at the meeting of German "Naturforscher und Aerzte" at Cologne in 1908,
has, however, only been completely fulfilled by Einstein's "Allgemeine
Relativitäts-theorie" of 1915, which incorporated gravitation into
the union. In the following pages an attempt is made to set forth,
without using any technical language, the leading ideas of that theory:
I will confine myself to the theory as published by Einstein in
November, 1915, which forms a consistent whole, complete in itself;
and I will not refer to later developments, which are still more
or less tentative, and not necessary for the understanding of the
theory. The mathematics used by Einstein is the so-called Absolute
Differential Calculus. It is not more difficult or recondite than that
used in other branches of theoretical physics, but it is somewhat
unfamiliar to most of us, because it is not generally taught in the
regular university courses. I will, however, in this essay abstain
from using any mathematics at all, at least, I will not be using it
openly. It is of course unavoidable to use at least the results of
the mathematical reasoning, if not the reasoning itself; but so long
as they are not put into formulas they will, it is hoped, not look
so formidable to the reader.
Referring to the quoted words of Minkowski, we may ask what is meant by
"reality." Physical science, like common sense, takes for granted that
there is a reality behind the phenomena, which is independent of the
person by whom, and the particular methods by which it is observed,
and which is also there when it is not observed. Strictly speaking,
all talk about what is not observed is metaphysics. Nevertheless
the physicist unhesitatingly believes that his laws are general, and
that the phenomena continue to happen according to them when nobody
is looking. And since it would be impossible to prove that they did
not, he is fully entitled to his belief. The observed phenomena are
the effects of the action of this reality, of which we assume the
existence, on the observer's senses--or apparatus, which are extended
and refined sense-organs. The laws governing the phenomena therefore
must convey some information regarding this reality. We shall never
by any means be able to know anything else about it but just these
laws. To all intents and purposes the laws are the reality, if we
eliminate from them all that refers to the observer alone. What refers
to the reality is called "absolute," and what involves reference to
the observer "relative." The elimination of the relative is one of
the things the theory of relativity has set out to do.
THE EXTERNAL WORLD AND ITS GEOMETRY
To describe the phenomena and derive laws from them, we locate them
in space and time. To do this we use geometry. Here it is that the
part contributed by the observer comes in. There are an infinite
number of geometries, and a priori there seems to be no reason to
choose one rather than the other. Taking geometry of two dimensions
as an example, we can draw figures on a piece of paper, and discuss
their properties, and we can also do so on the shell of an egg. But
we cannot draw the same figures on the egg as on the paper. The
ones will be distorted as compared with the others: the two surfaces
have a different geometry. Similarly it is not possible to draw an
accurate map of the earth on a sheet of paper, because the earth is
spherical and its representation on the flat paper is always more or
less distorted. The earth requires spherical geometry, which differs
from the flat, or Euclidean, geometry of the paper.
Up to a few years ago Euclidean (i.e. flat) geometry of three
dimensions had been exclusively used in physical theories. Why? Because
it is the true one, is the one answer generally given. Now a statement
about facts can be true or false, but a mathematical discipline is
neither true nor false; it can only be correct--i.e. consistent in
itself--or incorrect, and of course it always is correct. The assertion
that a certain geometry is the "true" one can thus only mean, that
it is the geometry of "true" space, and this again, if it is to have
any meaning at all, can only mean that it corresponds to the physical
"reality." Leaving aside the question whether this reality has any
geometry at all, we are confronted with the more immediately practical
consideration how we shall verify the asserted correspondence. There is
no other way than by comparing the conclusions derived from the laws
based upon our geometry, with observations. It thus appears that the
only justification for the use of the Euclidean geometry is its success
in enabling us to "draw an accurate map" of the world. As soon as any
other geometry is found to be more successful, that other must be used
in physical theories, and we may, if we like, call it the "true" one.
Accurate observations always consist of measures, determining the
position of material bodies in space. But the positions change,
and for a complete description we also require measures of time. An
important remark must be made here. Nobody has ever measured a
pure space-distance, nor a pure lapse of time. The only thing that
can be measured is the distance from a body at a certain point of
space and a certain moment of time, to a body (either the same or
another) at another point and another time. We can even go further
and say that time cannot be measured at all. We profess to measure
it by clocks. But a clock really measures space, and we derive the
time from its space-measures by a fixed rule. This rule depends
on the laws of motion of the mechanism of the clock. Thus finally
time is defined by these laws. This is so, whether as a "clock"
we use an ordinary chronometer, or the rotating earth, or an atom
emitting light-waves, or anything else that may be suggested. The
physical laws, of course, must be so adjusted that all these devices
give the same time. About the reality of time, if it has any, we
know nothing. All we know about time is that we want it. We cannot
adequately describe nature with the three space-coordinates alone,
we require a fourth one, which we call time. We might thus say with
some reason that the physical world has four dimensions. But so long
as it was found possible adequately to describe all known phenomena
by a space of three dimensions and an independent time, the statement
did not convey any very important information. Only after it had been
found out that the space-coordinates and the time are not independent,
did it acquire a real meaning.
As is well known the observation by which this was found out is the
famous experiment of Michelson and Morley. It led to the "special"
theory of relativity, which is the one referred to by Minkowski
in 1908. In it a geometry of four dimensions is used, not a mere
combination of a three-dimensional space and a one-dimensional time,
but a continuum of truly fourfold order. This time-space is not
Euclidean, since the time-component and the three space-components
are not on the same footing, but its fundamental formula has a
great resemblance to that of Euclidean geometry. We may call it
"pseudo-Euclidean."
This theory, which we need not explain here, was very satisfactory so
far as the laws of electromagnetism, and especially the propagation
of light, were concerned, but it did not include gravitation, and
mechanics generally. We then had this curious state of affairs, that
physicists actually believed in two different "realities." When they
were thinking of light they believed in Minkowski's time-space; when
they were thinking of gravitation they believed in the old Euclidean
space and independent time. This, of course, could not last. Attempts
were made so to alter Newton's law of gravitation that it would fit
into the four-dimensional world of the special relativity-theory,
but these only succeeded in making the law, which had been a model
of simplicity, extremely complicated, and, what was worse, it became
ambiguous.
It is Einstein's great merit to have perceived that gravitation is
of such fundamental importance, that it must not be fitted into a
ready-made theory, but must be woven into the space-time geometry from
the beginning. And that he not only saw the necessity of doing this,
but actually did it.
GRAVITATION AND ITS PLACE IN THE UNIVERSE
To see the necessity we must go back to Newton's system of
mechanics. Newton did two things (amongst others). He canonised
Galileo's system of mechanics into his famous "laws of motion,"
the most important of which is the law of inertia, which says that:
a body, that is not interfered with, moves in a straight line
with constant velocity.
The velocity, of course, can be nil, and the body at rest. This is
a perfectly general law, the same for all material bodies, whatever
their physical or chemical status. Newton took good care exactly to
define what he meant by uniform motion in a straight line, and for
this purpose he introduced the absolute Euclidean space and absolute
time as an essential part of his system of laws at the very beginning
of his great work. The other thing Newton did was to formulate the
law of gravitation. Gravitation was in his system considered as
an interference with the free, or inertial, motion of bodies, and
accordingly required a law of its own.
But gravitation has this in common with inertia, and in this it
differs from all other interferences, that it is perfectly general. All
material bodies are equally subjected to it, whatever their physical
or chemical status may be. But there is more. Gravitation and inertia
are actually indistinguishable from each other, and are measured
by the same number: the "mass". This was already remarked by Newton
himself, and from his point of view it was a most wonderful accidental
coincidence. If an apple falls from the tree, that which makes it fall
is its weight, which is the gravitational attraction by the earth,
diminished by the centrifugal force due to the earth's rotation and the
apple's inertia. In Newton's system the gravitational attraction is a
"real" force, whereas the centrifugal force is only "fictitious". But
the one is as real as the other. The most refined experiments,
already begun by Newton himself, have not succeeded in distinguishing
between them. Their identity is actually one of the best established
facts in experimental physics. From this identity of "fictitious," or
inertial, and "real," or gravitational, forces it follows that locally
a gravitational field can be artificially created or destroyed. Thus
inside a closed room which is falling freely, say a lift of which the
cable has been broken, bodies have no weight: a balance could be in
equilibrium with different weights in the two scales.
Having thus come to the conclusion that gravitation is not an
interference, but is identical with inertia, we are tempted to restate
the law of motion, so as to include both, thus:
Bodies which are not interfered with--do not move in straight
lines, but--fall.
Now this is exactly what Einstein did. Only the "falling" of
course requires a precise mathematical definition (like the uniform
motion in a straight line), and the whole gist of his theory is the
finding of that definition. In our earthly experience the falling
never lasts long, very soon something--the floor of the room, or
the earth itself--interferes. But in free space bodies go on falling
forever. The motion of the planets is, in fact, adequately described as
falling, since it consists in nothing else but obeying Newton's law of
gravitation together with his law of inertia. A body very far removed
from all other matter is not subjected to gravitation, consequently it
falls with constant velocity in a straight line according to the law
of inertia. The problem was thus to find a mathematical definition of
"falling," which would embrace the uniform straight-line motion very
far from all matter as well as the complex paths of the planets around
the sun, and of an apple or a cannon-ball on earth.
GRAVITATION AND SPACE-TIME
For the definition of the uniform rectilinear motion of pure inertia
Newton's Euclidean space and independent time were sufficient. For
the much more complicated falling under the influence of gravitation
and inertia together, evidently a more complicated geometry
would be needed. Minkowski's pseudo-Euclidean time-space also was
insufficient. Einstein accordingly introduced a general non-Euclidean
four-dimensional time-space, and enunciated his law of motion thus:
Bodies which are not interfered with move in geodesics.
A geodesic in curved space is exactly the same thing as a straight
line in flat space. We only call it by its technical name, because
the name "straight line" would remind us too much of the old Euclidean
space. If the curvature gets very small, or zero, the geodesic becomes
very nearly, or exactly, a straight line.
The problem has now become to assign to time-space such curvatures
that the geodesics will exactly represent the tracks of falling
bodies. Space of two dimensions can just be flat, like a sheet of
paper, or curved, like an egg. But in geometry of four dimensions
there are several steps from perfect flatness, or "pseudo-flatness," to
complete curvature. Now the law governing the curvature of Einstein's
time-space, i.e., the law of gravitation, is simply that it can never,
outside matter, be curved more than just one step beyond perfect
(pseudo-)flatness.
Since I have promised not to use any mathematics I can hardly convey
to the reader an adequate idea of the difficulty of the problem,
nor do justice to the elegance and beauty of the solution. It is,
in fact, little short of miraculous that this solution, which was
only adopted by Einstein because it was the simplest he could find,
does so exactly coincide in all its effects with Newton's law. Thus the
remarkably accurate experimental verification of this law can at once
be transferred to the new law. In only one instance do the two laws
differ so much that the difference can be observed, and in this case
the observations confirm the new law exactly. This is the well known
case of the motion of the perihelion of Mercury, whose disagreement
with Newton's law had puzzled astronomers for more than half a century.
Since Einstein's time-space includes Minkowski's as a particular case,
it can do all that the other was designed to do for electro-magnetism
and light. But it does more. The track of a pulse of light is also a
geodesic, and time-space being curved in the neighborhood of matter,
rays of light are no longer straight lines. A ray of light from a star,
passing near the sun, will be bent round, and the star consequently
will be seen in a different direction from where it would be seen if
the sun had not been so nearly in the way. This has been verified by
the observations of the eclipse of the sun of 1919 of May 29.
There is one other new phenomenon predicted by the theory, which falls
within the reach of observation with our present means. Gravitation
chiefly affects the time-component of the four-dimensional
continuum, in such a way that natural clocks appear to run slower
in a strong gravitational field than in a weak one. Thus, if we
make the hypothesis--which, though extremely probable, is still
a hypothesis--that an atom emitting or absorbing light-waves is a
natural clock, and the further hypothesis--still very probable, though
less so than the former--that there is nothing to interfere with its
perfect running, then an atom on the sun will give off light-waves
of smaller frequency than a similar atom in a terrestrial laboratory
emits. Opinions as yet differ as to whether this is confirmed or
contradicted by observations.
The great strength and the charm of Einstein's theory do however
not lie in verified predictions, nor in the explanation of small
outstanding discrepancies, but in the complete attainment of its
original aim: the identification of gravitation and inertia, and in
the wide range of formerly apparently unconnected subjects which it
embraces, and the broad view of nature which it affords.
Outside matter, as has been explained, the law of gravitation restricts
the curvature of time-space. Inside continuous matter the curvature
can be of any arbitrary kind or amount; the law of gravitation then
connects this curvature with measurable properties of the matter,
such as density, velocity, stress, etc. Thus these properties define
the curvature, or, if preferred, the curvature defines the properties
of matter, i.e. matter itself.
From these definitions the laws of conservation of energy, and of
conservation of momentum, can be deduced by a purely mathematical
process. Thus these laws, which at one time used to be considered
as the most fundamental ones of mechanics, now appear as simple
corollaries from the law of gravitation. It must be pointed out that
such things as length, velocity, energy, momentum, are not absolute,
but relative, i.e. they are not attributes of the physical reality,
but relations between this reality and the observer. Consequently
the laws of conservation are not laws of the real world, like the
law of gravitation, but of the observed phenomena. There is, however
one law which, already before the days of relativity, had come to be
considered as the most fundamental of all, viz: the principle of least
action. Now action is absolute. Accordingly this principle retains its
central position in Einstein's theory. It is even more fundamental than
the law of gravitation, since both this law, and the law of motion,
can be derived from it. The principle of least action, so far as we
can see at present, appears to be the law of the real world.
XI
THE PRINCIPLE OF GENERAL RELATIVITY
How Einstein, to a Degree Never Before Equalled, Isolates the External
Reality from the Observer's Contribution
BY E. T. BELL
UNIVERSITY OF WASHINGTON
SEATTLE
Einstein's general relativity is of such vast compass, being
coextensive with the realm of physical events, that in any brief
account a strict selection from its numerous aspects is prescribed. The
old, restricted principle being contained in the general, we shall
treat the latter, its close relations with gravitation, and the
significance of both for our knowledge of space and time. The essence
of Einstein's generalization is its final disentanglement of that
part of any physical event which is contributed by the observer from
that which is inherent in the nature of things and independent of
all observers.
The argument turns upon the fact that an observer must describe any
event with reference to some framework from which he makes measurements
of time and distance. Thus, suppose that at nine o'clock a ball is
tossed across the room. At one second past nine the ball occupies a
definite position which we can specify by giving the three distances
from the centre of the ball to the north and west walls and the
floor. In this way, refining our measurements, we can give a precise
description of the entire motion of the ball. Our final description
will consist of innumerable separate statements, each of which contains
four numbers corresponding to four measurements, and of these one
will be for time and three for distances at the time indicated.
Imagine now that a man in an automobile looks in and observes the
moving ball. Suppose he records the motion. To do so, he must refer to
a timepiece and some body of reference. Say he selects his wrist-watch,
the floor of his auto and two sides meeting in a corner. Fancy that
just as he begins his series of observations his auto starts bucking
and the main-spring of his watch breaks, so that he must measure
"seconds" by the crazy running-down of his watch, and distances with
reference to the sides of his erratic auto. Despite these handicaps
he completes a set of observations, each of which consists of a
time measured by his mad watch and three distances reckoned from
the sides of his bucking machine. Let us assume him to have been so
absorbed in his experiment that he noticed neither the disorders
of his watch nor the motion of his auto. He gives us his sets of
measurements. We remark that his seconds are only small fractions
of ours, also his norths and wests are badly mixed. If we interpret
his sets in terms of our stationary walls and sober clock we find
the curious paradox that the ball zigzagged across the room like an
intoxicated bee. He obstinately argues that we know no more than he
about how the ball actually moved. For we got a smooth description,
he asserts, by choosing an artificially simple reference framework,
having no necessary relations whatever to the ball. The crooked
path plotted from his observations proves, he declares, that the
ball was subject to varying forces of which we in the room suspected
nothing. He contends that our room was being jarred by a system of
forces which exactly compensated and smoothed out the real jaggedness
of path observed by himself. But if we know all about his watch and
auto we can easily apply necessary corrections to his measurements,
and, fitting the corrected set to our reference-framework of walls
and clock, recover our own smooth description.
For consistency we must carry our readjustments farther. The path
mapped from our measurements is a curve. Perhaps the curvature was
introduced by some peculiarity of our reference framework? Possibly
our own room is being accelerated upward, so that it makes the ball's
true path--whatever that may be--appear curved downward, just as the
autoist's zigzags made the path he mapped appear jagged. Tradition
attributes the downward curving to the tug of gravity. This force we
say accelerates the ball downward, producing the curved path. Is this
the only possible explanation? Let us see.
GRAVITATION AND ACCELERATION
Imagine a man in a room out of which he cannot see. He notices that
when he releases anything it falls to the floor with a constant
acceleration. Further he observes that all his objects, independently
of their chemical and physical properties, are affected in precisely
the same way. Now, he previously has experimented with magnets, and
has remarked that they attract certain bodies in essentially the same
way that the things which he drops are "attracted" to whatever is
beneath the floor. Having explained magnetic attraction in terms of
"forces," he makes his first hypothesis: (A) He and his room are in
a strong "field of force," which he designates gravitational. This
force pulls all things downward with a constant acceleration. Here
he notes a singular distinction between magnetic and gravitational
"forces": magnets attract only a few kinds of matter, notably iron;
the novel "force," if indeed a force at all, acts similarly upon all
kinds of matter. He makes another hypothesis: (B) His room and he
are being accelerated upward.
Either (A) or (B) describes the facts perfectly. By no experiment
can he discriminate between them. So he takes the great step, and
formulates the Equivalence Hypothesis:
A gravitational field of force is precisely equivalent in its effects
to an artificial field of force introduced by accelerating the
framework of reference, so that in any small region it is impossible
to distinguish between them by any experiment whatever.
Next reconsidering his magnetic "forces," he extends the equivalence
hypothesis to cover all manifestations of force: The effects
attributed to forces of any kind whatever can be described equally
well by saying that our reference frameworks are accelerated; and
moreover there is possible no experiment which will discriminate
between the descriptions.
If the accelerations are null, the frameworks are at rest or in
uniform motion relatively to one another. This special case is the
"restricted" principle of relativity, which asserts that it is
impossible experimentally to detect a uniform motion through the
ether. Being thus superfluous for descriptions of natural phenomena,
the ether may be abandoned, at least temporarily. The older physics
sought this absolute ether framework to which all motions could be
unambiguously referred, and failed to find it. The most exacting
experiments, notably that of Michelson-Morley, revealed no trace of
the earth's supposed motion through the ether. Fitzgerald accounted
for the failure by assuming that such motion would remain undetected if
every moving body contracted by an amount depending upon its velocity
in the direction of motion. The contraction for ordinary velocities
is imperceptible. Only when as in the case of the beta particles,
the velocity is an appreciable fraction of the velocity of light,
is the contraction revealed. This contraction follows immediately
from Einstein's generalization constructed upon the equivalence
hypothesis and the restricted relativity principle. We shall see that
the contraction inevitably follows from the actual geometry of the
universe. [2]
Let us return for a moment to the moving ball. Four measures,
three of distances and one of time, are required in specifying its
position with reference to some framework at each point and at each
instant. All of these measures can be summed up in one compendious
statement--the equations of motion showed how in changing from our
room to his accelerated auto we found a new summary, "transformed
equations," which seemed to indicate that the ball had traversed
a strong, variable field of force. Is there then in the chaos of
observational disagreements anything which is independent of all
observers? There is, but it is hidden at the very heart of nature.
PATHS THROUGH THE WORLD OF FOUR DIMENSIONS
To exhibit this, we must recall a familiar proposition of geometry:
the square on the longest side of a right-angled triangle is equal
to the sum of the squares on the other two sides. It has long been
known that from this alone all the metrical properties of Euclidean
space--the space in which for 2,000 years we have imagined we were
living--can be deduced. Metrical properties are those depending upon
measurement. Now, in the geometry of any space, Euclidean or not,
there is a single proposition of a similar sort which tells us how
to find the most direct distance between any two points that are
very close together. This small distance is expressed in terms of
the two sets of distance measurements by which the end-points are
located, just as two neighboring positions of our ball were located
by two sets of four measurements each. We say by analogy that two
consecutive positions of the ball are separated by a small interval of
time-space. From the formula for the very small interval of time-space
we can calculate mathematically all the metrical properties of the time
and space in which measurements for the ball's motion must be made. So
in any geometry mathematical analysis predicts infallibly the truth
about all facts depending upon measurements from the simple formula
of the interval between neighboring points. Thus, on a sphere the sum
of the angles of any triangle formed by arcs of great circles exceeds
180°, and this follows from the formula for the shortest ("geodesic")
distance between neighboring points on the spherical surface.
We saw that it takes four measurements, one for time and three for
distances, to fix an elementary event, viz., the position of the
centre of our ball at any instant. A system of all possible such sets
of four measurements each, constitutes what mathematicians call a
four-dimensional space. The study of the four-dimensional time-space
geometry, once its shortest-distance proposition is known, reveals all
those relations in nature which can be ascertained by measurements,
that is, experimentally. We have then to find this indispensable
proposition.
Imagine the path taken by a particle moving solely under the influence
of gravitation. This being the simplest possible motion of an actual
particle in the real world, it is natural to guess that its path
will be such that the particle moves from one point of time-space to
another by the most direct route. This in fact is verified by forming
the equations of the free particle's motion, which turn out to be
precisely those that specify a geodesic (most direct line) joining
the two points. On the (two-dimensional) surface of a sphere such a
line is the position taken by a string stretched between two points
on the surface, and this is the shortest distance on the surface
between them. But in the time-space geometry we find a remarkable
distinction: the interval between any two points of the path taken is
the longest possible, and between any two points there is only one
longest path. Translated into ordinary space and time this merely
asserts that the time taken between any two points on the natural
path is the longest possible.
Recall now that when the line-formula for any kind of space is known
all the metrical properties of that space are completely determined,
and combine with this what we have just found, namely, the equations
of motion of a particle subject only to gravitation are the same
equations as those which fix the line-formula for the four-dimensional
time-space. Since gravitation alone determines the motion of the
particle, and since this motion is completely described by the very
equations which fix all the metrical properties of time-space, it
follows that the metrical (experimentally determinable) properties
of time-space are equivalent to those of gravitation, in the sense
that each set of properties implies the other.
THE UNIVERSE OF SPACE-TIME
We have found the thing in nature which is independent of all
observers, and it turns out to be the very structure of time-space
itself. The motion of the free particle obviously is a thing
unconditioned by accidents of observation; the particle under the
influence of gravitation alone must go a way of its own. And if some
observer in an artificial field of force produced by the acceleration
of his reference framework describes the path as knotted, he merely is
foisting eccentricities of his own motion upon the direct path of the
particle. The conclusion is rational, for we believe that time-space
exists independently of any man's way of perceiving it.
Incidentally note that this space is that of the physical world. For
only by measurements of distances and times can we become aware of
our extension in time and space. If beyond this time-space geometry
of measurements there is some "absolute geometry," science can have
no concern with it, for never can it be revealed by the one exploring
device we possess--measurement.
We have followed a single particle. Let us now form a picture of
several. Any event can be analyzed into a multitude of coincidences
in time-space. For consider two moving particles--say electrons. If
they collide they both are in very approximately one place at the
same time. We imagine the path of an electron through time-space
plotted by a line (in four-dimensional space), which will deviate
from a "most direct" (geodesic) path if the electron is subjected
to forces. This is the "world-line" of the electron. If the world
lines of several electrons intersect at one point in time-space,
the intersection pictures the fact of their coincidence somewhere
and somewhen; for all their world-lines having a time-space point in
common, at some instant they must have been in collision. Each point
of a world-line pictures the position at a certain place at a certain
time; and it is the intersections of world-lines which correspond to
physical events. Of what lies between the intersections we have no
experimental knowledge.
Imagine the world-lines of all the electrons in the universe threading
time-space like threads in a jelly. The intersections of the tangle
are a complete history of all physical events. Now distort the
jelly. Clearly the mutual order of the intersections will be unchanged,
but the distances between them will be shortened or lengthened. To a
distortion of the jelly corresponds a special choice (by some observer)
of a reference framework for describing the order of events. He cannot
change the natural sequence of events. Again we have found something
which is independent of all observers.
We can now recapitulate our conclusions and state the principle of
relativity in its most general form.
(1) Observers describe events by measures of times and distances made
with regard to their frameworks of reference.
(2) The complete history of any event is summarized in a set of
equations giving the positions of all the particles involved at
every instant.
(3) Two possibilities arise. (A) Either these equations are the
same in form for all space-time reference frameworks, persisting
formally unchanged for all shifts of the reference scheme; or (B),
they subsist only when some special framework is used, altering their
form as they are referred to different frameworks. If (B) holds,
we naturally assume that the equations, and the phenomena which they
profess to represent, owe their existence to some peculiarity of the
reference framework. They do not, therefore, describe anything which is
inherent in the nature of things, but merely some idiosyncrasy of the
observer's way of regarding nature. If (A) holds, then obviously the
equations describe some real relation in nature which is independent
of all possible ways of observing and recording it.
(4) In its most general form the principle of relativity states that
those relations, and those alone, which persist unchanged in form
for all possible space-time reference frameworks are the inherent
laws of nature.
To find such relations Einstein has applied a mathematical method
of great power--the calculus of tensors--with extraordinary
success. This calculus threshes out the laws of nature, separating
the observer's eccentricities from what is independent of him, with
the superb efficiency of a modern harvester. The residue is a physical
geometry--or geometrical physics--of time-space, in which it appears
that the times and spaces contributed by the several observers'
reference frameworks are shadows of their own contrivings; while
the real, enduring universe is a fourfold order of time and space
indissolubly bound together. One observer separates this time-space
into his own "time" and "space" in one way, determined by his path
through the world of events; another, moving relatively to the first,
separates it differently, and what for one is time shades into space
for another.
This time-space geometry is non-Euclidean. It is "warped" (curved),
the amount of warping at any place being determined by the intensity
of the gravitational field there. Thus again gravitation is rooted in
the nature of things. In this sense it is not a force, but a property
of space. Wherever there is matter there is a gravitational field,
and hence a warping of space. Conversely, as long ago imagined by
Clifford, wherever there is a warping of space, there is matter;
and matter is resolved ultimately into wrinkles in time-space.
To visualize a warped space, consider a simple analogy. A man walks
away from a polished globe; his image recedes into the mirror-space,
shortening and thinning as it goes, and thinning (in the direction of
motion) faster than it shortens. Everything around him experiences
a like effect. If he tries to discover this by a footrule it
automatically shortens faster as he turns it into the horizontal
position, so his purpose eludes him. The mirror-space is warped in
the direction of the image's motion. So is our own. For all bodies,
as evidenced by the Fitzgerald contraction, shorten in the direction
of motion. And just as the image can never penetrate the mirror-space
a greater distance than half its radius, so probably time-space is
curved in such a way that our universe, like the surface of a sphere,
is finite in extent, but unbounded.
XII
FORCE VS. GEOMETRY
How Einstein Has Substituted the Second for the First in Connection
With the Cause of Gravitation
BY SAUL DUSHMAN
GENERAL ELECTRIC LABORATORIES SCHENECTADY, N. Y.
The theory of relativity represents a most strikingly original
conception of time and space, which was suggested by Einstein in order
to correlate with all our past experience certain observations made
in recent years. It is therefore extremely comprehensive in its scope;
it demands from us a radical revision in our notions of time and space;
it throws new light on the nature of mass and energy, and finally, it
furnishes a totally new conception of the old problem of gravitation.
The starting point of the theory is the familiar observation that
motion is always relative: that is, to define the motion of any object
we must always use some point of reference. Thus we speak of the
velocity of a train as 40 miles per hour with respect to the earth's
surface, but would find it impossible to determine its absolute speed,
or motion in space, since we know of no star whose position can be
spoken of as absolutely fixed. These and similar considerations have
led to the conclusion, pointed out by Newton and others, that it is
impossible by any mechanical experiments on the earth to measure its
velocity in space.
However, the results of observations on the phenomena of light and
electricity led to the revival of the same problem under another
form. As well known, there was evolved from these discoveries,
the theory that light and electrical energy are of the same nature,
and are in each case manifestations of wave-disturbances propagated
through a hypothetical medium, the ether, with a velocity of 186,000
miles per second.
The problem therefore arose as to whether the earth and all stellar
bodies move through this ether. In that case it ought to be possible
to measure the velocity of the earth with respect to this medium, and
under these conditions we could speak, in a sense, of absolute motion.
A large number of experiments has been tried with this end in view. The
most famous of these, and the one which stimulated the subsequent
development of the theory of relativity, was that carried out by
Michelson and Morley in 1887. To understand the significance of this
experiment we shall refer briefly to an analogous observation which
is quite familiar.
Does it take longer to swim to a point 1 mile up a stream and back
or to a point 1 mile across stream and back? The experienced swimmer
will answer that the up-and-down journey takes longer. If we assume
that the swimmer has a speed of 5 miles an hour in still water and
that the current is 3 miles an hour, we find that, while it requires
five-eighths hour to make the up-and-down journey, it takes only
one-half hour for the trip across stream and back. The ratio between
the times required for the two journeys is thus five-fourths, and if
this is written in the form
$$\frac{1}{\sqrt{1 - (\frac{3}{5})^2}}$$
it shows how the result depends upon the square of the ratio of the
speeds of the swimmer and the current.
Now the earth is moving in its orbit about the sun with a velocity
of 18 miles per second. If the earth moves through the ether and a
light-beam passes from one mirror to another and back again, the time
taken for this journey ought to be longer when the light-path is in
the direction of the earth's motion than when it is at right angles
to this direction. For we can consider the light as a swimmer having
a speed of 186,000 miles per second and travelling in a stream whose
current is 18 miles per second.
When Michelson and Morley tried the experiment they could not observe
any difference in the velocity of light in the two directions. The
experiment has since been repeated under various conditions, but
always with negative results.
Einstein's contribution to science consists in interpreting this
result as being in accord with Newton's ideas on mechanical relativity
in that it demonstrates the impossibility of measuring absolute
motion, not only by mechanical, but also by optical or electrical
experiments. Consequently the velocity of light must be regarded as
constant and independent of the motion of either source or observer.
THE RELATIVITY OF UNIFORM MOTION
Let us consider some of the consequences which follow from this
principle. An observer travelling with say one-half the velocity
of light in the same direction as a ray of light would find that the
latter has the usual velocity of 186,000 miles per second. Similarly an
observer travelling in the opposite direction to that of the light-ray,
with one-half the velocity of light, would obtain the same result.
Einstein has shown that these conclusions can be valid only if the
units of time and space used by the two observers depend upon their
relative motions. A careful calculation shows that the unit of length
used by either observer appears to the other observer contracted
when placed in the direction of their relative motion (but not,
when placed at right angles to this direction), and the unit of time
used by either observer appears to the other too great. Moreover, the
ratio of the units of length or of time varies with the square of the
relative speed of the two observers, according to a relation which is
similar to that mentioned above for the swimmer in the current. This
relation shows that as the relative speed approaches that of light
the discrepancy between the units increases.
Thus, for an observer moving past our earth with a velocity which is
nine-tenths that of light, a meter stick on the earth would be 44
centimeters as measured by him, while a second on our clocks would
be about two and a half seconds as marked by his clock. Similarly,
what he calls a meter length would, for us, be only 44 centimeters
and he would appear to us to be living about two and a half times
slower than we are. Each observer is perfectly consistent in his
measurements of time and space as long as he confines his observations
to his own system, but when he tries to make observations on another
system moving past his, he finds that the results which he obtains
do not agree with those obtained by the other observer.
It is not surprising that in accordance with this conclusion it also
follows that the mass of a body must increase with its velocity. For
low velocities the increase is so small that we cannot ever hope to
measure it, but as the velocity of light is approached the difference
becomes more and more appreciable and a body having the velocity of
light would possess infinite mass, which simply means that such a
velocity cannot be attained by any material object. This conclusion
has been experimentally confirmed by observations on the mass of
the extremely small negatively charged particles which are emitted
by radioactive elements. Some of these particles are ejected with
velocities which are over nine-tenths that of light, and measurements
show that the increase in mass is in accord with this theory.
The relativity theory also throws new light on the nature of mass
itself. According to this view, mass and energy are equivalent. The
absolute destruction of 1 gram of any substance, if possible, would
yield an amount of energy which is one hundred million times as much
as that obtained by burning the same mass of coal. Conversely, energy
changes are accompanied by changes in mass. The latter are ordinarily
so inappreciably small as to escape our most refined methods of
measurements, but in the case of the radioactive elements we actually
observe this phenomenon. From this standpoint, also, the laws of
conservation of energy and of mass are shown to be intimately related.
UNIVERSAL RELATIVITY
So far we have dealt with what has been designated as the special
theory of relativity. This, as we have seen, applies to uniform motion
only. In extending the theory to include non-uniform or accelerated
motion, Einstein has at the same time deduced a law of gravitation
which is much more general than that of Newton.
A body falling towards the earth increases in velocity as it falls. The
motion is said to be accelerated. We ascribe this increase in velocity
to a gravitational force exerted by the earth on all objects. As
shown by Newton, this force acts between all particles of matter in
the universe, and varies inversely as the square of the distance,
and directly as the product of the masses.
Of course, we have had a number of theories of gravitation, and
none of them have proven successful. Einstein, however, was the
first one to suggest a conception of gravitation which has proven
extremely significant. He points out that a gravitational force
is non-existent for a person falling freely with the acceleration
due to gravity. For this person there is no sensation of weight,
and if he were in a closed box which is also falling with the same
acceleration, he would be unable to decide as to whether his system
were falling or situated in interplanetary space where there is
no gravitational field. Furthermore, if he were to carry out any
optical or electrical experiments in this box he would observe the
same results as an experimenter on the earth. A ray of light would
travel in a straight line so far as this observer can perceive,
while an external observer would, of course, judge differently.
Einstein shows that this is equally true for all kinds of acceleration
including that due to rotation. In the case of a rotating body there
exists a centrifugal force which tends to make objects on the surface
fly outwards, but for an external observer this force does not exist
any more than gravity exists for the observer falling freely.
Thus we can draw the general conclusion that a gravitational field or
any other field of force may be eliminated by choosing an observer
moving with the proper acceleration. For this observer, however,
the laws of optics and electricity must be just as valid as for an
observer on the earth.
In postulating this equivalence hypothesis Einstein merely makes use
of the very familiar observation that, independently of the nature
of the material, all bodies possess the same acceleration in a given
field of force.
The problem which Einstein now sets out to solve is that of determining
the law which shall describe the motion of any system in a field of
force in such a general manner as to leave unaltered the fundamental
relations of electricity and optics.
In connection with the solution of this problem he finds it necessary
to discard the limitations placed on us by ordinary or Euclidean
geometry. In this manner geometrical concepts as well as those of
force are completely robbed of all notions of absoluteness, and the
goal of a general theory of relativity is attained.
THE GEOMETRY OF GRAVITATION
Let us consider a circular disc rotating with a uniform peripheral
speed. According to the deductions from the "special theory" of
relativity, an observer situated near the edge of this disc, but not
rotating with it, will observe that units of length measured along
the circumference of the disc are contracted. On the other hand,
measurements along the diameter, which is at right angles to the
direction of motion of the circumference, will show no contraction
whatever, and, consequently the observer will find that the ratio
of circumference to diameter has not the well known value 3.14159
... but exceeds this value, the difference being greater and greater
as the peripheral speed approaches that of light. That is, the laws
of ordinary geometry no longer hold true.
However, we know other cases in which the ordinary or Euclidean
geometry is not applicable. Thus suppose that on the surface of a
sphere we describe a series of concentric circles. Since the surface
is curved, we are not surprised at finding that the circumference of
any one of these circles is less than 3.14159 ... times the distance
across the circle as measured on the surface of the sphere. What
this means, therefore, is that we cannot use Euclidean geometry to
describe measurements on the surface of a sphere, and every schoolboy
knows this from comparing Mercator's projection of the earth's surface
with the actual representation on a globe.
When we come to think of it, the reason we realize all this is
because our sense of three dimensions enables us to differentiate
flat surfaces from those that are curved. Let us, however, imagine
a two-dimensional being living on the surface of a large sphere. So
long as his measurements are confined to relatively small areas he
will find it possible to describe all his measurements in terms of
Euclidean geometry. As, however, his area of operation increases
he will begin to observe greater and greater discrepancies. Being
unfamiliar with the existence of such a three-dimensional object as
a sphere, and therefore not realizing that he is on the surface of
one, our intelligent two-dimensional being will conclude that the
disturbance in his geometry is due to the action of a force, and by
means of plausible assumptions on the "law" of this force he will
reconcile his observations with the laws of plane geometry.
Now since an acceleration in a gravitational field is identical with
that due to centrifugal force produced by rotation, we concluded that
the geometry in a gravitational field must also be non-Euclidean. That
is, space in the neighborhood of matter is distorted or curved. The
curvature of space bears the same relation to three dimensions that
the curvature of a spherical surface bears to two dimensions, and
that is why we do not perceive it, any more than the intelligent
two-dimensional being would be aware of the distortion of his space
(or surface). Furthermore, like this being, we have assumed the
existence of a gravitational force to account for discrepancies in
our geometrical measurements.
The identification in this manner of gravitational effects with
geometrical curvature of space enables Einstein to derive a general
law for the path of any particle in a gravitational field, with respect
both to space and to time. Furthermore, the law expresses this motion
in terms which are independent of the relative motion and position
of the observer, and satisfies the condition that the fundamental
laws of physics be equally valid for all observers. The solution
of the problem involved the use of a new kind of higher calculus,
elaborated by two Italian mathematicians, Ricci and Levi-Civita. The
result is a law of motion which is extremely general in its validity.
For low velocities it approximates to Newton's solution, and in the
absence of a gravitational field it leads to the same conclusions as
the special theory of relativity. There are three deductions from this
law which have aroused a great deal of interest, and the confirmation
of two of these by actual observation must be regarded as striking
proof of Einstein's theory.
XIII
AN INTRODUCTION TO RELATIVITY
A Treatment in Which the Mathematical Connections of Einstein's Work
are Brought Out More Strongly and More Successfully Than Usual in a
Popular Explanation
BY HAROLD T. DAVIS, UNIVERSITY OF WISCONSIN, Madison, Wis.
One of the first questions which appears in philosophy is this: What
is the great reality that underlies space and time and the phenomena
of the physical universe? Kant, the philosopher, dismissed it as
a subjective problem, affirming that space and time are "a priori"
concepts beyond which we can say no more.
Then the world came upon some startling facts. In 1905 a paper appeared
by Professor Albert Einstein which asserted that the explanation of
certain remarkable discoveries in physics gave us a new conception
of this strange four-dimensional manifold in which we live. Thus,
the great difference between the space and time of philosophy and the
new knowledge is the objective reality of the latter. It rests upon
an amazing sequence of physical facts, and the generalized theory,
which appeared several years later, founded as it is upon the abstruse
differential calculus of Riemann, Christoffel, Ricci and Levi-Civita,
emerges from its maze of formulas with the prediction of real phenomena
to be sought for the in the world of facts.
We shall, therefore, approach the subject from this objective point
of view. Let us go to the realm of actual physical events and see how
the ideas of relativity gradually unfolded themselves from the first
crude wonderings of science to the stately researches that first
discovered the great ocean of ether and then penetrated in such a
marvelous manner into some of its most mysterious properties.
THE ELECTROMAGNETIC THEORY OF LIGHT
Suppose that we go out on a summer night and look into the dark depths
of the sky. A thousand bright specks are flashing there, blue, red,
yellow against the dark velvet of space. And as we look we must all
be impressed by the fact that such remote objects as the stars can be
known to us at all. How is it that light, that curious thing which
falls upon the optic nerve and transmits its pictures to the brain,
can ever reach us through the black regions of interstellar space? That
is the question which has for its answer the electromagnetic theory
of light.
The first theory to be advanced was Newton's "corpuscular" theory
which supposed that the stars are sending off into space little pellets
of matter so infinitesimally small that they can move at the rate of
186,000 miles a second without injuring even so delicate a thing as
the eye when they strike against it.
But in 1801, when Thomas Young made the very important discovery of
interference, this had to give way to the wave theory, first proposed
by Huyghens in the 17th century. The first great deduction from this,
of course, was the "luminiferous ether," because a wave without some
medium for its propagation was quite unthinkable. Certain peculiar
properties of the ether were at once evident, since we deduce that
it must fill all space and at the same time be so extremely tenuous
that it will not retard to any noticeable degree the motion through
it of material bodies like the planets.
But how light was propagated through the ether still remained a
perplexing problem and various theories were proposed, most prominent
among them being the "elastic solid" theory which tried to ascribe to
ether the properties of an elastic body. This theory, however, laid
itself open to serious objection on the ground that no longitudinal
waves had been detected in the ether, so that it began to appear
that further insight into the nature of light had to be sought for
in another direction.
This was soon forthcoming for in 1864 a new theory was
proposed by James Clerk Maxwell which seemed to solve all of the
difficulties. Maxwell had been working with the facts derived from
a study of electrical and magnetic phenomena and had shown that
electromagnetic disturbances were propagated through the ether at a
velocity identical with that of light. This, of course, might have been
merely a strange coincidence, but Maxwell went further and demonstrated
the interesting fact that an oscillating electric charge should give
rise to a wave that would behave in a manner identical with all of the
known properties of a light wave. One particularly impressive assertion
was that these waves, consisting of an alternating electric field
accompanied by an alternating magnetic field at right angles to it,
and hence called electromagnetic waves, would advance in a direction
perpendicular to the alternating fields. This satisfied the first
essential property of light rays, i.e., that they must be transverse
waves, and the ease with which it explained all of the fundamental
phenomena of optics and predicted a most striking interrelation
between the electrical and optical properties of material bodies,
gave it at once a prominent place among the various theories.
The electromagnetic theory, however, had to wait until 1888 for
verification when Heinrich Hertz, in a series of brilliant experiments,
succeeded in producing electromagnetic waves in the laboratory and
in showing that they possessed all of the properties predicted by
Maxwell. These waves moved with the velocity of light: they could be
reflected, refracted, and polarized: they exhibited the phenomenon
of interference and, in short, could not be distinguished from light
waves except for their difference in wave length.
THE MICHELSON-MORLEY EXPERIMENT
With the final establishment of the electromagnetic theory of light as
a fact of physics, we have at last endowed the ether with an actual
substantiality. The "empty void" is no longer empty, but a great
ocean of ether through which the planets and the suns turn without
ever being aware that it is there.
In 1881 A. A. Michelson undertook an experiment, originally suggested
by Maxwell, to determine the relative motion of our earth to the
ether ocean and six years later he repeated it with the assistance
of E. W. Morley. The experiment is now known as the Michelson-Morley
experiment and since it is the great physical fact upon which the
theory of relativity rests, it will be well for us to examine it
in detail.
Since we can scarcely think that our earth is privileged in the
universe and that it is at rest with respect to this great ether ocean
that fills space, we propose to discover how fast we are actually
moving. But the startling fact is that the experiment devised for this
purpose failed to detect any motion whatever of the earth relative
to the ether. [3]
The explanation of this very curious fact was given by both
H. A. Lorentz and G. F. Fitzgerald in what is now widely known under
the name of the "contraction hypothesis." It is nothing more nor less
than this:
Every solid body undergoes a slight change in dimensions, of the order
of ($v^2/c^2$), when it moves with a velocity $v$ through the ether.
The reason why the experiment failed, then, was not because the earth
was not moving through the ether, but because the instruments with
which the experiment was being conducted had shrunk just enough to
negative the effect that was being looked for. [4]
THE LORENTZ TRANSFORMATION
We can not at this point forebear introducing a little mathematics
to further emphasize the theory and the very logical nature of this
contraction hypothesis.
Let us suppose that we were on a world that was absolutely motionless
with respect to the ether and were looking at a ray of light. The
magnetic and electric fields which form the ray can be described by
means of four mathematical expressions which have come to bear the
name of "Maxwell's field equations." Now suppose that we ask ourselves
the question: How must these equations be changed so that they will
apply to a ray of light which is being observed by people on a world
that is moving with a velocity v through the ether?
The answer is immediate. From the Michelson-Morley experiment we know
that we can not tell how fast or how slowly we are moving with respect
to the ether. This means that no matter what world we may be upon,
the form of the Maxwell field equations will always be the same,
even though the second set of axes (or frame of reference) may be
moving with high velocity with respect to the first.
Starting from this hypothesis (called in technical language the
covariance of the equations with respect to a transformation of
coordinates), Lorentz found that the transformation which leaves the
field equations unchanged in form was the following:
$$x' = k(x - vt), y' = y, z' = z, t' = k(t - vx/c)$$
where $k$ is as on page 92.
And what, now, can be deduced from these very simple looking
equations? In the first place we see that the space of $x'$, $y'$,
$z'$, $t'$ is not our ordinary concept of space at all, but a space in
which time is all tangled up with length. To put it more concretely,
we may deduce from them the interesting fact that whenever an aviator
moves with respect to our earth, his shape changes, and if he were to
compare his watch with one on the earth, he would find that his time
had changed also. A sphere would flatten into an ellipse, a meter
stick would shorten up, a watch would slow down and all because, as
H. Minkowski has shown us from these very equations, we are really
living in a physical world quite different from the world of Euclid's
geometry in which we are accustomed to think we live.
A variety of objections has very naturally been made to this rather
radical hypothesis in an attempt to discredit the entire theory, but
it is easily seen that any result obtained through the field equations
must necessarily be in conformity with the theory of contraction, since
this theory is only the physical interpretation of that transformation
which leaves the field equations unaltered. Indeed, it is even possible
to postulate the Lorentz transformation together with the assumption
that each element of charge is a center of uniformly diverging tubes
of strain and derive the Maxwell field equations from this, which
shows from another point of view the truly fundamental nature of
the transformation.
THE FIRST THEORY OF RELATIVITY
The whole question of the ether had arrived at this very
interesting point when Professor Einstein in 1905 stated the theory
of relativity. He had noticed that the equations of dynamics as
formulated by Newton did not admit the Lorentz transformation, but
only the simple Galilean transformation:
$$x' = x - vt, y' = y, z' = z, t' = t\,.$$
Here, indeed, was a curious situation. Two physical principles,
that of dynamics and that of electromagnetism, were coexistent and
yet each one admitted a different transformation when the system of
reference was transferred to axes moving with constant velocity with
respect to the ether.
Now the electromagnetic equations and their transformation had been
shown to be in accord with experimental fact, whereas it had long
been felt that Newton's equations were only a first approximation
to the truth. For example, the elliptic orbit of a planet had been
observed by Leverrier to exhibit a disquieting tendency to rotate
in the direction of motion. This precession, which in the case of
Mercury was as large as 43'' per century, could not be accounted
for in any way by the ordinary Newtonian laws and was, consequently,
a very celebrated case of discordance in gravitational astronomy.
With this example clearly before him, Einstein took the great step
and said that the laws of dynamics and all other physical laws had to
be remade so that they, also, admit the Lorentz transformation. That
is to say,
The laws of physical phenomena, or rather the mathematical expressions
for these laws, are covariant (unchanged in form) when we apply the
Lorentz transformation to them.
The deductions from the Michelson-Morley experiment now seem to have
reached their ultimate conclusion.
One discordant fact in this new theory remained, however. That same
precession of the perihelion of Mercury which had first lead Einstein
to his theory remained unsettled. When the new approximations were
applied to the formula of orbital motion, a precession was, indeed,
obtained, but the computed value fell considerably below that of the
observed 43'' per century.
THE INCLUSION OF GRAVITATION
With the idea of investigating the problem from the very bottom,
Einstein now undertook a broader and more daring point of view. In
the first place he said that there is no apparent reason in the great
scheme of world events why any one special system of coordinates should
be fundamental to the description of phenomena, just as in the special
theory a ray of light would appear the same whether viewed from a
fixed system or a system moving with constant velocity with respect
to the ether. This makes the very broad assumption that no matter
what system of coordinates we may use, the mathematical expressions
for the laws of nature must be the same. In Einstein's own words,
then, the first principle of this more general theory of relativity
must be the following:
"The general laws of nature are expressed through equations which
hold for all systems of coordinates, that is, they are covariant with
respect to arbitrary substitutions." [5]
But this was not enough to include gravitation so Einstein
next formulated what he was pleased to call his "equivalence
hypothesis." This is best illustrated by an example. Suppose that
we are mounting in an elevator and wish to investigate the world
of events from our moving platform. We mount more and more rapidly,
that is with constant acceleration, and we appear to be in a strong
gravitational field due to our own inertia. Suppose, on the other
hand, that the elevator descends with an acceleration equal to that
of gravity. We would now feel certain that we were in empty space
because our own relative acceleration has entirely destroyed that of
the earth's gravitational field and all objects placed upon scales
in an elevator would apparently be without weight.
Applying this idea, then, Einstein decided to do away with gravitation
entirely by referring all events in a gravitational field to a new set
of axes which should move with constant acceleration with respect to
the first. In other words we are going to deal with a system moving
with uniform acceleration with respect to the ether, just as we
considered a system moving with uniform velocity in the special theory.
The next step in the construction of this complicated theory is to
reduce these two hypotheses to the language of mathematics and this
was accomplished by Einstein with the help of M. Grossmann by means
of the theory of tensors.
On account of the very great intricacy of the details, we must
content ourselves with the mere statement that this really involved
the generalization of the famous expressions known as Laplace's
and Poisson's equations, on the explicit assumption that these
two equations would still describe the gravitational field when we
are content to use a first approximation to the truth. The set of
ten differential equations which Einstein got as a result of his
generalization he called his field equations of gravitation. [6]
XIV
NEW CONCEPTS FOR OLD
What the World Looks Like After Einstein Has Had His Way With It
BY JOHN G. McHARDY, COMMANDER R.N.,
LONDON
"The new-created world, which fame in heaven
Long had foretold, a fabric wonderful,
Of absolute perfection."
Einstein's Theory of Relativity has led to determining a key law
of nature--the law of gravitation--which is also the basic law of
mechanics. Thus it embraces a whole realm of physics, and promises,
through the researches of Professor Weyl, to embrace another
realm--electro-dynamics. Its limitations are not yet reached, for
Einstein has already postulated therefrom a theory of a finite, yet
unbounded, universe. This essay, however, is mainly concerned with
mechanics, and electrical forces are not considered.
To have synthesised Newton's two great principles--his law of motion
and law of gravitation--interpreting in the process the empirical law
of equality of gravitational and inertial mass, is alone an immense
achievement; but Einstein's researches have opened up a new world
to the physicist and philosopher which is of greater importance. He
has given us a vision of the immaterial world, a geometrical or
mathematical vision, which is more satisfying than the "ether"
conceptions hitherto presented. The fabric of his vision is not
baseless. It is this fabric we shall consider, touching on certain
aspects of the Einstein theory in the endeavor to present an image in
miniature of his edifice of thought and to show the firmness of its
foundations. That they are well and truly laid was demonstrated by
the verification, from observations made during the solar eclipse in
1919, of Einstein's prediction of the displacement of a wave of light
in a gravitational field, showing light to have the property of weight.
The physical world is shown by Einstein to be a world of
"relations." Underlying it there is an absolute world of which physical
phenomena are the manifestation. "Give me matter and motion," says
Descartes, "and I will construct the world." "Give me a world in
which there are ordered relations," says the Relativist, "and I will
show you the behavior of matter therein" (mechanics). We first view
this underlying world as an abstraction, abstracting energy ("bound"
as in matter and electrons, "free" as in light), and its attribute
force. This abstraction we will call the "World-Frame." Later, we
will study the underlying world in connection with energy, and will
call this absolute world the "World-Fabric." The connection between
the geometrical character of the World-Frame and the geometrical
characters of the World-Fabric is the key to the law of gravitation.
THE WORLD-FRAME
This is our conception of a world, if such were possible, entirely free
from the influence of energy. We may conceive of it as an amorphous
immaterial something containing "point-events" (a point-event
being an instant of time at a point in space--a conception, not a
definition). These point-events have a fourfold order and definite
relation in this Frame, i.e. they can be specified by four variables
or coordinates in reference to some base called a reference system,
with respect to which they are forward or backward, right or left,
above or below, sooner or later. This shows the World-Frame to be
four-dimensional. Thus an aggregate of point-events (or an "event,"
which implies limited extension in space and limited duration
in time) [7] would have what we familiarly describe as length,
breadth, height and time. To express these metrical properties most
simply we must choose a four-dimensional reference system having
a particular form--rectilinear axes (Cartesian coordinates), and a
particular motion--uniform and rectilinear, i.e. unaccelerated, and
non-rotating with respect to the path of a light ray. We call this
an inertial system because Newton's Law of Inertia holds for such
a system alone. This system indicates how observers partition the
World-Frame into space and time. It restricts observers to uniform
rectilinear motion, and observations to bodies and light-pulses in
such motion. Thus gravitational and other forces are discounted,
and we obtain World-Frame conditions notwithstanding the fact that
observers are in the presence of energy.
Now the separation between point-events which have a definite relation
to each other must be absolute. The separation between two points in
a plane is defined by the unique distance between them (the straight
line joining them). Between point-events the analogue of this unique
distance, which we call the "separation-interval" (to indicate its
time-like and space-like nature), is also unique. Its unique and
absolute character give it great importance as thereby it is the same
for all observers regardless of their reference system.
If, in place of the rather cumbersome expression $X-x$ to indicate
the difference between the $x$-coordinates of two points, we employ
the more compact expression $dx$; if for the benefit of readers who
have a little algebra but no analysis we state explicitly that this
expression is a single symbol for a single quantity, and has nothing
to do with any product of two quantities $d$ and $x$; and if we
extend this notation to all our coordinates: then it is clear from
previous essays that the distance $S$ between two points in a plane
referred to a rectilinear system $OX$, $OY$, is given by the simple
equation $S^2 = (dx)^2 + (dy)^2$. Einstein and Minkowski show that
the value for the separation interval $\Omega$, the analogue of $S$,
referred to an inertial system is given by the equation
$$\Omega^2 = (dx)^2 + (dy)^2 + (dz)^2 - (dt)^2\,,$$
which is seen to be a modified extension to four dimensions of the
equation for $S$. We must measure $t$ in the same units as $x$, $y$,
$z$. By taking the constant velocity of light (300,000 kilometres
per second) as unit velocity, we can measure in length or time
indiscriminately. [8]
We will analyse briefly this equation as it epitomizes the Special
Theory of Relativity. If the World-Frame had been Euclidean the
equation would have been
$$\Omega^2 = (dx)^2 + (dy)^2 + (dz)^2 + (dt)^2$$
but this would not satisfy the "transformation equations" which
resulted from the Special Theory. These transformation equations
arose directly from a reconciliation between two observed facts; (a)
the observed agreement of all natural phenomena with the "Restricted
Principle of Relativity"--a principle which shows that absolute
rectilinear motion cannot be established--(as regards mechanics this
was recognized by Newton; the Michelson-Morley and other experiments
showed this principle also applied to optical and electro-dynamical
phenomena); and (b) the observed disagreement of optical and
electro-dynamical phenomena (notably the constancy of light velocity)
with the laws of dynamics as given by classical mechanics, e.g., in
regard to the compounding of relative velocities. Einstein effected
this reconciliation by detecting a flaw in classical mechanics. He
showed that by regarding space and time measurements as relative
to the observer--not absolute as Newton defined them--there was
nothing incompatible between the Principle of Relativity and the
laws of dynamics so modified. Newton's definitions were founded on
conception. Einstein's recognition of the relativity of space and
time is based on observation.
Equation (1) shows that the geometry of the World-Frame referred to
an inertial system is semi-Euclidean (hyperbolic), and that space and
time measurements are relative to the observer's inertial reference
system. The equation shows that the World-Frame has a certain
geometrical character which we distinguish as four-dimensional
"flatness." It is everywhere alike (homaloidal). Its flat character
is shown by the straight line nature of the separation-interval and
of the system to which it is most simply referred.
Thus we have found two absolute features in the World-Frame--(1) Its
geometrical character--"flatness"; (2) The separation-interval--which
can be expressed in terms of measurable variables called space and time
partitions, this partitioning being dependent on the observer's motion.
We are now in a position to explore the World-Fabric. Already we see
that, studied under inertial conditions (free of force), it agrees
with the World-Frame.
THE WORLD-FABRIC
The General Theory of relativity is largely concerned with the
investigation of the World-Fabric. Consider the World-Frame to be
disturbed. We may regard this disturbance, which manifests itself in
physical phenomena, as energy, or more correctly "action."
When energy is thwarted in its natural flow, force is manifested,
with which are associated non-uniform motions such as accelerations
and rotations. This disturbed World-Frame we distinguish as the
World-Fabric. It is found to have various non-Euclidean characters
differing from the simple "flat" character of the World-Frame according
to the degree of disturbance (action) in the region. Disturbance
gives the fabric a geometrical character of "curvature"; the more
considerable the disturbance, the greater the curvature. Thus an
empty region (not containing energy, but under its influence) has
less curvature than a region in which free energy abounds.
Our problem, after showing the relativity of force (especially
gravitational force), is to determine the law underlying the fabric's
geometrical character; to ascertain how the degree of curvature is
related to the energy influencing a region, and how the curvature of
one region is linked by differential equations to that of neighboring
regions. Such a law will be seen to be the law of gravitation.
We study the World-Fabric by considering tracks on which material
particles and light-pulses progress; we find such tracks regulated
and defined by the Fabric's curvature, and not, as hitherto supposed,
by attractive force inherent in matter. As a track is measurable by
summing the separation-intervals between near-by point-events on it,
all observers will agree which is the unique track between two distant
point-events. Einstein postulates that freely progressing bodies
will follow unique tracks, which are therefore called natural tracks
(geodesics).
If material bodies are prevented from following natural tracks by
contact with matter or other causes, the phenomenon of gravitational
force is manifested relative to them. Whenever the natural flow
of energy is interrupted force is born. For example, when the
piston interrupts the flow of steam, or golf ball flow of club,
force results--the interruption is mutual, and the force relative
to both. Likewise when the earth interrupts the natural track of a
particle (or observer) gravitational force is manifested relative
to both.
So long as a body moves freely no force is appreciated by it. A
falling aviator (neglecting air resistance) will not appreciate any
gravitational force. He follows a natural track, thereby freeing
himself from the force experienced in contact with matter. He
acquires an accelerating motion with respect to an inertial system. By
acquiring a particular accelerating motion an observer can annul any
force experienced in any small region where the field of force can
be considered constant.
Thus Einstein, interpreting the equality of gravitational and inertial
mass, showed that the same quality manifests itself according to
circumstances as "weight" or as inertia, and that all force is purely
relative and may be treated as one phenomenon (an interruption in
energy flow). This "Principle of Equivalence" shows that small portions
of the World-Fabric, observed from a freely moving particle (free of
force), could be treated as small portions of the World-Frame. [9]
If such observations were practicable, we could determine the Fabric
curvature by referring point-event measurements to equation (1). We
cannot observe from unique tracks but we can observe them from our
restrained situation. Their importance is now apparent, because, by
tracing them over a region, we are tracing something absolute in the
Fabric--its geometrical character. We study this curvature by exploring
separation-intervals on the tracks of freely moving bodies, relating
these separation-intervals to actual measurements in terms of space
and time components depending on the observer's reference system. The
law of curvature must be the law of gravitation. To illustrate the
lines on which Einstein proceeded to survey the World-Fabric from
the earth we will consider a similar but more simple problem--the
survey of the sea-surface curvature from an airship. We study this
curvature by exploring small distances on the tracks of ships (which we
must suppose can only move uniformly on unique tracks--arcs of great
circles), relating such distances to actual measurements in terms of
length and breadth components depending on the observer's reference
system. This two-dimensional surface problem can be extended to the
four-dimensional Fabric one.
We consider the surface to be covered by two arbitrarily drawn
intersecting series of curves: curves in one series not intersecting
each other, vide figure. This Gaussian system of coordinates is
appropriate only when the smaller the surface considered, the
more nearly it approximates to Euclidean conditions. It admits of
defining any point on the surface by two numbers indicating the curves
intersecting at that point. $P$ is defined by $x_1$, $x_2$. $P_1$
(very near $P$) is defined by $x_1 + dx_1$, $x_2 + dx_2$. The equation
for the minute distance $s$ between two adjacent points in such a
system is given by the general formula
$$s^2 = g_{11}d{x_1}^2 + g_{12}dx_1dx_2 + g_{22}d{x_2}^2\,.$$
The $g$'s may be constants or functions of $x_1$, $x_2$. Their
value is dependent on the observer's reference system and on the
geometrical character of the surface observed. The curves being
arbitrary, the formula is appropriate for any reference system,
or even if the observer does not know exactly what his reference
system is. (The Fabric observer does not know what his space and time
partitioning actually is because he is in a gravitational field). It
is the $g$'s which disclose the geometry of an observer's partitions,
and their values also contain a reflection of the character of the
region observed.
We find $s$ by direct exploration with a moving ship ($\Omega$ is found
by direct exploration with a freely moving particle); $dx_1$, $dx_2$
are the observed length and breadth measurement differences which we
have to relate to $s$. By making sufficient observations in a small
area and referring them to the general formula we can find the values
of the $g$'s for the observer's particular reference system. Different
values for $g$'s will be found if the observer changes his reference
system, but there is a limitation to the values so obtainable owing to
the part played by the surface itself, which is diffidently expressing
its intrinsic geometrical character in the $g$'s in each observation.
EINSTEIN'S RESULTS
Thus we approach the absolute character of the surface through
the relative nature of the observer's reference system. There is a
relationship common to all values of the $g$'s that belong to the
same curvature. This relationship is expressed by a differential
equation. It is this equation of curvature that the airship's observer
must find. Einstein's problem was similar, but he was concerned with
four dimensions, which entailed a general formula with ten $g$'s,
and he had to find a set of differential equations of the second
order to determine the law of Fabric curvature. He divided the Fabric
into regions: I. World-Frame--beyond influence of energy. II. Empty
region--free of energy, but under its influence. III. Region containing
free energy only. Each region has a characteristic curvature. By
means of an absolute differential calculus--a wonderful mathematical
scaffolding erected by Riemann, Christoffel and others--involving the
theory of tensors, he succeeded in finding such a set of equations. He
kept the following points in view: (1) The equations must not only
give the character of region II, but must satisfy the special case
of region I; (2) They must be independent of any partitioning system,
because the General Theory of Relativity demands that a law of nature
be in a form appropriate for all observers whatever their position
and motion; (3) They must be concerned with energy which is conserved,
not mass which the Special Theory showed dependent on velocity. This
set of differential equations which shows how the curvature of the
Fabric at any point links to the curvature at neighboring points is
the law of gravitation, a law which has been severely tested by the
practical observation of the solar eclipse already referred to. At a
first approximation these equations degenerate into Newton's Law. At
a second approximation they account for the motion of the perihelion
of Mercury, which had hitherto baffled astronomers. All the laws
of mechanics are deducible from this law of World-Fabric curvature,
i.e. conservation of energy (which includes conservation of mass since
we re-define mass as energy) and conservation of momentum (re-defined
by a relativist). It must be noted that this law and the General
Theory show that the velocity of light is not absolutely constant,
but, like everything else, a light-pulse is affected by the Fabric
curvature in a gravitational field. In conclusion we will contrast
some conspicuous differences in the old world view of classical
mechanics and the new view presented by Einstein.
1. A three-dimensional ether medium with variously conceived properties
which communicated the supposed inherent attractive force in matter
in some unexplained way, and transmitted electromagnetic waves,
has been replaced by a four-dimensional external World-Fabric, the
geometrical character of which controls the motion of matter (energy)
and accounts for all mechanical laws.
2. After separating the observer's subjective share in definitions
from nature's share in the things defined, space, time, and force,
hitherto regarded as absolute, have been shown to be purely relative
and dependent on the observer's track. Mass has also proved to be
relative to velocity unless re-defined as energy. As classical
mechanics bases all definitions on space, time, and mass units,
the relativity of such defined quantities is now apparent.
3. Newton's laws of motion, his law of gravitation, and the laws of
conservation, hitherto regarded as unrelated, are now synthesised in
a basic law of mechanics.
Einstein has not disturbed the electric theory of matter, and both the
old and new physics have in common the "Principle of Least Action." We
obtain a glimpse of this principle in the unique tracks pursued by
freely moving bodies, which may be regarded as tracks of least effort,
force only being manifested as an expression of the Fabric's resentment
when bodies depart from these natural tracks. Einstein has approached
nearer to the truth in regard to the laws underlying nature, and,
as always, this means a simplification. His theory, which entails
a readjustment of such fundamental conceptions as space and time,
opens up fresh fields to scientific investigation and to philosophic
thought. It reveals a bridge uniting the domains of physics and
philosophy, and it heralds a new era in the history of science.
XV
THE NEW WORLD
A Universe in Which Geometry Takes the Place of Physics, and Curvature
That of Force
BY GEORGE FREDERICK HEMENS, M.C., B.SC., LONDON
It is familiar knowledge that the line, the surface and ordinary
Euclidean space are to be regarded as spaces of one, two and three
dimensions respectively and readers of this journal are aware
that a hypothetical space of four dimensions has been closely
investigated. The most convenient space to study is the surface or
two-space, since we can regard it as embedded in a three-space. If a
surface is curved it is generally impossible to draw a straight line
on it, for as we see clearly, the "straightest" line is changing its
direction at every point. To describe this property accurately it
is necessary to ascribe to each point a magnitude which expresses
what happens to the direction of a short line in the region when
displaced a short distance parallel to itself. This is called the
direction-defining magnitude. Different sets of values of this
magnitude relate to surfaces of different curvatures.
A second fundamental property has recently been pointed out. There
is inherent in every part of a space a measure of length peculiar
to that particular region and which in general varies from region
to region. To describe this variation accurately it is necessary to
ascribe to each point another magnitude called the length-defining
magnitude, which expresses the change from each point to the next of
the unit of length. These two magnitudes define the surface completely.
Similarly, a space of any number of dimensions is defined completely
by a similar pair of magnitudes. A space is the "field" of such
a magnitude-pair and the nature of these magnitudes defines the
dimensions of the space. The four-space usually described is the
Euclidean member of an infinity of four-spaces.
When we look into a mirror we see a space differing from ordinary
space in that right and left are interchanged and this is described
mathematically by saying that if we locate points as usual by
specifying three distances $X_1$, $X_2$, $X_3$ of the point from three
mutually perpendicular planes, then a point $X_1$, $X_2$, $X_3$, in
actual space corresponds with a point $X_1$, $X_2$, $ -X_3$ in the
mirrored space: in other words the mirrored space is derived from
the real space by multiplying the $X_3$ coordinates by $-1$. If we
were to multiply by $\sqrt{-1}$ instead of $-1$ we should derive a
different space; in this case, however, we have no mirror to show us
what it looks like. Such a space is said to have one negative dimension
and it has the peculiar property that in the figure derived from the
right triangle of ordinary space the square of the "hypotenuse" equals
the difference and not the sum of the squares of the other two sides,
so that the length of a line may sometimes have to be represented by
the square-root of a negative number, a "complex" number.
In considering what at first sight may appear to be fantastic
statements made by this theory, it must be borne in mind that all our
knowledge of the external universe comes through our sense-impressions,
and our most confident statements about external things are really
of the nature of inferences from these sense-impressions and, being
inferences, liable to be wrong. So that if the theory says that a
stone lying on the ground is not a simple three-dimensional object, and
that its substance is not the same as its substance a moment before,
the matter is one for due consideration and not immediate disbelief.
The idea that the universe extends in time as well as in space is not
new, and fiction-writers have familiarized us with wonderful machines
in which travellers journey in time and are present at various stages
of the world's history. This conception of the universe, to which the
name "space-time" is usually applied, is adopted by the new theory
and assigned the status of a physical reality.
THE WORLD GEOMETRY
The fundamental creed of the new theory is that the space-time universe
constitutes a true four-dimensional space of one negative dimension,
this dimension being time. The variations from point to point of
the direction-defining and length-defining magnitudes generate the
geometrical properties of curvature, etc., and these are cognised by
the human mind as physical phenomena: our sense-impressions are nothing
more nor less than perceptions of the geometry of a fourspace. So
instead of inferring from our sense-impressions the existence of
matter, motion and the like as we are accustomed to do, we should with
equal justice infer the existence of a geometrical fourspace. Thus it
becomes necessary to prepare a dictionary in which the familiar things
of our world are identified with those geometrical properties of the
four-space which really constitute them, and in so doing parts of
our geometrical knowledge assume the guise of new physical knowledge.
Through the fourspace our consciousness travels, cognising a changing
three-dimensional section of it as it goes and thus giving rise
to time. It becomes aware that the fourspace is pleated or folded
along lines all running roughly in the same direction, and possibly
because this is the easiest direction to follow, it travels along the
lines. The direction of this motion is the negative dimension. Thus
consciousness is always aware of the nearly constant forms of the
cross-sections of the pleats along which it travels. These unvarying
forms constitute matter: matter is the form of a section through
a uniform pleat of the fourspace--a three-dimensional aspect of a
four-dimensional curvature; so that in strict accuracy we should
say that a stone is the shape or form of a changing section of a
four-dimensional object, the complete object being a long fold in the
fourspace. The physical interpretation of this conservation of form
of the cross-section is that matter is conserved. It is thus seen that
the conscious mind, by following these pleats, has so determined time
that the law of the conservation of matter must hold. The mathematical
treatment of the subject makes it clear that practically all other
physical laws similarly follow as a direct result of this choice of
time. The type of order prevailing in the physical universe, the laws
of gravitation, heat, motion and the rest are not directly imposed
by some external power, but are apparently chosen by mind itself.
In the neighborhood of these pleats the fourspace is still curved,
but to a smaller degree. This we cognise as energy or as a field
of force. Thus energy is seen to be the same kind of thing as
matter and would therefore be expected to have weight. This was
experimentally demonstrated in 1919 when light was in effect actually
weighed. Conversely, matter consists of energy; and it is calculated
that one liter of water contains sufficient energy to develop a million
horsepower for about four years. It is now believed that the sun's
energy is derived from the disintegration of the matter of which it
is made.
The method of establishing these identifications will be clear from the
following: We already knew that matter is made up of electrons and that
radiant energy is electromagnetic and before the advent of this theory
it was regarded as certain that practically all observed physical
phenomena except gravitation were manifestations of the electromagnetic
field. The new theory has confirmed this belief. It is found that
the gravitational and electromagnetic conditions of the universe are
completely defined if to each point of space-time a gravitational and
an electric potential are ascribed. These are magnitudes of the same
nature as the direction-defining and length-defining magnitudes which
must necessarily be associated with every point of space-time if it
is a true "space," and they are therefore identified with these. By
performing ordinary mathematical operations on these magnitudes
statements of fact clothed in mathematical form are obtained, which are
to be interpreted on the one hand as physical laws and on the other
as geometrical properties of the fourspace. Nearly all our physical
laws are derivable mathematically in this way, so that an extensive
identification is effected which has been fruitful of results.
It has been mentioned that a slight curvature is sometimes cognised
as force and as this identification appeared originally as a postulate
its history is interesting.
THE GENESIS OF THE THEORY
An experiment by Michelson and Morley (1887), on which the whole theory
is based, made it appear that if a man measures the velocity at which
light passes him he will get the same result whether he is stationary,
rushing to meet the light, or moving in the same direction as the
light. The solution was provided by Einstein in 1905. He suggested that
since we know the results of these determinations ought not to agree,
something must have happened to the clocks and measuring-rods used in
measuring the velocity so that the standards of length and time were
not the same in the three cases, the alterations being exactly such as
to make the velocity of light constant. This solution is universally
accepted as true and is the fundamental postulate. Thus the length of
a stick and the rate at which time passes will change as the velocity
of the person observing these things changes. If a man measured the
length of an aeroplane going past him at 161,000 miles per second it
would measure only half the length observed when stationary. If the
aeroplane were going with the velocity of light, its length would
vanish though its breadth and height would be unaltered. Similarly,
if of two twin brothers one were continually moving with reference
to the other their ages would gradually diverge, for time would go at
different rates for the two. If one moved with the velocity of light,
time would stand still for him while for the other it would go on as
usual. To get actually younger it would be necessary to move quicker
than light which is believed to be impossible. The velocity of light
is assumed to be the greatest velocity occurring in nature.
Evidently then if the distance in space and the interval in time
separating two given events, such as the firing of a gun and the
bursting of the shell, are measured by two observers in uniform
relative motion, their estimates will not agree. Consider now the
simple problem of measuring the distance between two points on an
ordinary drawing-board. If we draw two perpendicular axes, we can
define this distance by specifying the lengths of the projections
on the two axes of the line joining the points. If we choose two
different axes the projections will not be the same but will define
the same length. Similarly, in a Euclidean four-space the distance
between two points will be defined by the projections on the four
axes, but if these axes be rotated slightly, the projections will be
different, but will define the same length. Now, returning to the two
observers just mentioned, it was noticed by Minkowski in 1908 that
if the space measurements between the two events are split into the
usual three components, and if the time measurements are multiplied
by $\sqrt{-1}$, the difference between the two sets of measurements
is exactly the same as would have occurred had these two events
been points in a Euclidean fourspace, and two different observations
made of their distance apart using two sets of axes inclined to each
other. The velocity of light is made equal to 1 in this calculation by
a suitable choice of units. This discovery threw a vivid light on the
problem of space-time, showing that it is probably a true four-space
of one negative dimension, a simple derivative of the much-discussed
and now familiar Euclidean four-space.
Although this discovery gave a tremendous impetus to the progress
of the theory, it is probable that it holds a deeper significance
not yet revealed. It is probably a statement of the "stuff" of which
the four-space is made, and perhaps also of how it is made; but the
problem remains unsolved.
It thus becomes plain that our two observers are merely looking at the
same thing from different viewpoints. Each has just as much right as
the other to regard himself as being at rest in ordinary space (this is
the postulate of the relativity of uniform motion) and to regard his
time direction as a straight line in the four-space. The difference
is merely that the two time axes are inclined to each other. If,
however, one were moving with an acceleration with reference to the
other his path in the four-space will appear curved to the other,
though he himself, since he regards it as his time axis, will still
assume it to be straight. If there is a body moving in what one
observer sees to be a straight line, the other will, of course,
in general see it as curved, and following the usual custom, since
this body, without apparent reason, deviates from the straight path,
will say there must be some force acting on it. Thus the curvature
of his time axis, due to his accelerated motion, makes it appear
that there is round him a field of force, which causes freely moving
bodies to deviate from the straight path. Now if space-time is itself
inherently curved it is not generally possible for any line in it to
be straight any more than it is possible for any line on the surface
of a sphere to be straight. Hence, all axes must be curved, and all
observers, whatever their states of motion, must experience fields of
force which are of the same nature as those due to motion only. The
extra force experienced when a lift begins to rise is an example of
force due to pure motion: gravitation is the similar force due to an
inherent curvature of the four-space, and it was the postulate that
these forces were similar that made possible Einstein's solution of
the general problem of gravitation.
THE TIME DIAGRAM
The correlation of time with its geometrical analogue is of absorbing
interest. Representing velocity by the common method of plotting
a curve showing positions at various times and marking distances
horizontally and times vertically, the velocity of light being 1,
$MM'$ and $NN'$ will both represent this velocity. Since this is
assumed to be the greatest velocity occurring in nature, all other
possible velocities are represented by lines falling within the upper
and lower V's. Now this diagram correctly represents two dimensions
of Minkowski's Euclidean four-space so, transmuting to real but flat
four-space by multiplying times by $\sqrt{-1}$, it is seen that there
is a region outside which no effect can be propagated from $O$ since
that would involve the existence of a velocity greater than that of
light. This region represents the future of $O$. Similarly, $O$ can
only be affected by events within the region derived from the
downward-opening V, which therefore represents the past of $O$.
The region between the two represents events which may be either
simultaneous with $O$ or not, according to the velocity of the
observer at $O$. Thus in this theory an event dictated by free-will,
could affect points in its "future" region, but not in any other,
which agrees with experience and shows that the theory is not
essentially "determinist." If "free-will" is really free, the future
is not yet determined, and the fourspace must be in some way formed
by the will as time progresses.
The trains of thought inspired by Einstein's postulates have already
carried us to a pinnacle of knowledge unprecedented in the history of
man. On every hand, as we look out upon the universe from our new and
lofty standpoint, unexpected and enthralling vistas open up before us,
and we find ourselves confronting nature with an insight such as no
man has ever before dared aspire to.
It is completely unthinkable that this theory can ever be swept
aside. Apart from experimental verifications which, in point of fact,
lend it the strongest support, no one could work through the theory
without feeling that here, in truth, the inner workings of the universe
were laid bare before him. The harmony with nature is far too complete
for any doubt to arise of its truth.
XVI
THE QUEST OF THE ABSOLUTE
Modern Developments in Theoretical Physics, and the Climax Supplied
by Einstein
BY DR. FRANCIS D. MURNAGHAN,
JOHNS HOPKINS UNIVERSITY
BALTIMORE
We shall discuss the more important aspects of the theory popularly
known as the "Einstein Theory of Gravitation" and shall try to show
clearly that this theory is a natural outcome of ideas long held by
physicists in general. These ideas are:
(a) The impossibility of "action at a distance;" in other words we
find an instinctive repugnance to admit that one body can affect
another, remote from it, instantaneously and without the existence
of an intervening medium.