The special theory of relativity, which we have been considering
hitherto, solved completely a certain definite problem: to account for
the experimental fact that, when two bodies are in uniform relative
motion, all the laws of physics, both those of ordinary dynamics and
those connected with electricity and magnetism, are exactly the same
for the two bodies. “Uniform” motion, here, means motion in a straight
line with constant velocity. But although one problem was solved by
the special theory, another was immediately suggested: what if the
motion of the two bodies is not uniform? Suppose, for instance, that
one is the earth while the other is a falling stone. The stone has
an accelerated motion: it is continually falling faster and faster.
Nothing in the special theory enables us to say that the laws of
physical phenomena will be the same for an observer on the stone as for
one on the earth. This is particularly awkward, as the earth itself
is, in an extended sense, a falling body: It has at every moment
an acceleration[4] towards the sun, which makes it go round the sun
instead of moving in a straight line. As our knowledge of physics is
derived from experiments on the earth, we cannot rest satisfied with
a theory in which the observer is supposed to have no acceleration.
The general theory of relativity removes this restriction, and allows
the observer to be moving in any way, straight or crooked, uniformly
or with an acceleration. In the course of removing the restriction,
Einstein was led to his new law of gravitation, which we shall consider
presently. The work was extraordinarily difficult, and occupied him for
ten years. The special theory dates from 1905, the general theory from
1915.
[4] This does not mean that its velocity is increasing, but that it
is changing its direction. The only sort of motion which is called
“unaccelerated” is motion with uniform velocity _in a straight line_.
It is obvious from experiences with which we are all familiar that an
accelerated motion is much more difficult to deal with than a uniform
one. When you are in a train which is traveling steadily, the motion
is not noticeable so long as you do not look out of the window; but
when the brakes are applied suddenly you are precipitated forwards,
and you become aware that something is happening without having to
notice anything outside the train. Similarly in a lift everything
seems ordinary while it is moving steadily, but at starting and
stopping, when its motion is accelerated, you have odd sensations
in the pit of the stomach. (We call a motion “accelerated” when it
is getting slower as well as when it is getting quicker; when it is
getting slower the acceleration is negative.) The same thing applies
to dropping a weight in the cabin of a ship. So long as the ship is
moving uniformly, the weight will behave, relatively to the cabin,
just as if the ship were at rest: if it starts from the middle of
the ceiling, it will hit the middle of the floor. But if there is an
acceleration everything is changed. If the boat is increasing its
speed very rapidly, the weight will seem to an observer in the cabin
to fall in a curve directed towards the stern; if the speed is being
rapidly diminished, the curve will be directed towards the bow. All
these facts are familiar, and they led Galileo and Newton to regard an
accelerated motion as something radically different, in its own nature,
from a uniform motion. But this distinction could only be maintained by
regarding motion as absolute, not relative. If all motion is relative,
the earth is accelerated relatively to the lift just as truly as the
lift relatively to the earth. Yet the people on the ground have no
sensations in the pits of their stomachs when the lift starts to go
up. This illustrates the difficulty of our problem. In fact, though
few physicists in modern times have believed in absolute motion, the
technique of mathematical physics still embodied Newton’s belief in it,
and a revolution in method was required to obtain a technique free from
this assumption. This revolution was accomplished in Einstein’s general
theory of relativity.
It is somewhat optional where we begin in explaining the new ideas
which Einstein introduced, but perhaps we shall do best by taking the
conception of “interval.” This conception, as it appears in the special
theory of relativity, is already a generalization of the traditional
notion of distance in space and time; but it is necessary to generalize
it still further. However, it is necessary first to explain a certain
amount of history, and for this purpose we must go back as far as
Pythagoras.
Pythagoras, like many of the greatest characters in history, perhaps
never existed: he is a semi-mythical character, who combined
mathematics and priestcraft in uncertain proportions. I shall, however,
assume that he existed, and that he discovered the theorem attributed
to him. He was roughly a contemporary of Confucius and Buddha; he
founded a religious sect, which thought it wicked to eat beans,
and a school of mathematicians, who took a particular interest in
right-angled triangles. The theorem of Pythagoras (the forty-seventh
proposition of Euclid) states that the sum of the squares on the two
shorter sides of a right-angled triangle is equal to the square on
the side opposite the right angle. No proposition in the whole of
mathematics has had such a distinguished history. We all learned to
“prove” it in youth. It is true that the “proof” proved nothing, and
that the only way to prove it is by experiment. It is also the case
that the proposition is not _quite_ true—it is only approximately
true. But everything in geometry, and subsequently in physics, has been
derived from it by successive generalizations. The latest of these
generalizations is the general theory of relativity.
The theorem of Pythagoras was itself, in all probability, a
generalization of an Egyptian rule of thumb. In Egypt, it had been
known for ages that a triangle whose sides are 3, 4, and 5 units of
length is a right-angled triangle; the Egyptians used this knowledge
practically in measuring their fields. Now if the sides of a triangle
are 3, 4, and 5 inches, the squares on these sides will contain
respectively 9, 16, and 25 square inches; and 9 and 16 added together
make 25. Three times three is written “3²”; four times four, “4²”; five
times five, “5².” So that we have
3² + 4² = 5².
It is supposed that Pythagoras noticed this fact, after he had learned
from the Egyptians that a triangle whose sides are 3, 4 and 5 has a
right angle. He found that this could be generalized, and so arrived
at his famous theorem: In a right-angled triangle, the square on the
side opposite the right angle is equal to the sum of the squares on the
other two sides.
[Illustration]
Similarly in three dimensions: if you take a right-angled solid block,
the square on the diagonal (the dotted line in the figure) is equal to
the sum of the squares on the three sides.
This is as far as the ancients got in this matter.
[Illustration]
The next step of importance is due to Descartes, who made the theorem
of Pythagoras the basis of his method of analytical geometry. Suppose
you wish to map out systematically all the places on a plain—we will
suppose it small enough to make it possible to ignore the fact that
the earth is round. We will suppose that you live in the middle of the
plain. One of the simplest ways of describing the position of a place
is to say: starting from my house, go first such and such a distance
east, then such and such a distance north (or it may be west in the
first case, and south in the second). This tells you exactly where
the place is. In the rectangular cities of America, it is the natural
method to adopt: in New York you will be told to go so many blocks east
(or west) and then so many blocks north (or south). The distance you
have to go east is called _x_, and the distance you have to go north
is called _y_. (If you have to go west, _x_ is negative; if you have
to go south, _y_ is negative.) Let =O= be your starting point (the
“origin”); let =OM= be the distance you go east, and =MP= the distance
you go north. How far are you from home in a direct line when you reach
=P=? The theorem of Pythagoras gives the answer. The square on =OP= is
the sum of the squares on =OM= and =MP=. If =OM= is four miles, and
=MP= is three miles, =OP= is 5 miles. If =OM= is 12 miles and =MP= is 5
miles, =OP= is 13 miles, because 12² + 5² = 13². So that if you adopt
Descartes’ method of mapping, the theorem of Pythagoras is essential in
giving you the distance from place to place. In three dimensions the
thing is exactly analogous. Suppose that, instead of wanting merely
to fix positions on the plain, you want to fix stations for captive
balloons above it, you will then have to add a third quantity, the
height at which the balloon is to be. If you call the height _z_, and
if _r_ is the direct distance from =O= to the balloon, you will have
_r_² = _x_² + _y_² + _z_²,
and from this you can calculate _r_ when you know _x_, _y_, and _z_.
For example, if you can get to the balloon by going 12 miles east, 4
miles north, and then 3 miles up, your distance from the balloon in a
straight line is 13 miles, because 12 × 12 = 144, 4 × 4 = 16, 3 × 3 =
9, 144 + 16 + 9 = 169 = 13 × 13.
But now suppose that, instead of taking a small piece of the earth’s
surface which can be regarded as flat, you consider making a map of
the world. An accurate map of the world on flat paper is impossible.
A globe can be accurate, in the sense that everything is produced
to scale, but a flat map cannot be. I am not talking of practical
difficulties, I am talking of a theoretical impossibility. For example:
the northern halves of the meridian of Greenwich and the ninetieth
meridian of west longitude, together with the piece of the equator
between them, make a triangle whose sides are all equal and whose
angles are all right angles. On a flat surface, a triangle of that sort
would be impossible. On the other hand, it is possible to make a square
on a flat surface, but on a sphere it is impossible. Suppose you try on
the earth: walk 100 miles west, then 100 miles north, then 100 miles
east, then 100 miles south. You might think this would make a square,
but it wouldn’t, because you would not at the end have come back to
your starting point. If you have time, you may convince yourself of
this by experiment. If not, you can easily see that it must be so. When
you are nearer the pole, 100 miles takes you through more longitude
than when you are nearer the equator, so that in doing your 100 miles
east (if you are in the northern hemisphere) you get to a point further
east than that from which you started. As you walk due south after
this, you remain further east than your starting point, and end up at a
different place from that in which you began. Suppose, to take another
illustration, that you start on the equator 4,000 miles east of the
Greenwich meridian; you travel till you reach the meridian, then you
travel northwards along it for 4,000 miles, through Greenwich and up
to the neighborhood of the Shetland Islands; then you travel eastward
for 4,000 miles, and then 4,000 miles south. This will take you to the
equator at a point 4,000 miles further east than the point from which
you started.
In a sense, what we have just been saying is not quite fair, because,
except on the equator, traveling due east is not the shortest route
from a place to another place due east of it. A ship traveling (say)
from New York to Lisbon, which is nearly due east, will start by going
a certain distance northward. It will sail on a “great circle,” that
is to say, a circle whose centre is the centre of the earth. This
is the nearest approach to a straight line that can be drawn on the
surface of the earth. Meridians of longitude are great circles, and so
is the equator, but the other parallels of latitude are not. We ought,
therefore, to have supposed that, when you reach the Shetland Islands,
you travel 4,000 miles, not due east, but along a great circle which
lands you at a point due east of the Shetland Islands. This, however,
only reinforces our conclusion: you will end at a point even further
east of your starting point than before.
What are the differences between the geometry on a sphere and the
geometry on a plane? If you make a triangle on the earth, whose sides
are great circles, you will not find that the angles of the triangle
add up to two right angles: they will add up to rather more. The amount
by which they exceed two right angles is proportional to the size of
the triangle. On a small triangle such as you could make with strings
on your lawn, or even on a triangle formed by three ships which can
just see each other, the angles will add up to so little more than two
right angles that you will not be able to detect the difference. But
if you take the triangle made by the equator, the Greenwich meridian,
and the ninetieth meridian, the angles add up to _three_ right angles.
And you can get triangles in which the angles add up to anything up to
six right angles. All this you could discover by measurements on the
surface of the earth, without having to take account of anything in the
rest of space.
The theorem of Pythagoras also will fail for distances on a sphere.
From the point of view of a traveler bound to the earth, the distance
between two places is their great circle distance, that is to say, the
shortest journey that a man can make without leaving the surface of
the earth. Now suppose you take three bits of great circles which make
a triangle, and suppose one of them is at right angles to another—to
be definite, let one be the equator and one a bit of the meridian of
Greenwich going northward from the equator. Suppose you go 3,000 miles
along the equator, and then 4,000 miles due north; how far will you
be from your starting point, estimating the distance along a great
circle? If you were on a plane, your distance would be 5,000 miles,
as we saw before. In fact, however, your great circle distance will be
considerably less than this. In a right-angled triangle on a sphere,
the square on the side opposite the right angle is less than the sum of
the squares on the other two sides.
These differences between the geometry on a sphere and the geometry on
a plane are intrinsic differences; that is to say, they enable you to
find out whether the surface on which you live is like a plane or like
a sphere, without requiring that you should take account of anything
outside the surface. Such considerations led to the next step of
importance in our subject, which was taken by Gauss, who flourished a
hundred years ago. He studied the theory of surfaces, and showed how to
develop it by means of measurements on the surfaces themselves, without
going outside them. In order to fix the position of a point in space,
we need three measurements; but in order to fix the position of a point
on a surface we need only two—for example, a point on the earth’s
surface is fixed when we know its latitude and longitude.
Now Gauss found that, whatever system of measurement you adopt,
and whatever the nature of the surface, there is always a way of
calculating the distance between two not very distant points of the
surface, when you know the quantities which fix their positions.
The formula for the distance is a generalization of the formula of
Pythagoras: it tells you the square of the distance in terms of the
squares of the differences between the measure quantities which fix
the points, and also the product of these two quantities. When you
know this formula, you can discover all the intrinsic properties of
the surface, that is to say, all those which do not depend upon its
relations to points outside the surface. You can discover, for example,
whether the angles of a triangle add up to two right angles, or more,
or less, or more in some cases and less in others.
But when we speak of a “triangle,” we must explain what we mean,
because on most surfaces there are no straight lines. On a sphere, we
shall replace straight lines by great circles, which are the nearest
possible approach to straight lines. In general, we shall take,
instead of straight lines, the lines that give the shortest route on
the surface from place to place. Such lines are called “geodesics.”
On the earth, the geodesics are great circles. In general, they are
the shortest way of traveling from point to point if you are unable
to leave the surface. They take the place of straight lines in the
intrinsic geometry of a surface. When we inquire whether the angles of
a triangle add up to two right angles or not, we mean to speak of a
triangle whose sides are geodesics. And when we speak of the distance
between two points, we mean the distance along a geodesic.
The next step in our generalizing process is rather difficult: it is
the transition to non-Euclidean geometry. We live in a world in which
space has three dimensions, and our empirical knowledge of space is
based upon measurement of small distances and of angles. (When I speak
of small distances, I mean distances that are small compared to those
in astronomy; all distances on the earth are small in this sense.) It
was formerly thought that we could be sure _à priori_ that space is
Euclidean—for instance, that the angles of a triangle add up to two
right angles. But it came to be recognized that we could not prove this
by reasoning; if it was to be known, it must be known as the result
of measurements. Before Einstein, it was thought that measurements
confirm Euclidean geometry within the limits of exactitude attainable;
now this is no longer thought. It is still true that we can, by what
may be called a natural artifice, cause Euclidean geometry to _seem_
true throughout a small region, such as the earth; but in explaining
gravitation Einstein is led to the view that over large regions where
there is matter we cannot regard space as Euclidean. The reasons for
this will concern us later. What concerns us now is the way in which
non-Euclidean geometry results from a generalization of the work of
Gauss.
There is no reason why we should not have the same circumstances in
three-dimensional space as we have, for example, on the surface of a
sphere. It might happen that the angles of a triangle would always
add up to more than two right angles, and that the excess would be
proportional to the size of the triangle. It might happen that the
distance between two points would be given by a formula analogous
to what we have on the surface of a sphere, but involving three
quantities instead of two. Whether this does happen or not, can only
be discovered by actual measurements. There are an infinite number of
such possibilities.
This line of argument was developed by Riemann, in his dissertation
“On the hypotheses which underlie geometry” (1854), which applied
Gauss’s work on surfaces to different kinds of three-dimensional
spaces. He showed that all the essential characteristics of a kind
of space could be deduced from the formula for small distances. He
assumed that, from the small distances in three given directions
which would together carry you from one point to another not far from
it, the distances between the two points could be calculated. For
instance, if you know that you can get from one point to another by
first moving a certain distance east, then a certain distance north,
and finally a certain distance straight up in the air, you are to be
able to calculate the distance from the one point to the other. And
the rule for the calculation is to be an extension of the theorem of
Pythagoras, in the sense that you arrive at the square of the required
distance by adding together multiples of the squares of the component
distances, together possibly with multiples of their products. From
certain characteristics in the formula, you can tell what sort of
space you have to deal with. These characteristics do not depend upon
the particular method you have adopted for determining the positions of
points.
In order to arrive at what we want for the theory of relativity, we
now have one more generalization to make: we have to substitute the
“interval” between events for the distance between points. This takes
us to space-time. We have already seen that, in the special theory
of relativity, the square of the interval is found by subtracting
the square of the distance between the events from the square of the
distance that light would travel in the time between them. In the
general theory, we do not assume this special form of interval, except
at a great distance from matter. Elsewhere, we assume to begin with a
general form, like that which Riemann used for distances. Moreover,
like Riemann, Einstein only assumes his formula for _neighboring_
events, that is to say, events which have only a small interval
between them. What goes beyond these initial assumptions depends upon
observation of the actual motion of bodies, in ways which we shall
explain in later chapters.
We may now sum up and re-state the process we have been describing.
In three dimensions, the position of a point relatively to a fixed
point (the “origin”) can be determined by assigning three quantities
(“co-ordinates”). For example, the position of a balloon relatively to
your house is fixed if you know that you will reach it by going first
a given distance due east, then another given distance due north,
then a third given distance straight up. When, as in this case, the
three co-ordinates are three distances all at right angles to each
other, which, taken successively, transport you from the origin to the
point in question, the square of the direct distance to the point in
question is got by adding up the squares of the three co-ordinates. In
all cases, whether in Euclidean or in non-Euclidean spaces, it is got
by adding multiples of the squares and products of the co-ordinates
according to an assignable rule. The co-ordinates may be any quantities
which fix the position of a point, provided that neighboring points
must have neighboring quantities for their co-ordinates. In the general
theory of relativity, we add a fourth co-ordinate to give the time, and
our formula gives “interval” instead of spatial distance; moreover we
assume the accuracy of our formula for small distances only. We assume
further that, at great distances from matter, the formula approximates
more and more closely to the formula for interval which is used in the
special theory.
We are now at last in a position to tackle Einstein’s theory of
gravitation.