The reasons for accepting Einstein’s law of gravitation rather than
Newton’s are partly empirical, partly logical. We will begin with the
former.
Einstein’s law of gravitation gives very nearly the same results
as Newton’s, when applied to the calculation of the orbits of the
planets and their satellites. If it did not, it could not be true,
since the consequences deduced from Newton’s law have been found to be
almost exactly verified by observation. When, in 1915, Einstein first
published his new law, there was only one empirical fact to which he
could point to show that his theory was better than Newton’s. This was
what is called the “motion of the perihelion of Mercury.”
The planet Mercury, like the other planets, moves round the sun in
an ellipse, with the sun in one of the foci. At some points of its
orbit it is nearer to the sun than at other points. The point where
it is nearest to the sun is called its “perihelion.” Now it was found
by observation that, from one occasion when Mercury is nearest to the
sun until the next, Mercury does not go exactly once round the sun,
but a little bit more. The discrepancy is very small; it amounts to
an angle of forty-two seconds in a century. That is to say, in each
year the planet has to move rather less than half a second of angle
after it has finished a complete revolution from the last perihelion
before it reaches the next perihelion. This very minute discrepancy
from Newtonian theory had puzzled astronomers. There was a calculated
effect due to perturbations caused by the other planets, but this small
discrepancy was the residue after allowing for these perturbations.
Einstein’s theory accounted for this residue, as well as for its
absence in the case of the other planets. (In them it exists, but is
too small to be observed.) This was, at first, his only empirical
advantage over Newton.
His second success was more sensational. According to orthodox
opinion, light in a vacuum ought always to travel in straight lines.
Not being composed of material particles, it ought to be unaffected
by gravitation. However, it was possible, without any serious breach
with old ideas, to admit that, in passing near the sun, light might be
deflected out of the straight path as much as if it were composed of
material particles. Einstein, however, maintained, as a deduction from
his law of gravitation, that light would be deflected twice as much as
this. That is to say, if the light of a star passed very near the sun,
Einstein maintained that the ray from the star would be turned through
an angle of just under one and three-quarters seconds. His opponents
were willing to concede half of this amount. Now it is not every day
that a star almost in line with the sun can be seen. This is only
possible during a total eclipse, and not always then, because there may
be no bright stars in the right position. Eddington points out that,
from this point of view, the best day of the year is May 29, because
then there are a number of bright stars close to the sun. It happened
by incredible good fortune that there was a total eclipse of the sun
on May 29, 1919—the first year after the armistice. Two British
expeditions photographed the stars near the sun during the eclipse,
and the results confirmed Einstein’s prediction. Some astronomers
who remained doubtful whether sufficient precautions had been taken
to insure accuracy were convinced when their own observations in a
subsequent eclipse gave exactly the same result. Einstein’s estimate of
the amount of the deflection of light by gravitation is therefore now
universally accepted.
The third experimental test is on the whole favorable to Einstein,
though the quantities concerned are so small that it is only just
possible to measure them, and the result is therefore not decisive. But
successive investigations have made it more and more probable that the
small effect predicted by Einstein really occurs. Before explaining the
effect in question, a few preliminary explanations are necessary. The
spectrum of an element consists of certain lines of various shades of
light, separated by a prism, and emitted by the element when it glows.
They are the same (to a very close approximation) whether the element
is in the earth or the sun or a star. Each line is of some definite
shade of color, with some definite wave length. Longer wave lengths are
towards the red end of the spectrum, shorter ones towards the violet
end. When the source of light is moving towards you, the apparent wave
lengths grow shorter, just as waves at sea come quicker when you are
traveling against the wind. When the source of light is moving away
from you, the apparent wave lengths grow longer, for the same reason.
This enables us to know whether the stars are moving towards us or away
from us. If they are moving towards us, all the lines in the spectrum
of an element are moved a little toward violet; if away from us, toward
red. You may notice the analogous effect in sound any day. If you are
in a station and an express comes through whistling, the note of the
whistle seems much more shrill while the train is approaching you than
when it has passed. Probably many people think the note has “really”
changed, but in fact the change in what you hear is only due to the
fact that the train was first approaching and then receding. To people
in the train, there was no change of note. This is _not_ the effect
with which Einstein is concerned. The distance of the sun from the
earth does not change much; for our present purposes, we may regard
it as constant. Einstein deduces from his law of gravitation that
any periodic process which takes place in an atom in the sun (whose
gravitation is very intense) must, as measured by our clocks, take
place at a slightly slower rate than it would in a similar atom on the
earth. The “interval” involved will be the same in the sun and on the
earth, but the same interval in different regions does not correspond
to exactly the same time; this is due to the “hilly” character of
space-time which constitutes gravitation. Consequently any given line
in the spectrum ought, when the light comes from the sun, to seem to
us a little nearer the red end of the spectrum than if the light came
from a source on the earth. The effect to be expected is very small—so
small that there is still some slight uncertainty as to whether it
exists or not. But it now seems highly probable that it exists.
No other measurable differences between the consequences of Einstein’s
law and those of Newton’s have hitherto been discovered. But the above
experimental tests are quite sufficient to convince astronomers that,
where Newton and Einstein differ as to the motions of the heavenly
bodies, it is Einstein’s law that gives the right results. Even if
the empirical grounds in favor of Einstein stood alone, they would be
conclusive. Whether his law represents the exact truth or not, it is
certainly more nearly exact than Newton’s, though the inaccuracies in
Newton’s were all exceedingly minute.
But the considerations which originally led Einstein to his law were
not of this detailed kind. Even the consequence about the perihelion of
Mercury, which could be verified at once from previous observations,
could only be deduced after the theory was complete, and could not
form any part of the original grounds for inventing such a theory.
These grounds were of a more abstract logical character. I do not
mean that they were not based upon observed facts, and I do not mean
that they were _à priori_ fantasies such as philosophers indulged in
formerly. What I mean is that they were derived from certain general
characteristics of physical experience, which showed that Newton _must_
be wrong and that something like Einstein’s law _must_ be substituted.
The arguments in favor of the relativity of motion are, as we saw in
earlier chapters, quite conclusive. In daily life, when we say that
something moves, we mean that it moves relatively to the earth. In
dealing with the motions of the planets, we consider them as moving
relatively to the sun, or to the center of mass of the solar system.
When we say that the solar system itself is moving, we mean that it is
moving relatively to the stars. There is no physical occurrence which
can be called “absolute motion.” Consequently the laws of physics must
be concerned with relative motions, since these are the only kind that
occur.
We now take the relativity of motion in conjunction with the
experimental fact that the velocity of light is the same relatively
to one body as relatively to another, however the two may be moving.
This leads us to the relativity of distances and times. This in turn
shows that there is no objective physical fact which can be called “the
distance between two bodies at a given time,” since the time and the
distance will both depend on the observer. Therefore Newton’s law of
gravitation is logically untenable, since it makes use of “distance at
a given time.”
This shows that we cannot rest content with Newton, but it does not
show what we are to put in his place. Here several considerations
enter in. We have in the first place what is called “the equality
of gravitational and inertial mass.” What this means is as follows:
When you apply a given force[6] to a heavy body, you do not give it
as much acceleration as you would to a light body. What is called the
“inertial” mass of a body is measured by the amount of force required
to produce a given acceleration. At a given point of the earth’s
surface, the “mass” is proportional to the “weight.” What is measured
by scales is rather the mass than the weight: the weight is defined as
the force with which the earth attracts the body. Now this force is
greater at the poles than at the equator, because at the equator the
rotation of the earth produces a “centrifugal force” which partially
counteracts gravitation. The force of the earth’s attraction is also
greater on the surface of the earth than it is at a great height or at
the bottom of a very deep mine. None of these variations are shown by
scales, because they affect the weights used just as much as the body
weighed; but they are shown if we use a spring balance. The mass does
not vary in the course of these changes of weight.
[6] Although “force” is no longer to be regarded as one of the
fundamental concepts of dynamics, but only as a convenient way of
speaking, it can still be employed, like “sunrise” and “sunset,”
provided we realize what we mean. Often it would require very
roundabout expressions to avoid the term “force.”
The “gravitational” mass is differently defined. It is capable of two
meanings. We may mean (1), the way a body responds in a situation
where gravitation has a known intensity, for example, on the surface
of the earth, or on the surface of the sun; or (2), the intensity of
the gravitational force produced by the body, as, for example, the sun
produces stronger gravitational forces than the earth does. Newton
says that the force of gravitation between two bodies is proportional
to the product of their masses. Now let us consider the attraction of
different bodies to one and the same body, say the sun. Then different
bodies are attracted by forces which are proportional to their masses,
and which, therefore, produce exactly the same acceleration in all of
them. Thus if we mean “gravitational mass” in sense (1), that is to
say, the way a body responds to gravitation, we find that “the equality
of inertial and gravitational mass,” which sounds formidable, reduces
to this: that in a given gravitational situation, all bodies behave
exactly alike. As regards the surface of the earth, this was one of
the first discoveries of Galileo. Aristotle thought that heavy bodies
fall faster than light ones; Galileo showed that this is not the case,
when the resistance of the air is eliminated. In a vacuum, a feather
falls as fast as a lump of lead. As regards the planets, it was Newton
who established the corresponding facts. At a given distance from the
sun, a comet, which has a very small mass, experiences exactly the
same acceleration towards the sun as a planet experiences at the same
distance. Thus the way in which gravitation affects a body depends only
upon where the body is, and in no degree upon the nature of the body.
This suggests that the gravitational effect is a characteristic of the
locality, which is what Einstein makes it.
As for the gravitational mass in sense (2), _i.e._, the intensity of
the force produced by a body, this is no longer _exactly_ proportional
to its inertial mass. The question involves some rather complicated
mathematics, and I shall not go into it.[7]
[7] See Eddington, _The Mathematical Theory of Relativity_, Cambridge
University Press, 2d edition, p. 128.
We have another indication as to what sort of thing the law of
gravitation _must_ be, if it is to be a characteristic of a
neighborhood, as we have seen reason to suppose that it is. It must
be expressed in some law which is unchanged when we adopt a different
kind of co-ordinates. We saw that we must not, to begin with, regard
our co-ordinates as having any physical significance: they are merely
systematic ways of naming different parts of space-time. Being
conventional, they cannot enter into physical laws. That means to say
that, if we have expressed a law correctly in terms of one set of
co-ordinates, it must be expressed by the same formula in terms of
another set of co-ordinates. Or, more exactly, it must be possible
to find a formula which expresses the law, and which is unchanged
however we change the co-ordinates. It is the business of the theory
of tensors to deal with such formulæ. And the theory of tensors shows
that there is one formula which obviously suggests itself as being
possibly the law of gravitation. When this possibility is examined,
it is found to give the right results; it is here that the empirical
confirmations come in. But if Einstein’s law had not been found to
agree with experience, we could not have gone back to Newton’s law. We
should have been compelled by logic to seek some law expressed in terms
of “tensors,” and therefore independent of our choice of co-ordinates.
It is impossible without mathematics to explain the theory of
tensors; the non-mathematician must be content to know that it is the
technical method by which we eliminate the conventional element from
our measurements and laws, and thus arrive at physical laws which are
independent of the observer’s point of view. Of this method, Einstein’s
law of gravitation is the most splendid example.