this apparently insurmountable difficulty, however,
Einstein shows in his Special Theory that any unaccelerated (i.e.,
uniformly moving) frame of reference is as suitable as any other for
the mathematical expression of physical laws. This is accomplished by
regarding the universe and all objects therein as existing in four
dimensions, viz., length, breadth, thickness and _time_--the
latter altering the length unit according to the relative velocities of
the reference frames.
It is naturally impossible to mentally visualize or graphically portray
more than three dimensions, but they can be mathematically conceived.
We shall endeavor presently to show how time takes its place alongside
the ordinary three dimensions in the true geometry of the universe,
possessing the value of a fourth co-ordinate or dimension. In reality
it supersedes in importance the other three in the sense that it
possesses the illusive quality of automatically correcting or adjusting
physical values which otherwise would be inconsistently altered by the
velocities of our frames of reference.
Irrespective of whether we can visualize the matter or not, it will be
necessary for the reader to divest himself of all previous conceptions
of time and of space as universally absolute, separate and unvarying
in their unit length if he would comprehend the Einstein Theory of
Relativity. He must guard against the notion that time and space are
independent elements that should be measured separately. He must
adjust himself to the relativistic viewpoint that time and space are so
interlinked that either, when taken alone, becomes meaningless except
by analogy. When, therefore, we measure the distance between bodies or
the dimensions of the bodies themselves, we are not calculating the
miles or units of _space_ merely, but of _space-time_.
We will not attempt but will purposely avoid, in a work of this scope,
the setting out of algebraic equations, believing they would not tend
to make the essay _popularly_ readable however much they may
appeal to the mathematical student. Therefore it must suffice here to
state that the numerical value of time required for a light ray to
traverse a given distance, together with the relative velocity between
the frame of reference and the object, become considerable physical
factors in calculation where great distances and enormous velocities
are being dealt with. These items, however, are infinitesimal when
merely earthly distances and ordinary velocities are involved.
Einstein’s equations, therefore, may be said to have no practical
bearing upon the ordinary things with which we have to do in daily
experience--we may continue to use our yardstick and our pocket
timepiece exactly as before. But this in no wise diminishes the
fundamental importance of the matter. Scientific interest rests not in
the _amount_ of variance from accustomed laws, but rather in the
fact that a variance exists and why.
=Universe a Four-Dimensional Continuum=
We shall at this point merely touch upon the space-time character of
the universe as Einstein sees it, leaving the subject for treatment in
appropriate order later on. Einstein did not originate the geometry
which he uses; he has simply made a masterful application of the work
of Riemann, Minkowski and others in the outworking of his theory.
Various geometers, notably Minkowski, departing from the beaten path of
Euclid, had come to view the universe as a four-dimensional continuum
in which space and time are inseparably interlinked, and Einstein saw
in this a solution of several phenomenal problems which had arisen in
recent years to which the laws of Newton appeared inapplicable.
The term _continuum_ denotes a continuity of units. The geometer
speaks of a straight line as a one-dimensional continuum, because it
consists of a continuity of _points_ extending in one direction. A
plane, likewise, is termed a two-dimensional continuum for the reason
that it is a continuity of _lines_ laid side by side producing an
area of length and also breadth. Accordingly the whole spatial universe
has been long regarded as a three-dimensional continuum, i. e., a
continuity of _planes_ piled one on top of another, extending from
infinity to infinity.
As for _time_, it has been commonly looked upon as something
entirely separate and apart from space. Humanity has habitually
regarded it as an independent one-dimensional continuum, i. e., a
continuity of _instants_, each being of universally definite
duration, beginning where its predecessor ended and ending where its
successor begins, and thus flowing on forever regardless of motion,
location or any physical condition.
=Time and Space Inseparable=
But what ground have we for regarding space and time as independent and
unrelated continua? Does not _time_ thrust itself upon us at every
turn, wherever we undertake a measurement in space? This might not be
so if we could regard space as motionless and then really measure it
from point to point. But absolute space cannot be measured. The best
that can be done is to measure from one body of matter to another
body of matter--and all matter is in motion and continuously changing
position, hence the entrance of time into all physical calculation.
Everything in the universe is somewhere at some time and somewhere else
at some other time. Thus it is seen that time must intersect space at
every point wherever moving matter is involved. And if for any reason
whatsoever the time units are shortened or lengthened then the points
which they intersect are in reality distorted irrespective of what our
conception of such a state of affairs might be.
Instead, therefore, of regarding the universe as a continuity of
immovable points it is in reality a continuity of _events_ in
each one of which _time_ is an essentially governing factor.
Consequently the term _event_ in a four-dimensional continuum is
analogous to the term _point_ in a three-dimensional continuum.
Something does not necessarily need to _happen_ at each
_event_ in the continuum in order to constitute them “events.”
On the contrary, the continuum of events exists as a background for
phenomena, and when happenings occur in any region whatsoever, the
events (time and space points combined) are there, ready to give forth
their testimony to the mathematician when he calls for his location
data.
The fact that we cannot diagram such a continuum in no sense detracts
from its reality. A combined space and time is no less real than
the conceptual independent space and time which it supplants. There
are various other continua in the world about us which cannot be
represented by lines and angles or physical models. There is, for
instance, the continuum of color, reaching from ultra violet to ultra
red with its infinite number of graduating hues in between. Then there
is the continuum of sound which cannot be visualized in any degree.
The scale of musical notes is a perfect continuity, extending from
the lowest to the highest audible sound and beyond, yet we cannot
_see_ a continuum of this character any more than we can visualize
the four-dimensional continuum of space-time.
From the foregoing it will be seen that the difficulty of making clear
in few words the substance of the Einstein theory is due to its radical
departure from our ordinary concepts of things. Many pages might be
utilized to fully define such apparently simple terms as time, space,
distance, straight line, etc., and after these are seen from the
viewpoint of the relativist, then the reader is equipped to proceed
with his study of relativity, but not before.
The aforementioned handicap has been apparent to every writer who has
attempted to make the theory of relativity popularly readable. The
following pages will discuss these terms in an applied fashion and
will attempt to give the necessary foundation knowledge without which
it would be impossible to appreciate the theory as expounded later in
this essay and in other works on the same subject. The reader will
undoubtedly find it profitable to review the foregoing paragraphs after
he has perused the arguments which follow.
=Relativity and Its Effects=
Relativity, as applied to motion of all matter and systems of matter in
the universe, stands opposed to the idea of _absolute velocity_.
If nothing is stationary in the entire universe from which we may
determine the _actual_ rate of motion of bodies, then the best we
can do to describe the velocity of anything is to say that it moves
at such and such a speed _relative_ to something else which is
also moving at some unknown rate except as it is related to some other
moving body.
We may say that the earth is traveling at the rate of over eighteen
miles per second in its annual trip around the sun. But this does not
represent the actual velocity of our earth. It describes merely our
motion relative to the sun; but who shall say how rapidly our sun is
moving through space[1] and carrying us with it, just as we carry
the moon with us as we revolve around the sun. We know the sun is
apparently approaching a distant star cluster, but we cannot determine
whether our system is moving toward it or whether the cluster is moving
toward us, or both. We may be, in fact, chasing it through the heavens
as a dog chases a rabbit, and gaining on it a trifle each century; or
it may be really chasing us. All we know about it is that the distance
between the two systems is growing gradually less.
Then, again, who knows but that the entire stellar universe, including
not only our solar system but all other systems as well, may be
revolving about one common center located in the remote regions
of space? And if so, in what general direction does it revolve?
These reflections immediately convince us that all motion is purely
relative; that no velocity can be looked upon as being absolute. Hence
our eighteen-mile-per-second velocity around the sun is probably
infinitesimal in comparison to our _actual_ speed through space,
if such could really be determined by some stationary standard.
Einstein did not originate the doctrine of relativity; it has been
a much discussed philosophic subject for centuries and particularly
of the nineteenth century. What he did, however, was to formulate a
particular theory concerning it which co-ordinates and satisfies the
observed laws of nature and accounts for discrepancies which have
long troubled mathematicians and scientists who have based their
calculations on the theories of the past, notably Newton’s laws.
=Mercury’s Perihelion=
A striking example of such a discrepancy which Einstein has accounted
for, is the unusual yearly advancement of the perihelion of Mercury’s
orbit. Due to gravity, all planets revolve about the sun in ellipses
rather than in perfect circles, with the sun a trifle to one side of
the center of such ellipse. This brings the planet nearer to the sun
at one end of the ellipse than at the other. The near end of the orbit
is called the _perihelion_, while the distant end is called the
_aphelion_. See Fig. 1.
Newton’s law would indicate that if our spherical sun had but one
planet revolving around it, the orbit of that planet would never change
its position unless disturbed by some outside cause, its perihelion
and aphelion being fixed. But where there are more than one planet in
a system, a slow annual advance of the perihelion would be produced.
The amount of such advance is easily calculated; hence it has been
an astonishment to astronomers to find that the perihelion of Mercury
actually advances 42 minutes (that is, seven-tenths of a degree) per
century more than Newton’s law allows for. Einstein, however simply
points out that at perihelion a planet is moving with greater velocity
than at aphelion because of its relative nearness to the sun, and
that its velocity (the _time_ element) must be reckoned with in
addition to the Newtonian gravitational advance. He computed that this
should increase the advance of Mercury’s perihelion by 43 minutes per
century, which most fully accounts for the observed discrepancy.
[Illustration: FIG. 1.]
Opponents of Einstein have attempted to account for the aforementioned
discrepancy on the ground that the sun is not a perfect sphere, and
that its equatorial diameter exceeds its polar diameter sufficiently
to add the required amount to the attraction at perihelion. But this
involves other difficulties, as for instance, a change of 3 minutes
per century in the inclination of the orbit, which manifestly does not
exist. The orbits of the other planets in our solar system are not
sufficiently eccentric to reveal any marked difference between Newton’s
and Einstein’s calculated results. But Einstein’s success in connection
with Mercury has placed his theory upon a very satisfactory foundation.
We have seen that all motion is relative. The same is true of
_time_ because motion and time are inseparable. But even if this
were not so, where would we find an absolute standard or universal unit
of time any more than an absolute rate of motion of matter? We on earth
count time according to the rotation of the earth on its axis, and we
call the period of rotation a day, but the other planets in our solar
system have days of very different length from ours, some shorter and
some longer.
All heavenly bodies possess their respective time standards, all
different from ours and different from each other. Which shall be
taken as the _absolute_ standard? There is no universal standard.
Time is not an absolute quantity; it is relative even as motion is
relative. A “perfect timekeeper” if suddenly transferred from earth to
Jupiter would immediately be seen to keep a different time due to the
differences in velocity of the two planets.
=The Universal Unit=
We have already emphasized that conceptual time, as an independent
one-dimensional continuum, is fictitious. It does not really exist as
such, but is a component part of space-time. The question naturally
arises in our minds: if time does not exist in and of itself, and if
there is no universal time unit, then how is it possible for Einstein
or anybody else to make a calculation in which time is involved and
arrive at any definite conclusion? The answer is that there is a
universal unit, but not a universal _time_ unit. This true unit
is the _separation-interval_ between events in the space-time
continuum. It is a combination of distance and time.
Such a combination unit may be partially illustrated by a crude
analogy. Suppose we are calculating the distance between two points
in a plane. We would first describe a triangle and let the hypotenuse
of the triangle connect the two points in question. But somebody else
might erect a different triangle from ours which would describe the
distance between the two points equally well. The triangles would have
the same hypotenuse, but their respective bases and altitudes would be
dissimilar, as shown in Figure 2.
Let us think of the bases as representing time and the altitudes
as representing space, while the hypotenuse stands for our
separation-interval. Even as we may have many base-altitude
combinations for our common hypotenuse, so we may have numerous
combinations of space and time for the same separation-interval. In
certain combinations the space element is greater than in others,
while the time element is correspondingly smaller, and vice versa,
although the separation-interval, like our hypotenuse, remains a
constant.
[Illustration: FIG. 2.]
It may be somewhat easier for the reader to appreciate the
non-existence of a universal unit of length than it is for him to
comprehend the unreality of independent time. No unit of length can
be taken as a universal standard of measure, because measurements
are relative and dependent upon the motion of the observer and his
reference frame, or upon the velocity of the object relative thereto.
This has been scientifically established by experiments made with
particles emitted by radioactive substances whose velocities range
from 20,000 to 170,000 miles per second.
Lorentz and Fitzgerald, previous to Einstein, had suggested that all
moving matter suffers a physical contraction in the direction of its
motion, but their theory is not particularly convincing. To Einstein
belongs the credit of postulating upon this subject in a manner that
agrees with experiment and satisfactorily answers several phenomenal
questions. He points out that there is an _apparent_ contraction
which is proportional to the relative velocity between object and
observer, but that this “contraction” does not exist if the observer
happens to be moving along with the object which he is measuring.
To illustrate: If an ocean liner measures 1,000 feet in length while
lying at the pier, theoretically it would be a trifle less than 1,000
feet while under way if viewed by an observer on shore. If, however,
the measurement were taken by an observer aboard the moving ship,
using the same yardstick that was used at the pier, the result would
still be 1,000 feet. Einstein’s contention is that the ship undergoes
no physical shortening such as Lorentz and Fitzgerald supposed, but
it is simply the victim of a phenomenon of observation. The apparent
contraction, however, holds good for all object or bodies of matter in
exact ratio to their velocity relative to the observer as specified
in Einstein’s mathematical formula, and is just as real in practical
calculation as though it were a physical factor.
=Motion and Contraction=
It is within the realm of possibility, of course, that some degree of
physical contraction does result from motion, on the theory that the
electro-magnetic forces operating between the atoms and molecules of
matter undergo a change due to velocity. If, therefore, the yardstick
and everything else aboard the moving ship suffered a physical
contraction exactly proportional to the length of the ship itself, then
there would be no way of detecting it by any measurement taken aboard
the vessel.
It hardly appears reasonable, however, that materials of different
density would undergo the same proportional contraction, as for
instance a wooden yardstick and the steel sides of a ship, inasmuch
as they are of entirely different molecular composition. The Einstein
theory therefore proposes an observational variation rather than a
physical contraction of the object, and shows that it equally exists
whether we regard ourselves as at rest and the object as moving away
from us, or whether we consider the object as stationary and ourselves
as speeding past it.
Let us suppose our observational instruments are lifted from their
fixed position on shore and placed aboard a railway train and carried
in the same direction and at the same velocity as the coastwise vessel.
In this event the ship would measure full 1,000 feet in length just as
it did when we measured it at the pier, because the observer under the
conditions stated would be at rest relative to the moving ship. But if
our train carried us faster than the ship, then the ship would again
begin to measure short because our _relative_ motion would be the
same as though we were stationary and the ship were moving away from us
in the opposite direction.
The foregoing illustration is merely theoretical, however, because the
variation is too small for observation in cases of small distances
and low velocities. Nevertheless, when the velocity approximates that
of light rays the apparent contraction becomes plainly visible. Some
of the particles emitted by radioactive substances possess a velocity
of about nine-tenths that of light and in such cases the amount of
apparent shortening which they undergo can be computed because it is
very great. And should the velocity become equal to that of light rays
(i. e., 186,300 miles a second), then, says Einstein, the observed
length of the particle would be reduced to zero.
Einstein does not claim that relative velocities greater than 186,300
miles a second cannot be attained, but he does contend that velocities
greater than that relative to an observer cannot be observed. Thus if
a body of matter were moving away from an observer at only half the
velocity of light, and the observer himself should suddenly become
accelerated in the opposite direction until the relative velocity
between the two became equal to or greater than that of light, then the
observed _length_ of the body in the direction of motion would be
zero, although its width would be unaffected.
=“Absolute Length” Fictitious=
All this leads us ultimately to the conclusion that there is no such
thing as determining the _absolute_ dimensions of anything,
because relativity of motion and the time element are undeniable
factors in all measurements. The assertion that these are infinitesimal
so far as the quantities we ordinarily have to deal with are concerned
does not alter the fact that “_absolute_ length” is a fictitious
phrase. For this reason we cannot reckon the _absolute distance_
between any two conceptual points in the universe; we must calculate in
units of _space and time_ combined or else accept the fact that
our conclusions are simply of local and not of universal significance.
We may, for instance, make a measurement of the distance between
Neptune and the sun according to Euclidean geometry, obtaining a
certain result. But if we were on another planet and there chose a
reference frame which has a wholly different relative velocity to the
frame of reference which we used here on earth, and with this new
reference frame we measured the identically same space between Neptune
and the sun we would obtain a very different result. Which measurement
would be correct? Neither would be correct if by that term we mean the
_absolute_ distance if the universe were motionless and unwarped
by matter.
Hence in measuring great distances involving enormous velocities
we cannot ignore the principle of relativity and hope to obtain
universally accurate results. It is because of this oversight that
Euclid’s geometry is found to be inadequate in such cases. But as
already suggested, there is a geometry that is universal in its
application, in which _time_ enters as a fourth dimension. The
measurements taken according to such a geometry do not, therefore,
represent distance merely, but a blending of distance and time. This is
the geometry which Einstein employs. It will be discussed in greater
detail further on.
=The Special Theory of Relativity=
We will now consider Einstein’s original theory of 1905, which has
subsequently been called the “Special” or “Restricted” Theory of
Relativity, before attempting an examination of his “General Principle”
which he announced ten years later. By so doing we will find it a
stepping stone of much worth, leading us naturally to the General
Theory which will be discussed later.
The Special Theory is summarized in two postulates (propositions),
one relating to uniform, straight-ahead motion, and the other to the
velocity of light rays. It is the combination of these two propositions
that necessitates the interrelation of time and space. If both are
true, and the weight of evidence seems to be on Einstein’s side, then
the space-time geometry already alluded to must be correct. In any
event our new conception of space and time as heretofore mentioned has
doubtless prepared our minds to appreciate the postulates which we will
now paraphrase in non-technical terms as follows:
(1) EVERY LAW OF NATURE MAY BE STATED IN A FORM WHICH WILL HOLD
GOOD BETWEEN ALL OBSERVERS AND OBJECTS PROVIDED THE OBSERVERS AND
OBJECTS ARE EACH MOVING IN A STRAIGHT LINE AND WITH UNIFORM VELOCITY;
AND UNDER THESE CONDITIONS NO OBSERVER COULD POSSIBLY DETECT HIS OWN
MOTION BY ANY LOCAL EXPERIMENT WHATSOEVER, UNLESS HE MAKES REFERENCE
TO OBJECTS OUTSIDE HIS OWN MOVING SYSTEM.
(2) LIGHT RAYS IF UNOBSTRUCTED HAVE AN OBSERVED CONSTANT VELOCITY
IRRESPECTIVE OF THE RELATIVE VELOCITY BETWEEN THE OBSERVER AND THE
SOURCE OF LIGHT.
On the surface there is nothing unusual about the first postulate;
it appears to be simply another way of setting forth the mechanical
principle of relativity announced by Newton. But it really involves
more than that, because so long as we regard all bodies and measured
lengths and times as rigid and unchanged by the motion of our reference
frames, then it is not true that every law of nature holds good between
observer and objects as set forth in the postulate. Electro-magnetic
laws, for instance, are an exception, for they really do change their
form in proportion to the relative motion of our reference frames.
This first postulate simply demands a restatement of the laws of nature
to make them harmonize with the principle of relativity _in toto_.
As already observed, many inconsistencies arise in regard to time,
distances and dimensions if we hold to our old conceptions that units
of time and of lengths are absolute quantities which cannot vary under
any circumstance or condition. We know by experiment that they do vary,
hence the need of a restatement of nature’s laws to account for the
facts resulting from motion. This postulate, however, concerns only
one kind of motion, viz., uniform and rectilinear motion. Rotating and
accelerated and generally irregular motions are dealt with under the
General Theory which will be considered later.
=Laws of Nature Not Unalterable=
In this connection it is well to bear in mind that a “law of nature,”
as the term is commonly used, is not some God-given formula that cannot
be altered. It is but a human description of the operation of nature,
based on observed facts. As our powers of observation and knowledge
increase it sometimes occurs that our “laws of nature” are found to be
inadequate and need revision. That is precisely what Einstein calls
for. In mathematics we may stipulate that “things equal to the same
thing are equal to each other,” but physically speaking the phrase is
wholly ambiguous. Things observed to be equal while in uniform relative
motion lose their equality when their velocities vary.
It will be noted that Einstein in his first postulate also specifies
that when an observer is moving straight ahead at uniform velocity he
will be unable to detect his own motion and will believe himself to be
at rest unless he performs an experiment on some outside object that
is not moving along with him. The truth of this is apparent to anyone
who has been aboard a smoothly running train while entering or leaving
a station, and has been unable to determine whether it is his train or
the train on the next track that is in motion, or both, until finally
his train ceases to move uniformly and he experiences a jolt as it
suddenly comes to a halt, or is thrown against the back of his seat as
it begins to speed up. Or perhaps while the train was moving uniformly
he performed an observational experiment on some outside object, such
as a house or a telegraph pole, and thereby arrived at the conclusion
that his train is moving rather than the one alongside it.
Similarly we on earth are prone to consider ourselves as at rest and
the heavenly bodies as moving around us, and for untold centuries
mankind never stopped to question that conception. Even since we
have come to know that we are merely one of the millions of rotating
heavenly spheres which go to make up a universe, we are disinclined
to admit that ours is not in some sense a favored position, or that
an earthly reference frame is not somehow intrinsically better. But
reflection convinces us that this is not so; and since it is not so,
we would do well to put our science on a universal rather than upon a
purely local basis by altering whatever age-old conception is necessary
to make it agree with the principle of relativity.
=Paradoxical Behavior of Light=
It is the second postulate rather than the first that is astounding,
because it substantially states that light rays from a given source
will reach an observer who is running away from the rays just as
quickly, i. e., at the same velocity, as they would reach another
observer who is moving toward the rays. If an explosion should take
place on the sun at this moment it would, of course, take the light
about eight minutes to reach the earth, traveling at a constant
velocity of 186,300 miles a second. Einstein’s declaration is that
if two observers are on opposite sides of the rotating earth, one
revolving away from the sun and the other revolving toward it, the
instruments of each observer will indicate that the rays from the flash
are traveling past him at exactly 186,300 miles a second regardless of
whether he is traveling toward or away from the sun.
Ordinary concept would assume that in the one case the observer’s own
velocity toward the sun should shorten the time it takes the light
to reach him and thereby make it appear to him that the light is
traveling faster than 186,300 miles a second, while in the other case
we would suppose that the observer’s instruments would indicate that
the light is traveling slower than 186,300 miles a second due to his
own velocity away from the sun which should lengthen the period of
transit of the rays in overtaking him. But actual experiment appears to
contradict this and to corroborate Einstein’s postulate, paradoxical
though it seems to our accustomed concepts. Only by viewing the
universe as a four dimensional (not as a three dimensional) continuum
can the matter be understood.
It is not, of course, any more strange that when _light_ waves
once start on their journey, the velocity of those waves would
thereafter be unaffected by the movement of the source from which they
originated, than that waves of _water_ would not be increased or
retarded by any forward or backward movement of the ship after the
waves have been started on their course across the lake. Newton knew
that the velocity of light past an observer is not increased by reason
of the source of light moving toward the observer. But what he did not
see was that _neither is the velocity of light increased by reason of
the observer moving toward the light_.
In other words, after admitting the principle of relativity in a
mechanical sense, practically as stated in Einstein’s first postulate,
Newton then denied or at least overlooked that principle when it
touched the subject of the propagation of light. Take, for instance,
the case of light rays reaching us from certain distant stars. We
cannot surely know whether the source of light is traveling toward us,
or whether we are moving toward the source of light. Newton would say
that the velocity of light would be unaffected in the first instance,
but increased in the second. Einstein says it makes no real difference
which way it is, because the principle of relativity is universal,
all motion is relative, and the universe is so constructed that the
velocity of light always appears constant to all observers irrespective
of their motion or of the motion of the source of light relative to
them.
If, however, we were traveling away from a given source of light at
a greater velocity than light itself can travel it would certainly
not be true that light, under such a condition, would register a
constant velocity. But such a rate of motion is manifestly impossible
of physical attainment, light being considered to possess the maximum
velocity greater than which no material body can travel. In other
words, 186,300 miles a second is the limiting velocity beyond which
physical phenomena does not reach, because all matter would evidently
suffer complete dissolution, being reduced to the state of free
electrons, by the time such a velocity would be attained. Light is
matter in the free electronic state, and the velocity of such a form of
matter is known to be 186,300 miles a second, as has been stated.
Light, being a form of matter, is acted upon by gravitation. This was
first demonstrated during the solar eclipse of May 29, 1919, when
photographs were taken of various stars whose light at that season had
to pass very near the sun in order to reach us. Three months later
photographs were taken of these same stars after they had moved from
their former positions. Their normal relative motion being known, it
was easy to determine from the two sets of photographs whether their
displacement was entirely due to that motion. It was found that their
displacement was considerably less than it would have been under normal
conditions, thereby indicating that the light from these stars had been
deflected as it passed near the sun’s rim, making it appear to the
observer that these stars then occupied positions which they did not
occupy. See Fig. 3.
[Illustration: FIG. 3.]
=Electronic Structure of Matter=
Light and electricity both travel at the same velocity, and in the
final analysis they are the same form of matter. All ponderable matter
is made up of molecules, and molecules consist of atoms of varying
elements, except where the substance is elementary throughout, in which
case the molecules are made up of a certain number of atoms of the same
element. An atom is an aggregation of negatively charged particles of
electricity, called electrons, which revolve and vibrate at enormous
rates around a central nucleus of “protons” carrying a positive charge.
If we conceive of a handful of sand as whirling around and around at
such an enormous velocity as to appear as a globe several feet in
diameter we have a visualization of the ultra microscopic atom and its
comparative dimension to that of the individual electrons and protons
which compose it.
Electrons and protons are manifestly identical in all form of matter,
but all atoms do not contain the same number of them. An atom of
hydrogen contains only one electron and one proton, while atoms of
heavier matter contain a vast number. It is the _number of electrons
and protons per atom_, together with their respective vibratory
rates, that constitute the difference between elements. Hence every
known form of matter, if reduced to the electronic state, would be
found to consist of the same original stuff. Light, therefore, may be
termed _free electrons_, i. e., electrons which are not bound into
atoms, and consequently glide off in every direction at the highest
possible velocity of which matter is capable, viz., 186,300 miles per
second.
Thus it may be seen that all other forms of matter, since they consist
of electrons in the atomic or “bound” state, could not possess a
velocity equal to that of _free_ electrons. But electrons do not
become “free” simply by reason of matter undergoing a change of form.
Water, for instance, may be changed into a solid (ice) or into gas
(steam), but the electrons continue to revolve around their nucleus
in true atomic order, and the atoms continue to hold together in the
molecular state, the only change being in their rate of vibration which
affects the degree of elasticity between the molecules and between the
atoms. Hence whether water be in the solid, liquid or gaseous state,
the molecules thereof will be found intact, each consisting of two
atoms of hydrogen and one atom of oxygen (H₂O).
When matter is being burned up, producing a flame or a glow of light,
this phenomenon indicates that some of the electrons have been freed
from their atomic condition and have started off on their journey in
the form of light rays, while another portion of them remain bound as
atoms but undergo a change of form, becoming either gas or ashes.
=Michelson-Morley Experiment=
Returning now to Einstein’s second postulate, that the velocity of
unobstructed light rays appears to be constant to all observers
irrespective of the relative velocity between the observer and the
source of light. The evidence on which this theory is based was first
produced by Michelson and Morley in 1887 who at that time undertook an
experiment to ascertain if possible the velocity of the earth relative
to the ether. The experiment revealed that light registers a constant
velocity, whether it travels in the direction of the earth’s rotation,
or against the rotation, or at right angles thereto. The experiment
was repeated many times with different apparatus and under various
conditions, but always with the same result.
It is interesting to note the details of this experiment. Light is
known to travel in waves, because rays coming against each other
from opposite directions can be made to “interfere” in precisely the
same manner as waves in water. If waves in the ocean come together
from different directions, one of two things will happen; they will
either unite and produce a larger wave, or else they will strike in
a manner to measurably kill off both waves. If the crests of the two
waves coincide they reinforce each other, but if they strike at right
angles, the destruction of the wave motion results. This is called
“interference.”
Light rays conduct themselves in the same manner, thereby revealing
their wave-like nature. This is not contradictory to the idea that
light rays really consist of individual electrons, freed from atomic
attraction. Possibly each separate electron which goes to make up a
ray of light vibrates in a wave-like manner, possessing a wave motion
within itself. Waves of light are exceedingly minute and we do not
ordinarily witness any reinforcement or interference of light rays
about us, because there is no occasion for them to “pile up.” However,
in laboratory experiments, interference of light waves has been
produced, and to whatever extent the interference kills off the wave
motion, to that extent darkness ensues irrespective of the brightness
of the light at its source.
Michelson and Morley worked on the theory that if they sent rays of
light from west to east (the direction of the earth’s rotation) and
then reflected them back over their course it should take longer
to make the eastward trip than the westward, because in the first
instance the earth is carrying the objective point away from the light
while in the latter instance it would be rushing to meet the oncoming
reflected rays. Under this condition there should be a noticeable
_interference_ of the light waves due to the difference in
distance and time involved in making the two halves of the round
trip. But to the amazement of all there was no interference whatever,
notwithstanding the fact that the apparatus was ten times larger than
it needed to be to reveal such interference of the waves had it really
occurred.
The conclusion reached by Einstein as a result of this experiment is
that since light rays consist of matter in its basic or electronic
state, freed from atomic attraction, they therefore possess the
limiting velocity of which matter is capable. Hence they could not
travel more rapidly than 186,300 miles a second even if given a quick
send-off, nor would our traveling toward the light affect its apparent
velocity to us--unless it were possible for us to be traveling forward
more rapidly than light itself can travel. This would undoubtedly be
impossible, inasmuch as any physical body would necessarily consist
of electrons in the atomic or “bound” state and therefore could not
possess the mobility that free electrons would enjoy. The universe,
then, being a four-dimensional continuum, is so constructed that the
velocity of light always appears constant to all observers within it.
This is what Einstein means when he postulates that light _in
vacuuo_ (i. e., unobstructed) possesses a constant velocity
irrespective of the relative velocity of observer and source of light.
That is, it is constant so far as the observer is concerned. Thus if
a flash should occur on any heavenly body and we were moving toward
the flash at say 40,000 miles a second and another observer were
moving away from it at say 60,000 miles a second, the experiment of
each observer would indicate that the light has reached him at exactly
186,300 miles a second, although according to Euclid’s conception of
space the light has been obliged to travel 100,000 miles a second
faster to reach the one observer than the other. But Euclid’s
conception is faulty, as will be seen shortly.
How, then, would it be possible for the light rays to possess the same
apparent velocity per second for the two observers? It would not be
possible if “time” and “distance” are absolute quantities having the
same meaning for all observers. But if “seconds” and “miles” mean one
thing to observer “A” and a totally different thing to observer “B,”
then the apparent contradiction of facts becomes harmonious. This
is the essence of the doctrine of relativity. Observer “A” himself
does not use the terms “seconds” and “miles” consistently, i. e., as
unvarying quantities, nor does anyone. They mean one thing today and
something else tomorrow, depending upon what we are measuring and the
relative velocity between the observer and the object. The observer
is not aware of this inconsistency. To him there is no inconsistency
whatever. Nevertheless, only by acknowledging the varying quantities of
time and of space, and admitting the geometry which combines the two
into one unit, can the Michelson-Morley experiment and other similar
observations be understood and explained.
=Simultaneity a Meaningless Term=
We have been taught that the true length of a moving body is “the
distance between _simultaneous_ positions of its end points”--a
very good definition, but impossible of application for the reason
that we cannot determine the _simultaneous_ positions of any two
points in the universe. Simultaneity is a meaningless term so long as
the absolute velocity of the observer and the absolute velocity of the
object being measured are unknown. We may know the _relative_
velocity between them, but that is not sufficient. The two may be
relatively at rest--but for all we know the entire universe may be
speeding through space at thousands of miles a second in either one
direction or another.
We may _see_ two events occur at the same instant, but that does
not prove that they actually _occurred_ simultaneously. Before we
could compute the exact time of the occurrence of either of the events
we must know the direction in which, and the velocity at which the
universe as a whole is moving, together with any and all velocities of
the observer at the moment. This knowledge we do not possess. Until the
_absolute_ velocity of bodies can be determined the question of
simultaneity must remain unsolved.
=The General Principle of Relativity=
When in 1905 Einstein published the foregoing postulates which are
limited to uniform, rectilinear motion he may have considered that
it would be expecting too much to look for a general principle of
relativity such as would hold good for all kinds of rotating and
irregular motions and by which observers of different and variable
velocities might agree as to the reality of things under their
observation. Concluding, however, that the universe must surely be
constructed in a consistent manner he finally set out to find some rule
or principle by means of which an observer in one region would be seen
to possess no advantage over an observer in any other region of the
great expanse in arriving at accurate conclusions.
Of course Einstein hardly expects to go to the Pleiades or to
Betelguese and from there take measurements and make calculations; he
is doubtless content to make all his observations from this earth.
But how may he be sure that observations made from a reference frame
located in this particular region of the universe will be true to the
reality since it is manifest that observers located elsewhere and using
different reference frames must necessarily reach conclusions different
from ours if they employed our accustomed laws? Maybe they would be
much nearer the reality than we! What right have _we_ to assume a
monopoly on truth! None whatever until we can formulate nature’s laws
in a manner that will hold good for every part of the universe alike.
Until we are able to do this our science must be like the vain efforts
of the unskilled fisherman who harpoons for fish. Ignorant of the
trick that water plays on the line of sight he strikes directly at
the spot where he “sees” the fish and always misses his prey. The
skilled harpooner, on the contrary, understands the law of refraction
of light rays in water, and knows how to allow for this refraction;
hence he strikes a little this side of where the fish appears to be
and is rewarded with success. He is guided by a proven law and thereby
ascertains the true location of the fish, whereas the other man follows
“blind” observation which is quite frequently deceptive.
Einstein’s “General Principle of Relativity” is not, in fact, a mere
generalization of the Special Theory in the sense that it simply
enlarges upon the two postulates which we have already considered.
On the contrary it handles the subject of Relativity from quite a
new standpoint, and therefore might be said to belong to an entirely
different school of thought. It does not lend itself to visualization
as readily as does the Special Theory, and is consequently more
difficult of explanation and comprehension. However, what we have
already learned concerning Relativity will materially aid us in
understanding what follows, for the two theories are, after all,
dealing with the same general subject matter. We shall therefore
endeavor to link the two phases of the subject in a logical and
consistent manner.
We know, as a matter of fact, that “uniform, straight-ahead motion”
which Einstein in his original theory assumed to exist, is an ideality
that does not appear in nature, because all motion with which we are
familiar is to some extent irregular, nor does any material object
move in a perfectly straight line. But realizing the necessity for a
standard from which to proceed, Einstein properly enough assumed a
standard of absolute perfection and absolute simplicity of motion,
even though it does not actually exist anywhere around us. In exactly
the same manner Euclidean geometry assumes and deals with theoretical
points, lines and planes which have no material existence in fact.
As set forth in Einstein’s first postulate of the Special Theory, an
observer on a uniformly moving system could not possibly detect the
motion of his system without making reference to some outside object.
In the case of bodies or systems moving irregularly (i. e., with
acceleration) however, an observer thereon would detect “forces” acting
upon himself and upon all other objects on his system, due, of course,
to the acceleration. Recalling the illustration of the moving train:
so long as it is moving with perfect uniformity an observer thereon
would not know he is in motion at all until he made a comparison
with some outside object. But if the train suddenly slows down he is
thrown forward in his seat; if it speeds up he is thrown backward.
This force is called _inertia_. Now if we had never experienced
it before and were put aboard a noiseless and uniformly moving car
from which we could not see out we would be unable to interpret these
strange “forces” that we would feel as the motion of the car became
accelerated. We would probably attempt to explain them as some sort
of magnetic attraction, exactly as we are accustomed to explain the
“force” of gravity.
=Gravitation and Inertia=
In the General Principle of Relativity Einstein deals with these forces
(inertial and gravitational) and attributes them to a common cause,
viz., acceleration of motion, and has put the matter upon a consistent
mathematical basis which at once accounts for certain discrepancies
long observed in Euclidean geometry and in Newton’s laws. It is obvious
enough that where there is no acceleration of motion there could be no
centrifugal or inertial force exhibited: but we have been accustomed
to looking upon gravitation as something entirely different--as a
mysterious drawing power or attractive force that is somehow inherent
in matter. But gravitation is non-existent if we fall with the proper
acceleration. To use Einstein’s own illustration: if we were in a
closed room poised somewhere in gravitational space, and began to
fall with the acceleration common to that field, there would be no
gravitational effects to be observed. Objects released by our hand
would not fall but would remain where they are, and we could raise
ourselves from the floor and stand midway between the floor and the
ceiling as easily as upon the floor itself.
Again assume we are in a closed room poised in space, in a region
remote from any gravitational field whatsoever. Then suppose we began
to rise with a constant acceleration. Forthwith we would feel our feet
pressing against the floor. Objects released from our hand would strike
the floor by reason of the floor rising up to meet them, and in all
respects the effects would be identical with that of gravitation. In
other words we would have created an artificial gravitational field,
and it would be due to our accelerated motion.
The characteristics of gravitation and inertia are identical. No
amount of insulation or screening will diminish the “pull” of gravity
on anything. Furthermore, gravity acts on every kind and quantity of
matter alike, so that if a feather weighing less than an ounce and
a pig of lead weighing a ton were held side by side at the top of a
great vacuum tube and allowed to drop at the same instant, the feather
would reach bottom within the same time as the lead, each falling at an
acceleration of approximately 32 feet per second. It is the resistance
of the air that retards the fall of light materials, such as a feather,
but in a vacuum there is no resistance and gravity is found to act on
all matter to the same degree under such conditions. The same is true
of inertia _in vacuuo_.
When this relationship between the two forces is recognized we are
prepared to believe Einstein when he states that inertial force and
gravitational force are due to a common cause, viz., acceleration. This
does not mean that our earth, for instance, is being accelerated in all
directions at once, expanding out to meet “falling” objects such as in
the case of the artificial gravitational field mentioned in the above
paragraph. It does mean, however, that the falling objects themselves
are accelerated, but as will be presently seen this acceleration is not
due to any attractive force exerted by a “center of gravity” but rather
to a warped condition of space which surrounds all bodies of matter.
Neither Newton nor Einstein have attempted to analyze the structure
of matter and on this basis explain the phenomenon of gravitation.
Newton evidently believed, however, that every particle of matter
exerts a drawing force upon every other particle of matter, hence he
formulated his law which specifies this attraction between bodies as
being directly proportional to the product of their mass and inversely
proportional to the square of the distance between them. But he did not
attempt to make clear what that “drawing force” is, or _why_ it is
inherent in all matter, nor did he explain how or through what medium
or mechanism it operates.
Newton contented himself with merely dealing with the phenomenon
of gravitation in the abstract. So does Einstein, but with this
difference: the latter denies the existence of any mechanism whatever
in connection with gravitational force so far as any attractive power
from within is concerned, and accounts for it on purely geometrical
grounds. This is the most difficult phase of the Einstein theory
for the layman to grasp, for the reason that it involves the whole
structure of non-Euclidean geometry with which the public is generally
unfamiliar.
=Non-Euclidean Geometry=
Euclid, the famous Greek mathematician, in the third century B. C.
published the first systematic treatise on geometry (the science of
space and its measurement), and his axioms and theorems are generally
taught in our high schools and colleges today. Euclid proceeded upon
the simple theory that all space consists of points, lines and planes.
He defined a _point_ as that which has position but not size; a
_line_ (continuity of points) as possessing length but no breadth
or thickness; and a _plane_ (continuity of lines) as having length
and breadth, but no thickness. They are simply abstract terms having
no physical existence in nature, except as they exist in our minds.
Nevertheless they have proved themselves convenient in measurement and
calculation.
But when mathematicians, after centuries of earnest effort, were unable
to prove Euclid’s postulate concerning parallel lines, it occurred
to some of them that possibly the whole Euclidean system rests upon
a faulty foundation. Then it was that Saccheri in Italy, Legendre
in France, Gauss in Germany, Bolyai in Hungary and Lobatschewsky in
Russia, all masters of Euclidean geometry, conceived of other methods
of decomposing space than that proposed in Euclid’s _Elements_.
Thus it was that early in the nineteenth century, almost simultaneously
in many countries, did many non-Euclidean geometric works come to be
published. These were of the same general character or form, commonly
called Hyperbolic geometry. Each of them is as consistent in itself
as is the geometry of Euclid. But to Riemann belongs the credit of
formulating a geometry which in the light of Einstein is seen to
approach much nearer to the reality of nature than does the Euclidean
or any other system.
Riemann produced his general work along this line in 1854 which was
far ahead of his time. He actually prophesied the connection of
geometry with matter, and had he possessed a little more vision he
would doubtless have worked out the details as well as the principles
underlying gravitation in much the same manner as Einstein has done.
Riemann’s efforts in the field of non-Euclidean geometry has materially
aided Einstein in the development of the present theory. Minkowski’s
work was utilized by Einstein to much profit in the outworking of the
Special Theory, particularly his clarification of _time_ as a
fourth dimension.
=Time as a Fourth Dimension=
[Illustration: FIG. 4.]
It is natural for us to think of all matter as possessing but three
dimensions--length, breadth and thickness--and we have been accustomed
to making our measurements of matter and of space on that basis. Using
the formula of Pythagoras we have ascertained the distance between any
two points in a _plane_ (a two-dimensional area) by extracting the
square root of the sum of the squares of the co-ordinate axes, i. e.,
the base and the altitude as in the accompanying diagram. See Figure 4.
If point A is 8 miles south and 6 miles west of point B then A and B
are 10 miles apart, thus:
The square of 8 is 64
The square of 6 is 36
---
The square root of 100 is 10
Likewise, the distance between any two points in a three-dimensional
region (as from an upper to the remotest lower corner of a room) is
generally considered to be “the square root of the sum of the squares
of the three sides” (Fig. 5).
Thus if the distance O to X is 12 feet and X to Y is also 12 feet,
while Y to Z is 14 feet, then the straight diagonal distance from O
directly through the room to point Z is 22 feet, because the sum of the
squares of the three sides (144 + 144 + 196) yields a total of 484, and
the square root of that number is 22. This simple formula will hold
good for all ordinary measurements, but for great distances in space
a slight correction is found necessary because of the little trick
that light rays are prone to factor, i. e., the numerical value of the
interval of time required for a light ray to traverse play upon us. We
must subtract the _time_ from the distance. Hence if our cube
were large enough to fill a goodly portion of the universe we would
no longer say that the diagonal distance from O to Z is
√(x^2 + y^2 + z^2) but rather √(x^2 + y^2 + z^2 - t^2), ----t,
of course, representing _time_.
[Illustration: FIG. 5.]
Now recall what we learned in the preceding pages, that the velocity of
light always appears to be the same to all observers irrespective of
the relative velocity between the observer and the source of light. It
is manifest, therefore, that in making measurements the time factor (t)
_really_ represents one quantity for one observer and a totally
different quantity for another observer notwithstanding the fact that
it appears to be a constant to all observers. Inasmuch as the velocity
of light does appear to be constant to all observers its actual
stretching or contracting of units is not manifest. Therefore the
corrected equation as given above (the subtraction of the _time_
element) holds good for all observers irrespective of their motion.
The point of interest to the non-Euclidean geometer in connection with
any measurement, be it remembered, is not the abstract _distance_
between points, because distance is not a constant and is not
determinable unless we know the _absolute_ velocity of the
observer and of the points being measured, which knowledge we do not
possess. What we should look for, then, is the _distance and time
combined_, or the _separation-interval_ as it is aptly called.
The time factor automatically corrects the units for each observer, no
matter what his motion may be, and thus the separation-interval appears
a constant.
The foregoing illustrates how time takes its place alongside the
ordinary three dimensions of space, and is in reality a fourth
dimension, although it is not a thing that can be visualized as we can
visualize the length or breadth or thickness of any object. In the
following paragraphs we shall examine further into the geometry of the
universe, particularly as it relates to the phenomenon of gravitation.
=Geometry with a Physical Meaning=
Certain news dispatches and book reviews have erroneously reported
Professor Einstein as having said “only twelve men in the world can
understand the Principle of Relativity.” The statement becomes absurd
in view of the scores of volumes now in print, all of which set forth
more or less clearly the details of the Einstein theory. What he
alluded to in the remark so generally misquoted and misconstrued is his
mathematical equations (calculus of tensors). He questioned if there
are more than a dozen mathematicians in the world who are familiar with
this abstruse differential calculus because it is not generally taught
in the university text books.
This calculus is a veritable maze of formulæ, really invented by
Riemann and Cristoffel, but systematized by the celebrated Italian
mathematicians, Ricci and Levi-Cevita, and is impractical for any
ordinary use. This is why so few mathematicians have familiarized
themselves with it. Einstein, however, found it invaluable in dealing
with such complex geometrical problems as his theory produced.
Briefly, the non-Euclidean geometer deals with _surfaces_ rather
than planes, and his fundamental postulates are sufficiently broad
to apply to all regular surfaces whether they be planes, spheres,
cylinders, conicoids or even spheroids or ellipsoids. He considers
a “straight” line as being the shortest distance between two points
_on a surface_, hence if the surface is curved the “straight”
line connecting any two points thereon will also be curved. This
_shortest_ distance between points is called a _geodesic_. If
the surface happens to be a _plane_ then the geodesics connecting
points thereon are really straight lines in the Euclidean sense, but
this would not be true for any other kind of a surface. Thus it is seen
that Euclidean geometry is simply a limiting case of this more general
geometry.
Geometers of the elliptic or spherical school, including Einstein,
declare that in nature there is no such thing as a purely Euclidean
straight line such as may be prolonged in opposite directions to
infinity. On the contrary they hold that any “straight” line if
prolonged sufficiently would return upon itself, because the universe
is so constructed. In other words, what we ordinarily call a straight
line is but an arc of a near infinite circle which possesses the
_least possible curvature_. Magazine writers in an endeavor to
make clear this portion of the theory of Relativity have strikingly
declared that “according to Einstein a man might look through a
telescope in any direction whatsoever and behold the back of his
neck.” This jest, though omitting essential facts, is not without
geometrical foundation. If we possessed a near infinite telescope and
should live for a near infinite period of time to enable the rays of
light to traverse this near infinite circle, then, if there were no
obstructions along our line of sight, we might be rewarded with a
round trip view of the rear portion of our body--though the simpler
method would be to use two ordinary mirrors.
All this, however, has an important bearing upon Einstein’s
interpretation of gravitation. Not only does he contend for
Lobatschewsky’s “curvature of space” but he also holds that surrounding
every body of matter there is a _special_ space-curvature (four
dimensional), the degree of which depends upon the body’s observed
mass. This special curvature or “warp” of space constitutes the
“gravitational field” surrounding all large bodies of matter and causes
the acceleration of falling particles in that field. This distortion of
space increases in proportion to the mass of the body causing it, and
decreases with the distance from that body until ultimately it becomes
_nil_ or practically so in a region remote from all matter.
Perhaps the nearest approach to a visualization of this space-curvature
(which constitutes a gravitational field) is to consider the lines of
force in a magnetic field. The reader is doubtless familiar with the
age-old experiment of placing file dust on a thin sheet of cardboard or
plate of glass and then holding a horseshoe magnet underneath with the
two poles touching the sheet or plate. Immediately the filings arrange
themselves into curved lines between the poles as shown in Fig. 6.
This experiment indicates that between the poles of a magnet are
constant lines of force, invisible to sight but manifesting themselves
when attractable particles are in or near their path. The earth,
likewise, is a great magnet, having one magnetic pole in upper Canada
above Hudson Bay, about 70° north latitude, and another pole in the
Antarctic Ocean south of Australia. Between these two magnetic poles
continually flow these invisible curved lines of force, just as with
our horseshoe magnet in Figure 6. These lines of force are the cause
of compasses pointing in a northerly and southerly direction. Other
planets undoubtedly possess magnetic poles similar to those of the
earth.
[Illustration: FIG. 6.]
=The Background of Gravitation=
Now let us conceive of invisible geometric lines pervading the entire
spatial universe somewhat analogous to these lines of force in a
local magnetic field. To each point in these lines let us ascribe an
electric and gravitational potential (remembering, of course, that we
are dealing with _four_-dimensional space), and we have before
us, in a nearly visualized, sense, the background of the new theory of
gravitation.
Einstein was the first to present the subject of gravitation from this
viewpoint. For centuries up to this time geometry and physics were
considered as belonging to entirely different schools of thought, but
under the master hand of Einstein the two sciences have been welded
together into one. As Freundlich puts it, “quantities which hitherto
had only a purely geometrical import, for the first time became
animated with physical meaning.” Thus “empty space” is no longer empty,
even though the existence of the ether be denied. When the study of
free electrons has sufficiently advanced, it may be seen that these
elementary particles of electricity, or energy-particles, freed from
atomic or mass attraction, play an important role in gravitational
phenomena.
Figure 7 represents in a crude fashion the special curvature of space
in the region of a large body of matter, for instance our earth, with
the points (events) situated at finite instead of infinite nearness to
each other for sake of illustration. It will be readily seen that the
distortion of the geometric lines would necessarily alter the relative
positions of the point-potentials.
Any falling body moves in a geodesic, i. e., from one point to the
_next nearest_ point in space-time.[2] In an undisturbed region,
remote from matter, the points (events) may be considered as so
arranging themselves that any four neighboring ones would constitute
practically a square. It may then be seen that the easiest path for
falling bodies would be to follow the _sides_ of the squares
because by so doing they would be following the _geodesic_ or
shortest distance between points. (See Fig. 7.) But in the region of
a large body of matter the lines of points become so distorted that
the _diagonal_ of any four neighbor points becomes the geodesic.
Then the path of the falling particle will accordingly deviate. It will
always follow the geodesic, or easiest path.
[Illustration: FIG. 7.]
This causes the falling particle to take a direction which points
toward the center of the gravitational field--but the center of gravity
is exerting no drawing or attractive force as Newton supposed. Gravity
is thus seen to be not an external _drawing_ power operating
between bodies of matter, but an inherent order of nature in space.
The acceleration as well as the direction of the falling particle is
accounted for by this theory. As the separation-interval between points
becomes shorter--due to the constantly accentuated distortion as the
large body is approached--the falling particle would be correspondingly
accelerated. The distortion being constantly increased the acceleration
would likewise be constant.
Newton, in his law of inertia, postulated that any particle of matter
at rest will forever remain at rest if not disturbed, but when once
set in motion it will continue to move at uniform velocity in a
straight line unless interfered with by outside force. Einstein, on
the contrary, holds that any particle of matter if left to itself will
_move_ (let us say _fall_, if you please) in the easiest
direction (i. e., in a geodesic) at constant velocity unless it
encounters a gravitational field (a distorted region), in which event
it will become accelerated, and will also, if necessary, change its
direction, in obedience to the principle of least action. In other
words, it is natural for matter to possess energy, therefore natural
for it to be in motion and unnatural for it to be at rest. And the
contention has this much in its favor: every particle of matter in the
universe, from the infinitesimal electron to the more gigantic sun
and super-system of outer space, _is moving_, so far as our most
modern observations extend. Nothing has yet been discovered to be at
rest.
[Footnote 1: When either “space” or “time” is mentioned independently
of the other in this treatise it may be understood that the terms are
used in the ordinary conceptual sense for purposes of simplicity.]
[Footnote 2: The reader must bear in mind that four-dimensional, not
the ordinary three-dimensional, space is here discussed. The author
has endeavored, however, to treat the matter in such a manner as to
approach a visualization of this otherwise quite complex subject.]
=TRANSCRIBER’S NOTES=
Simple typographical errors have been silently corrected; unbalanced
quotation marks were remedied when the change was obvious, and
otherwise left unbalanced.
Punctuation, hyphenation, and spelling were made consistent when a
predominant preference was found in the original book; otherwise they
were not changed.
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