Before tackling Einstein’s new law, it is as well to convince
ourselves, on logical grounds, that Newton’s law of gravitation cannot
be quite right.
Newton said that between any two particles of matter there is a force
which is proportional to the product of their masses and inversely
proportional to the square of their distance. That is to say, ignoring
for the present the question of mass, if there is a certain attraction
when the particles are a mile apart, there will be a quarter as much
attraction when they are two miles apart, a ninth as much when they
are three miles apart, and so on: the attraction diminishes much
faster than the distance increases. Now, of course, Newton, when he
spoke of the distance, meant the distance at a given time: He thought
there could be no ambiguity about time. But we have seen that this
was a mistake. What one observer judges to be the same moment on the
earth and the sun, another will judge to be two different moments.
“Distance at a given moment” is therefore a subjective conception,
which can hardly enter into a cosmic law. Of course, we could make
our law unambiguous by saying that we are going to estimate times as
they are estimated by Greenwich Observatory. But we can hardly believe
that the accidental circumstances of the earth deserve to be taken so
seriously. And the estimate of distance, also, will vary for different
observers. We cannot, therefore, allow that Newton’s form of the law of
gravitation can be quite correct, since it will give different results
according to which of many equally legitimate conventions we adopt.
This is as absurd as it would be if the question whether one man had
murdered another were to depend upon whether they were described by
their Christian names or their surnames. It is obvious that physical
laws must be the same whether distances are measured in miles or in
kilometers, and we are concerned with what is essentially only an
extension of the same principle.
Our measurements are conventional to an even greater extent than
is admitted by the special theory of relativity. Moreover, every
measurement is a physical process carried out with physical material;
the result is certainly an experimental datum, but may not be
susceptible of the simple interpretation which we ordinarily assign to
it. We are, therefore, not going to assume to begin with that we know
how to measure anything. We assume that there is a certain physical
quantity, called “interval,” which is a relation between two events
that are not widely separated; but we do not assume in advance that we
know how to measure it, beyond taking it for granted that it is given
by some generalization of the theorem of Pythagoras such as we spoke of
in the preceding chapter.
We do assume, however, that events have an _order_, and that this order
is four-dimensional. We assume, that is to say, that we know what we
mean by saying that a certain event is nearer to another than to a
third, so that before making accurate measurements we can speak of the
“neighborhood” of an event; and we assume that, in order to assign the
position of an event in space-time, four quantities (co-ordinates) are
necessary—_e.g._ in our former case of an explosion on an airship,
latitude, longitude, altitude and time. But we assume nothing about the
way in which these co-ordinates are assigned, except that neighboring
co-ordinates are assigned to neighboring events.
The way in which these numbers, called co-ordinates, are to be assigned
is neither wholly arbitrary nor a result of careful measurement—it
lies in an intermediate region. While you are making any continuous
journey, your co-ordinates must never alter by sudden jumps. In America
one finds that the houses between (say) Fourteenth Street and Fifteenth
Street are likely to have numbers between 1400 and 1500, while those
between Fifteenth Street and Sixteenth Street have numbers between
1500 and 1600, even if the 1400’s were not used up. This would not do
for our purposes, because there is a sudden jump when we pass from one
block to the next. Or again we might assign the time co-ordinate in the
following way: take the time that elapses between two successive births
of people called Smith; an event occurring between the births of the
3000th and the 3001st Smith known to history shall have a co-ordinate
lying between 3000 and 3001; the fractional part of its co-ordinate
shall be the fraction of a year that has elapsed since the birth of the
3000th Smith. (Obviously there could never be as much as a year between
two successive additions to the Smith family.) This way of assigning
the time co-ordinate is perfectly definite, but it is not admissible
for our purposes, because there will be sudden jumps between events
just before the birth of a Smith and events just after, so that in a
continuous journey your time co-ordinate will not change continuously.
It is assumed that, independently of measurement, we know what a
continuous journey is. And when your position in space-time changes
continuously, each of your four co-ordinates must change continuously.
One, two, or three of them may not change at all; but whatever change
does occur must be smooth, without sudden jumps. This explains what is
_not_ allowable in assigning co-ordinates.
To explain all the changes that are legitimate in your co-ordinates,
suppose you take a large piece of soft india-rubber. While it is in an
unstretched condition, measure little squares on it, each one-tenth
of an inch each way. Put in little tiny pins at the corners of the
squares. We can take as two of the co-ordinates of one of these pins
the number of pins passed in going to the right from a given pin
until we come just below the pin in question, and then the number of
pins we pass on the way up to this pin. In the figure, let =O= be the
pin we start from and =P= the pin to which we are going to assign
co-ordinates. =P= is in the fifth column and the third row, so its
co-ordinates in the plane of the india-rubber are to be 5 and 3.
[Illustration: Fig. 1.]
[Illustration: Fig. 2.]
Now take the india-rubber and stretch it and twist it as much as
you like. Let the pins now be in the shape they have in Fig. 2. The
divisions now no longer represent distances according to our usual
notions, but they will still do just as well as co-ordinates. We may
still take =P= as having the co-ordinates 5 and 3 in the plane of the
india-rubber; and we may still regard the india-rubber as being in a
plane, even if we have twisted it out of what we should ordinarily
call a plane. Such continuous distortions do not matter.
To take another illustration: instead of using a steel measuring rod to
fix our co-ordinates, let us use a live eel, which is wriggling all the
time. The distance from the tail to the head of the eel is to count as
one from the point of view of co-ordinates, whatever shape the creature
may be assuming at the moment. The eel is continuous, and its wriggles
are continuous, so it may be taken as our unit of distance in assigning
co-ordinates. Beyond the requirement of continuity, the method of
assigning co-ordinates is purely conventional, and therefore a live eel
is just as good as a steel rod.
We are apt to think that, for really careful measurements, it is better
to use a steel rod than a live eel. This is a mistake: not because
the eel tells us what the steel rod was thought to tell, but because
the steel rod really tells no more than the eel obviously does. The
point is, not that eels are really rigid, but that steel rods really
wriggle. To an observer in just one possible state of motion, the eel
would appear rigid, while the steel rod would seem to wriggle just
as the eel does to us. For everybody moving differently both from
this observer and ourselves, both the eel and the rod would seem to
wriggle. And there is no saying that one observer is right and another
wrong. In such matters, what is seen does not belong solely to the
physical process observed, but also to the standpoint of the observer.
Measurements of distances and times do not directly reveal properties
of the things measured, but relations of the things to the measurer.
What observation can tell us about the physical world is therefore more
abstract than we have hitherto believed.
It is important to realize that geometry, as taught in schools since
Greek times, ceases to exist as a separate science, and becomes merged
in physics. The two fundamental notions in elementary geometry were
the straight line and the circle. What appears to you as a straight
road, whose parts all exist now, may appear to another observer to
be like the flight of a rocket, some kind of curve whose parts come
into existence successively. The circle depends upon measurement of
distances, since it consists of all the points at a given distance
from its center. And measurement of distances, as we have seen, is
a subjective affair, depending upon the way in which the observer
is moving. The failure of the circle to have objective validity was
demonstrated by the Michelson-Morley experiment, and is thus, in a
sense, the starting point of the whole theory of relativity. Rigid
bodies, which we need for measurement, are only rigid for certain
observers; for others, they will be constantly changing all their
dimensions. It is only our obstinately earth-bound imagination that
makes us suppose a geometry separate from physics to be possible.
That is why we do not trouble to give physical significance to our
co-ordinates from the start. Formerly, the co-ordinates used in physics
were supposed to be carefully measured distances; now we realize
that this care at the start is thrown away. It is at a later stage
that care is required. Our co-ordinates now are hardly more than a
systematic way of cataloguing events. But mathematics provides, in
the method of tensors, such an immensely powerful technique that we
can use co-ordinates assigned in this apparently careless way just
as effectively as if we had applied the whole apparatus of minutely
accurate measurement in arriving at them. The advantage of being
haphazard at the start is that we avoid making surreptitious physical
assumptions, which we can hardly help making, if we suppose that our
co-ordinates have initially some particular physical significance.
We assume that, if two events are close together (but not necessarily
otherwise), there is an interval between them which can be calculated
from the differences between their co-ordinates by some such formula
as we considered in the preceding chapter. That is to say, we take the
squares and products of the differences of co-ordinates, we multiply
them by suitable amounts (which in general will vary from place to
place), and we add the results together. The sum obtained is the
square of the interval. We do not assume in advance that we know the
amounts by which the squares and products must be multiplied; this
is going to be discovered by observing physical phenomena. We know,
however, certain things. We know that the old Newtonian physics is
very nearly accurate when our co-ordinates have been chosen in a
certain way. We know that the special theory of relativity is still
more nearly accurate for suitable co-ordinates. From such facts we can
infer certain things about our new co-ordinates, which, in a logical
deduction, appear as postulates of the new theory.
As such postulates we take:
1. That every body travels in a geodesic in
space-time, except in so far as electromagnetic
forces act upon it.
2. That a light ray travels so that the interval
between two parts of it is zero.
3. That at a great distance from gravitating matter,
we can transform our co-ordinates by mathematical
manipulation so that the interval shall be what it
is in the special theory of relativity; and that
this is approximately true wherever gravitation is
not very powerful.
Each of these postulates requires some explanation.
We saw that a geodesic on a surface is the shortest line that can be
drawn on the surface from one point to another; for example, on the
earth the geodesics are great circles. When we come to space-time,
the mathematics is the same, but the verbal explanations have to be
rather different. In the general theory of relativity, it is only
neighboring events that have a definite interval, independently of
the route by which we travel from one to the other. The interval
between distant events depends upon the route pursued, and has to be
calculated by dividing the route into a number of little bits and
adding up the intervals for the various little bits. If the interval
is space-like, a body cannot travel from one event to the other;
therefore when we are considering the way bodies move, we are confined
to time-like intervals. The interval between neighboring events, when
it is time-like, will appear as the time between them for an observer
who travels from the one event to the other. And so the whole interval
between two events will be judged by a person who travels from one to
the other to be what his clocks show to be the time that he has taken
on the journey. For some routes this time will be longer, for others
shorter; the more slowly the man travels, the longer he will think he
has been on the journey. This must not be taken as a platitude. I am
not saying that if you travel from London to Edinburgh you will take
longer if you travel more slowly. I am saying something much more odd.
I am saying that if you leave London at 10 A.M. and arrive in Edinburgh
at 6.30 P.M. Greenwich time, the more slowly you travel the longer
you will take—if the time is judged by your watch. This is a very
different statement. From the point of view of a person on the earth,
your journey takes eight and a half hours. But if you had been a ray
of light traveling round the solar system, starting from London at 10
A.M., reflected from Jupiter to Saturn, and so on, until at last you
were reflected back to Edinburgh and arrived there at 6.30 P.M., you
would judge that the journey had taken you exactly no time. And if you
had gone by any circuitous route, which enabled you to arrive in time
by traveling fast, the longer your route the less time you would judge
that you had taken; the diminution of time would be continual as your
speed approached that of light. Now I say that when a body travels, if
it is left to itself, it chooses the route which makes the time between
two stages of the journey as long as possible; if it had traveled from
one event to another by any other route, the time, as measured by its
own clocks, would have been shorter. This is a way of saying that
bodies left to themselves do their journeys as slowly as they can; it
is a sort of law of cosmic laziness. Its mathematical expression is
that they travel in geodesics, in which the total interval between any
two events on the journey is _greater_ than by any alternative route.
(The fact that it is greater, not less, is due to the fact that the
sort of interval we are considering is more analogous to time than to
distance.) For example, if a person could leave the earth and travel
about for a time and then return, the time between his departure and
return would be less by his clocks than by those on the earth: the
earth, in its journey round the sun, chooses the route which makes
the time of any bit of its course by its clocks longer than the time
as judged by clocks which move by a different route. This is what is
meant by saying that bodies left to themselves move in geodesics in
space-time.
We assume that the body considered is not acted upon by electromagnetic
forces. We are concerned at present with the law of gravitation, not
with the effects of electromagnetism. These effects have been brought
into the framework of the general theory of relativity by Weyl,[5] but
for the present we will ignore his work. The planets, in any case,
are not subject, as wholes, to appreciable electromagnetic forces; it
is only gravitation that has to be considered in accounting for their
motions, with which we are concerned in this chapter.
[5] See his _Space, Time, Matter_, Methuen, 1922.
Our second postulate, that a light ray travels so that the interval
between two parts of it is zero, has the advantage that it does not
have to be stated only for _small_ distances. If each little bit of
interval is zero, the sum of them all is zero, and so even distant
parts of the same light ray have a zero interval. The course of a light
ray is also a geodesic according to the definition. Thus we now have
two empirical ways of discovering what are the geodesics in space-time,
namely light rays and bodies moving freely. Among freely-moving
bodies are included all which are not subject to constraints or to
electromagnetic forces, that is to say, the sun, stars, planets and
satellites, and also falling bodies on the earth, at least when they
are falling in a vacuum. When you are standing on the earth, you are
subject to electromagnetic forces: the electrons and protons in the
neighborhood of your feet exert a repulsion on your feet which is just
enough to overcome the earth’s gravitation. This is what prevents you
from falling through the earth, which, solid as it looks, is mostly
empty space.
The third postulate, which relates the general to the special theory,
is very useful. It is not necessary for the application of the special
theory to a limited region that there should be no gravitation in the
region; it is enough if the intensity of gravitation is practically the
same throughout the region. This enables us to apply the special theory
within any small region. How small it will have to be, depends upon the
neighborhood. On the surface of the earth, it would have to be small
enough for the curvature of the earth to be negligible. In the spaces
between the planets, it need only be small enough for the attraction
of the sun and the planets to be sensibly constant throughout the
region. In the spaces between the stars it might be enormous—say half
the distance from one star to the next—without introducing measurable
inaccuracies.
At a great distance from gravitating matter, we can so choose our
co-ordinates as to obtain a Euclidean space; this is really only
another way of saying that the special theory of relativity applies. In
the neighborhood of matter, although we can make our space Euclidean
in any small region, we cannot do so throughout any region within
which gravitation varies sensibly—at least, if we do, we shall have
to abandon the view that bodies move in geodesics. In the neighborhood
of a piece of matter, there is, as it were, a hill in space-time;
this hill grows steeper and steeper as it gets nearer the top, like
the neck of a champagne bottle. It ends in a sheer precipice. Now by
the law of cosmic laziness which we mentioned earlier, a body coming
into the neighborhood of the hill will not attempt to go straight
over the top, but will go round. This is the essence of Einstein’s
view of gravitation. What a body does, it does because of the nature
of space-time in its own neighborhood, not because of some mysterious
force emanating from a distant body.
An analogy will serve to make the point clear. Suppose that on a dark
night a number of men with lanterns were walking in various directions
across a huge plain, and suppose that in one part of the plain there
was a hill with a flaring beacon on the top. Our hill is to be such
as we have described, growing steeper as it goes up, and ending in a
precipice. I shall suppose that there are villages dotted about the
plain, and the men with lanterns are walking to and from these various
villages. Paths have been made showing the easiest way from any one
village to any other. These paths will all be more or less curved, to
avoid going too far up the hill; they will be more sharply curved when
they pass near the top of the hill than when they keep some way off
from it. Now suppose that you are observing all this, as best you can,
from a place high up in a balloon, so that you cannot see the ground,
but only the lanterns and the beacon. You will not know that there is a
hill, or that the beacon is at the top of it. You will see that people
turn out of the straight course when they approach the beacon, and
that the nearer they come the more they turn aside. You will naturally
attribute this to an effect of the beacon; you may think that it is
very hot and people are afraid of getting burnt. But if you wait for
daylight you will see the hill, and you will find that the beacon
merely marks the top of the hill and does not influence the people with
lanterns in any way.
Now in this analogy the beacon corresponds to the sun, the people with
lanterns correspond to the planets and comets, the paths correspond
to their orbits, and the coming of daylight corresponds to the coming
of Einstein. Einstein says that the sun is at the top of a hill, only
the hill is in space-time, not in space. (I advise the reader not to
try to picture this, because it is impossible.) Each body, at each
moment, adopts the easiest course open to it, but owing to the hill the
easiest course is not a straight line. Each little bit of matter is at
the top of its own little hill, like the cock on his own dung-heap.
What we call a big bit of matter is a bit which is at the top of a big
hill. The hill is what we know about; the bit of matter at the top is
assumed for convenience. Perhaps there is really no need to assume it,
and we could do with the hill alone, for we can never get to the top of
any one else’s hill, any more than the pugnacious cock can fight the
peculiarly irritating bird that he sees in the looking glass.
I have given only a qualitative description of Einstein’s law of
gravitation; to give its exact quantitative formulation is impossible
without more mathematics than I am permitting myself. The most
interesting point about it is that it makes the law no longer the
result of action at a distance: the sun exerts no force on the planets
whatever. Just as geometry has become physics, so, in a sense, physics
has become geometry. The law of gravitation has become the geometrical
law that every body pursues the easiest course from place to place, but
this course is affected by the hills and valleys that are encountered
on the road.