Space Time and Gravitation/Chapter 5

by Arthur Eddington

 

CHAPTER V

KINDS OF SPACE

The danger of asserting dogmatically that an axiom based on the experience of a limited region holds universally will now be to some extent apparent to the reader. It may lead us to entirely overlook, or when suggested at once reject, a possible explanation of phenomena. The hypothesis that space is not homaloidal [flat], and again that its geometrical character may change with the time, may or may not be destined to play a great part in the physics of the future; yet we cannot refuse to consider them as possible explanations of physical phenomena, because they may be opposed to the popular dogmatic belief in the universality of certain geometrical axioms—a belief which has risen from centuries of indiscriminating worship of the genius of Euclid.

W. K. Clifford (and K. Pearson), Common Sense of the Exact Sciences.

On any surface it requires two independent numbers or "coordinates" to specify the position of a point. For this reason a surface, whether flat or curved, is called a two-dimensional space. Points in three-dimensional space require three, and in four-dimensional space-time four numbers or coordinates.

To locate a point on a surface by two numbers, we divide the surface into meshes by any two systems of lines which cross one another. Attaching consecutive numbers to the lines, or better to the channels between them, one number from each system will identify a particular mesh; and if the subdivision is sufficiently fine any point can be specified in this way with all the accuracy needed. This method is used, for example, in the Post Office Directory of London for giving the location of streets on the map. The point (4, 2) will be a point in the mesh where channel No. 4 of the first system crosses channel No. 2 of the second. If this indication is not sufficiently accurate, we must divide channel No. 4 into ten parts numbered 4.0, 4.1, etc. The subdivision must be continued until the meshes are so small that all points in one mesh can be considered identical within the limits of experimental detection.

The diagrams, Figs. 10, 11, 12, illustrate three of the many kinds of mesh-systems commonly used on a flat surface.

If we speak of the properties of the triangle formed by the points (1, 2), (3, 0), (4, 4), we shall be at once asked, What meshsystem are you using? No one can form a picture of the triangle until that information has been given. But if we speak of the properties of a triangle whose sides are of lengths 2, 3, 4 inches, anyone with a graduated scale can draw the triangle, and follow our discussion of its properties. The distance between two points can be stated without referring to any mesh-system. For this reason, if we use a mesh-system, it is important to find formulae connecting the absolute distance with the particular system that is being used.

In the more complicated kinds of mesh-systems it makes a great simplification if we content ourselves with the formulae for very short distances. The mathematician then finds no difficulty in extending the results to long distances by the process called integration. We write $ds$ for the distance between two points

close together, $x_{1}$ and $x_{2}$ for the two numbers specifying the location of one of them, $dx_{1}$ and $dx_{2}$ for the small differences of these numbers in passing from the first point to the second. But in using one of the particular mesh-systems illustrated in the diagrams, we usually replace $x_{1}$, $x_{2}$ by particular symbols sanctioned by custom, viz. $(x_{1},x_{2})$ becomes $(x,y)$, $(r,\theta )$, $(\xi ,\eta )$ for Figs. 10, 11, 12, respectively.

The formulae, found by geometry, are:

⁠For rectangular coordinates $(x,y)$, Fig. 10, $ds^{2}=dx^{2}+dy^{2}$. ⁠For polar coordinates $(r,\theta )$, Fig. 11, $ds^{2}=dr^{2}+r^{2}d\theta ^{2}$. ⁠For oblique coordinates $(\xi ,\eta )$, Fig. 12, $ds^{2}=d\xi ^{2}-2\kappa \,d\xi \,d\eta +d\eta ^{2}$, where $\kappa$ is the cosine of the angle between the lines of partition. As an example of a mesh-system on a curved surface, we may take the lines of latitude and longitude on a sphere.

For latitude and longitude $(\beta ,\lambda )$$ds^{2}=d\beta ^{2}+\cos ^{2}\beta \,d\lambda ^{2}$.

These expressions form a test, and in fact the only possible test, of the kind of coordinates we are using. It may perhaps seem inconceivable that an observer should for an instant be in doubt whether he was using the mesh-system of Fig. 10 or Fig. 11. He sees at a glance that Fig. 11 is not what he would call a rectangular mesh-system. But in that glance, he makes measures with his eye, that is to say he determines $ds$ for pairs of points, and he notices how these values are related to the number of intervening channels. In fact he is testing which formula for $ds$ will fit. For centuries man was in doubt whether the earth was flat or round—whether he was using plane rectangular coordinates or some kind of spherical coordinates. In some cases an observer adopts his mesh-system blindly and long afterwards discovers by accurate measures that $ds$ does not fit the formula he assumed—that his mesh-system is not exactly of the nature he supposed it was. In other cases he deliberately sets himself to plan out a mesh-system of a particular variety, say rectangular coordinates; he constructs right angles and rules parallel lines; but these constructions are all measurements of the way the $x$-channels and $y$-channels ought to go, and the rules of construction reduce to a formula connecting his measures $ds$ with $x$ and $y$.

The use of special symbols for the coordinates, varying according to the kind of mesh-system used, thus anticipates a knowledge which is really derived from the form of the formulae. In order not to give away the secret prematurely, it will be better to use the symbols $x_{1}$, $x_{2}$ in all cases. The four kinds of coordinates already considered then give respectively the relations,${\begin{aligned}ds^{2}&=d{x_{1}}^{2}+d{x_{2}}^{2}&{\text{(rectangular),}}\\ds^{2}&=d{x_{1}}^{2}+{x_{1}}^{2}d{x_{2}}^{2}&{\text{(polar),}}\\ds^{2}&=d{x_{1}}^{2}-2\kappa \,dx_{1}dx_{2}+d{x_{2}}^{2}&{\text{(oblique),}}\\ds^{2}&=d{x_{1}}^{2}+\cos ^{2}x_{1}d{x_{2}}^{2}&{\text{(latitude and longitude).}}\\\end{aligned}}$ If we have any mesh-system and want to know its nature, we must make a number of measures of the length $ds$ between adjacent points $(x_{1},x_{2})$ and $(x_{1}+dx_{1},x_{2}+dx_{2})$ and test which formula fits. If, for example, we then find that $ds^{2}$ is always equal to $d{x_{1}}^{2}+{x_{1}}^{2}d{x_{2}}^{2}$, we know that our mesh-system is like that in Fig. 11, $x_{1}$ and $x_{2}$ being the numbers usually denoted by the polar coordinates $r$, $\theta$. The statement that polar coordinates are being used is unnecessary, because it adds nothing to our knowledge which is not already contained in the formula. It is merely a matter of giving a name; but, of course, the name calls to our minds a number of familiar properties which otherwise might not occur to us.

For instance, it is characteristic of the polar coordinate system that there is only one point for which $x_{1}$ (or $r$) is equal to 0, whereas in the other systems $x_{1}=0$ gives a line of points. This is at once apparent from the formula; for if we have two points for which $x_{1}=0$ and $x_{1}dx_{1}=0$, respectively, then $d{x_{1}}^{2}+{x_{1}}^{2}d{x_{2}}^{2}=0$. The distance $ds$ between the two points vanishes, and accordingly they must be the same point.

The examples given can all be summed up in one general expression $ds^{2}=g_{11}d{x_{1}}^{2}+2g_{12}dx_{1}d{x_{2}}+g_{22}d{x_{2}}^{2}$, where $g_{11}$, $g_{12}$, $g_{22}$ may be constants or functions of $x_{1}$ and $x_{2}$. For instance, in the fourth example their values are 1, 0, $\cos ^{2}x_{1}$. It is found that all possible mesh-systems lead to values of $ds^{2}$ which can be included in an expression of this general form; so that mesh-systems are distinguished by three functions of position $g_{11}$, $g_{12}$, $g_{22}$ which can be determined by making physical measurements. These three quantities are sometimes called potentials.

We now come to a point of far-reaching importance. The formula for $ds^{2}$ teaches us not only the character of the mesh-system, but the nature of our two-dimensional space, which is independent of any mesh-system. If $ds^{2}$ satisfies any one of the first three formulae, then the space is like a flat surface; if it satisfies the last formula, then the space is a surface curved like a sphere. Try how you will, you cannot draw a mesh-system on a flat (Euclidean) surface which agrees with the fourth formula.

If a being limited to a two-dimensional world finds that his measures agree with the first formula, he can make them agree with the second or third formulae by drawing the meshes differently. But to obtain the fourth formula he must be translated to a different world altogether.

We thus see that there are different kinds of two-dimensional space, betrayed by different metrical properties. They are naturally visualised as different surfaces in Euclidean space of three dimensions. This picture is helpful in some ways, but perhaps misleading in others. The metrical relations on a plane sheet of paper are not altered when the paper is rolled into a cylinder—the measures being, of course, confined to the two-dimensional world represented by the paper, and not allowed to take a short cut through space. The formulae apply equally well to a plane surface or a cylindrical surface; and in so far as our picture draws a distinction between a plane and a cylinder, it is misleading. But they do not apply to a sphere, because a plane sheet of paper cannot be wrapped round a sphere. A genuinely two-dimensional being could not be cognisant of the difference between a cylinder^[1] and a plane; but a sphere would appear as a different kind of space, and he would recognise the difference by measurement.

Of course there are many kinds of mesh-systems, and many kinds of two-dimensional spaces, besides those illustrated in the four examples. Clearly it is not going to be a simple matter to discriminate the different kinds of spaces by the values of the $g$'s. There is no characteristic, visible to cursory inspection, which suggests why the first three formulae should all belong to the same kind of space, and the fourth to a different one. Mathematical investigation has discovered what is the common link between the first three formulae. The $g_{11}$, $g_{12}$, $g_{22}$ satisfy in all three cases a certain differential equation^[2]; and whenever this differential equation is satisfied, the same kind of space occurs.

No doubt it seems a very clumsy way of approaching these intrinsic differences of kinds of space to introduce potentials which specifically refer to a particular mesh-system, although the mesh-system can have nothing to do with the matter. It is worrying not to be able to express the differences of space in a purer form without mixing them up with irrelevant differences of potential. But we have neither the vocabulary nor the imagination for a description of absolute properties as such. All physical knowledge is relative to space and time partitions; and to gain an understanding of the absolute it is necessary to approach it through the relative. The absolute may be defined as a relative which is always the same no matter what it is relative to^[3]. Although we think of it as self-existing, we cannot give it a place in our knowledge without setting up some dummy to relate it to. And similarly the absolute differences of space always appear as related to some mesh-system, although the mesh-system is only a dummy and has nothing to do with the problem.

The results for two dimensions can be generalised, and applied to four-dimensional space-time. Distance must be replaced by interval, which it will be remembered, is an absolute quantity, and therefore independent of the mesh-system used. Partitioning space-time by any system of meshes, a mesh being given by the crossing of four channels, we must specify a point in space-time by four coordinate numbers, $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$. By analogy the general formula will be ${\begin{aligned}ds^{2}&=g_{11}d{x_{1}}^{2}+g_{22}d{x_{2}}^{2}+g_{33}d{x_{3}}^{2}+g_{44}d{x_{4}}^{2}+2g_{12}dx_{1}dx_{2}\\&\qquad {}+2g_{13}dx_{1}dx_{3}+2g_{14}dx_{1}dx_{4}+2g_{23}dx_{2}dx_{3}\\&\qquad {}+2g_{24}dx_{2}dx_{4}+2g_{34}dx_{3}dx_{4}\quad {\text{........................... (3).}}\end{aligned}}$ The only difference is that there are now ten $g$'s, or potentials, instead of three, to summarise the metrical properties of the mesh-system. It is convenient in specifying special values of the potentials to arrange them in the standard form ${\begin{matrix}g_{11}&g_{12}&g_{13}&g_{14}\\&g_{22}&g_{23}&g_{24}\\&&g_{33}&g_{34}\\&&&g_{44}\end{matrix}}$ The space-time already discussed at length in Chapter iii corresponded to the formula (2), p. 75, $ds^{2}=-dx^{2}-dy^{2}-dz^{2}+dt^{2}$. Here ($x$, $y$, $z$, $t$) are the conventional symbols for ($x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$) when this special mesh-system is used, viz. rectangular coordinates and time. Comparing with (3) the potentials have the special values ${\begin{matrix}-1&0&0&0\\&-1&0&0\\&&-1&0\\&&&+1\end{matrix}}$ These are called the "Galilean values." If the potentials have these values everywhere, space-time may be called "flat," because the geometry is that of a plane surface drawn in Euclidean space of five dimensions. Recollecting what we found for two dimensions, we shall realise that a quite different set of values of the potentials may also belong to flat space-time, because the meshes may be drawn in different ways. We must clearly understand that

(1) The only way of discovering what kind of space-time is being dealt with is from the values of the potentials, which are determined practically by measurements of intervals,

(2) Different values of the potentials do not necessarily indicate different kinds of space-time,

(3) There is some complicated mathematical property common to all values of the potentials which belong to the same space-time, which is not shared by those which belong to a different kind of space-time. This property is expressed by a set of differential equations.

It can now be deduced that the space-time in which we live is not quite flat. If it were, a mesh-system could be drawn for which the $g$'s have the Galilean values, and the geometry with respect to these partitions of space and time would be that discussed in Chapter iii. For that geometry the geodesics, giving the natural tracks of particles, are straight lines.

Thus in flat space-time the law of motion is that (with suitably chosen coordinates) every particle moves uniformly in a straight line except when it is disturbed by the impacts of other particles. Clearly this is not true of our world; for example, the planets do not move in straight lines although they do not suffer any impacts. It is true that if we confine attention to a small region like the interior of Jules Verne's projectile, all the tracks become straight lines for an appropriate observer, or, as we generally say, he detects no field of force. It needs a large region to bring out the differences of geometry. That is not surprising, because we cannot expect to tell whether a surface is flat or curved unless we consider a reasonably large portion of it.

According to Newtonian ideas, at a great distance from all matter beyond the reach of any gravitation, particles would all move uniformly in straight lines. Thus at a great distance from all matter space-time tends to become perfectly flat. This can only be checked by experiment to a certain degree of accuracy, and there is some doubt as to whether it is rigorously true. We shall leave this afterthought to Chapter x, meanwhile assuming with Newton that space-time far enough away from everything is flat, although near matter it is curved. It is this puckering near matter which accounts for its gravitational effects.

Just as we picture different kinds of two-dimensional space as differently curved surfaces in our ordinary space of three-dimensions, so we are now picturing different kinds of four-dimensional space-time as differently curved surfaces in a Euclidean space of five dimensions. This is a picture only^[4]. The fifth dimension is neither space nor time nor anything that can be perceived; so far as we know, it is nonsense. I should not describe it as a mathematical fiction, because it is of no great advantage in a mathematical treatment. It is even liable to mislead because it draws distinctions, like the distinction be tween a plane and a roll, which have no meaning. It is, like the notion of a field of force acting in space and time, merely introduced to bolster up Euclidean geometry, when Euclidean geometry has been found inappropriate. The real difference between the various kinds of space-time is that they have different kinds of geometry, involving different properties of the $g$'s. It is no explanation to say that this is because the surfaces are differently curved in a real Euclidean space of five dimensions. We should naturally ask for an explanation why the space of five dimensions is Euclidean; and presumably the answer would be, because it is a plane in a real Euclidean space of six dimensions, and so on ad infinitum.

The value of the picture to us is that it enables us to describe important properties with common terms like "pucker" and "curvature" instead of technical terms like "differential invariant." We have, however, to be on our guard, because analogies based on three-dimensional space do not always apply immediately to many-dimensional space. The writer has keen recollections of a period of much perplexity, when he had not realised that a four-dimensional space with "no curvature" is not the same as a "flat" space! Three-dimensional geometry does not prepare us for these surprises.

Picturing the space-time in the gravitational field round the earth as a pucker, we notice that we cannot locate the pucker at a point; it is "somewhere round" the point. At any special point the pucker can be pressed out flat, and the irregularity runs off somewhere else. That is what the inhabitants of Jules Verne's projectile did; they flattened out the pucker inside the projectile so that they could not detect any field of force there; but this only made things worse somewhere else, and they would find an increased field of force (relative to them) on the other side of the earth.

What determines the existence of the pucker is not the values of the $g$'s at any point, or, what comes to the same thing, the field of force there. It is the way these values link on to those at other points—the gradient of the $g$'s, and more particularly the gradient of the gradient. Or, as has already been said, the kind of space-time is fixed by differential equations.

Thus, although a gravitational field of force is not an absolute thing, and can be imitated or annulled at any point by an acceleration of the observer or a change of his mesh-system, nevertheless the presence of a heavy particle does modify the world around it in an absolute way which cannot be imitated artificially. Gravitational force is relative; but there is this more complex character of gravitational influence which is absolute.

The question must now be put, Can every possible kind of space-time occur in an empty region in nature? Suppose we give the ten potentials perfectly arbitrary values at every point; that will specify the geometry of some mathematically possible space-time. But could that kind of space-time actually occur—by any arrangement of the matter round the region?

The answer is that only certain kinds of space-time can occur in an empty region in nature. The law which determines what kinds can occur is the law of gravitation.

It is indeed clear that, since we have reduced the theory of fields of force to a theory of the geometry of the world, if there is any law governing fields of force (including the gravitational field), that law must be of the nature of a restriction on the possible geometries of the world.

The choice of $g$'s in any special problem is thus arrived at by a three-fold sorting out: (1) many sets of values can be dismissed because they can never occur in nature, (2) others, while possible, do not relate to the kind of space-time present in the problem considered, (3) of those which remain, one set of values relates to the particular mesh-system that has been chosen. We have now to find the law governing the first discrimination. What is the criterion that decides what values of the $g$'s give a kind of space-time possible in nature?

In solving this problem Einstein had only two clues to guide him.

(1) Since it is a question of whether the kind of space-time is possible, the criterion must refer to those properties of the $g$'s which distinguish different kinds of space-time, not to those which distinguish different kinds of mesh-system in the same space-time. The formulae must therefore not be altered in any way, if we change the mesh-system.

(2) We know that flat space-time can occur in nature (at great distances from all gravitating matter). Hence the criterion must be satisfied by any values of the $g$'s belonging to flat space-time.

It is remarkable that these slender clues are sufficient to indicate almost uniquely a particular law. Afterwards the further test must be applied—whether the law is confirmed by observation.

The irrelevance of the mesh-system to the laws of nature is sometimes expressed in a slightly different way. There is one type of observation which, we can scarcely doubt, must be independent of any possible circumstances of the observer, namely a complete coincidence in space and time. The track of a particle through four-dimensional space-time is called its world-line. Now, the world-lines of two particles either intersect or they do not intersect; the standpoint of the observer is not involved. In so far as our knowledge of nature is a knowledge of intersections of world-lines, it is absolute knowledge independent of the observer. If we examine the nature of our observations, distinguishing what is actually seen from what is merely inferred, we find that, at least in all exact measurements, our knowledge is primarily built up of intersections of world-lines of two or more entities, that is to say their coincidences. For example, an electrician states that he has observed a current of 5 milliamperes. This is his inference: his actual observation was a coincidence of the image of a wire in his galvanometer with a division of a scale. A meteorologist finds that the temperature of the air is 75°; his observation was the coincidence of the top of the mercury-thread with division 75 on the scale of his thermometer. It would be extremely clumsy to describe the results of the simplest physical experiment entirely in terms of coincidence. The absolute observation is, whether or not the coincidence exists, not when or where or under what circumstances the coincidence exists; unless we are to resort to relative knowledge, the place, time and other circumstances must in their turn be described by reference to other coincidences. But it seems clear that if we could draw all the world-lines so as to show all the intersections in their proper order, but otherwise arbitrary, this would contain a complete history of the world, and nothing within reach of observation would be omitted.

Let us draw such a picture, and imagine it embedded in a jelly. If we deform the jelly in any way, the intersections will still occur in the same order along each world-line and no additional intersections will be created. The deformed jelly will represent a history of the world, just as accurate as the one originally drawn; there can be no criterion for distinguishing which is the best representation.

Suppose now we introduce space and time-partitions, which we might do by drawing rectangular meshes in both jellies. We have now two ways of locating the world-lines and events in space and time, both on the same absolute footing. But clearly it makes no difference in the result of the location whether we first deform the jelly and then introduce regular meshes, or whether we introduce irregular meshes in the undeformed jelly. And so all mesh-systems are on the same footing.

This account of our observational knowledge of nature shows that there is no shape inherent in the absolute world, so that when we insert a mesh-system, it has no shape initially, and a rectangular mesh-system is intrinsically no different from any other mesh-system.

Returning to our two clues, condition (1) makes an extraordinarily clean sweep of laws that might be suggested; among them Newton's law is swept away. The mode of rejection can be seen by an example; it will be sufficient to consider two dimensions. If in one mesh-system $(x,y)$ $ds^{2}=g_{11}dx^{2}+2g_{12}dx\,dy+g_{22}dy^{2}$, and in another system $(x^{\prime },y^{\prime })$ $ds^{2}={g_{11}}^{\prime }{dx}^{\prime 2}+2{g_{12}}^{\prime }dx^{\prime }dy^{\prime }+{g_{22}}^{\prime }dy^{\prime 2}$, the same law must be satisfied if the unaccented letters are throughout replaced by accented letters. Suppose the law $g_{11}=g_{22}$ is suggested. Change the mesh-system by spacing the $y$-lines twice as far apart, that is to say take $y^{\prime }={\tfrac {1}{2}}y$, with $x^{\prime }=x$. Then ${\begin{aligned}ds^{2}&=g_{11}dx^{2}+2g_{12}dx\,dy+g_{22}dy^{2}\\&=g_{11}dx^{\prime 2}+4g_{12}dx^{\prime }dy^{\prime }+4g_{22}dy^{\prime 2}{\text{,}}\end{aligned}}$ so that ${\begin{aligned}{g_{11}}^{\prime }&=g_{11}&{g_{22}}^{\prime }&=4g_{22}\end{aligned}}$. And if $g_{11}$ is equal to $g_{22}$, ${g_{11}}^{\prime }$ cannot be equal to ${g_{22}}^{\prime }$.

After a few trials the reader will begin to be surprised that any possible law could survive the test. It seems so easy to defeat any formula that is set up by a simple change of mesh-system. Certainly it is unlikely that anyone would hit on such a law by trial. But there are such laws, composed of exceedingly complicated mathematical expressions. The theory of these is called the "theory of tensors," and had already been worked out by the pure mathematicians Riemann, Christoffel, Ricci, Levi-Civita who, it may be presumed, never dreamt of a physical application for it.

One law of this kind is the condition for flat space-time, which is generally written in the simple, but not very illuminating, form $B_{\nu \sigma \mu }^{\rho }=0$ ...........................(4). The quantity on the left is called the Riemann-Christoffel tensor, and it is written out in a less abbreviated form in the Appendix^[5]. It must be explained that the letters $\rho$, $\nu$, $\sigma$, $\mu$ indicate gaps, which are to be filled up by any of the numbers 1, 2, 3, 4, chosen at pleasure. (When the expression is written out at length, the gaps are in the suffixes of the $x$'s and $g$'s.) Filling the gaps in different ways, a large number of expressions, $B_{111}^{1}$, $B_{123}^{4}$, $B_{432}^{1}$, etc., are obtained. The equation (4) states that all of these are zero. There are 4⁴, or 256, of these expressions altogether, but many of them are repetitions. Only 20 of the equations are really necessary; the others merely say the same thing over again.

It is clear that the law (4) is not the law of gravitation for which we are seeking, because it is much too drastic. If it were a law of nature, then only flat space-time could exist in nature, and there would be no such thing as gravitation. It is not the general condition, but a special case—when all attracting matter is infinitely remote.

But in finding a general condition, it may be a great help to know a special case. Would it do to select a certain number of the 20 equations to be satisfied generally, leaving the rest to be satisfied only in the special case? Unfortunately the equations hang together; and, unless we take them all, it is found that the condition is not independent of the mesh-system. But there happens to be one way of building up out of the 20 conditions a less stringent set of conditions independent of the mesh-system. Let $G_{11}=B_{111}^{1}+B_{112}^{2}+B_{113}^{3}+B_{114}^{4}$, and, generally $G_{\mu \nu }=B_{\mu \nu 1}^{1}+B_{\mu \nu 2}^{2}+B_{\mu \nu 3}^{3}+B_{\mu \nu 4}^{4}$, then the conditions $G_{\mu \nu }=0$ ...........................(5), will satisfy our requirements for a general law of nature.

This law is independent of the mesh-system, though this can only be proved by elaborate mathematical analysis. Evidently, when all the $B$'s vanish, equation (5) is satisfied; so, when flat space-time occurs, this law of nature is not violated. Further it is not so stringent as the condition for flatness, and admits of the occurrence of a limited variety of non-Euclidean geometries. Rejecting duplicates, it comprises 10 equations; but four of these can be derived from the other six, so that it gives six conditions, which happens to be the number required for a law of gravitation^[6].

The suggestion is thus reached that $G_{\mu \nu }=0$ may be the general law of gravitation. Whether it is so or not can only be settled by experiment. In particular, it must in ordinary cases reduce to something so near the Newtonian law, that the remarkable confirmation of the latter by observation is accounted for. Further it is necessary to examine whether there are any exceptional cases in which the difference between it and Newton's law can be tested. We shall see that these tests are satisfied.

What would have been the position if this suggested law had failed? We might continue the search for other laws satisfying the two conditions laid down; but these would certainly be far more complicated mathematically. I believe too that they would not help much, because practically they would be indistinguishable from the simpler law here suggested—though this has not been demonstrated rigorously. The other alternative is that there is something causing force in nature not comprised in the geometrical scheme hitherto considered, so that force is not purely relative, and Newton's super-observer exists.

Perhaps the best survey of the meaning of our theory can be obtained from the standpoint of a ten-dimensional Euclidean continuum, in which space-time is conceived as a particular four-dimensional surface. It has to be remarked that in ten dimensions there are gradations intermediate between a flat surface and a fully curved surface, which we shall speak of as curved in the "first degree" or "second degree^[7]." The distinction is something like that of curves in ordinary space, which may be curved like a circle, or twisted like a helix; but the analogy is not very close. The full "curvature" of a surface is a single quantity called $G$, built up out of the various terms $G_{\mu \nu }$ in somewhat the same way as these are built up out of $B_{\mu \nu \sigma }^{\rho }$.The following conclusions can be stated.

If

$B_{\mu \nu \sigma }^{\rho }=0$

(20 conditions)

space-time is flat. This is the state of the world at an infinite distance from all matter and all forms of energy.

If

$G_{\mu \nu }=0$

(6 conditions)

space-time is curved in the first degree. This is the state of the world in an empty region—not containing matter, light or electromagnetic fields, but in the neighbourhood of these forms of energy.

If

$G=0$

(1 condition)

space-time is curved in the second degree. This is the state of the world in a region not containing matter or electrons (bound energy), but containing light or electromagnetic fields (free energy).

If

$G{\text{ is not zero}}$

space-time is fully curved. This is the state of the world in a region containing continuous matter.

According to current physical theory continuous matter does not exist, so that strictly speaking the last case never arises. Matter is built of electrons or other nuclei. The regions lying between the electrons are not fully curved, whilst the regions inside the electrons must be cut out of space-time altogether. We cannot imagine ourselves exploring the inside of an electron with moving particles, light-waves, or material clocks and measuring-rods; hence, without further definition, any geometry of the interior, or any statement about space and time in the interior, is meaningless. But in common life, and frequently in physics, we are not concerned with this microscopic structure of matter. We need to know, not the actual values of the $g$'s at a point, but their average values through a region, small from the ordinary standpoint but large compared with the molecular structure of matter. In this macroscopic treatment molecular matter is replaced by continuous matter, and uncurved space-time studded with holes is replaced by an equivalent fully curved space-time without holes.

It is natural that our senses should have developed faculties for perceiving some of these intrinsic distinctions of the possible states of the world around us. I prefer to think of matter and energy, not as agents causing the degrees of curvature of the world, but as parts of our perceptions of the existence of the curvature.

It will be seen that the law of gravitation can be summed up in the statement that in an empty region space-time can be curved only in the first degree.

1. One should perhaps rather say a roll, to avoid any question of joining the two edges.
2. Appendix, Note 4.
3. Cf. p. 31, where a distinction was drawn between knowledge which does not particularise the observer and knowledge which does not postulate an observer at all.
4. A fifth dimension suffices for illustrating the property here considered; but for an exact representation of the geometry of the world, Euclidean space of ten dimensions is required. We may well ask whether there is merit in Euclidean geometry sufficient to justify going to such extremes.
5. Note 5.
6. Isolate a region of empty space-time; and suppose that everywhere outside the region the potentials are known. It should then be possible by the law of gravitation to determine the nature of space-time in the region. Ten differential equations together with the boundary-values would suffice to determine the ten potentials throughout the region; but that would determine not only the kind of space-time but the mesh-system, whereas the partitions of the mesh-system can be continued across the region in any arbitrary way. The four sets of partitions give a four-fold arbitrariness; and to admit of this, the number of equations required is reduced to six.
7. This is not a recognised nomenclature.