plane is merely one special case. Certain of the features met in that
case are general. If we agree that we know what we mean by distance,
we find that on every surface there is a shortest distance between
two points, together with a series of lines or curves along which
such distances are taken. These lines or curves we call geodesics. On
the plane the geodesic is the straight line. On surfaces in general
the geodesic, whatever its particular and peculiar shape, plays
the same rôle that is played by the straight line in the plane;
it is the secondary element of the geometry, the surface itself
and all other surfaces of its type are the tertiary elements. And
it is a fact that we can take all the possible spheres, or all the
possible French-horn surfaces, and conceive of space as we know it
being broken down by analysis into these surfaces instead of into
planes. The only reason we habitually decompose space into planes is
because it comes natural to us to think that way. But geometric points,
lines and surfaces must be recognized as abstractions without actual
existence, for all of them lack one or more of the three dimensions
which such existence implies. These figures exist in our minds but
not in the external world about us. So any decomposition of space
into geometric elements is a phenomenon of the mind only; it has no
parallel and no significance in the external world, and is made in
one way or in another purely at our pleasure. There isn't a true,
honest-to-goodness geometrical plane in existence any more than there
is an honest-to-goodness spherical surface: so on intrinsic grounds
one decomposition is as reasonable as another.
Certain of the most fundamental postulates are obeyed by all
surfaces. As we attempt to discriminate between surfaces of different
types, and get, for instance, a geometry that shall be valid for
spheres and ellipsoids but not for conicoids in general, we must do
so by bringing in additional postulates that embody the necessary
restrictions. A characteristic shared by planes, spheres, and various
other surfaces is that the geodesics can be freely slid along upon
themselves and will coincide with themselves in all positions when
thus slid; with a similar arrangement for the surface itself. But
the plane stands almost unique among surfaces in that it does not
force us to distinguish between its two sides; we can turn it over
and still it will coincide with itself; and this property belongs
also to the straight line. It does not belong to the sphere, or to
the great circles which are the geodesics of spherical geometry;
when we turn one of these over, through the three-dimensional space
that surrounds it, we find that the curvature lies in the wrong way
to make superposition possible. If we postulate that superposition be
possible under such treatment, we throw out the sphere and spherical
geometry; if we postulate that superposition be only by sliding the
surface upon itself we admit that geometry--as Saccheri failed to
see, as Lobatchewsky realized, and as Riemann showed at great length
in rehabilitating the "obtuse-angled hypothesis." Lobatchewsky's
acute-angled geometry is realized on a surface of the proper sort,
which admits of unrestricted superposition; but it is not the sort
of a surface that I care to discuss in an article of this scope.
Euclidean geometry is the natural and easy one, I suppose, because it
makes it easy to stop with three dimensions. If we take a secondary
element, a geodesic, which is "curved" in the Euclidean sense, we
get a tertiary element, a surface, which is likewise curved. Then
unless we are to make an altogether abrupt and unreasonable break,
we shall find that just as the curved geodesic generated a curved
surface, the curved surface must give rise to a "curved space"; and
just as the curved geodesic needed a second dimension to curve into,
and the curved surface a third, so the curved three-space requires
a fourth. Once started on this sort of thing, there doesn't really
seem to be any end.
EUCLIDEAN OR NON-EUCLIDEAN
Nevertheless, we must face the possibility that the space we live
in, or any other manifold of any sort whatever with which we deal on
geometric principles, may turn out to be non-Euclidean. How shall we
finally determine this? By measures--the Euclidean measures the angles
of an actual triangle and finds the sum to be exactly 180 degrees;
or he draws parallel lines of indefinite extent and finds them to
be everywhere equally distant; and from these data he concludes that
our space is really Euclidean. But he is not necessarily right.
We ask him to level off a plot of ground by means of a plumb
line. Since the line always points to the earth's center, the "level"
plot is actually a very small piece of a spherical surface. Any test
conducted on this plot will exhibit the numerical characteristics of
the Euclidean geometry; yet we know the geometry of this surface is
Riemannian. The angle-sum is really greater than 180 degrees; lines
that are everywhere equidistant are not both geodesics.
The trouble, of course, is that on this plot we deal with
so minute a fraction of the whole sphere that we cannot make
measurements sufficiently refined to detect the departure from
Euclidean standards. So it is altogether sensible for us to ask:
"Is the universe of space about us really Euclidean in whatever of
realized geometry it presents to us? Or is it really non-Euclidean,
but so vast in size that we have never yet been able to extend our
measures to a sufficiently large portion of it to make the divergence
from the Euclidean standard discernible to us?"
This discussion is necessarily fragmentary, leaving out much that
the writer would prefer to include. But it is hoped that it will
nevertheless make it clear that when the contestants in the Einstein
competition speak of a non-Euclidean universe as apparently having
been revealed by Einstein, they mean simply that to Einstein has
occurred a happy expedient for testing Euclideanism on a smaller
scale than has heretofore been supposed possible. He has devised a
new and ingenious sort of measure which, if his results be valid,
enables us to operate in a smaller region while yet anticipating that
any non-Euclidean characteristics of the manifold with which we deal
will rise above the threshold of measurement. This does not mean that
Euclidean lines and planes, as we picture them in our mind, are no
longer non-Euclidean, but merely that these concepts do not quite so
closely correspond with the external reality as we had supposed.
As to the precise character of the non-Euclideanism which is revealed,
we may leave this to later chapters and to the competing essayists. We
need only point out here that it will not necessarily be restricted
to the matter of parallelism. The parallel postulate is of extreme
interest to us for two reasons; first because historically it was the
means by which the possibilities and the importance of non-Euclidean
geometry were forced upon our attention; and second because it happens
to be the immediate ground of distinction between Euclidean geometry
and two of the most interesting alternatives. But Euclidean geometry
is characterized, not by a single postulate, but by a considerable
number of postulates. We may attempt to omit any one of these so that
its ground is not specifically covered at all, or to replace any one
of them by a direct alternative. We might conceivably do away with the
superposition postulate entirely, and demand that figures be proved
equivalent, if at all, by some more drastic test. We might do away
with the postulate, first properly formulated by Hilbert, on which our
ideas of the property represented in the word "between" depend. We
might do away with any single one of the Euclidean postulates, or
with any combination of two or more of them. In some cases this would
lead to a lack of categoricity and we should get no geometry at all;
in most cases, provided we brought a proper degree of astuteness to
the formulation of alternatives for the rejected postulates, we should
get a perfectly good system of non-Euclidean geometry: one realized,
if at all, by other elements than the Euclidean point, line and plane,
and one whose elements behave toward one another differently from
the Euclidean point, line and plane.
Merely to add definiteness to this chapter, I annex here the
statement that in the geometry which Einstein builds up as more
nearly representing the true external world than does Euclid's, we
shall dispense with Euclid's (implicit) assumption, underlying his
(explicitly stated) superposition postulate, to the effect that the
act of moving things about does not affect their lengths. We shall
at the same time dispense with his parallel postulate. And we shall
add a fourth dimension to his three--not, of course, anything in the
nature of a fourth Euclidean straight line perpendicular, in Euclidean
space, to three lines that are already perpendicular to each other,
but something quite distinct from this, whose nature we shall see
more exactly in the next chapter. If the present chapter has made it
clear that it is proper for us to do this, and has prevented anyone
from supposing that the results of doing it must be visualized in a
Euclidean space of three dimensions or of any number of dimensions,
it will have served its purpose.
VI
THE SPACE-TIME CONTINUUM
Minkowski's World of Events, and the Way It Fits Into Einstein's
Structure
BY THE EDITOR, EXCEPT AS NOTED
Seeking a basis for the secure formulation of his results, and
especially a means for expressing mathematically the facts of the
dependence which he had found to exist between time and space, Einstein
fell back upon the prior work of Minkowski. It may be stated right
here that the idea of time as a fourth dimension is not particularly
a new one. It has been a topic of abstract speculation for the best
part of a century, even on the part of those whose notions of the
fourth dimension were pretty closely tied down to the idea of a
fourth dimension of Euclidean point-space, which would be marked by
a fourth real line, perpendicular to the other three, and visible
to us if we were only able to see it. Moreover, every mathematician,
whether or not he be inclined to this sort of mental exercise, knows
well that whenever time enters his equations at all, it does so on an
absolutely equal footing with each of his space coordinates, so that
as far as his algebra is concerned he could never distinguish between
them. When the variables $x$, $y$, $z$, $t$ come to the mathematician
in connection with some physical investigation, he knows before he
starts that the first three represent the dimensions of Euclidean
three-space and that the last stands for time. But if the algebraic
expressions of such a problem were handed to him independently of
all physical tie-up, he would never be able to tell, from them alone,
whether one of the four variables represented time, or if so, which
one to pick out for this distinction.
It was Minkowski who first formulated all this in a form susceptible
of use in connection with the theory of relativity. His starting point
lies in the distinction between the point and the event. Mr. Francis
has brought this out rather well in his essay, being the only
competitor to present the Euclidean geometry as a real predecessor
of Newtonian science, rather than as a mere part of the Newtonian
system. I think his point here is very well taken. As he says,
Euclid looked into the world about him and saw it composed of
points. Ignoring all dynamic considerations, he built up in his
mind a static world of points, and constructed his geometry as a
scientific machine for dealing with this world in which motion played
no part. It could to be sure be introduced by the observer for his
own purposes, but when so introduced it was specifically postulated
to be a matter of no moment at all to the points or lines or figures
that were moved. It was purely an observational device, intended
for the observer's convenience, and in the bargain a mental device,
calling for no physical action and the play of no force. So far as
Euclid in his daily life was obliged to take cognizance of the fact
that in the world of work-a-day realities motion existed, he must,
as a true Greek, have looked upon this as a most unfortunate deviation
of the reality from his beautiful world of intellectual abstraction,
and as something to be deplored and ignored. Even in their statuary
the Greeks clung to this idea. A group of marvelous action, like
the Laocoon, they held to be distinctly a second rate production,
a prostitution of the noble art; their ideal was a figure like the
majestic Zeus--not necessarily a mere bust, be it understood, but
always a figure in repose without action. Their statuary stood for
things, not for action, just as their geometry stood for points,
not for events.
Galileo and Newton took a different viewpoint. They were interested
in the world as it is, not as it ought to be; and if motion appears
to be a fundamental part of that world, they were bound to include
it in their scheme. This made it necessary for them to pay much more
attention to the concept of time and its place in the world than did
the Greeks. In the superposition process, and even when he allowed
a curve to be generated by a moving point, the sole interest which
Euclid had in the motion was the effect which was to be observed upon
his static figures after its completion. In this effect the rate of
the motion did not enter. So all questions of velocity and time are
completely ignored, and we have in fact the curious spectacle of
motion without time.
To Galileo and Newton, on the other hand, the time which it took a
body to pass from one point of its path to another was of paramount
importance. The motion itself was the object of their study, and they
recognized the part played by velocity. But Galileo and Newton were
still sufficiently under the influence of Euclid to fit the observed
phenomena of motion, so far as they could, upon Euclid's static world
of points. This they effected by falling in with the age-old procedure
of regarding time and space as something entirely disassociated and
distinct. The motion of an object--in theory, of a point--was to be
recorded by observing its successive positions. With each of these
positions a time was to be associated, marking the instant at which
the point attained that position. But in the face of this association,
space and time were to be maintained as entirely separate entities.
THE FOUR-DIMENSIONAL WORLD OF EVENTS
This severe separation of time and space Minkowski has now questioned,
with the statement that the elements of which the external
world is composed, and which we observe, are not points at all,
but are events. This calls for a revision of our whole habit of
thought. It means that the perceptual world is four-dimensioned,
not three-dimensioned as we have always supposed; and it means, at
the very least, that the distinction between time and space is not
so fundamental as we had supposed.
[This should not impress us as strange or incomprehensible. What
do we mean when we say that a plane is two-dimensional? Simply that
two coordinates, two numbers, must be given to specify the position
of any point of the plane. Similarly for a point in the space of our
accustomed concepts we must give three numbers to fix the position--as
by giving the latitude and longitude of a point on the earth and its
height above sea-level. So we say this space is three-dimensional. But
a material body is not merely somewhere; it is somewhere now,]182 or
was somewhere yesterday, or will be somewhere tomorrow. The statement
of position for a material object is meaningless unless we at the
same time specify the time at which it held that position. [If I
am considering the life-history of an object on a moving train,
I must give three space-coordinates and one time-coordinate to fix
each of its positions.]182 And each of its positions, with the time
pertaining to that position, constitutes an event. The dynamic,
ever-changing world about us, that shows the same aspect at no two
different moments, is a world of events; and since four measures or
coordinates are required to fix an event, we say this world of events
is four-dimensional. If we wish to test out the soundness of this
viewpoint, we may well do so by asking whether the naming of values
for the four coordinates fixes the event uniquely, as the naming of
three under the old system fixes the point uniquely.
Suppose we take some particular event as the one from which to measure,
and agree upon the directions to be taken by our space axes, and make
any convention about our time-axis which subsequent investigation
may show to be necessary. Certainly then the act of measuring so many
miles north, and so many west, and so many down, and so many seconds
backward, brings us to a definite time and place--which is to say,
to a definite event. Perhaps nothing "happened" there, in the sense
in which we usually employ the word; but that is no more serious
than if we were to locate a point with reference to our familiar
space coordinate system, and find it to lie in the empty void of
interstellar space, with no material body occupying it. In this
second case we still have a point, which requires, to insure its
existence and location, three coordinates and nothing more; in the
first case we still have an event, which requires for its existence
and definition four coordinates and nothing more. It is not an event
about which the headline writers are likely to get greatly excited;
but what of that? It is there, ready and waiting to define any physical
happening that falls upon it, just as the geometer's point is ready
and waiting to define any physical body that chances to fall upon it.
A CONTINUUM OF POINTS
It is now in order to introduce a word, which I shall have to confess
the great majority of the essayists introduce, somewhat improperly,
without explanation. But when I attempt to explain it, I realize
quite well why they did this. They had to have it; and they didn't
have space in their three thousand words to talk adequately about
it and about anything else besides. The mathematician knows very
well indeed what he means by a continuum; but it is far from easy
to explain it in ordinary language. I think I may do best by talking
first at some length about a straight line, and the points on it.
If the line contains only the points corresponding to the integral
distances 1, 2, 3, etc., from the starting point, it is obviously
not continuous--there are gaps in it vastly more inclusive than the
few (comparatively speaking) points that are present. If we extend
the limitations so that the line includes all points corresponding
to ordinary proper and improper fractions like 1/4 and 17/29 and
1633/7--what the mathematician calls the rational numbers--we shall
apparently fill in these gaps; and I think the layman's first impulse
would be to say that the line is now continuous. Certainly we cannot
stand now at one point on the line and name the "next" point, as we
could a moment ago. There is no "next" rational number to 116/125,
for instance; 115/124 comes before it and 117/126 comes after it,
but between it and either, or between it and any other rational
number we might name, lie many others of the same sort. Yet in spite
of the fact that the line containing all these rational points is now
"dense" (the technical term for the property I have just indicated),
it is still not continuous; for I can easily define numbers that
are not contained in it--irrational numbers in infinite variety
like $\sqrt{2}$; or, even worse, the number pi = 3.141592 ... which
defines the ratio of the circumference of a circle to the diameter,
and many other numbers of similar sort.
If the line is to be continuous, there may be no holes in it at all;
it must have a point corresponding to every number I can possibly
name. Similarly for the plane, and for our three-space; if they are
to be continuous, the one must contain a point for every possible
pair of numbers $x$ and $y$, and the other for every possible set
of three numbers $x$, $y$ and $z$, that I can name. There may be no
holes in them at all.
A line is a continuum of points. A plane is a continuum of points. A
three-space is a continuum of points. These three cases differ only in
their dimensionality; it requires but one number to determine a point
of the first continuum, two and three respectively in the second and
third cases. But the essential feature is not that a continuum shall
consist of points, or that we shall be able to visualize a pseudo-real
existence for it of just the sort that we can visualize in the case
of line, plane and point. The essential thing is merely that it shall
be an aggregate of elements numerically determined in such a way as to
leave no holes, but to be just as continuous as the real number system
itself. Examples, however, aside from the three which I have used,
are difficult to construct of such sort that the layman shall grasp
them readily; so perhaps, fortified with the background of example
already presented, I may venture first upon a general statement.
THE CONTINUUM IN GENERAL
Suppose we have a set of "elements" of some sort--any sort. Suppose
that these elements possess one or more fundamental identifying
characteristics, analogous to the coordinates of a point, and
which, like these coordinates, are capable of being given numerical
values. Suppose we find that no two elements of the set possess
identically the same set of defining values. Suppose finally--and
this is the critical test--that the elements of the set are such that,
no matter what numerical values we may specify, it we do specify the
proper number of defining magnitudes we define by these an actual
element of the set, that corresponds to this particular collection
of values. Our elements then share with the real number system
the property of leaving no holes, of constituting a continuous
succession in every dimension which they possess. We have then a
continuum. Whatever its elements, whatever the character of their
numerical identifiers, whatever the number n of these which stands for
its dimension, there may be no holes or we have no continuum. There
must be an element for every possible combination of n numbers
we can name, and no two of these combinations may give the same
element. Granted this condition, our elements constitute a continuum.
As I have remarked, it is not easy to cite examples of continua which
shall mean anything to the person unaccustomed to the term. The
totality of carbon-oxygen-nitrogen-hydrogen compounds suggested
by one essayist as an example is not a continuum at all, for the
set contains elements corresponding only to integer values of the
numbers which tell us how many atoms of each substance occur in the
molecule. We cannot have a compound containing $\sqrt2$ carbon atoms,
or $3\pi$ oxygen atoms. Perhaps the most satisfactory of the continua,
outside the three Euclidean space-continua already cited, [is the
manifold of music notes. This is four-dimensional; each note has
four distinctions--length, pitch, intensity, timbre--to distinguish
it perfectly, to tell how long, how high, how loud, how rich.]263
We might have a little difficulty in reducing the characteristic of
richness to numerical expression, but presumably it could be done;
and we should then be satisfied that every possible combination of four
values, $l$, $p$, $i$, $t$ for these four identifying characteristics
would give us a musical effect, and one to be confused with no other.
There is in the physical world a vast quantity of continua of one
sort or another. The music-note continuum brings attention to the
fact that not all of these are such that their elements make their
appeal to the visual sense. This remark is a pertinent one; for we are
by every right of heritage an eye-minded race, and it is frequently
necessary for us to be reminded that so far as the external world is
concerned, the verdict of every other sense is entirely on a par with
that of sight. The things which we really see, like matter, and the
things which we abstract from these visual impressions, like space,
are by no means all there is to the world.
EUCLIDEAN AND NON-EUCLIDEAN CONTINUA
If we are dealing with a continuum of any sort whatever having one
or two or three dimensions, we are able to represent it graphically
by means of the line, the plane, or the three-space. The same set of
numbers that defines an element of the given continuum likewise defines
an element of the Euclidean continuum of the same dimensionality;
so the one continuum corresponds to the other, element for element,
and either may stand for the other. But if we have a continuum of four
or more dimensions, this representation breaks down in the absence
of a real, four-dimensional Euclidean point-space to serve as a
picture. This does not in the least detract from the reality of the
continuum which we are thus prevented from representing graphically
in the accustomed fashion.
The Euclidean representation, in fact, may in some cases be
unfortunate--it may be so entirely without significance as to be
actually misleading. For in the Euclidean continuum of points, be
it line, plane or three-space, there are certain things which we
ordinarily regard as secondary derived properties, but which possess
a great deal of significance none the less.
In particular, in the Euclidean plane and in Euclidean three-space,
there is the distance between two points. I have indicated, in
the chapter on non-Euclidean geometry, that the parallel postulate
of Euclid, which distinguishes his geometry from others, could be
replaced by any one of numerous other postulates. Grant Euclid's
postulate and you can prove any of these substitutes; grant any of
the substitutes and you can prove Euclid's postulate. Now it happens
that there is one of these substitutes to which modern analysis has
given a position of considerable importance. It is merely our good
old friend the Pythagorean theorem, that the square on the hypotenuse
equals the sum of the squares on the sides; but it is dressed in new
clothes for the present occasion.
Mr. Francis' discussion of this part of the subject, and especially his
figure, ought to make it clear that this theorem can be considered
as dealing with the distance between any two points. When we so
consider it, and take it as the fundamental, defining postulate of
Euclidean geometry which distinguishes this geometry from others,
we have a statement of considerable content. We have, first, that the
characteristic property of Euclidean space is that the distance between
two points is given by the square root of the sum of the squares of
the coordinate-differences for these points--by the expression
$$D = \sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2}\,,$$
where the large letters represent the coordinates of the one point
and the small ones those of the other. We have more than this,
however; we have that this distance is the same for all observers,
no matter how different their values for the individual coordinates
of the individual points. And we have, finally, as a direct result of
looking upon the thing from this viewpoint, that the expression for $D$
is an "invariant"; which simply means that every observer may use the
same expression in calculating the value of $D$ in terms of his own
values for the coordinates involved. The distance between two points
in our space is given numerically by the square root of the sum of the
squares of my coordinate-differences for the two points involved; it
is given equally by the square root of the sum of the squares of your
coordinate-differences, or those of any other observer whatsoever. We
have then a natural law--the fundamental natural law characterizing
Euclidean space. If we wish to apply it to the Euclidean two-space (the
plane) we have only to drop out the superfluous coordinate-difference;
if we wish to see by analogy what would be the fundamental natural law
for a four-dimensional Euclidean space, we have only to introduce under
the radical a fourth coordinate-difference for the fourth dimension.
If we were not able to attach any concrete meaning to the expression
for $D$ the value of all this would be materially lessened. Consider,
for instance, the continuum of music notes. There is no distance
between different notes. There is of course significance in talking
about the difference in pitch, in intensity, in duration, in timbre,
between two notes; but there is none in a mode of speech that implies
a composite expression indicating how far one note escapes being
identical with another in all four respects at once. The trouble,
of course, is that the four dimensions of the music-note continuum
are not measurable in terms of a common unit. If they were, we should
expect to measure their combination more or less absolutely in terms
of this same unit. We can make measurements in all three dimensions
of Euclidean space with the same unit, with the same measuring
rod in fact. [This presents a peculiarity of our three-space which
is not possessed by all three-dimensional manifolds. Riemann has
given another illustration in the system of all possible colors,
composed of arbitrary proportions of the three primaries, red, green
and violet. This system forms a three-dimensional continuum; but we
cannot measure the "distance" or difference between two colors in
terms of the difference between two others.]130
Accordingly, in spite of the fact that the Euclidean three-space
gives us a formal representation of the color continuum, and in
spite of the fact that the hypothetical four-dimensional Euclidean
space would perform a like office for the music-note continuum, this
representation would be without significance. We should not say that
the geometry of these two manifolds is Euclidean. We should realize
that any set of numerical elements can be plotted in a Euclidean
space of the appropriate dimensionality; and that accordingly, before
allowing such a plot to influence us to classify the geometry of the
given manifold as Euclidean, we must pause long enough to ask whether
the rest of the Euclidean system fits into the picture. If the square
root of the sum of the squares of the coordinate-differences between
two elements possesses significance in the given continuum, and if it
is invariant between observers of that continuum who employ different
bases of reference, then and only then may we allege the Euclidean
character of the given continuum.
If under this test the given continuum fails of Euclideanism, it is
in order to ask what type of geometry it does present. If it is of
such character that the "distance" between two elements possesses
significance, we should answer this question by investigating that
distance in the hope of discovering a non-Euclidean expression for it
which will be invariant. If it is not of such character, we should seek
some other characteristic of single elements or groups of elements,
of real physical significance and of such sort that the numerical
expression for it would be invariant.
If the continuum with which we have to do is one in which the
"distance" between two elements possesses significance, and if it
turns out that the invariant expression for this distance is not
the Pythagorean one, but one indicating the non-Euclideanism of
our continuum, we say that this continuum has a "curvature." This
means that, if we interpret the elements of our continuum as points
in space (which of course we may properly do) and if we then try to
superpose this point-continuum upon a Euclidean continuum, it will not
"go"; we shall be caught in some such absurdity as trying to force a
sphere into coincidence with a plane. And of course if it won't go,
the only possible reason is that it is curved or distorted, like
the sphere, in such a way as to prevent its going. It is unfortunate
that the visualizing of such curvature requires the visualizing of
an additional dimension for the curved continuum to curve into; so
that while we can picture a curved surface easily enough, we can't
picture a curved three-space or four-space. But that is a barrier to
visualization alone, and in no sense to understanding.
OUR WORLD OF FOUR DIMENSIONS
It will be observed that we have now a much broader definition of
non-Euclideanism than the one which served us for the investigation
of Euclid's parallel postulate. If we may at pleasure accept
this postulate or replace it by another and different one, we
may presumably do the same for any other or any others of Euclid's
postulates. The very statement that the distance between elements of
the continuum shall possess significance, and shall be measurable by
considering a path in the continuum which involves other elements,
is an assumption. If we discard it altogether, or replace it by one
postulating that some other joint property of the elements than
their distance be the center of interest, we get a non-Euclidean
geometry. So for any other of Euclid's postulates; they are all
necessary for a Euclidean system, and in the absence of any one of
them we get a non-Euclidean system.
Now the four-dimensional time-space continuum of Minkowski is plainly
of a sort which ought to make susceptible of measurement the separation
between two of its events. We can pass from one element to another
in this continuum--from one event to another--by traversing a path
involving "successive" events. Our very lives consist in doing just
this: we pass from the initial event of our career to the final
event by traversing a path leading us from event to event, changing
our time and space coordinates continuously and simultaneously in the
process. And while we have not been in the habit of measuring anything
except the space interval between two events and the time interval
between two events, separately, I think it is clear enough that,
considered as events, as elements in the world of four dimensions,
there is a less separation between two events that occur in my office
on the same day than between two which occur in my office a year
apart; or between two events occurring 10 minutes apart when both
take place in my office than when one takes place there and one in
London or on Betelgeuse.
It is not at all unreasonable, a priori, then, to seek a numerical
measure for the separation, in space-time of four-dimensions, of
two events. If we find it, we shall doubtless be asked just what
its subjective significance to us is. This must be answered with
some circumspection. It will presumably be something which we cannot
observe with the visual sense alone, or it would have forced itself
upon our attention thousands of years ago. It ought, I should think,
to be something that we would sense by employing at the same time
the visual sense and the sense of time-passage. In fact, I might
very plausibly insist that, by my very remarks about it in the above
paragraph, I have sensed it.
Minkowski, however, was not worried about this phase of the matter. He
had only to identify the invariant expression for distance; sensing
it could wait. He found, of course, that this expression was not the
Euclidean expression for a four-dimensional interval. He had discarded
several of the Euclidean assumptions and could not expect that the
postulate governing the metric properties of Euclid's space would
persist. Especially had he violated the Euclidean canons in discarding,
with Einstein, the notion that nothing which may happen to a measuring
rod in the way of uniform translation at high velocity can affect
its measures. So he had to be prepared to find that his geometry was
non-Euclidean; yet it is surprising to learn how slightly it deviates
from that of Euclid. Without any extended discussion to support the
statement, we may say that he found that when two observers measure the
time- and the space-coordinates of two events, using the assumptions
and therefore the methods of Einstein and hence subjecting themselves
to the condition that their measures of the pure time-interval and of
the pure space-interval between these events will not necessarily be
the same, they will discover that they both get the same value for
the expression
$$S = \sqrt{(X - x)^2 + (Y - y)^2 + (Z - z)^2 - (CT - Ct)^2}\,.$$
If our acceptance of this as the numerical measure of the separation
in space-time between the two events should lead to contradiction we
could not so accept it. No contradiction arises however and we may
therefore accept it. And at once the mathematician is ready with some
interpretative remarks.
THE CURVATURE OF SPACE-TIME
The invariant expression for separation, it will be seen, is in
the same form as that of the Euclidean four-dimensional invariant
save for the minus sign before the time-difference (the appearance
of the constant $C$ in connection with the time coordinate $t$ is
merely an adjustment of units; see page 153). This tells us that not
alone is the geometry of the time-space continuum non-Euclidean in its
methods of measurement, but also in its results, to the extent that it
possesses a curvature. It compares with the Euclidean four-dimensional
continuum in much the same way that a spherical surface compares with
a plane. As a matter of fact, a more illuminating analogy here would
be that between the cylindrical surface and the plane, though neither
is quite exact. To make this clear requires a little discussion of
an elementary notion which we have not yet had to consider.
Our three-dimensional existence often reduces, for all practical
purposes, to a two-dimensional one. The objects and the events of
a certain room may quite satisfactorily be defined by thinking
of them, not as located in space, but as lying in the floor of
the room. Mathematically the justification for this viewpoint
is got by saying that we have elected to consider a slice of our
three-dimensional world of the sort which we know as a plane. When we
consider this plane and the points in it, we find that we have taken
a cross-section of the three-dimensional world. A line in that world
is now reduced, for us, to a single point--the point where it cuts
our plane; a plane is reduced to a line--the line where it cuts our
plane; the three-dimensional world itself is reduced to our plane
itself. Everything three-dimensional falls down into its shadow in
our plane, losing in the process that one of the three dimensions
which is not present in our plane.
For simplicity's sake it is usual to take a cross-section of space
parallel to one of our coordinate axes. We think of our three
dimensions as extending in the directions of those axes; and it is
easier to take a horizontal or vertical section which shall simply
wipe out one of these dimensions than to take an oblique section which
shall wipe out a dimension that consists partly of our original length,
and partly of our original width, and partly of our original height.
If we have a four-dimensional manifold to begin with, we may equally
shake out one of the four dimensions, one of the four coordinates,
and consider the three-dimensional result of this process as a
cross-section of the original four-dimensional continuum. And where, in
cross-sectioning a three-dimensioned world, we have but three choices
of a coordinate to eliminate, in cross-sectioning a world of four
dimensions we have four choices. By dropping out either the $x$, or the
$y$, or the $z$, or the $t$, we get a three-dimensioned cross-section.
Now our accustomed three-dimensional space is strictly Euclidean. When
we cross-section it, we get a Euclidean plane no matter what the
direction in which we make the cut. Likewise the Euclidean plane is
wholly Euclidean, because when we cross-section it in any direction
whatever we get a Euclidean line. A cylindrical surface, on the other
hand, is neither wholly Euclidean nor wholly non-Euclidean in this
matter of cross-sectioning. If we take a section in one direction we
get a Euclidean line and if we take a section in the other direction
we get a circle (if the cylindrical surface be a circular one). And
of course if we take an oblique section of any sort, it is neither
line nor circle, but a compromise between the two--the significant
thing being that it is still not a Euclidean line.
The space-time continuum presents an analogous situation. When
we cross-section it by dropping out any one of the three space
dimensions, we get a three-dimensional complex in which the distance
formula is still non-Euclidean, retaining the minus sign before
the time-difference and therefore retaining the geometric character
of its parent. But if we take our cross-section in such a way as to
eliminate the time coordinate, this peculiarity disappears. The signs
in the invariant expression are then all plus, and the cross-section
is in fact our familiar Euclidean three-space.
If we set up a surface geometry on a sphere, we find that the
elimination of one dimension leaves us with a line-geometry that is
still non-Euclidean since it pertains to the great circles of the
sphere rather than to Euclidean straight lines. In shaking Minkowski's
continuum down into a three-dimensional one by eliminating any one of
his coordinates, if we eliminate either the $x$, the $y$ or the $z$,
we have left a three-dimensional geometry in which the disturbing minus
sign still occurs in the distance-formula, and which is therefore still
non-Euclidean. If we omit the $t$, this does not occur. We see, then,
that the time dimension is the disturbing factor, the one which gives
to space-time its non-Euclidean character so far as the possession
of curvature is concerned. And we see that this curvature is not the
same in all directions, and in one direction is actually zero--whence
the attempted analogy with a cylinder instead of with a sphere.
Many writers on relativity try to give the space-time continuum an
appeal to our reason and a character of inevitableness by insisting on
the lack of any fundamental distinction between space and time. The
very expression for the space-time invariant denies this. Time is
distinguishable from space. The three dimensions of space are quite
indistinguishable--we can interchange them without affecting the
formula, we can drop one out and never know which is gone. But the
very formula singles out time as distinct from space, as inherently
different in some way. It is not so inherently different as we have
always supposed; it is not sufficiently different to offer any obstacle
to our thinking in terms of the four-dimensional continuum. But while
we can group space and time together in this way, [this does not mean
at all that space and time cease to differ. A cook may combine meat
with potatoes and call the product hash, but meat and potatoes do
not thereby become identical.]223
THE QUESTION OF VISUALIZATION
To the layman there is a great temptation to say that while,
mathematically speaking, the space-time continuum may be a great
simplification, it does not really represent the external world. To
be sure, you can't see the space-time continuum in precisely the same
way that you can the three-dimensional space continuum, but this is
only because Einstein finds the time dimension to be not quite freely
interchangeable with the space dimension. Yet you do perceive this
space-time continuum, in the manner appropriate for its perception;
and it would be just as sensible to throw out the space continuum
itself on the ground that perception of the two is not of exactly
the same sort, as to throw out the space-time continuum on this
ground. With appropriate conventions, either may stand as the mental
picture of the external world; it is for us to choose which is the
more convenient and useful image. Einstein tells us that his image
is the better, and tells us why.
Before we look into this, we must let him tell us something more
about the geometry of his continuum. What he tells us is, in its
essentials, just this. The observer in a pure space continuum of three
dimensions finds that as he changes his position, his right-and-left,
his backward-and-forward, and his up-and-down are not fixed directions
inherent in nature, but are fully interchangeable. The observers, in
the adjoined sketch, whose verticals are as indicated by the arrows,
find very different vertical and horizontal components for the distance
between the points $O$ and $P$; a similar situation would prevail if
we used all three space directions. The statement analogous to this
for Einstein's four-dimensional continuum of space and time combined
is that, as observers change their relative motion, their time axes
take slightly different directions, so that what is purely space or
purely time for the one becomes space with a small component in the
time direction, or time with a small component in the space direction,
for the other. This it will be seen explains fully why observers in
relative motion can differ about space and time measurements. We
should not be at all surprised if the two observers of the figure
reported different values for horizontals and verticals; we should
realize that what was vertical for one had become partly horizontal
for the other. It is just so, says Einstein, with his observers of
time and space who are in relative motion to one another; what one
sees as space the other sees as partly time, because their time axes
do not run quite parallel.
The natural question here, of course, is "Well, where are their
time axes?" If you know what to look for, of course, you ought to
be able to perceive them in just the way you perceive ordinary time
intervals--with the reservation that they are imaginary, after all,
just like your space axes, and that you must only expect to see them
in imagination. If you look for a fourth axis in Euclidean three-space
to represent your time axis, you will of course not find it. But you
will by all means agree with me that your time runs in a definite
direction; and this it is that defines your time axis. Einstein adds
that if you and I are in relative motion, my time does not run in
quite the same direction as yours.
How shall we prove it? Well, how would we prove it if he told us that
our space axes did not run in precisely the same direction? Of course
we could not proceed through direct measures upon the axes themselves;
we know these are imaginary. What we should do would be to strike out,
each of us, a very long line indeed in what seemed the true horizontal
direction; and we should hope that if we made them long enough,
and measured them accurately enough, we should be able to detect any
divergence that might exist. This is precisely what we must do with
our time axes if we wish to verify Einstein's statement that they are
not precisely parallel; and what better evidence could we demand of
the truth of this statement than the evidence already presented--that
when we measure our respective time components between two events,
we get different results?
WHAT IT ALL LEADS TO
The preceding chapters have been compiled and written with a view to
putting the reader in a state of mind and in a state of informedness
which shall enable him to derive profit from the reading of the actual
competing essays which make up the balance of the book. For this
purpose it has been profitable to take up in detail the preliminaries
of the Special Theory of Relativity, and to allow the General Theory
to go by default, in spite of the fact that it is the latter which
constitutes Einstein's contribution of importance to science. The
reason for this is precisely the same as that for taking up Euclidean
geometry and mastering it before proceeding to the study of Newtonian
mechanics. The fundamental ideas of the two theories, while by no
means identical, are in general terms the same; and the conditions
surrounding their application to the Special Theory are so very
much simpler than those which confront us when we apply them to the
more general case, that this may be taken as the controlling factor
in a popular presentation. We cannot omit the General Theory from
consideration, of course; but we can omit it from our preliminary
discussion, and leave its development to the complete essays which
follow, and which in almost every case give it the larger half of their
space which its larger content demands. In the process of the slow
and difficult preparation of the lay mind for the assimilation of an
altogether new set of fundamental ideas, it is altogether desirable to
give the Special Theory, with its simpler applications of these ideas,
a place out of proportion to its importance in Einstein's completed
structure; and this we have therefore done.
The Special Theory, postulating the relativity of uniform motion and
deducing the consequences of that relativity, is often referred to
as a "special case" of the General Theory, in which this restriction
of uniformity is removed. This is not strictly speaking correct. The
General Theory, when we have formulated it, will call our attention to
something which we really knew all the time, but to which we chose not
to give heed--that in the regions of space to which we have access,
uniform motion does not exist. All bodies in these regions are under
the gravitational influence of the other bodies therein, and this
influence leads to accelerated motion. Nothing in our universe can
possibly travel at uniform velocity; the interference of the rest of
the bodies in the universe prevents this.
Obviously, we ought not to apply the term "special case" to a case that
never occurs. Nevertheless, this case is of extreme value to us in our
mental processes. Many of the motions with which we are concerned are
so nearly at constant velocity that we find it convenient to treat
them as though they were uniform, either ignoring the resulting
error or correcting for it at the end of our work. In many other
cases we are able to learn what actually occurs under accelerated
motion by considering what would have occurred under uniform motion
were such a thing possible. Science is full of complications which
we unravel in this fashion. The physicist deals with gas pressures
by assuming temperatures to be constant, though he knows temperature
never is constant; and in turn he deals with temperatures by assuming
pressures to be constant. After this, he is able to predict what will
happen when, as in nature, pressures and temperatures are varying
simultaneously. By using as a channel of attack the artificially simple
case that never occurs, we get a grip on the complex case that gives
us a true picture of the phenomenon. And because in actual nature we
can come as close as we please to this artificial case, by supposing
the variable factor to approach constancy, so when we assume it to be
absolutely constant we speak of the result as the limiting case. This
situation does not occur, but is the limiting case for those that
do occur.
When, in the matter of motion, we abandon the artificial, limiting
case of uniform velocity and look into the general, natural one of
unrestricted motion, we find that the structure which we have built up
to deal with the limiting case provides us with many of the necessary
ideas and viewpoints. This is what we expect--in it lies the value
of the limiting case. We shall see that the relativity of time and
space, established for the limiting case, holds good in the general
one. We shall see that the idea of the four-dimensional space-time
continuum as representing the external world persists, forming the
whole background of the General Theory much more definitely than
in the Special Theory. Incidentally we shall see that the greater
generality of the case under consideration will demand a greater degree
of generality in the geometry of this continuum, a non-Euclideanism of
a much more whole-hearted type than that of the Special Theory. But
all the revisions of fundamental concepts which we have been at such
pains to make for the sake of the Special Theory will remain with us
in the General. With this we may consider our preliminary background
as established, and give our attention to the essayists, who will
try to take us more deeply into the subject than we have yet gone,
without losing us in its intricacies.
VII
RELATIVITY
The Winning Essay in the Contest for the Eugene Higgins $5,000 Prize
BY LYNDON BOLTON
SENIOR EXAMINER, BRITISH PATENT OFFICE LONDON
The reader is probably acquainted with the method of specifying
positions of points in a plane by their distances from two mutually
perpendicular lines, or if the points are in space by their distances
from three mutually perpendicular planes like adjacent sides of a
flat-sided box. The method is in fact in common use for exhibiting
relations between quantities by graphs or diagrams. These sets of axes,
as they are called, together with any scales used for measuring, must
be supposed rigid, otherwise the events or points which they are used
to specify are indefinite. The lengths which locate any point with
reference to a set of axes are called its coordinates.
When such systems are used for physical purposes, they must be
supplemented by clocks to enable the times at which events occur to
be determined. The clocks must be synchronized, and must go at the
same rate, but it must suffice here to state that this is possible
without indicating how these conditions can be attained. A system of
axes with its clocks will hereinafter be called a Frame of Reference,
and every observer will be supposed to be provided with such a frame
partaking of his motion. All the objects which partake of an observer's
motion will be called his system.
It is a question whether among all possible frames of reference
any one frame or class of frames is more suited than another for
the mathematical statement of physical laws. This is for experience
to decide, and a Principle of Relativity is a statement embodying
the answer.
THE MECHANICAL PRINCIPLE OF RELATIVITY
It has been ascertained that all such frames are equally suitable for
the mathematical statement of general mechanical laws, provided that
their motion is rectilinear and uniform and without rotation. This
fact is comprehended in the general statement that all unaccelerated
frames of reference are equivalent for the statement of the general
laws of mechanics. This is the mechanical principle of relativity.
It is well recognized however that the laws of dynamics as hitherto
stated involve the assumptions that the lengths of rigid bodies are
unaffected by the motion of the frame of reference, and that measured
times are likewise unaffected; that is to say that any length measured
on his own system by either of two relatively moving observers appears
the same to both observers, or that lengths of objects and rates of
clocks do not alter whatever the motion relative to an observer. These
assumptions seem so obvious that it is scarcely perceived that they
are assumptions at all. Yet this is the case, and as a matter of fact
they are both untrue.
THE SPECIAL PRINCIPLE OF RELATIVITY
Although all unaccelerated frames of reference are equivalent for
the purposes of mechanical laws, this is not the case for physical
laws generally as long as the above suppositions are adhered
to. Electromagnetic laws do alter their form according to the motion
of the frame of reference; that is to say, if these suppositions are
true, electromagnetic agencies act in different ways according to the
motion of the system in which they occur. There is nothing a priori
impossible in this, but it does not agree with experiment. The motion
of each locality on the earth is continually changing from hour to
hour but no corresponding changes occur in electromagnetic actions. It
has however been ascertained that on discarding these suppositions
the difficulty disappears, and electromagnetic laws retain their
form under all circumstances of unaccelerated motion. According
to the theory of relativity, the correct view which replaces these
suppositions is deducible from the following postulates:
(1) By no experiment conducted on his own system can an observer
detect the unaccelerated motion of his system.
(2) The measure of the velocity of light in vacuo is unaffected
by relative motion between the observer and the source of light.
Both these postulates are well established by experiment. The first
may be illustrated by the familiar difficulty of determining whether a
slowly moving train one happens to be sitting in, or an adjacent one,
is in motion. The passenger has either to wait for bumps (that is,
accelerations) or else he has to look out at some adjacent object which
he knows to be fixed, such as a building (that is, he has to perform
an experiment on something outside his system), before he can decide.
The second postulate is an obvious consequence of the wave theory of
light. Just as waves in water, once started by a ship, travel through
the water with a velocity independent of the ship, so waves in space
travel onward with a speed bearing no relation to that of the body
which originated them. The statement however is based on experiment,
and can be proved independently of any theory of light.
It is not difficult to deduce from these postulates certain remarkable
conclusions relating to the systems of two observers, A and B, in
relative motion, among them the following:
(1) Objects on B's system appear to A to be shorter in the
direction of relative motion than they appear to B.
(2) This opinion is reciprocal. B thinks that A's measurements
on A's system are too great.
(3) Similarly for times: each observer thinks that the other's
clocks have a slower rate than his own, so that B's durations
of time appear shorter to B than to A, and conversely.
(4) Events which appear simultaneous to A do not in general appear
so to B, and conversely.
(5) Lengths at right angles to the direction of motion are
unaffected.
(6) These effects vary with the ratio of the relative velocity
to that of light. The greater the relative velocity, the
greater the effects. They vanish if there is no relative
velocity.
(7) For ordinary velocities the effects are so small as to escape
notice. The remarkable point however is their occurrence rather
than their magnitude.
(8) The observers similarly form different estimates of the
velocities of bodies on each other's systems. The velocity of
light however appears the same to all observers.
Taking into account these revised views of lengths and times the
mechanical principle of relativity may be extended to physical laws
generally as follows: All unaccelerated frames of reference are
equivalent for the statement of the general laws of physics. In this
form the statement is called the Special, or Restricted, Principle
of Relativity, because it is restricted to unaccelerated frames of
reference. Naturally the laws of classical mechanics now require some
modification, since the suppositions of unalterable lengths and times
no longer apply.
THE FOUR DIMENSIONAL CONTINUUM
Lengths and times therefore have not the absolute character formerly
attributed to them. As they present themselves to us they are relations
between the object and the observer which change as their motion
relative to him changes. Time can no longer be regarded as something
independent of position and motion, and the question is what is the
reality? The only possible answer is that objects must be regarded as
existing in four dimensions, three of these being the ordinary ones of
length, breadth and thickness, and the fourth, time. The term "space"
is applicable only by analogy to such a region; it has been called a
"continuum," and the analogue of a point in ordinary three-dimensional
space has been appropriately called an "event." By "dimension" must be
understood merely one of four independent quantities which locate an
event in this continuum. In the nature of the case any clear mental
picture of such a continuum is impossible; mankind does not possess
the requisite faculties. In this respect the mathematician enjoys
a great advantage. Not that he can picture the thing mentally any
better than other people, but his symbols enable him to abstract the
relevant properties from it and to express them in a form suitable
for exact treatment without the necessity of picturing anything,
or troubling whether or not the properties are those on which others
rely for their conceptions.
GRAVITATION AND ACCELERATION
The limitation of statements of general law to uniformly moving systems
is hardly satisfactory. The very concept of general law is opposed
to the notion of limitation. But the difficulties of formulating a
law so that the statement of it shall hold good for all observers,
whose systems may be moving with different and possibly variable
accelerations, are very great. Accelerations imply forces which
might be expected to upset the formulation of any general dynamical
principles, and besides, the behavior of measuring rods and clocks
would be so erratic as to render unmeaning such terms as rigidity
and measured time, and therefore to preclude the use of rigid scales,
or of a rigid frame of reference which is the basis of the foregoing
investigation.
The following example taken from Einstein will make this clear,
and also indicate a way out of the difficulty. A rotating system is
chosen, but since rotation is only a particular case of acceleration
it will serve as an example of the method of treating accelerated
systems generally. Moreover, as it will be seen, the attribution of
acceleration to the system is simply a piece of scaffolding which
can be discarded when the general theory has been further developed.
Let us note the experiences of an observer on a rotating disk which
is isolated so that the observer has no direct means of perceiving
the rotation. He will therefore refer all the occurrences on the disk
to a frame of reference fixed with respect to it, and partaking of
its motion.
He will notice as he walks about on the disk that he himself and all
the objects on it, whatever their constitution or state, are acted upon
by a force directed away from a certain point upon it and increasing
with the distance from that point. This point is actually the center
of rotation, though the observer does not recognize it as such. The
space on the disk in fact presents the characteristic properties of a
gravitational field. The force differs from gravity as we know it by
the fact that it is directed away from instead of toward a center,
and it obeys a different law of distance; but this does not affect
the characteristic properties that it acts on all bodies alike, and
cannot be screened from one body by the interposition of another. An
observer aware of the rotation of the disk would say that the force
was centrifugal force; that is, the force due to inertia which a body
always exerts when it is accelerated.
Next suppose the observer to stand at the point of the disk where
he feels no force, and to watch someone else comparing, by repeated
applications of a small measuring rod, the circumference of a circle
having its center at that point, with its diameter. The measuring rod
when laid along the circumference is moving lengthwise relatively
to the observer, and is therefore subject to contraction by his
reckoning. When laid radially to measure the diameter this contraction
does not occur. The rod will therefore require a greater proportional
number of applications to the circumference than to the diameter,
and the number representing the ratio of the circumference of the
circle to the diameter thus measured will therefore be greater than
3.14159+, which is its normal value. Moreover the relative velocity
decreases as the center is approached, so that the contraction of the
measuring rod is less when applied to a smaller circle; and the ratio
of the circumference to the diameter, while still greater than the
normal, will be nearer to it than before, and the smaller the circle
the less the difference from the normal. For circles whose centers
are not at the point of zero force the confusion is still greater,
since the velocities relative to the observer of points on them now
change from point to point. The whole scheme of geometry as we know
it is thus disorganized. Rigidity becomes an unmeaning term since
the standards by which alone rigidity can be tested are themselves
subject to alteration. These facts are expressed by the statement
that the observer's measured space is non-Euclidean; that is to say,
in the region under consideration measurements do not conform to the
system of Euclid.
The same confusion arises in regard to clocks. No two clocks will in
general go at the same rate, and the same clock will alter its rate
when moved about.
THE GENERAL PRINCIPLE OF RELATIVITY
The region therefore requires a space-time geometry of its own, and
be it noted that with this special geometry is associated a definite
gravitational field, and if the gravitational field ceases to exist,
for example if the disk were brought to rest, all the irregularities
of measurement disappear, and the geometry of the region becomes
Euclidean. This particular case illustrates the following propositions
which form the basis of this part of the theory of relativity:
(1) Associated with every gravitational field is a system of
geometry, that is, a structure of measured space peculiar to
that field.
(2) Inertial mass and gravitational mass are one and the same.
(3) Since in such regions ordinary methods of measurement fail,
owing to the indefiniteness of the standards, the systems of
geometry must be independent of any particular measurements.
(4) The geometry of space in which no gravitational field exists
in Euclidean. [1]
The connection between a gravitational field and its appropriate
geometry suggested by a case in which acceleration was their common
cause is thus assumed to exist from whatever cause the gravitational
field arises. This of course is pure hypothesis, to be tested by
experimental trial of the results derived therefrom.
Gravitational fields arise in the presence of matter. Matter is
therefore presumed to be accompanied by a special geometry, as though
it imparted some peculiar kink or twist to space which renders the
methods of Euclid inapplicable, or rather we should say that the
geometry of Euclid is the particular form which the more general
geometry assumes when matter is either absent or so remote as to
have no influence. The dropping of the notion of acceleration is
after all not a very violent change in point of view, since under
any circumstances the observer is supposed to be unaware of the
acceleration. All that he is aware of is that a gravitational field
and his geometry coexist.
The prospect of constructing a system of geometry which does not depend
upon measurement may not at first sight seem hopeful. Nevertheless this
has been done. The system consists in defining points not by their
distances from lines or planes (for this would involve measurement)
but by assigning to them arbitrary numbers which serve as labels
bearing no relation to measured distances, very much as a house
is located in a town by its number and street. If this labeling
be done systematically, regard being had to the condition that the
label-numbers of points which are close together should differ from
one another by infinitesimal amounts only, it has been found that
a system of geometry can actually be worked out. Perhaps this will
appear less artificial when the fact is called to mind that even
when standards of length are available no more can be done to render
lengths of objects amenable to calculation than to assign numbers to
them, and this is precisely what is done in the present case. This
system of labeling goes by the name of "Gaussian coordinates" after
the mathematician Gauss who proposed it.
It is in terms of Gaussian coordinates that physical laws must be
formulated if they are to have their widest generality, and the general
principle of relativity is that all Gaussian systems are equivalent for
the statement of general physical laws. For this purpose the labeling
process is applied not to ordinary space but to the four dimensional
space-time continuum. The concept is somewhat difficult and it may
easily be aggravated into impossibility by anyone who thinks that he
is expected to visualize it. Fortunately this is not necessary; it is
merely one of these irrelevancies to which those who are unaccustomed
to think in symbols are liable.
It will now be seen that among physical laws the law of gravitation
stands pre-eminent, for it is gravitating matter which determines
the geometry, and the geometry determines the form of every other
law. The connection between the geometry and gravitation is the law
of gravitation. This law has been worked out, with the result that
Newton's law of the inverse square is found to be approximate only,
but so closely approximate as to account for nearly all the motions of
the heavenly bodies within the limits of observation. It has already
been seen that departure from the Euclidean system is intensified
by rapidity of motion, and the movements of these bodies are usually
too slow for this departure to be manifest. In the case of the planet
Mercury the motion is sufficiently rapid, and an irregularity in its
motion which long puzzled astronomers has been explained by the more
general law.
Another deduction is that light is subject to gravitation. This has
given rise to two predictions, one of which has been verified. The
verification of the other is as yet uncertain, though the extreme
difficulty of the necessary observations may account for this.
Since light is subject to gravitation it follows that the constancy
of the velocity of light assumed in the earlier part of this
paper does not obtain in a gravitational field. There is really no
inconsistency. The velocity of light is constant in the absence of
gravitation, a condition which unaccelerated motion implies. The
special principle of relativity is therefore a limiting case of the
general principle.
VIII
THE NEW CONCEPTS OF TIME AND SPACE
The Essay in Behalf of Which the Greatest Number of Dissenting Opinions
Have Been Recorded
BY MONTGOMERY FRANCIS NEW YORK
We have all had experiences, on trains and boats, illustrating our
inability to tell, without looking off to some external body, whether
we are at rest or moving uniformly; and when we do so look, to tell,
without reference to the ground or some other point external to both
systems, whether ours or the other be the seat of motion. Uniform
motion must be relative, because we find nowhere in the universe a
body in the unique state of absolute rest from which alone absolute
motion might be measured.
True, the wave theory of light with its homogeneous space-filling
ether seemed to provide a reference standard for the concept of
absolute motion, and for its measurement by experiment with light
rays. But when Michelson and Morley looked for this absolute motion
they found no trace of it. To the physicist, observational student
of the external world, nothing exists save observationally; what he
can never observe is not there. So: I. By no means whatever may we
regard uniform straight-line motion as other than relative.
As a further direct consequence of the Michelson-Morley experiment
we have: II. Light in a vacuum presents the same velocity, $C =
186,330$ miles per second, to all observers whatever their velocity
of relative motion. In addition to being experimentally established,
this is necessary to support I, for if light will distinguish between
our velocities, its medium is necessarily a universal standard for
absolute motion. But it is contrary to common sense to suppose that
if I pass you at 100 miles per hour, the same light impulse can pass
us both at the same speed, $C$. We feel, instinctively, that space and
time are not so constituted as to make this possible. But the fact has
been repeatedly demonstrated. And when common sense and fundamental
concepts clash with facts, it is not the facts that must yield. We
have survived such crises, notably one where we had to change the
fundamental concept of up-and-down; if another one is here, says
Einstein, let us meet it.
This the Special Theory of Relativity does. It accepts Postulates
I and II above; their consequences it deduces and interprets. For
extensive demonstration of these I lack space, and this has been
satisfactorily done by others so it is not my chief duty; but clearly
they will be startling. For the very ray of light which refuses to
recognize our relative motion is the medium through which I must
observe your system and you mine.
It turns out that I get different values for lengths and time intervals
in your system than you get, and vice versa. And we are both right! For
me to accept your "correction" were for me to admit that you are
at absolute rest and I in absolute motion, that your measure of
light velocity is right and mine wrong: admissions barred by the
postulates. We have nothing to correct; we can only recognize the
reason for the discrepancy; and knowing our relative velocity, each
can calculate from his own results what the other's will be. We find,
of course, that at ordinary velocities the discrepancy is many times
too small for detection; but at relative velocities at all comparable
with that of light it rises above the observational horizon.
To inquire the "true" length is meaningless. Chicago is east of Denver,
west of Pittsburgh, south of Milwaukee; we do not consider this
contradictory, or demand the "true" direction of Chicago. Einstein
finds that the concept of length, between points in space or events
in time, does not as we had supposed represent an intrinsic property
of the points or the events. Like direction, it is merely a relation
between these and the observer--a relation whose value changes with
the observer's velocity relative to the object. If our ideas of the
part played in the world by time and space do not permit us to believe
this, we must alter these ideas. Let us see how we may do this.
A WORLD OF POINTS
To deal with points in a plane the mathematician draws two
perpendicular lines, and locates any point, as $P$, by measuring its
distances, $X$ and $Y$, from these "coordinate axes." The directions
of his axes acquire for him a peculiar significance, standing out
above other directions; he is apt to measure the distances $X-x$ and
$Y-y$ between the points $P$ and $Q$ in these directions, instead of
measuring the single distance $PQ$. We do the same thing when we say
that the railroad station is five blocks north and two east.
The mathematician visualizes himself as an observer, located on his
coordinate framework. For another observer on another framework, the
horizontal and vertical distances $X'-x'$ and $Y'-y'$ between $P$ and
$Q$ are different. But for both, the distance from $P$ direct to $Q$
is the same. In each case the right triangle tells us that:
$$PQ = \sqrt{(X-x)^2 + (Y-y)^2} = \sqrt{(X'-x')^2 + (Y'-y')^2}$$
Imagine an observer so dominated by his coordinate system that he
knows no way of relating $P$ with $Q$ save by their horizontal and
vertical separation. His whole scheme of things would be shattered
by the suggestion that other observers on other reference frames find
different horizontal and vertical components. We have to show him the
line $PQ$. We have to convince him that this length is the absolute
property enjoyed by his pair of points; that horizontals and verticals
are merely relations between the points and the observer, result of
the observer's having analyzed the distance $PQ$ into two components;
that different observers effect this decomposition differently; that
this seems not to make sense to him only because of his erroneous
concept of a fundamental difference between verticals and horizontals.
THE FOUR-DIMENSIONAL WORLD OF EVENTS
We too have created a distinction in our minds corresponding to no
sufficient reality. Our minds seize on time as inherently separable
from space. We see the world made up of things in a continuum of three
space dimensions; to make this dead world live there runs through it
a one-dimensional time continuum, imposed from without, unrelated.
But did you ever observe anything suggesting the presence of time
in the absence of space, or vice versa? No; these vessels of the
universe always occur together. Association of the space dimensions
into a manifold from which time is excluded is purely a phenomenon
of the mind. The space continuum cannot begin to exist until the time
dimension is supplied, nor can time exist without a place to exist in.
The external world that we observe is composed, not of points,
but of events. If a point lacks position in time it does not exist;
give it this position and it becomes an event. This world of events
is four-dimensional--which means nothing more terrifying than that
you must make four measures to locate an event. It does not mean,
at all, that you must visualize four mutually perpendicular lines in
your accustomed three-space or in a four-space analogous to it. If
this world of four dimensions seems to lack reality you will be
able to exhibit no better reality for your old ideas. Time belongs,
without question; and not as an afterthought, but as part of the
world of events.
To locate an event we use four measures: $X$, $Y$ and $Z$ for space,
$T$ for time. Using the same reference frame for time and space, we
locate a second event by the measures $x$, $y$, $z$, $t$. Minkowski
showed that the quantity
$$\sqrt{(X-x)^2 + (Y-y)^2 + (Z-z)^2 - (CT-Ct)^2}$$
is the same for all observers, no matter how different their $x$'s,
$y$'s, $z$'s and $t$'s; just as in the plane the quantity
$$\sqrt{(X-x)^2 + (Y-y)^2}$$
is the same for all observers, no matter how different their $x$'s
and $y$'s.
Such a quantity, having the same value for all observers, is
absolute. In the plane it represents the true, absolute distance
between the points--their intrinsic property. In dealing with events it
represents the true, absolute "interval," in time and space together
between the events. It is not space, nor time, but a combination of
the two. We have always broken it down into separate space and time
components. In this we are as naive as the plane observer who could
not visualize the distance $PQ$ until it was split into separate
horizontals and verticals. He understood with difficulty that another
observer, employing a different reference frame because in different
position, would make the decomposition differently. We understand with
difficulty that another observer, employing a different reference
frame because in uniform motion relative to us, will decompose the
"interval" between events into time and space components different
from ours. Time and space are relative to the observer; only the
interval representing space-time is absolute. So common sense stands
reconciled to the Special Theory of Relativity.
SUCCESSIVE STEPS TOWARD GENERALITY
Is then our laboriously acquired geometry of points in a
three-dimensional space to go into the discard? By no means. Jeans,
investigating the equilibrium of gaseous masses, found the general
case too difficult for direct attack. So he considered the case where
the masses involved are homogeneous and incompressible. This never
occurs; but it throws such light on the general case as to point the
way toward attack on it.
Euclidean geometry excludes motion, save that engineered by the
observer; and then the time is immaterial. Time does not enter at all;
the three space dimensions suffice. This simple case never occurs
where matter exists; but its conclusions are of value in dealing with
more general cases.
When we look into a world alleged to be that of Euclid and find motion,
we may retain the Euclidean concept of what constitutes the world and
invent a machinery to account for the motion; or we may abandon the
Euclidean world, as inadequate, in favor of a more general one. We
have adopted the second alternative.
Newton's laws tells us that a body free to move will do so, proceeding
in a straight line at uniform velocity until interfered with. We
do not ask, nor does the theory tell us, whence comes the initial
motion. There is no machinery to produce it; it is an inherent property
of Newton's world--assured by the superposition of the time continuum
upon Euclid's world to make Newton's, accepted without question along
with that world itself.
But Newton saw that his world of uniform motion, like Euclid's,
was never realized. In the neighborhood of one particle a second is
interfered with, forced to give up its uniform motion and acquire a
constant acceleration. This Newton explained by employing the first
of the alternatives mentioned above. He tells us that in connection
with all matter there exists a force which acts on other matter in a
certain way. He does not display the actual machinery through which
this "force" works, because he could not discover any machinery;
he had to stop with his brilliant generalization of the observed
facts. And all his successors have failed to detect the slightest
trace of a machinery of gravitation.
Einstein asks whether this is not because the machinery is
absent--because gravitation, like position in Euclid's world and
motion in Newton's, is a fundamental property of the world in which it
occurs. His point of attack here lay in precise formulation of certain
familiar facts that had never been adequately appreciated. These
facts indicate that even accelerated motion is relative, in spite of
its apparently real and absolute effects.
GRAVITATION AND ACCELERATION
An observer in a closed compartment, moving with constant acceleration
through empty space, finds that the "bottom" of his cage catches up
with objects that he releases; that it presses on his feet to give
him the sensation of weight, etc. It displays all the effects that
he would expect if it were at rest in a gravitational field. On the
other hand, if it were falling freely under gravitational influence,
its occupant would sense no weight, objects released would not leave
his hand, the reaction from his every motion would change his every
position in his cage, and he could equally well assume himself at
rest in a region of space free from gravitational action. Accelerated
motion may always be interpreted, by the observer on the system,
as ordinary force effects on his moving system, or as gravitational
effects on his system at rest.
An alternative statement of the Special Theory is that the observed
phenomena of uniform motion may equally be accounted for by supposing
the object in motion and the observer with his reference frame at
rest, or vice versa. We may similarly state the General Theory: The
observed phenomena of uniformly accelerated motion may in every case be
explained on a basis of stationary observer and accelerated objective,
or of stationary objective with the observer and his reference system
in accelerated motion. Gravitation is one of these phenomena. It
follows that if the observer enjoy properly accelerated axes (in
time-space, of course), the absolute character of the world about him
must be such as to present to him the phenomenon of gravitation. It
remains only to identify the sort of world, of which gravitation as
it is observed would be a fundamental characteristic.
Euclid's and Newton's systems stand as first and second approximations
to that world. The Special Relativity Theory constitutes a correction
of Newton, presumably because it is a third approximation. We must
seek in it those features which we may most hopefully carry along,
into the still more general case.
Newton's system retained the geometry of Euclid. But Minkowski's
invariant expression tells us that Einstein has had to abandon this;
for in Euclidean geometry of four dimensions the invariant takes
the form:
$$\sqrt{(X-x)^2 + (Y-y)^2 + (Z-z)^2 + (T-t)^2}\,,$$
analogous to that of two and three dimensions. It is not the presence
of the constant $C$ in Minkowski's formula that counts; this is merely
an adjustment so that we may measure space in miles and time in the
unit that corresponds to a mile. It is the minus sign where Euclidean
geometry demands a plus that makes Minkowski's continuum non-Euclidean.
The editor has told us what this statement means. I think he has made
it clear that when we speak of the geometry of the four-dimensional
world, we must not read into this term the restrictions surrounding
the kind of geometry we are best acquainted with--that of the
three-dimensional Euclidean continuum. So I need only point out that
if we are to make a fourth (and we hope, final) approximation to the
reality, its geometry must preserve the generality attained by that
of the third step, if it goes no further.
EINSTEIN'S TIME-SPACE WORLD
Einstein accordingly examined the possible non-Euclidean geometries
of four dimensions, in search of one displaying fundamental
characteristics which, interpreted in terms of space-time, would
lead to the observed facts of gravitation. The mathematics of this
investigation is that part of his work which, we are told, but twelve
men can follow; so we may only outline his conclusions.
If we assume that in the neighborhood of matter the world of
space-time is non-Euclidean, and that its curvature or distortion or
non-Euclideanism is of a certain type already known to mathematicians;
that the curvature of this world in the neighborhood of matter
increases with the mass, and decreases as the distance from the matter
increases; and that every particle of matter that is not interfered
with travels through space-time in the most direct path possible in
that continuum; then the observed facts of gravitation are accounted
for as an inherent geometric property of this space-time world. We
usually say that the presence of matter distorts this world, and
that this distortion gives the track of particles through the region
affected its non-uniform character.
Gravitation then is not a force at all; it is the fundamental nature of
things. A body free to move through the world must follow some definite
path. Euclid says it will stand still; Newton that it will traverse
a straight line in three-space at uniform time-rate; Einstein that it
will move in a "geodesic" through time-space--in every-day language,
that it will fall.
The numerical consequences of Einstein's theory are, within the
limits of observation, the same as those of Newton's for all bodies
save one--Mercury. This planet shows a small deviation from the
path predicted by Newton's law; Einstein's theory gives its motion
exactly. Again, when modern research showed that light must be
affected by gravitation, Einstein's theory, because of the extreme
velocity of light, deviates from Newton's, where the speed is less a
determining factor; and observations of starlight deflected by the
sun during the eclipse were in much better accord with Einstein's
theory than Newton's. Moreover, the Special Theory predicts that mass
is an observational variable like length and duration. Radioactive
emanations have a velocity high enough to give appreciable results
here, and the prediction is verified, tending to support the general
theory by supporting its limiting case.
We like always to unify our science; and seldom, after effecting
a unification, are we forced to give it up. Einstein for the first
time brings mechanical, electromagnetic and gravitational phenomena
within one structure. This is one reason why physicists are so open
minded toward his theory--they want it to be true.
THE LAYMAN'S LAST DOUBT
The final answer to any series of questions is inevitably "because the
world is so constructed." The things we are content to leave on that
basis are those to which we are accustomed, and which we therefore
think we understand; those for which this explanation leaves us
unsatisfied are those which are new and unfamiliar. Newton told us
that the world of three-dimensional space with one-dimensional time
superposed was so constructed that bodies left to themselves would go
on forever in a straight line at constant speed. We think we understand
this, but our understanding consists merely of the unspoken query,
"Why, of course; what is there to prevent?" The Greeks, an intelligent
people, looked at this differently; they would have met Newton with
the unanimous demand "Why so; what is there to keep them going?" So
if, in seeking an explanation of anything, we come sooner than we
had expected to the finality "Because the world is so constructed,"
let us not feel that we have been cheated.
IX
THE PRINCIPLE OF RELATIVITY
A Statement of What it is All About, in Ideas of One Syllable
BY HUGH ELLIOT
CHISLEHURST, KENT, ENGLAND
The invariance of the laws of nature was one of the most popular themes
of nineteenth century philosophy. For it was not till last century that
general acceptance was accorded to the doctrine of the "Uniformity
of Law," adumbrated in ancient times by Epicurus and Lucretius. It
is now a cardinal axiom of science that the same cause in the same
conditions is always followed by the same effect. There exists in
nature no indeterminate element; all things are governed by fixed laws,
and the discovery of these laws is the main business of science.
It is necessary to guard against reading into this statement an
erroneous idea of the content of a "law of nature." Such a law
is of course not an enactment of any sort; and it is not even to
be thought of as an actual explanation of the how and why of the
phenomena with which it has to do. It really is nothing but an
expression of our belief in the pronouncement of the preceding
paragraph, that like conditions do produce like results. It is
a prediction based on past experience, and is of value merely in
that past experience leads us to credit its accuracy. The composite
essay beginning on page 19 discusses this question of the reality
of natural laws, and should be consulted in connection with the
present contribution.--Editor.
This great philosophic principle was derived of course from the study
of natural science; i.e., from observations and experiments conducted
upon the earth. Their comprehensiveness is therefore limited by the
fact that the observer is always in a state of rest, or nearly so,
as compared with the earth. All observers upon the earth are moving
through space at the same velocity; and it was possible to argue that
the uniformity of law might only hold good, when experiments were
conducted at this velocity. An observer moving at very different
velocity might discover that the laws of nature under these new
conditions were somewhat different.
Such a view could indeed never be very plausible, for motion is only a
relative conception. Imagine a universe consisting of infinite "empty"
space, in which there is poised a single material body. How shall
we determine whether this body is at rest, or whether it is moving
at high or low velocity through space? It is never getting nearer
to anything or farther from anything, since there is no other body
for it to get nearer to or farther from. If we say it is moving at a
uniform velocity of a thousand miles a second, our statement really
has no significance. We have no more reason for affirming that it is
in motion than we have for affirming that it is at rest. In short,
there is no such thing as absolute motion; the conception of motion
only arises when there are two or more bodies changing their position
relatively to one another. This is what is meant by the relativity
of motion. It seemed therefore improbable that the laws of nature
would be different if the observer were moving at high velocity;
for the movement of the observer is not an absolute quantity, but
merely a statement of his relation to other bodies, and if there were
no other bodies, the statement itself would be meaningless.
THE BEHAVIOR OF LIGHT
Now among the established laws of nature is that which specifies
the velocity of light moving through a vacuum. If the laws of nature
are invariable, this velocity will always be the same. But consider
what would happen under the following circumstances: Suppose that
we are at rest, and that an observer on another body flies past us
at 150,000 miles a second. Suppose that at the moment he passes, a
piece of flint projecting from him grazes a piece of steel projecting
from us, giving rise to a spark; and that we both thereupon set about
to measure the velocity of the light so produced. After one second,
we should find that the light had traveled about 186,000 miles away,
and since during this second the other observer had traveled 150,000
miles, we should infer that the light traveling in his direction was
only about 36,000 miles ahead of him. We should also infer that he
would find this out by his experiment, and that he would estimate the
velocity of light as only 36,000 miles a second in his own direction,
and 336,000 miles a second in the opposite direction. But if this is
so, then that law of nature which specifies the velocity of light
is quite different for him and for us: the laws of nature must be
dependent upon the observer's motion--a conclusion which appears
incompatible with the idea of the relativity of motion.
And it so happens that it is also contradictory to experimental
conclusions. Experiments undertaken to settle the point show that
each observer finds the same velocity for the light of the spark;
and after one second, each observer finds that the light has traveled
186,000 miles from himself. But how is it possible that when it has
traveled 186,000 miles in the same direction as the other observer
who himself has moved 150,000 miles meanwhile, he should still think
it 186,000 miles ahead of him? That is the initial paradox; and since
there has been no room for error in the experiments, we are forced
to conclude that there was something wrong in the assumptions and
preconceptions with which we started.
SPACE AND TIME
There can in fact be only one interpretation. If we each find that the
light has moved the same number of miles in the same number of seconds,
then we must be meaning something different when we speak of miles and
seconds. We are speaking in different languages. Some subsidence has
occurred in the foundations of our systems of measurement. We are each
referring to one and the same objective fact; but since we describe
it quite differently, and at first sight incompatibly, some profound
alteration must have occurred in our perceptions--all unsuspected by
ourselves. It has been shown precisely what this alteration is. A body
moving at high velocity must become flattened in the direction of its
motion; all its measuring apparatus, when turned in that direction,
is shortened, so that no hint of the flattening can be obtained from
it. Furthermore, the standards of time are lengthened out, and clocks
go slower. The extent of this alteration in standards of space and time
is stated in the equations of the so-called Lorentz transformation.
Objection might be urged to the above paragraph on the ground that the
connection of the observer with the variability of measured lengths
and times is not sufficiently indicated, and that this variability
therefore might be taken as an intrinsic property of the observed
body--which of course it is not.--Editor.
We are accustomed to describe space as being of three dimensions, and
time as being of one dimension. As a matter of fact, both space and
time are "ideas," and not immediate sense-perceptions. We perceive
matter; we then infer a universal continuum filled by it, which
we call space. If we had no knowledge of matter, we should have no
conception of space. Similarly in the case of time: we perceive one
event following another, and we then invent a continuum which we call
time, as an abstraction based on the sequence of events. We do not see
space, and we do not see time. They are not real things, in the sense
that matter is real, and that events are real. They are products of
imagination: useful enough in common life, but misleading when we
try to look on the universe as a whole, free from the artificial
divisions and landmarks which we introduce into it for practical
convenience. Hence it is perhaps not so surprising after all that
in certain highly transcendental investigations, these artificial
divisions should cease to be a convenience, and become a hindrance.
Take for instance our conception of time. It differs from our
conception of space in that it has only one dimension. In space, there
is a right and left, an up and down, a before and after. But in time
there is only before and after. Why should there be this limitation of
the time-factor? Merely because that is the verdict of all our human
experience. But is our human experience based on a sufficiently broad
foundation to enable us to say that, under all conditions and in all
parts of the universe, there can be only one time-direction? May not
our belief in the uniformity of time be due to the uniformity of the
motion of all observers on the earth? Such in fact is the postulate of
relativity. We now believe that, at velocities very different from our
own, the standard of time would also be different from ours. From our
point of view, that different standard of time would not be confined
to the single direction fore and aft, as we know it, but would also
have in it an element of what we might call right and left. True,
it would still be of only one dimension, but its direction would
differ from the direction of our time. It would still run like a
thread through the universe, but not in the direction which we call
straight forward. It would have a slant in it, and the angle of the
slant depends upon the velocity of motion. It does not follow that
because we are all traveling in the same direction down the stream of
time, therefore that stream can only flow in the direction which we
know. "Before" and "after" are expressions which, like right and left,
depend upon our personal situation. If we were differently situated,
if to be precise we were moving at very high velocity, we should,
so to speak, be facing in a new direction and "before" and "after"
would imply a different direction of progress from that with which
we are now familiar.
THE WORLD OF REALITY
But, after all, the objective universe is the same old universe
however fast we are moving about in it, and whatever way we are
facing. These details merely determine the way we divide it up into
space and time. The universe is not affected by any arbitrary lines
which we draw through it for our personal convenience. For practical
purposes, we ascribe to it four dimensions, three in space and one in
time. Clearly if the time direction is altered, all dimensions both
of space and time must have different readings. If, for instance,
the time direction slopes away to the left, as compared with ours,
then space measurements to right and left must be correspondingly
altered. An analogy will simplify the matter.
Suppose we desire to reach a point ten miles off in a roughly
northeasterly direction. We might do so by walking six miles due east
and then eight miles due north. We should then be precisely ten miles
from where we started. But suppose our compass were out of order, so
that its north pole pointed somewhat to the west of north. Then in
order to get to our destination, we might have to walk seven miles
in the direction which we thought was east, and a little more than
seven miles in the direction which we thought was north. We should
then reach the same point as before. Both observers have walked
according to their lights, first due east and then due north, and
both have reached the same point: the one observer is certain that
the finishing point is six miles east of the starting-point, while
the other is sure it is seven miles.
Now we on the earth are all using a compass which points in the same
direction as regards time. But other observers, on bodies moving with
very different velocity, have a compass in which the time-direction is
displaced as compared with ours. Hence our judgments of distances will
not be alike. In our analogy, the northerly direction corresponds to
time, and the easterly direction to space; and so long as we use the
same compass we do not differ in our measurements of distances. But
for any one who has a different notion of the time-direction, not
only time intervals but space distances will be judged differently.
In short, the universe is regarded as a space-time continuum of four
dimensions. A "point" in space-time is called an "event"--that which
occurs at a specified moment and at a specified place. The distance
between two points in space-time is called their "interval." All
observers will agree as to the magnitude of any interval, since it
is a property of the objective universe; but they will disagree as
to its composition in space and time separately. In short, space
and time are relative conceptions; their relativity is a necessary
consequence of the relativity of motion. The paradox named at the
outset is overcome; for the two observers measuring the velocity of
the light produced as they passed one another, were using different
units of space and time. And hence emerges triumphant the Special
Principle of Relativity, which states that the laws of nature are
the same for all observers, whether they are in a state of rest or
of uniform motion in a straight line.
ACCELERATED MOTION
Uniform motion in a straight line is however a very special kind of
motion. Our experience in ordinary life is of motions that are neither
uniform nor in a straight line; both speed and direction of motion
are altering. The moving body is then said to undergo "acceleration":
which means either that its speed is increasing or diminishing, or
that its direction of motion is changing, or both. If we revert to our
former supposition of a universe in which there is only a single body
in "empty" space, we clearly cannot say whether it has acceleration
any more than whether it is moving, there being no outside standard
of comparison; and the General Principle of Relativity asserts the
invariance of the laws of nature for all states of motion of the
observer. In this case, however, a difference might be detected
by an observer on the moving body itself. It would be manifested
to him as the action of a force; such for instance as we feel when
a train in which we are traveling is increasing or reducing speed,
or when, without changing speed, it is rounding a corner. The force
dies away as soon as the velocity becomes uniform. Thus acceleration
reveals itself to us under the guise of action by a force. Force and
acceleration go together, and we may either say that the acceleration
is due to the force, or the impression of force to the acceleration.
Now when we are traveling with accelerated motion, we have quite a
different idea of what constitutes a straight line from that which
we had when at rest or in uniform motion. If we are moving at uniform
velocity in an airplane and drop a stone to the earth it will appear
to us in the airplane to fall in a straight line downward, while to
an observer on the earth it will appear to describe a parabola. This
is due to the fact that the stone gathers speed as it falls; it is
subject to the acceleration associated with gravity. Acceleration
obliterates the fundamental difference between a straight and
curved line. Unless we know what is the absolute motion of the
stone, and the two observers, we cannot say whether the line is
"really" a straight or a curved line. Since absolute motion is an
illegitimate conception, it follows that there is no such thing as
"really" straight or "really" curved. These are only appearances
set up as a consequence of our relative motions with respect to the
bodies concerned. If there were no such thing as acceleration--if the
stone fell to the earth at uniform velocity--then an observer on the
earth or anywhere else would agree that it fell in a straight line;
and straight lines would always be straight lines.
Under these circumstances, Euclidean geometry would be absolutely
true. But if we are in a state of acceleration, then what we think are
straight lines are "really" curved lines, and Euclidean geometry, based
on the assumption that its lines are straight, must founder when tested
by more accurate measurements. And in point of fact we are in a state
of acceleration: for we are being acted upon by a force--namely, the
force of gravitation. Wherever there is matter, there is gravitation;
wherever there is gravitation there is acceleration; wherever there
is acceleration Euclidean geometry is inaccurate. Hence in the space
surrounding matter a different geometry holds the field; and bodies
in general move through such space in curved lines.
Different parts of space are thus characterized by different
geometrical properties. All bodies in the universe proceed on
their established courses through space and time. But when they
come to distorted geometrical areas, their paths naturally seem
to us different from when they were moving through less disturbed
regions. They exhibit the difference by acquiring an acceleration;
and we explain the acceleration by alleging the existence of a force,
which we call the force of gravitation. But their motions can in fact
be perfectly predicted if we know the geometry of the space through
which they are traveling. The predictions so based have in fact proved
more accurate than those based on the law of gravitation.
X
SPACE, TIME AND GRAVITATION
An Outline of Einstein's Theory of General Relativity
BY W. DE SITTER
PROFESSOR OF ASTRONOMY IN THE UNIVERSITY OF LEYDEN
"Henceforth space by itself and time by itself shall sink to mere
shadows, and only a union of the two shall preserve reality."