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.