CHAPTER VITHE NEW CONCEPTION OF GRAVITATION
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 greatdeal 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.
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 ofEuclid, 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 incontact.” 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 aconvenientrepresentation 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 torealise 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 judgmentsa 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 thea prioridogmatism that one so often encounters amongstthese 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 bythe 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.
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 thereisa 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 isnaturalto 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 observerin 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 andproportions 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.
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 theproportions 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 tofollow 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 thegeodeticalof 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, thegeodetical, 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 waythe 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, admittingthat 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 analmostidentical 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 isalmostidentical 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 ellipseswhich—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 inscience 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.