THE NEW CONCEPTION OF GRAVITATION
_Geometry and reality—Euclid’s geometry and
others—Contingency of Poincaré’s criterion—The
real universe is not Euclidean but Riemannian—The
avatars of the number π—The point of view of the
drunken man—Straight and geodetic lines—The new
law of universal attraction—Explanation of the
anomaly of the planet Mercury—Einstein’s theory of
gravitation._
Does the universe conform to the laws of geometry? It is a question
that has been much discussed by philosophers and scholars, but the
deviation of light owing to its weight now enables us to approach it
with confidence.
In our schools we are taught a magnificent series of geometrical
theorems, all solidly interconnected, the principal of which were
created by the great Greek genius, Euclid. That is why classical
geometry is known as Euclidean geometry. Its theorems are based upon a
certain number of axioms and postulates, though these are really only
affirmations or definitions.
The most important of these definitions is: “A straight line is the
shortest distance between two points.” That seems to schoolboys quite
simple, because they know that the youth who amuses himself by running
in a zigzag on the racing track will be the last to reach the tape; and
at the sports ground one is not in a mood or has not time to bother
about the validity of the axioms of geometry. What is the precise
meaning of this definition of a straight line? There has been a great
deal of discussion of that point. Henri Poincaré has written a number
of fine and profound pages on it, yet his conclusions are not entirely
without an element of uncertainty.
In practice we all know what we mean by a straight line: it is the
line that we make by means of a good ruler. But how do we know that a
ruler is good and correct? By holding it up before the eye, and seeing
that both ends of it and all the intermediate points in its edge merge
together when we look along it. That is how a carpenter tells if a
board is smoothly planed. In a word, in practice we mean by a straight
line the line which is taken by the eye of the rifleman looking along
his sights.
All this amounts to saying that a straight line is the direction
in which a ray of light travels. However we look at the matter, we
always come back to the same point—to say that the edge of an object
is straight means that the delimiting line coincides in its whole
length with a ray of light.[10] We may therefore say that practically a
straight line is the path followed by light in a homogeneous medium.
[10] It goes without saying that in all this we assume that the
luminous ray travels in a homogeneous medium.
And that gives rise to a question. Is the world in which we live, the
universe, in conformity with Euclid’s geometry? Is it Euclidean?
It must be understood that Euclid’s geometry is not the only one
that has been created. In the nineteenth century there were bold
and profound mathematicians—Riemann, Bolyay, Lobatchewski, even
Poincaré—who founded new and different and rather strange geometries.
They are just as logical and coherent as the classical geometry of
Euclid, but they are based upon different axioms and postulates—in a
word, different definitions.
For instance, “parallels” are said to be two straight lines, being in
the same plane, which can never meet. The geometry which we learned
in our boyhood says: “Through a given point there can be only one
straight line parallel to a given straight line.” This is said to be
Euclid’s postulate. Riemann, however, does not admit this and wishes
to replace it by: “Through a given point there cannot be any straight
line parallel to a given straight line”—that is to say, any line which
never meets it. Upon this Riemann founds a quite consistent system of
geometry.
Who will venture to say that Euclid’s geometry is true and that of
Riemann false? As theoretical ideal constructions they are both equally
true.
* * * * *
A question that we may legitimately ask is: Does the real universe
correspond to the classical geometry of Euclid or to that of Riemann?
It was long believed that it corresponded to Euclid’s geometry.
Poincaré himself, speaking of Euclid’s system, said:
“It is, and will remain, the most convenient, (1)
because it is the simplest; (2) because it agrees
very well with the properties of natural solids,
the bodies with which our limbs and our eyes are
concerned, and out of which we make our measuring
instruments.”
When people used to say in earlier ages that the earth is flat,
they argued pretty much as Poincaré does: “This theory is the most
convenient, (1) because it is the simplest; (2) because it agrees very
well with the properties of the natural objects with which we are in
contact.” But when men came into touch with more remote objects, when
navigators and astronomers multiplied these remote objects, the idea of
a flat earth ceased to be the most convenient, the simplest, and the
best suited to the facts of experience. Then appeared the idea that the
earth is round, and this was found infinitely more convenient, simpler,
and better adapted to the material universe.
“Convenience,” which Poincaré makes a criterion of scientific truth, is
a contingent and elastic thing. A point of view may be convenient in
London and not in Bedford. A theory may be convenient in an area of a
hundred yards and no longer convenient for an area of a hundred million
miles.
The hypothesis of a flat earth has been replaced by the theory of
the earth’s rotundity. The stationary earth has been replaced by a
revolving globe. In the same way, it seems that in our time Euclid’s
geometry must give way to another as a _convenient_ representation
of the real world.
Can there be, in our universe, our space, a parallel to a straight
line? That is to say, is it true that two straight lines being in the
same plane will never meet? The real meaning of the question is: Is it
impossible for two luminous rays, travelling in empty space and being
in what (for each fraction of the rays) we will call the same plane,
ever to meet? _The answer to this question is in the negative._
As these two luminous rays are bent out of their paths in space by the
gravitation of the stars, and as they are differently affected in this
way because they are at different distances from the stars, it follows
necessarily that they will cease to be parallel (in the Euclidean sense
of the word) and will finally meet; or at least that they cease to
realise the first condition of parallelism—coexistence—in the same
local plane.
In a word, if we consider the matter, not within the ridiculously
limited field of experiment in the laboratory, but in the vast field
of celestial space, the real universe is not Euclidean, because in it
light does not travel in a straight line.
Kant regarded the truths—to be accurate, the deductive
affirmations—of the Euclidean geometry as “synthetic judgments _a
priori_,” or self-evident propositions. As we have seen, Kant was
wrong, not only from the point of view of theoretical geometry, but
also from the point of view of real geometry. The etymology of the word
“geometry” (which means “measuring the earth”) is enough of itself to
show that it was originally, and chiefly, a practical science. That is
a sufficient justification for our asking which geometry is most in
accord with the real universe.
Gauss, a profound thinker, asked the question long ago, in the last
century, and he made certain delicate experiments to measure if the sum
of the angles of a triangle is really equal to two right angles, as
the Euclidean geometry says. With this view he took a vast triangle,
the apices of which were formed by the highest peaks of three widely
separated mountains. One of them was the famous Brocken. With his
assistants he took simultaneous sights of each peak in relation to
the other two, and he found that the sum of the three angles of the
triangle only differed from 180 degrees to an extent that might be put
down to error in observation.
There were many philosophers who ridiculed Gauss and his experiments.
With the _a priori_ dogmatism that one so often encounters amongst
these people they said that his measurements, even if they had had
a different result, would have proved nothing to the detriment of
Euclid’s theorems, but would merely have shown that some disturbing
cause bent the luminous rays between the three apices of the triangle.
This is true, but it does not matter.
If Gauss had found that the sum of the angles of the triangle in
question was larger than two right angles, it would have proved that
real geometry is not the geometry of Euclid. The question which Gauss
asked was profound and reasonable. The philosophers who ridiculed it
might have been challenged to define real straight lines, natural
straight lines, in any other terms than those of the passage of light.
Gauss did not find the sum of the angles different from two right
angles because his measurements were not sufficiently precise. If they
had been much more rigorous, or if he could have used a much larger
triangle—with the earth, Jupiter in opposition, and another planet as
its apices—he would have found a considerable difference.
The real universe is not Euclidean. It is only approximately Euclidean
in those parts of space where light travels in a straight line: that is
to say, in the parts which are far from any gravitational mass, such as
that in which, on an earlier page, we left Jules Verne’s projectile.
There are many other reasons why the universe, in consequence of
gravitation, does not conform to the laws of Euclid’s geometry.
For instance, in the Euclidean geometry the extent of the circumference
has a well-known proportion to its diameter, and this is indicated by
the Greek letter π. This proportion, expressing how many times the
diameter is contained in the circumference, is equal to 3·14159265
... etc., but I pass over the rest, as π has an infinite number of
decimals. We then ask: In practice is the proportion of circumferences
to their diameters really equal to the classic value of π? For
instance, is this precisely the proportion of the earth’s circumference
to its diameter?[11] Einstein says that it is not, and he gives us the
following proof. Imagine two very clever and quick and wizard-like
surveyors setting out to measure the circumference and diameter of the
earth at the Equator. They both use the same scales of measurement.
They begin measuring at the same moment, and they start from the same
point on the Equator. But one goes westward and the other eastward,
and their speeds are equal, and such that the one who goes westward
keeps up with the earth’s rotation, and thus sees the sun all day
long stationary at the same height above the horizon. In music-halls,
for instance, one sometimes sees an acrobat walking on a rolling ball
and keeping to the top of the ball, because the pace of his steps is
exactly equal and contrary to the displacement of the spherical surface.
[11] We are, of course, imagining the earth as perfectly circular,
without irregularities.
A stationary observer in space—on the sun, let us say—would thus see
our surveyor who is going westward, stationary right opposite to him.
On the other hand, the surveyor who goes eastward will seem to him to
go round the earth, and twice as quickly as if he had remained at the
starting-point.
When each of our surveyors, both going at the same speed, has finished
his task of measuring the round of the earth, will they both have the
same result? Evidently not. As the super-observer in the sun will see,
the yard of the surveyor who travels eastward is shortened by velocity
in virtue of the Fitzgerald-Lorentz contraction. On the other hand,
the yard of the surveyor who travels westward does not experience this
contraction, as the super-observer on the sun, in reference to whom he
remains stationary, would see.
Consequently the two surveyors reach different figures for the earth’s
circumference, the one who travels westward finding a result a few
yards less than that of the other. Yet it is obvious that when they
proceed to measure the earth’s diameter, travelling at the same speed,
the two observers will reach the same figure for it.
Hence the π which expresses the proportion of the earth’s circumference
to its diameter on the ground of actual measurement differs according
as the measurer travels in the direction of the earth’s rotation or
in the opposite direction. Therefore, as the real values of π are
different, they cannot be the unique and quite definite figure of
classical geometry. Therefore the real universe does not conform to
this geometry.
These differences, in the illustration we have given, are due to the
earth’s rotation. From the standpoint of gravitation the earth’s
rotation has centrifugal effects which modify the centripetal influence
of weight. We have seen, moreover, that for the surveyor whose speed
equals that of the earth’s rotation the value of π is smaller than for
the observer whose speed seems to be double that of the rotation. Thus
the effects of weight being the reverse of those of rotation, or of
centrifugal force, it follows (it would be just as easy to prove this
as the preceding) that the effect of weight is to give π something less
than its classical value.
In a word, in the universe real circumferences traced upon gravitating
masses, such as stars, are, in proportion to their diameters, less than
they are in the Euclidean geometry.
The difference is generally very slight, it is true. But there
_is_ a difference. If we put a mass of a thousand kilogrammes
in the centre of a circle that is ten metres in diameter, the figure
π will differ in reality from its Euclidean value by less than
one-thousand-million-billionth.
In the neighbourhood of such formidable masses of matter as the stars
are, the difference may be far greater, as we shall see. This is the
origin of the divergences between Newton’s law of gravitation and that
of Einstein: divergences which observation has settled in favour of the
latter. But we will not anticipate.
* * * * *
We showed in a previous chapter that the real universe of the
Relativists is a four-dimensional continuum—not three-dimensional, as
classic science thought—and that in this continuum distances in time
and space are relative. The only thing that has a value independent
of the conditions of observation—that has an absolute, or at least
objective, value—is what we called the “Interval” of events, the
synthesis of the spatial and chronological data.
Yet, in spite of its four dimensions, the universe, as we discussed it
in connection with the Michelson experiment and the Special Relativity
which this discloses, was nevertheless a Euclidean continuum, in which
the classical geometry was verified, and light travelled in a straight
line. As we have just seen, we have to recant this. The universe not
only has four dimensions, but it is not Euclidean.
With what geometry does the universe accord best—or most conveniently,
to use the language of Poincaré? Probably that of Riemann. When we take
the compasses and draw a small circle on a sheet of paper spread on
the table, the radius of the circle is found by the distance between
the points of the compasses, and the circle is Euclidean. But if we
draw the circle on an egg, the fixed point of the compasses being
stuck in the top of the egg, and again get the radius by the distance
between the points, the circle we have now drawn is not Euclidean.
The proportion of the circumference to the radius as thus defined is
smaller than π, just as it is smaller than π when the circle is traced
round a massive star.
Well, there is the same difference between the non-Euclidean real
universe and a Euclidean continuum as there is between our flat sheet
of paper and the surface of the egg, taking into account the fact that
these surfaces have only two dimensions while the universe has four.
Two-dimensional space may be flat like the sheet of paper or curved
like the surface of the egg. By leaving the sheet of paper flat or
rolling it up we can make the geometry of the figures drawn on it
correspond with or differ from the Euclidean geometry. In just the same
way space with more than two dimensions may or may not be Euclidean.
As a matter of fact, the universe is, as we saw, only approximately
Euclidean in those regions which are remote from all heavy masses. It
is not Euclidean, but curved or warped in the vicinity of the stars;
and the curvature is the greater in proportion as we approach the stars.
Hence the geometry of curved space, as founded by Riemann, seems to be
the best adapted to the real universe. It is the one used by Einstein
in his calculations.
* * * * *
When we sought to prove, on a previous page, that rays of light fall
just as projectiles of the same velocity would, we used the following
argument:
Since the “Interval” of two events is the same for two observers moving
at uniform and different velocities, it is _natural_ to think that
it will be the same for a third observer whose velocity increases from
that of the first to that of the second—that is to say, whose velocity
is uniformly accelerated.
There is, in fact, no reason why the passengers in a train which runs
at a uniform speed of sixty miles an hour should observe an “invariant”
element in phenomena just as do those in another train moving at
half the speed, yet this “invariant” should cease to be such for the
passengers in a third train which passes gradually from the velocity
of the first train to that of the second. To admit the contrary would
be to grant a privileged position in the universe to the first two and
others like them. If there is any estate in the world that has had its
unjust privileges suppressed by the new physics, it is the study of the
material world.
This privilege of observers moving at a uniform velocity would be the
less justified as, if we go to the root of the matter, it is very
difficult to say exactly what a uniform movement is.
What do we mean when we say that a train has a uniform velocity of
sixty miles an hour? We mean that the train has this velocity in
reference to the rails or the ground. But in reference to an observer
in a balloon, or who passes in another train, the velocity has not
the same value, and it may cease to be a uniform velocity. We know
only relative movements, or, to be quite accurate, movements relative
to some material object or other. According to our choice of this
object, this standard of comparison, the same velocity may be uniform
or accelerated. In the long run, it is clear, we should have to have
recourse to Newton’s hypothesis of absolute space to be able to say
whether a given velocity is really uniform or accelerated.
That is the profound reason why the Einsteinian “Interval” of things,
the invariable quantity or “Invariant,” must be the same for all
observers whatever be their velocity, and in particular for observers
moving at velocities equivalent, in a given place, to the effects of
gravitation.
But in that case the inferences we draw from the Michelson experiment,
in regard to the aspect of phenomena for observers in uniform different
movements of translation, no longer suffice to explain to us the
whole of reality. They need to be completed in such fashion that the
universal invariant, the “Interval” of things, remains the same for an
observer who is moving in any way whatever.
If I pass along a street at some unheard-of speed, but with a uniform
motion, its general aspect may, on account of the contraction caused
by my velocity, be a little different from what it would seem to me
if I were stationary.[12] The houses, for instance, will seem narrower
in proportion to their height. Nevertheless the general aspect and
proportions of objects will be much the same in both cases, and they
will have something in common. Thus the gas-lights will seem to me
thinner, but they will be straight.
[12] It goes without saying that we assume the observer to have a
retina with instantaneous impressions.
It will be quite otherwise if the observer’s movements are varied:
if, for instance, we imagine him a drunken giant, reeling about at
a prodigious speed. For such an observer the street will have quite
a new aspect. The gas-jets will no longer be straight, but zigzag,
reproducing in an inverse way the zigzags which he himself makes as he
reels along. This is so true that caricaturists generally represent the
trees and lamp-posts and houses seen by a drunken man by ridiculously
waving lines.
Our observer will be convinced that objects really have the zigzag
forms which he sees, and that the forms change at every step he takes.
Try to tell him that it is he who is dancing, not the objects; that it
is he who is not walking straight, not the dog he has on leash. He will
not believe it—and from the point of view of General Relativity he is
neither more nor less right than you.
Yet there is something in the aspect of the world that must be common
to the drunkard and the drinker of water.
If the whole universe were suddenly plunged in a mass of gelatine
which has set, and one were to squeeze or alter the shape in any way
of this gelatinous mass, there would still be something unchanged in
the coagulated stuff. What is this something? And what is the calculus
to use for it? The answer to these questions was the last stage for
Einstein to cover in order to establish the equations of gravitation
and General Relativity.
* * * * *
Here it was the penetrating genius of Henri Poincaré that indicated the
path. It is very necessary to insist on this, as justice has not been
done in the matter to the great French mathematician.
If all the bodies in the universe were to be simultaneously dilated,
and to an identical extent, we should have no means of knowing it. Our
instruments and our own bodies being similarly dilated, we should not
perceive this formidable historical and cosmic event. It would not
distract us for a moment from the trivialities of the hour.
What is more, not only will it be unrecognisable if worlds are modified
in such a fashion as to alter the scale of lengths and time, but it
would be impossible to distinguish between two worlds, if one single
point of the first corresponds to each point of the second; if to each
object or event of the one world there corresponds one of the same
character, placed exactly in the same position, in the other. Now the
successive and diverse deformations which we impose upon the gelatinous
mass in which we metaphorically enclosed our entire universe in an
earlier paragraph give us precisely indistinguishable worlds from
this point of view. Poincaré has the distinction of first calling our
attention to this and proving that the relativity of things must be
understood in this very broad sense.
The amorphous and plastic continuum in which we place the universe
has a certain number of properties which are exempt from all idea of
measurement. The study of these properties is the work of a special
geometry, a qualitative geometry. The theorems of this geometry have
this peculiarity, that they would still be true even if the figures
were copied by a clumsy draughtsman who made gross errors in the
proportions and substituted irregular and wavy lines for straight lines.
This is the geometry which, as Poincaré ably indicated, must be used
for the four-dimensional and, according to its regions, more or less
Euclidean continuum which is the Einsteinian universe. It is precisely
this geometry which states what there is in common between the forms of
objects seen by the drunken man and those seen by the water-drinker.
It is along this route, or a route analogous to this, that Einstein
at last reached success. The universe being a more or less warped
continuum, he proposed to apply to it the geometry created by Gauss for
the study of surfaces of variable curvature: a geometry generalised by
Riemann. It is by means of this special geometry that we express the
fact that the “Interval” of events is an invariant.
Here is an illustration which will, I think, lead us to the heart of
the problem of gravitation and to the solution of it.
* * * * *
Let us consider a surface of variable curvature—for instance, the
surface of any large district with its hills, mountains, and valleys.
When we travel in this region, we can proceed in a straight line as
long as we are on the level plain. A straight line on a level plain
has the remarkable feature of being the shortest distance between two
points. It has also this peculiarity, that it is the only one of its
kind and its length, whereas we may draw a great number of lines that
are not straight uniting the two points, longer than the straight line
but all of equal length.
But we have reached the hilly district. It is now impossible for us to
follow a straight line from one point to another if there is a hill
between them. Whatever path we take, it will be curved. But amongst the
various possible paths which lead from one point to the other on the
farther side of the hill, there is one—and only one, as a rule—which
is shorter than any of the others, as we could prove by means of a
tape. This shortest path, the only one of its kind, is what is called
the _geodetical_ of the surface covered.
In the same way no vessel can go in a straight line if it is sailing
from Lisbon to New York. It must follow a curved path, because the
earth is round. But amongst the possible curved paths there is a
privileged one which is shorter than the others: the one which follows
the direction of the great circle of the earth. In going from Lisbon
to New York, though they are nearly in the same latitude, vessels
are careful not to head straight westward, in the direction of the
parallels. They sail a little to the north-west, so that when they
reach New York they come from the north-east, having followed pretty
closely a terrestrial great circle. On our globe, as on all spheres,
the _geodetical_, the shortest route between two points, is the
arc of a great circle passing through the two points.
Now the “Interval” of two points in the four-dimensional universe
precisely represents the geodetical, the minimum path of progress
between the two points traced in the universe. Where the universe
is curved, the geodetic is a curved line. Where the universe is
approximately Euclidean, it is a straight line.
I may be told that it is very difficult to imagine as curved a
three-dimensional space, and still more a four-dimensional. I
agree. We have already seen that it is difficult enough to imagine
four-dimensional space even when it is not curved.
But what does that prove? There are many other things in nature which
we cannot visualise or form a mental picture of. The Hertz waves,
the X-rays, and the ultra-violet waves exist all the same, though we
cannot imagine them, or at least only by giving them a visible form
which does not belong to them. It is just one of our human infirmities
that we cannot conceive what we cannot picture to ourselves. Hence our
tendency to—if one may use an inelegant but expressive word—visualise
everything.
Let us therefore return to our geodetics. These we can very well
picture to ourselves, because in the universe, in spite of its four
dimensions, they are lines of only one dimension, like all other lines
that we know.
* * * * *
The existence of geodetics, of shortest-distance lines, will now
beautifully explain to us the connection between inertia and weight,
which did not appear in the Euclidean world of classic science. Hence
the Newtonian distinction between the principle of inertia and the
force of gravitation.
We Relativists find this distinction no longer necessary. Material
masses, like light, travel in a straight line when they are far
from a gravitational field, and in a curved line when they are near
gravitational masses. In virtue of symmetry a free material point can
only follow a geodetic in the universe.
If we now reflect that the force of gravitation introduced by Newton
does not exist—such action at a distance is very problematical—and
that in empty space there are only objects freely left to themselves,
we are driven to the following conclusion, which unites in a simple way
the previously separated sisters, inertia and weight: _Every moving
body freely left to itself in the universe describes a geodetic._
Far from the massive stars this geodetic is a straight line, because
there the universe is almost Euclidean. Near the stars it is a curved
line, because there the universe is not Euclidean. A fine conception,
combining in a single rule the principle of inertia and the law of
gravitation! A brilliant synthesis of mechanics and gravitation,
putting an end to the schism which so long kept them separate and
non-corresponding sciences!
In this bold and simple theory gravitation is not a force. The planets
have curved paths because near the sun, just as in the neighbourhood
of every concentration of matter the universe is curved or warped.
The shortest path from one point to another is a line that only seems
straight to us—poor pygmies that we are—because we measure it with
very small rods and over small distances. If we could follow the line
over millions of miles, and during a sufficient period, we should find
it curved.
In a word—to use an illustration that must be regarded only as an
analogy—the planets describe curved paths because they follow the
shortest path in a curved universe, just as at a sports ground cyclists
have no need to turn the handles when they reach the corner, but pedal
straight on, because the slope of the ground compels them of itself to
turn. In the sports ground, as in the solar system, the curvature is
greater in proportion as the machine is nearer to the inner edge of the
track.
All that now remains is to assign to the universe, to space-time, such
a curvature at its various points that the geodetics will exactly
represent the paths of the planets and of falling bodies, admitting
that the curvature of the universe is caused at each point by the
presence or vicinity of material masses.
In this calculation we have to take into account the fact that the
“Interval”—that is to say, the part of the geodetic between two points
that are very near each other—must be an invariant whoever may be
the observer. In this way the same geodetic will be a curved or even
wavy line for the drunken man we introduced and a straight line for a
stationary observer. The length of the line is the same, whether it
appears straight or curved.
Taking all this into account, and doing prodigies of mathematical
skill of which we have sufficiently indicated the object, Einstein
has succeeded in expressing the law of gravitation in a completely
invariant form.
In calculating, on the ground of Newton’s law, the “Interval” of
two astronomical events—for instance, the successive falls of two
meteorites into the sun—we should find that the “Interval” has not
precisely the same value for observers who are moving at different
velocities.
With the new form given to the law by Einstein the difference
disappears. The two laws, however, differ little from each other, as
was to be expected in view of the accuracy with which astronomers found
Newton’s law verified during a couple of centuries. The improvement
made in Newton’s law by Einstein means, in a word (and to use the old
language of the Euclidean universe), that we consider the law accurate
with the reserve that the distances of the planets from the sun are
measured by a scale which decreases slightly in length as the sun is
approached.
* * * * *
It is surprising that Newton and Einstein agree in expressing the
movements of gravitating stars in an _almost_ identical form,
because their starting-points are very different.
Newton starts from the hypothesis of absolute space, the empirical
laws of the motions of the planets expressed in Kepler’s laws, and
the belief that gravitational attraction is a force proportional to
mass. Einstein, on the other hand, in making his calculations starts
from the conditions of invariance which we indicated. He starts, in a
sense, from the philosophical principle or postulate or impulse to hold
that the laws of nature are invariant and independent of the point of
view—irrelative, if I may use the word.
Einstein even abandons the hypothesis which ascribed the curving of
gravitational paths to a distinct force of attraction. Yet, starting
from a point of view so different from that of Newton, and one that
seems at first less overloaded with hypotheses, Einstein reaches a law
of gravitation which is _almost_ identical with Newton’s.
This “almost” is of immense interest, because it enables us to test
which is the accurate law, that of Newton or that of Einstein. They
give the same results when there is question of velocities that are
feeble in comparison with that of light, but their results differ a
little when there is question of very high velocities. We have already
seen that, near the sun, light itself is bent out of its course in
exact conformity with Einstein’s law, and in a way that Newton’s law
did not predict as such.
But there is another divergence between the two laws. According to the
Newtonian law the planets revolving round the sun describe ellipses
which—neglecting the small perturbations due to the other
planets—have a rigorously fixed position.
Suppose we put on a table a slice of lemon cut through the longer
diameter of the fruit, and imagine that the chief stars, the
northern constellations, are painted on the vaulted roof of the vast
hemispherical room in the middle of which we place our table. The slice
of lemon has very nearly the form of an ellipse, and, if we take one of
the pips to represent the sun, it will stand for the orbit of one of
our planets. Newton’s law says that—after making due corrections—the
planetary orbit keeps a fixed position relatively to the stars as long
as the planet continues to revolve. This means that the slice of lemon
remains stationary.
Einstein’s law says, on the contrary, that the orbital ellipse turns
very slowly amongst the stars while the planet traverses it. This means
that our slice of lemon must turn slightly on the table, in such wise
that the two ends of the lemon do not remain opposite the same stars
painted on the wall.
If we calculate, in virtue of Einstein’s law, the extent to which the
elliptical orbits of the planets must thus turn, we find it so small as
to be impossible of observation except in the case of one planet, the
swiftest of all, Mercury.
Mercury revolves completely round the sun in about eighty-eight days,
and Einstein’s law shows that its orbit must at the same time turn by a
small angle which amounts to forty-three seconds of an arc (43″) at the
end of a century. Small as this quantity is, the refined methods of the
modern astronomer can easily measure it.
As a matter of fact, it had been noticed during the last century that
Mercury was the only one of the planets to show a slight anomaly in its
movements, which could not be explained by Newton’s law. Le Verrier
made prodigious calculations in connection with it, as he thought that
the anomaly might be due to the attraction of an unknown body lying
between Mercury and the sun. He hoped that he would thus discover, by
calculation, an intra-Mercurial planet, just as he had discovered the
trans-Uranian planet Neptune.
But no one ever observed his planet, and the anomaly of Mercury
continued to be the despair of astronomers. Now, in what did the
anomaly consist? Precisely in an abnormal rotation of the planetary
orbit; a rotation which Le Verrier’s calculations showed to be
forty-three seconds of an arc in a century. That is exactly the figure
that we deduce, without using any hypothesis, from Einstein’s law of
gravitation!
It is true that, according to the recent calculations of Grossmann, the
astronomical observations collected by Newcomb give as the recorded
value of the secular displacement of the perihelion of Mercury, not
43″ as Le Verrier believed, but 38″ at the most. The agreement with
Einstein’s theoretical result is, therefore, not perfect (which would
have been extraordinary), but it is striking, and is within the limits
of possible error of observation.
Einstein’s law is just as exact as Newton’s for the slower planets. For
faster bodies, the motion of which can be observed with a higher degree
of precision, Newton’s law is wrong, and Einstein’s triumphs once more.
* * * * *
This improvement of what had been considered perfect—the work of
Newton—is a great victory for the human mind. Astronomy and celestial
mechanics derive additional precision and power of forecast from it. We
can now follow the golden orbs, on the triumphal wings of calculation,
better than we could before, or antedate their movements by centuries.
But there is another test of Einstein’s law of gravitation. If it is
sound, the duration of a phenomenon increases, according to Einstein,
when the gravitational field becomes more intense. It follows that the
duration of the vibration of a given atom must be longer on the sun
than on the earth. The wave-lengths of the spectral lines of the same
chemical element ought to be a little greater in sunlight than in light
which originates on the earth. Recent observations tend to confirm
this, but the verification is less satisfactory than in the case of
Mercury because other causes may intervene to modify the wave-lengths.
On the whole, the powerful synthesis which Einstein calls the theory
of General Relativity, which we have here rapidly outlined, is a lofty
and beautiful mental construction as well as a superb instrument of
exploration.
To know is to forecast. This theory forecasts, and better than its
predecessors did. For the first time it combines gravitation and
mechanics. It shows how matter imposes upon the external world a
curvature or warping of which gravitation is but a symptom: just as the
weeds one sees floating on the sea are but indications of the current
which bears them along.
Whatever modifications it may undergo in the future—for everything in
science is open to improvement—it has shown us a little more of the
harmony that is born of unity in the laws of nature.
But I have sufficiently shown that if I have succeeded in enabling the
reader to understand—to feel, at least—these matters without invoking
the aid of the pure light which geometry pours upon the invisible.