Space Time and Gravitation/Chapter 6

by Arthur Eddington

 

CHAPTER VI

THE NEW LAW OF GRAVITATION AND
THE OLD LAW

I don't know what I may seem to the world, but, as to myself, I seem to have been only as a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

Sir Isaac Newton.

Was there any reason to feel dissatisfied with Newton's law of gravitation?

Observationally it had been subjected to the most stringent tests, and had come to be regarded as the perfect model of an exact law of nature. The cases, where a possible failure could be alleged, were almost insignificant. There are certain unexplained irregularities in the moon's motion; but astronomers generally looked—and must still look—in other directions for the cause of these discrepancies. One failure only had led to a serious questioning of the law; this was the discordance of motion of the perihelion of Mercury. How small was this discrepancy may be judged from the fact that, to meet it, it was proposed to amend square of the distance to the 2.00000016 power of the distance. Further it seemed possible, though unlikely, that the matter causing the zodiacal light might be of sufficient mass to be responsible for this effect.

The most serious objection against the Newtonian law as an exact law was that it had become ambiguous. The law refers to the product of the masses of the two bodies; but the mass depends on the velocity—a fact unknown in Newton's day. Are we to take the variable mass, or the mass reduced to rest? Perhaps a learned judge, interpreting Newton's statement like a last will and testament, could give a decision; but that is scarcely the way to settle an important point in scientific theory.

Further distance, also referred to in the law, is something relative to an observer. Are we to take the observer travelling with the sun or with the other body concerned, or at rest in the aether or in some gravitational medium?

Finally is the force of gravitation propagated instantaneously, or with the velocity of light, or some other velocity? Until comparatively recently it was thought that conclusive proof had been given that the speed of gravitation must be far higher than that of light. The argument was something like this. If the Sun attracts Jupiter towards its present position $S$, and Jupiter attracts the Sun towards its present position $J$, the two forces are in the same line and balance. But if the Sun attracts Jupiter towards its previous position $S^{\prime }$, and Jupiter attracts the Sun towards its previous position $J^{\prime }$, when the force of attraction started out to cross the gulf, then the two forces

give a couple. This couple will tend to increase the angular momentum of the system, and, acting cumulatively, will soon cause an appreciable change of period, disagreeing with observation if the speed is at all comparable with that of light. The argument is fallacious, because the effect of propagation will not necessarily be that $S$ is attracted in the direction towards $J^{\prime }$. Indeed it is found that if $S$ and $J$ are two electric charges, $S$ will be attracted very approximately towards $J$ (not $J^{\prime }$) in spite of the electric influence being propagated with the velocity of light^[1]. In the theory given in this book, gravitation is propagated with the speed of light, and there is no discordance with observation.

It is often urged that Newton's law of gravitation is much simpler than Einstein's new law. That depends on the point of view; and from the point of view of the four-dimensional world Newton's law is far more complicated. Moreover, it will be seen that if the ambiguities are to be cleared up, the statement of Newton's law must be greatly expanded.

Some attempts have been made to expand Newton's law on the basis of the restricted principle of relativity (p. 20) alone. This was insufficient to determine a definite amendment. Using the principle of equivalence, or relativity of force, we have arrived at a definite law proposed in the last chapter. Probably the question has arisen in the reader's mind, why should it be called the law of gravitation? It may be plausible as a law of nature; but what has the degree of curvature of space-time to do with attractive forces, whether real or apparent?

A race of flat-fish once lived in an ocean in which there were only two dimensions. It was noticed that in general fishes swam in straight lines, unless there was something obviously interfering with their free courses. This seemed a very natural behaviour. But there was a certain region where all the fish seemed to be bewitched; some passed through the region but changed the direction of their swim, others swam round and round indefinitely. One fish invented a theory of vortices, and said that there were whirlpools in that region which carried everything round in curves. By-and-by a far better theory was proposed; it was said that the fishes were all attracted towards a particularly large fish a—sun-fish—which was lying asleep in the middle of the region; and that was what caused the deviation of their paths. The theory might not have sounded particularly plausible at first; but it was confirmed with marvellous exactitude by all kinds of experimental tests. All fish were found to possess this attractive power in proportion to their sizes; the law of attraction was extremely simple, and yet it was found to explain all the motions with an accuracy never approached before in any scientific investigations. Some fish grumbled that they did not see how there could be such an influence at a distance; but it was generally agreed that the influence was communicated through the ocean and might be better understood when more was known about the nature of water. Accordingly, nearly every fish who wanted to explain the attraction started by proposing some kind of mechanism for transmitting it through the water.

But there was one fish who thought of quite another plan. He was impressed by the fact that whether the fish were big or little they always took the same course, although it would naturally take a bigger force to deflect the bigger fish. He therefore concentrated attention on the courses rather than on the forces. And then he arrived at a striking explanation of the whole thing. There was a mound in the world round about where the sun-fish lay. Flat-fish could not appreciate it directly because they were two-dimensional; but whenever a fish went swimming over the slopes of the mound, although he did his best to swim straight on, he got turned round a bit. (If a traveller goes over the left slope of a mountain, he must consciously keep bearing away to the left if he wishes to keep to his original direction relative to the points of the compass.) This was the secret of the mysterious attraction, or bending of the paths, which was experienced in the region.

The parable is not perfect, because it refers to a hummock in space alone, whereas we have to deal with hummocks in space-time. But it illustrates how a curvature of the world we live in may give an illusion of attractive force, and indeed can only be discovered through some such effect. How this works out in detail must now be considered.

In the form $G_{\mu \nu }=0$, Einstein's law expresses conditions to be satisfied in a gravitational field produced by any arbitrary distribution of attracting matter. An analogous form of Newton's law was given by Laplace in his celebrated expression $\nabla ^{2}V=0$. A more illuminating form of the law is obtained if, instead of putting the question what kinds of space-time can exist under the most general conditions in an empty region, we ask what kind of space-time exists in the region round a single attracting particle? We separate out the effect of a single particle, just as Newton did. We can further simplify matters by introducing some definite mesh-system, which, of course, must be of a type which is not inconsistent with the kind of space-time found.

We need only consider space of two dimensions—sufficient for the so-called plane orbit of a planet—time being added as the third dimension. The remaining dimension of space can always be added, if desired, by conditions of symmetry. The result of long algebraic calculations ^[2] is that, round a particle $ds^{2}=-{\frac {1}{\gamma }}\,dr^{2}-r^{2}d\theta ^{2}+\gamma \,dt^{2}$ ..................(6) where $\gamma =1-{\tfrac {2m}{r}}$.

The quantity $m$ is the gravitational mass of the particle—but we are not supposed to know that at present. $r$ and $\theta$ are polar coordinates, the mesh-system being as in Fig. 11 ; or rather they are the nearest thing to polar coordinates that can be found in space which is not truly flat.

The fact is that this expression for $ds^{2}$ is found in the first place simply as a particular solution of Einstein's equations of the gravitational field; it is a variety of hummock (apparently the simplest variety) which is not curved beyond the first degree. There could be such a state of the world under suitable circumstances. To find out what those circumstances are, we have to trace some of the consequences, find out how any particle moves when $ds^{2}$ is of this form, and then examine whether we know of any case in which these consequences are found observationally. It is only after having ascertained that this form of $ds^{2}$ does correspond to the leading observed effects attributable to a particle of mass $m$ at the origin that we have the right to identify this particular solution with the one we hoped to find.

It will be a sufficient illustration of this procedure, if we indicate how the position of the matter causing this particular solution is located. Wherever the formula (6) holds good there can be no matter, because the law which applies to empty space is satisfied. But if we try to approach the origin ($r=0$), a curious thing happens. Suppose we take a measuring-rod, and, laying it radially, start marking off equal lengths with it along a radius, gradually approaching the origin. Keeping the time $t$ constant, and $d\theta$ being zero for radial measurements, the formula (6) reduces to $ds^{2}=-{\frac {1}{\gamma }}\,dr^{2}$ or $dr^{2}=-\gamma \,ds^{2}$. We start with $r$ large. By-and-by we approach the point where $r=2m$. But here, from its definition, </math>\gamma</math> is equal to 0. So that, however large the measured interval $ds$ may be, $dr=0$. We can go on shifting the measuring-rod through its own length time after time, but $dr$ is zero; that is to say, we do not reduce $r$. There is a magic circle which no measurement can bring us inside. It is not unnatural that we should picture something obstructing our closer approach, and say that a particle of matter is filling up the interior.

The fact is that so long as we keep to space-time curved only in the first degree, we can never round off the summit of the hummock. It must end in an infinite chimney. In place of the chimney, however, we round it off with a small region of greater curvature. This region cannot be empty because the law applying to empty space does not hold. We describe it therefore as containing matter—a procedure which practically amounts to a definition of matter. Those familiar with hydrodynamics may be reminded of the problem of the irrotational rotation of a fluid; the conditions cannot be satisfied at the origin, and it is necessary to cut out a region which is filled by a vortex-filament.

A word must also be said as to the coordinates $r$ and $t$ used in (6). They correspond to our ordinary notion of radial distance and time—as well as any variables in a non-Euclidean world can correspond to words which, as ordinarily used, presuppose a Euclidean world. We shall thus call $r$ and $t$, distance and time. But to give names to coordinates does not give more information—and in this case gives considerably less information—than is already contained in the formula for $ds^{2}$. If any question arises as to the exact significance of $r$ and $t$ it must always be settled by reference to equation (6).

The want of flatness in the gravitational field is indicated by the deviation of the coefficient $\gamma$ from unity. If the mass $m=0$, $\gamma =1$, and space-time is perfectly flat. Even in the most intense gravitational fields known, the deviation is extremely small. For the sun, the quantity $m$, called the gravitational mass, is only 1.47 kilometres^[3], for the earth it is 5 millimetres. In any practical problem the ratio $2m/r$ must be exceedingly small. Yet it is on the small corresponding difference in $\gamma$ that the whole of the phenomena of gravitation depend.

The coefficient $\gamma$ appears twice in the formula, and so modifies the flatness of space-time in two ways. But as a rule these two ways are by no means equally important. Its appearance as a coefficient of $dt^{2}$ produces much the most striking effects. Suppose that it is wished to measure the interval between two events in the history of a planet. If the events are, say 1 second apart in time, $dt$ = 1 second = 300,000 kilometres. Thus $dt^{2}$ = 90,000,000,000 sq. km. Now no planet moves more than 50 kilometres in a second, so that the change $dr$ associated with the lapse of 1 second in the history of the planet will not be more than 50 km. Thus $dr^{2}$ is not more than 2500 sq. km. Evidently the small term $2m/r$ has a much greater chance of making an impression where it is multiplied by $dt^{2}$ than where it is multiplied by $dr^{2}$.

Accordingly as a first approximation, we ignore the coefficient of $dr^{2}$, and consider only the meaning of $ds^{2}=-dr^{2}-r^{2}\,d\theta ^{2}+(1-2m/r)\,dt^{2}$ .........(7). We shall now show that particles situated in this kind of space-time will appear to be under the influence of an attractive force directed towards the origin.

Let us consider the problem of mapping a small portion of this kind of world on a plane.

It is first necessary to define carefully the distinction which is here drawn between a "picture" and a "map." If we are given the latitudes and longitudes of a number of places on the earth, we can make a picture by taking latitude and longitude as vertical and horizontal distances, so that the lines of latitude and longitude form a mesh-system of squares; but that does not give a true map. In an ordinary map of Europe the lines of longitude run obliquely and the lines of latitude are curved. Why is this? Because the map aims at showing as accurately as possible all distances in their true proportions^[4]. Distance is the important thing which it is desired to represent correctly. In four dimensions interval is the analogue of distance, and a map of the four-dimensional world will aim at showing all the intervals in their correct proportions. Our natural picture of space-time takes $r$ and $t$ as horizontal and vertical distances, e.g. when we plot the graph of the motion of a particle; but in a true map, representing the intervals in their proper proportions, the $r$ and $t$ lines run obliquely or in curves across the map.

The instructions for drawing latitude and longitude lines ($\beta$, $\lambda$) on a map, are summed up in the formula for $ds$, p. 79, $ds^{2}=d\beta ^{2}+\cos ^{2}\beta \,d\lambda ^{2}$

and similarly the instructions for drawing the $r$ and $t$ lines are given by the formula (7).

The map is shown in Fig. 14. It is not difficult to see why the $t$-lines converge to the left of the diagram. The factor $1-2m/r$ decreases towards the left where $r$ is small; and consequently any change of $t$ corresponds to a shorter interval, and must be represented in the map by a shorter distance on the left. It is less easy to see why the $r$-lines take the courses shown; by analogy with latitude and longitude we might expect them to be curved the other way. But we discussed in Chapter iii how the slope of the time-direction is connected with the slope of the space-direction; and it will be seen that the map gives approximately diamond-shaped partitions of the kind represented in Fig. 6^[5].

Like all maps of curved surfaces, the diagram is only accurate in the limit when the area covered is very small.

It is important to understand clearly the meaning of this map. When we speak in the ordinary way of distance from the sun and the time at a point in the solar system, we mean the two variables $r$ and $t$. These are not the result of any precise measures with scales and clocks made at a point, but are mathematical variables most appropriate for describing the whole solar system.

They represent a compromise, because it is necessary to deal with a region too large for accurate representation on a plane map. We should naturally picture them as rectangular coordinates partitioning space-time into square meshes, as in Fig. 15; but such a picture is not a true map, because it does not represent in their true proportions the intervals between the various points in the picture. It is not possible to draw any map of the whole curved region without distortion; but a small enough portion can be represented without distortion if the partitions of equal $r$ and $t$ are drawn as in Fig. 14. To get back from the true map to the customary picture of $r$ and $t$ as perpendicular space and time, we must strain Fig. 14 until all the meshes become squares as in Fig. 15.

Now in the map the geometry is Euclidean and the tracks of all material particles will be straight lines. Take such a straight track $PQ$, which will necessarily be nearly vertical, unless the velocity is very large. Strain the figure so as to obtain the customary representation of $r$ and $t$ (in Fig. 15), and the track $PQ$ will become curved—curved towards the left, where the sun lies. In each successive vertical interval (time), a successively greater progress is made to the left horizontally (space). Thus the velocity towards the sun increases. We say that the particle is attracted to the sun.

The mathematical reader should find no difficulty in proving from the diagram that for a particle with small velocity the acceleration towards the sun is approximately $m/r^{2}$, agreeing with the Newtonian law.

Tracks for very high speeds may be affected rather differently. The track corresponding to a wave of light is represented by a straight line at 45° to the horizontal in Fig. 14. It would require very careful drawing to trace what happens to it when the strain is made transforming to Fig. 15; but actually, whilst becoming more nearly vertical, it receives a curvature in the opposite direction. The effect of the gravitation of the sun on a light-wave, or very fast particle, proceeding radially is actually a repulsion!

The track of a transverse light-wave, coming out from the plane of the paper, will be affected like that of a particle of zero velocity in distorting from Fig. 14 to Fig. 15. Hence the sun's influence on a transverse light-wave is always an attraction. The acceleration is simply $m/r^{2}$ as for a particle at rest.

The result that the expression found for the geometry of the gravitational field of a particle leads to Newton's law of attraction is of great importance. It shows that the law, $G_{\mu \nu }=0$, proposed on theoretical grounds, agrees with observation at least approximately. It is no drawback that the Newtonian law applies only when the speed is small; all planetary speeds are small compared with the velocity of light, and the considerations mentioned at the beginning of this chapter suggest that some modification may be needed for speeds comparable with that of light.

Another important point to notice is that the attraction of gravitation is simply a geometrical deformation of the straight tracks. It makes no difference what body or influence is pursuing the track, the deformation is a general discrepancy between the "mental picture" and the "true map" of the portion of space-time considered. Hence light is subject to the same disturbance of path as matter. This is involved in the Principle of Equivalence; otherwise we could distinguish between the acceleration of a lift and a true increase of gravitation by optical experiments; in that case the observer for whom light-rays appear to take straight tracks might be described as absolutely unaccelerated and there could be no relativity theory. Physicists in general have been prepared to admit the likelihood of an influence of gravitation on light similar to that exerted on matter; and the problem whether or not light has "weight" has often been considered.

The appearance of $\gamma$ as the coefficient of $dt^{2}$ is responsible for the main features of Newtonian gravitation; the appearance of ${\tfrac {1}{\gamma }}$ as the coefficient of $dr^{2}$ is responsible for the principal deviations of the new law from the old. This classification seems to be correct; but the Newtonian law is ambiguous and it is difficult to say exactly what are to be regarded as discrepancies from it. Leaving aside now the time-term as sufficiently discussed, we consider the space-terms alone^[6] $ds^{2}={\frac {1}{\gamma }}\,dr^{2}+r^{2}d\theta ^{2}$

The expression shows that space considered alone is non-Euclidean in the neighbourhood of an attracting particle. This is something entirely outside the scope of the old law of gravitation. Time can only be explored by something moving, whether a free particle or the parts of a clock, so that the non-Euclidean character of space-time can be covered up by introducing a field of force, suitably modifying the motion, as a convenient fiction. But space can be explored by static methods; and theoretically its non-Euclidean character could be ascertained by sufficiently precise measures with rigid scales.

If we lay our measuring scale transversely and proceed to measure the circumference of a circle of nominal radius $r$, we see from the formula that the measured length $ds$ is equal to $r\,d\theta$, so that, when we have gone right round the circle, $\theta$ has increased by $2\pi$ and the measured circumference is $2\pi r$. But when we lay the scale radially the measured length $ds$ is equal to $dr/{\sqrt {\gamma }}$, which is always greater than $dr$. Thus, in measuring a diameter, we obtain a result greater than $2r$, each portion being greater than the corresponding change of $r$.

Thus if we draw a circle, placing a massive particle near the centre so as to produce a gravitational field, and measure with a rigid scale the circumference and the diameter, the ratio of the measured circumference to the measured diameter will not be the famous number $\pi =3.141592653589793238462643383279\ldots$ but a little smaller. Or if we inscribe a regular hexagon in this circle its sides will not be exactly equal to the radius of the circle. Placing the particle near, instead of at, the centre, avoids measuring the diameter through the particle, and so makes the experiment a practical one. But though practical, it is not practicable to determine the non-Euclidean character of space in this way. Sufficient refinement of measures is not attainable. If the mass of a ton were placed inside a circle of 5 yards radius, the defect in the value of $\pi$ would only appear in the twenty-fourth or twenty-fifth place of decimals.

It is of value to put the result in this way, because it shows that the relativist is not talking metaphysics when he says that space in the gravitational field is non-Euclidean. His statement has a plain physical meaning, which we may some day learn how to test experimentally. Meanwhile we can test it by indirect methods.

Suppose that a plane field is uniformly studded with hurdles. The distance between any two points will be proportional to the number of hurdles that must be passed over in getting from one point to the other by the straight route—in fact the minimum number of hurdles. We can use counts of hurdles as the equivalent of distance, and map the field by these counts. The map can be drawn on a plane sheet of paper without any inconsistency since the field is plane. Let us now dismiss from our minds all idea of distances in the field or straight lines in the field, and assume that distances on the map merely represent the minimum number of hurdles between two points; straight lines on the map will represent the corresponding routes. This has the advantage that if an earthquake occurs, deforming the field, the map will still be correct. The path of fewest hurdles will still cross the same hurdles as before the earthquake; it will be twisted out of the straight line in the field; but we should gain nothing by taking a straighter course, since that would lead through a region where the hurdles are more crowded. We do not alter the number of hurdles in any path by deforming it.

This can be illustrated by Figs. 14 and 15. Fig. 14 represents the original undistorted field with the hurdles uniformly placed. The straight line $PQ$ represents the path of fewest hurdles from $P$ to $Q$, and its length is proportional to the number of hurdles. Fig. 15 represents the distorted field, with $PQ$ distorted into a curve; but $PQ$ is still the path of fewest hurdles from $P$ to $Q$, and the number of hurdles in the path is the same as before. If therefore we map according to hurdle-counts we arrive at Fig. 14 again, just as though no deformation had taken place.

To make any difference in the hurdle-counts, the hurdles must be taken up and replanted. Starting from a given point as centre, let us arrange them so that they gradually thin out towards the boundaries of the field. Now choose a circle with this point as centre;—but first, what is a circle? It has to be defined in terms of hurdle-counts; and clearly it must be a curve such that the minimum number of hurdles between any point on it and the centre is a constant (the radius). With this definition we can defy earthquakes. The number of hurdles in the circumference of such a circle will not bear the same proportion to the number in the radius as in the field of uniform hurdles; owing to the crowding near the centre, the ratio will be less. Thus we have a suitable analogy for a circle whose circumference is less than $\pi$ times its diameter.

This analogy enables us to picture the condition of space round a heavy particle, where the ratio of the circumference of a circle to the diameter is less than $\pi$. Hurdle-counts will no longer be accurately mappable on a plane sheet of paper, because they do not conform to Euclidean geometry.

Now suppose a heavy particle wishes to cross this field, passing near but not through the centre. In Euclidean space, with the hurdles uniformly distributed, it travels in a straight line, i.e. it goes between any two points by a path giving the fewest hurdle jumps. We may assume that in the non-Euclidean field with rearranged hurdles, the particle still goes by the path of least effort. In fact, in any small portion we cannot distinguish between the rearrangement and a distortion; so we may imagine that the particle takes each portion as it comes according to the rule, and is not troubled by the rearrangement which is only visible to a general survey of the whole field^[7].

Now clearly it will pay not to go straight through the dense portion, but to keep a little to the outside where the hurdles are sparser—not too much, or the path will be unduly lengthened. The particle's track will thus be a little concave to the centre, and an onlooker will say that it has been attracted to the centre. It is rather curious that we should call it attraction, when the track has rather been avoiding the central region; but it is clear that the direction of motion has been bent round in the way attributable to an attractive force.

This bending of the path is additional to that due to the Newtonian force of gravitation which depends on the second appearance of $\gamma$ in the formula. As already explained it is in general a far smaller effect and will appear only as a minute correction to Newton's law. The only case where the two rise to equal importance is when the track is that of a light-wave, or of a particle moving with a speed approaching that of light; for then $dr^{2}$ rises to the same order of magnitude as $dt^{2}$.

To sum up, a ray of light passing near a heavy particle will be bent, firstly, owing to the non-Euclidean character of the combination of time with space. This bending is equivalent to that due to Newtonian gravitation, and may be calculated in the ordinary way on the assumption that light has weight like a material body. Secondly, it will be bent owing to the non-Euclidean character of space alone, and this curvature is additional to that predicted by Newton's law. If then we can observe the amount of curvature of a ray of light, we can make a crucial test of whether Einstein's or Newton's theory is obeyed.

This separation of the attraction into two parts is useful in a comparison of the new theory with the old; but from the point of view of relativity it is artificial. Our view is that light is bent just in the same way as the track of a material particle moving with the same velocity would be bent. Both causes of bending may be ascribed either to weight or to non-Euclidean space-time, according to the nomenclature preferred. The only difference between the predictions of the old and new theories is that in one case the weight is calculated according to Newton's law of gravitation, in the other case according to Einstein's.

There is an alternative way of viewing this effect on light according to Einstein's theory, which, for many reasons is to be preferred. This depends on the fact that the velocity of light in the gravitational field is not a constant (unity) but becomes smaller as we approach the sun. This does not mean that an observer determining the velocity of light experimentally at a spot near the sun would detect the decrease; if he performed Fizeau's experiment, his result in kilometres per second would be exactly the same as that of a terrestrial observer. It is the coordinate velocity that is here referred to, described in terms of the quantities $r$, $\theta$, $t$, introduced by the observer who is contemplating the whole solar system at the same time.

It will be remembered that in discussing the approximate geometry of space-time in Fig. 3, we found that certain events like $P$ were in the absolute past or future of $O$, and others like $P^{\prime }$ were neither before nor after $O$, but elsewhere. Analytically the distinction is that for the interval $OP$, $ds^{2}$ is positive; for $OP^{\prime }$, $ds^{2}$ is negative. In the first case the interval is real or "time-like"; in the second it is imaginary or "space-like." The two regions are separated by lines (or strictly, cones) in crossing which $ds^{2}$ changes from positive to negative; and along the lines themselves $ds$ is zero. It is clear that these lines must have important absolute significance in the geometry of the world. Physically their most important property is that pulses of light travel along these tracks, and the motion of a light-pulse is always given by the equation $ds=0$.

Using the expression for $ds^{2}$ in a gravitational field, we accordingly have for light $0=-{\frac {1}{\gamma }}\,dr^{2}--r^{2}d\theta ^{2}+\gamma \,dt^{2}$. For radial motion, $d\theta =0$, and therefore $\left({\frac {dr}{dt}}\right)^{2}=\gamma ^{2}$. For transverse motion, $dr=0$, and therefore $\left({\frac {r\,d\theta }{dt}}\right)^{2}=\gamma$. Thus the coordinate velocity of light travelling radially is $\gamma$, and of light travelling transversely is ${\sqrt {\gamma }}$, in the coordinates chosen.

The coordinate velocity must depend on the coordinates chosen; and it is more convenient to use a slightly different system in which the velocity of light is the same in all directions^[8], viz. $\gamma$ or $1-2m/r$. This diminishes as we approach the sun—an illustration of our previous remark that a pulse of light proceeding radially is repelled by the sun.

The wave-motion in a ray of light can be compared to a succession of long straight waves rolling onward in the sea. If the motion of the waves is slower at one end than the other, the whole wave-front must gradually slew round, and the direction in which it is rolling must change. In the sea this happens when one end of the wave reaches shallow water before the other, because the speed in shallow water is slower. It is well known that this causes waves proceeding diagonally across a bay to slew round and come in parallel to the shore; the advanced end is delayed in the shallow water and waits for the other. In the same way when the light waves pass near the sun, the end nearest the sun has the smaller velocity and the wave-front slews round; thus the course of the waves is bent.

Light moves more slowly in a material medium than in vacuum, the velocity being inversely proportional to the refractive index of the medium. The phenomenon of refraction is in fact caused by a slewing of the wave-front in passing into a region of smaller velocity. We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light. To give the velocity $1-2m/r$, the refractive index must be $1/(1-2m/r)$, or, very approximately, $1+2m/r$. At the surface of the sun, $r=697,000$ km., $m=1.47$ km., hence the necessary refractive index is 1.00000424. At a height above the sun equal to the radius it is 1.00000212.

Any problem on the paths of rays near the sun can now be solved by the methods of geometrical optics applied to the equivalent refracting medium. It is not difficult to show that the total deflection of a ray of light passing at a distance $r$ from the centre of the sun is (in circular measure) ${\frac {4m}{r}}$, whereas the deflection of the same ray calculated on the Newtonian theory would be ${\frac {2m}{r}}$.

For a ray grazing the surface of the sun the numerical value of this deflection is

1".75 (Einstein's theory),
0".87 (Newtonian theory).

1. Appendix, Note 6.
2. Appendix, Note 7.
3. Appendix, Note 8.
4. This is usually the object, though maps are sometimes made for a different purpose, e.g. Mercator's Chart.
5. The substitution $x=r+{\tfrac {1}{2}}t^{2}m/r$, $y=t(1-m/r)$, gives $ds*=-dx\,z+dy*$, if squares of $m$ are negligible. The map is drawn with $x$ and $y$ as rectangular coordinates.
6. We change the sign of $ds^{2}$, so that $ds$, when real, means measured space instead of measured time.
7. There must be some absolute track, and if absolute significance can only be associated with hurdle-counts and not with distances in the field, the path of fewest hurdles is the only track capable of absolute definition.
8. This is obtained by writing $r+m$ instead of $r$, or diminishing the nominal distance of the sun by 1 1/2 kilometres. This change of coordinates simplifies the problem, but can, of course, make no difference to anything observable. After we have traced the course of the light ray in the coordinates chosen, we have to connect the results with experimental measures, using the corresponding formula for $ds^{2}$. This final connection of mathematical and experimental results is, however, comparatively simple, because it relates to measuring operations performed in a terrestrial observatory where the difference of $\gamma$ from unity is negligible.