Space Time and Gravitation/Chapter 11

by Arthur Eddington

 

CHAPTER XI

ELECTRICITY AND GRAVITATION

Thou shalt not have in thy bag divers weights, a great and a small.
Thou shalt not have in thine house divers measures, a great and a small.
But thou shalt have a perfect and just weight, a perfect and just measure shalt thou have.

Book of Deuteronomy.

The relativity theory deduces from geometrical principles the existence of gravitation and the laws of mechanics of matter. Mechanics is derived from geometry, not by adding arbitrary hypotheses, but by removing unnecessary assumptions, so that a geometer like Riemann might almost have foreseen the more important features of the actual world. But nature has in reserve one great surprise—electricity.

Electrical phenomena are not in any way a misfit in the relativity theory, and historically it is through them that it has been developed. Yet we cannot rest satisfied until a deeper unity between the gravitational and electrical properties of the world is apparent. The electron, which seems to be the smallest particle of matter, is a singularity in the gravitational field and also a singularity in the electrical field. How can these two facts be connected? The gravitational field is the expression of some state of the world, which also manifests itself in the natural geometry determined with measuring appliances; the electric field must also express some state of the world, but we have not as yet connected it with natural geometry. May there not still be unnecessary assumptions to be removed, so that a yet more comprehensive geometry can be found, in which gravitational and electrical fields both have their place?

There is an arbitrary assumption in our geometry up to this point, which it is desirable now to point out. We have based everything on the "interval," which, it has been said, is some thing which all observers, whatever their motion or whatever their mesh-system, can measure absolutely, agreeing on the result. This assumes that they are provided with identical standards of measurement—scales and clocks. But if $A$ is in motion relative to $B$ and wishes to hand his standards to $B$ to check his measures, he must stop their motion; this means in practice that he must bombard his standards with material molecules until they come to rest. Is it fair to assume that no alteration of the standard is caused by this process? Or if $A$ measures time by the vibrations of a hydrogen atom, and space by the wave-length of the vibration, still it is necessary to stop the atom by a collision in which electrical forces are involved.

The standard of length in physics is the length in the year 1799 of a bar deposited at Paris. Obviously no interval is ever compared directly with that length; there must be a continuous chain of intermediate steps extending like a geodetic triangulation through space and time, first along the past history of the scale actually used, then through intermediate standards, and finally along the history of the Paris metre itself. It may be that these intermediate steps are of no importance—that the same result is reached by whatever route we approach the standard; but clearly we ought not to make that assumption without due consideration. We ought to construct our geometry in such a way as to show that there are intermediate steps, and that the comparison of the interval with the ultimate standard is not a kind of action at a distance.

To compare intervals in different directions at a point in space and time does not require this comparison with a distant standard. The physicist's method of describing phenomena near a point $P$ is to lay down for comparison (1) a mesh-system, (2) a unit of length (some kind of material standard), which can also be used for measuring time, the velocity of light being unity. With this system of reference he can measure in terms of his unit small intervals $PP^{\prime }$ running in any direction from $P$, summarising the results in the fundamental formula $ds^{2}=g_{11}\,{dx_{1}}^{2}+g_{22}\,{dx_{2}}^{2}+\ldots +2g_{12}\,dx_{1}\,dx_{2}+\ldots$ If now he wishes to measure intervals near a distant point $Q$, he must lay down a mesh-system and a unit of measure there. He naturally tries to simplify matters by using what he would call the same unit of measure at $P$ and $Q$, either by transporting a material rod or some equivalent device. If it is immaterial by what route the unit is carried from $P$ to $Q$, and replicas of the unit carried by different routes all agree on arrival at $Q$, this method is at any rate explicit. The question whether the unit at $Q$ defined in this way is really the same as that at $P$ is mere metaphysics. But if the units carried by different routes disagree, there is no unambiguous means of identifying a unit at $Q$ with the unit at $P$. Suppose $P$ is an event at Cambridge on March 1, and $Q$ at London on May 1; we are contemplating the possibility that there will be a difference in the results of measures made with our standard in London on May 1, according as the standard is taken up to London on March 1 and remains there, or is left at Cambridge and taken up on May 1. This seems at first very improbable; but our reasons for allowing for this possibility will appear presently. If there is this ambiguity the only possible course is to lay down (1) a mesh-system filling all the space and time considered, (2) a definite unit of interval, or gauge, at every point of space and time. The geometry of the world referred to such a system will be more complicated than that of Riemann hitherto used; and we shall see that it is necessary to specify not only the 10 $g$'s, but four other functions of position, which will be found to have an important physical meaning.

The observer will naturally simplify things by making the units of gauge at different points as nearly as possible equal, judged by ordinary comparisons. But the fact remains that, when the comparison depends on the route taken, exact equality is not definable; and we have therefore to admit that the exact standards are laid down at every point independently.

It is the same problem over again as occurs in regard to mesh-systems. We lay down particular rectangular axes near a point $P$; presently we make some observations near a distant point $Q$. To what coordinates shall the latter be referred? The natural answer is that we must use the same coordinates as we were using at $P$. But, except in the particular case of flat space, there is no means of defining exactly what coordinates at $Q$ are the same as those at $P$. In many cases the ambiguity may be too trifling to trouble us; but in exact work the only course is to lay down a definite mesh-system extending throughout space, the precise route of the partitions being necessarily arbitrary. We now find that we have to add to this by placing in each mesh a gauge whose precise length must be arbitrary. Having done this the next step is to make measurements of intervals (using our gauges). This connects the absolute properties of the world with our arbitrarily drawn mesh-system and gauge-system. And so by measurement we determine the $g$'s and the new additional quantities, which determine the geometry of our chosen system of reference, and at the same time contain within themselves the absolute geometry of the world—the kind of space-time which exists in the field of our experiments.

Having laid down a unit-gauge at every point, we can speak quite definitely of the change in interval-length of a measuring-rod moved from point to point, meaning, of course, the change compared with the unit-gauges. Let us take a rod of interval-length $l$ at $P$, and move it successively through the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$; and let the result be to increase its length in terms of the gauges by the amount $\lambda l$. The change depends as much on the difference of the gauges at the two points as on the behaviour of the rod; but there is no possibility of separating the two factors. It is clear that $\lambda$ will not depend on $l$, because the change of length must be proportional to the original length—unless indeed our whole idea of measurement by comparison with a gauge is wrong^[1]. Further it will not depend on the direction of the rod either in its initial or final positions because the interval-length is independent of direction. (Of course, the space-length would change, but that is already taken care of by the $g$'s.) $\lambda$ can thus only depend on the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$, and we may write it $\lambda =\kappa _{1}\,dx_{1}+\kappa _{2}\,dx_{2}+\kappa _{3}\,dx_{3}+\kappa _{4}\,dx_{4}$, so long as the displacements are small. The coefficients $\kappa _{1}$, $\kappa _{2}$, $\kappa _{3}$, $\kappa _{4}$ apply to the neighbourhood of $P$, and will in general be different in different parts of space.

This indeed assumes that the result is independent of the order of the displacements $dx_{1}$, $dx_{2}$, $dx_{3}$, $dx_{4}$—that is to say that the ambiguity of the comparison by different routes disappears in the limit when the whole route is sufficiently small. It is parallel with our previous implicit assumption that although the length of the track from a point $P$ to a distant point $Q$ depends on the route, and no definite meaning can be attached to the interval between them without specifying a route, yet in the limit there is a definite small interval between $P$ and $Q$ when they are sufficiently close together.

To understand the meaning of these new coefficients $\kappa$ let us briefly recapitulate what we understand by the $g$'s. Primarily they are quantities derived from experimental measurements of intervals, and describe the geometry of the space and time partitions which the observer has chosen. As consequential properties they describe the field of force, gravitational, centrifugal, etc., with which he perceives himself surrounded. They relate to the particular mesh-system of the observer; and by altering his mesh-system, he can alter their values, though not entirely at will. From their values can be deduced intrinsic properties of the world—the kind of space-time in which the phenomena occur. Further they satisfy a definite condition—the law of gravitation—so that not all mathematically possible space-times and not all arbitrary values of the $g$'s are such as can occur in nature.

All this applies equally to the $\kappa$'s, if we substitute gauge-system for mesh-system, and some at present unknown force for gravitation. They can theoretically be determined by interval-measurement; but they will be more conspicuously manifested to the observer through their consequential property of describing some kind of field of force surrounding him. The $\kappa$'s refer to the arbitrary gauge-system of the observer; but he cannot by altering his gauge-system alter their values entirely at will. Intrinsic properties of the world are contained in their values, unaffected by any change of gauge-system. Further we may expect that they will have to satisfy some law corresponding to the law of gravitation, so that not all arbitrary values of the $\kappa$'s are such as can occur in nature.

It is evident that the $\kappa$'s must refer to some type of phenomenon which has not hitherto appeared in our discussion; and the obvious suggestion is that they refer to the electromagnetic field. This hypothesis is strengthened when we recall that the electromagnetic field is, in fact, specified at every point by the values of four quantities, viz. the three components of electromagnetic vector potential, and the scalar potential of electrostatics. Surely it is more than a coincidence that the physicist needs just four more quantities to specify the state of the world at a point in space, and four more quantities are provided by removing a rather illogical restriction on our system of geometry of natural measures.

[The general reader will perhaps pardon a few words addressed especially to the mathematical physicist. Taking the ordinary unaccelerated rectangular coordinates $x$, $y$, $z$, $t$, let us write $F$, $G$, $H$, $-\Phi$ for $\kappa _{1}$, $\kappa _{2}$, $\kappa _{3}$, $\kappa _{4}$, then ${\frac {dl}{l}}=\lambda =F\,dx+G\,dy+H\,dz-\Phi \,dt$. From which, by integration, $\log l+{\text{const.}}=\int (F\,dx+G\,dy+H\,dz-\Phi \,dt)$.

The length $l$ will be independent of the route taken if $F\,dx+G\,dy+H\,dz-\Phi \,dt$ is a perfect differential. The condition for this is ${\begin{aligned}{\frac {\partial H}{\partial y}}-{\frac {\partial G}{\partial z}}&=0{\text{,}}&{\frac {\partial F}{\partial z}}-{\frac {\partial H}{\partial x}}&=0{\text{,}}&{\frac {\partial G}{\partial x}}-{\frac {\partial F}{\partial y}}&=0{\text{,}}\\-{\frac {\partial \Phi }{\partial x}}-{\frac {\partial F}{\partial t}}&=0{\text{,}}&-{\frac {\partial \Phi }{\partial y}}-{\frac {\partial G}{\partial t}}&=0{\text{,}}&-{\frac {\partial \Phi }{\partial z}}-{\frac {\partial H}{\partial t}}&=0{\text{.}}\end{aligned}}$ If $F$, $G$, $H$, $\Phi$ are the potentials of electromagnetic theory, these are precisely the expressions for the three components of magnetic force and the three components of electric force, given in the text-books. Thus the condition that distant intervals can be compared directly without specifying a particular route of comparison is that the electric and magnetic forces are zero in the intervening space and time.

It may be noted that, even when the coordinate system has been defined, the electromagnetic potentials are not unique in value; but arbitrary additions can be made provided these additions form a perfect differential. It is just this flexibility which in our geometrical theory appears in the form of the arbitrary choice of gauge-system. The electromagnetic forces on the other hand are independent of the gauge-system, which is eliminated by "curling."]

It thus appears that the four new quantities appearing in our extended geometry may actually be the four potentials of electromagnetic theory; and further, when there is no electromagnetic field our previous geometry is valid. But in the more general case we have to adopt the more general geometry in which there appear fourteen coefficients, ten describing the gravitational and four the electrical conditions of the world.

We ought now to seek the law of the electromagnetic field on the same lines as we sought for the law of gravitation, laying down the condition that it must be independent of mesh-system and gauge-system since it seeks to limit the possible kinds of world which can exist in nature. Happily this presents no difficulty, because the law expressed by Maxwell's equations, and universally adopted, fulfils the conditions. There is no need to modify it fundamentally as we modified the law of gravitation. We do, however, generalise it so that it applies when a gravitational field is present at the same time—not merely, as given by Maxwell, for flat space-time. The deflection of electromagnetic waves (light) by a gravitational field is duly contained in this generalised law.

Strictly speaking the laws of gravitation and of the electromagnetic field are not two laws but one law, as the geometry of the $g$'s and the $\kappa$'s is one geometry. Although it is often convenient to separate them, they are really parts of the general condition limiting the possible kinds of metric that can occur in empty space.

It will be remembered that the four-fold arbitrariness of our mesh-system involved four identities, which were found to express the conservation of energy and momentum. In the new geometry there is a fifth arbitrariness, namely that of the selected gauge-system. This must also give rise to an identity; and it is found that the new identity expresses the law of conservation of electric charge.

A grasp of the new geometry may perhaps be assisted by a further comparison. Suppose an observer has laid down a line of a certain length and in a certain direction at a point $P$, and he wishes to lay down an exactly similar line at a distant point $Q$. If he is in flat space there will be no difficulty; he will have to proceed by steps, a kind of triangulation, but the route chosen is of no importance. We know definitely that there is just one direction at $Q$ parallel to the original direction at $P$; and it is in ordinary geometry supposed that the length is equally determinate. But if space is not flat the case is different. Imagine a two-dimensional observer confined to the curved surface of the earth trying to perform this task. As he does not appreciate the third dimension he will not immediately perceive the impossibility; but he will find that the direction which he has transferred to $Q$ differs according to the route chosen. Or if he went round a complete circuit he would find on arriving back at $P$ that the direction he had so carefully tried to preserve on the journey did not agree with that originally drawn^[2]. We describe this by saying that in curved space, direction is not integrable; and it is this non-integrability of direction which characterises the gravitational field. In the case considered the length would be preserved throughout the circuit; but it is possible to conceive a more general kind of space in which the length which it was attempted to preserve throughout the circuit, as well as the direction, disagreed on return to the starting point with that originally drawn. In that case length is not integrable; and the non-integrability of length characterises the electromagnetic field. Length associated with direction is called a vector; and the combined gravitational and electric field describe that influence of the world on our measurements by which a vector carried by physical measurement round a closed circuit changes insensibly into a different vector.

The welding together of electricity and gravitation into one geometry is the work of Prof. H. Weyl, first published in 1918^[3]. It appears to the writer to carry conviction, although up to the present no experimental test has been proposed. It need scarcely be said that the inconsistency of length for an ordinary circuit would be extremely minute^[4], and the ordinary manifestations of the electromagnetic field are the consequential results of changes which would be imperceptible to direct measurement. It will be remembered that the gravitational field is likewise perceived by the consequential effects, and not by direct interval-measurement.

But the theory does appear to require that, for example, the time of vibration of an atom is not quite independent of its previous history. It may be assumed that the previous histories of terrestrial atoms are so much alike that there are no significant differences in their periods. The possibility that the systematic difference of history of solar and terrestrial atoms may have an effect on the expected shift of the spectral lines on the sun has already been alluded to. It seems doubtful, however, whether the effect could attain the necessary magnitude.

It may seem difficult to identify these abstract geometrical qualities of the world with the physical forces of electricity and magnetism. How, for instance, can the change in the length of a rod taken round a circuit in space and time be responsible for the sensations of an electric shock? The geometrical potentials ($\kappa$) obey the recognised laws of electromagnetic potentials, and each entity in the physical theory—charge, electric force, magnetic element, light, etc.—has its exact analogue in the geometrical theory; but is this formal correspondence a sufficient ground for identification? The doubt which arises in our minds is due to a failure to recognise the formalism of all physical knowledge. The suggestion "This is not the thing I am speaking of, though it behaves exactly like it in all respects" carries no physical meaning. Anything which behaves exactly like electricity must manifest itself to us as electricity. Distinction of form is the only distinction that physics can recognise; and distinction of individuality, if it has any meaning at all, has no bearing on physical manifestations.

We can only explore the world with apparatus, which is itself part of the world. Our idealised apparatus is reduced to a few simple types—a neutral particle, a charged particle, a rigid scale, etc. The absolute constituents of the world are related in various ways, which we have studied, to the indications of these test-bodies. The main features of the absolute world are so simple that there is a redundancy of apparatus at our disposal; and probably all that there is to be known could theoretically be found out by exploration with an uncharged particle. Actually we prefer to look at the world as revealed by exploration with scales and clocks—the former for measuring so-called imaginary intervals, and the latter for real intervals; this gives us a unified geometrical conception of the world. Presumably, we could obtain a unified mechanical conception by taking the moving uncharged particle as standard indicator; or a unified electrical conception by taking the charged particle. For particular purposes one test-body is generally better adapted than others. The gravitational field is more sensitively explored with a moving particle than a scale. Although the electrical field can theoretically be explored by the change of length of a scale taken round a circuit, a far more sensitive way is to use a little bit of the scale—an electron. And in general for practical efficiency, we do not use any simple type of apparatus, but a complicated construction built up with a view to a particular experiment. The reason for emphasising the theoretical interchangeability of test-bodies is that it brings out the unity and simplicity of the world; and for that reason there is an importance in characterising the electromagnetic condition of the world by reference to the indications of a scale and clock, however inappropriate they may be as practical test-bodies.

Weyl's theory opens up interesting avenues for development. The details of the further steps involve difficult mathematics; but a general outline is possible. As on Einstein's more limited theory there is at any point an important property of the world called the curvature; but on the new theory it is not an absolute quantity in the strictest sense of the word. It is independent of the observer's mesh-system, but it depends on his gauge. It is obvious that the number expressing the radius of curvature of the world at a point must depend on the unit of length; so we cannot say that the curvatures at two points are absolutely equal, because they depend on the gauges assigned at the two points. Conversely the radius of curvature of the world provides a natural and absolute gauge at every point; and it will presumably introduce the greatest possible symmetry into our laws if the observer chooses this, or some definite fraction of it, as his gauge. He, so to speak, forces the world to be spherical by adopting at every point a unit of length which will make it so. Actual rods as they are moved about change their lengths compared with this absolute unit according to the route taken, and the differences correspond to the electromagnetic field. Einstein's curved space appears in a perfectly natural manner in this theory; no part of space-time is flat, even in the absence of ordinary matter, for that would mean infinite radius of curvature, and there would be no natural gauge to determine, for example, the dimensions of an electron—the electron could not know how large it ought to be, unless it had something to measure itself against.

The connection between the form of the law of gravitation and the total amount of matter in the world now appears less mysterious. The curvature of space indirectly provides the gauge which we use for measuring the amount of matter in the world.

Since the curvature is not independent of the gauge, Weyl does not identify it with the most fundamental quantity in nature. There is, however, a slightly more complicated invariant which is a pure number, and this is taken to be Action^[5]. We can thus mark out a definite volume of space and time, and say that the action within it is 5, without troubling to define coordinates or the unit of measure! It might be expected that the action represented by the number 1 would have specially interesting properties; it might, for instance, be an atom of action and indivisible. Experiment has isolated what are believed to be units of action, which at least in many phenomena behave as indivisible atoms called quanta; but the theory, as at present developed, does not permit us to represent the quantum of action by the number 1. The quantum is a very minute fraction of the absolute unit.

When we come across a pure number having some absolute significance in the world it is natural to speculate on its possible interpretation. It might represent a number of discrete entities; but in that case it must necessarily be an integer, and it seems clear that action can have fractional values. An angle is commonly represented as a pure number, but it has not really this character; an angle can only be measured in terms of a unit of angle, just as a length is measured in terms of a unit of length. I can only think of one interpretation of a fractional number which can have an absolute significance, though doubtless there are others. The number may represent the probability of something, or some function of a probability. The precise function is easily found. We combine probabilities by multiplying, but we combine the actions in two regions by adding; hence the logarithm of a probability is indicated. Further, since the logarithm of a probability is necessarily negative, we may identify action provisionally with minus the logarithm of the statistical probability of the state of the world which exists.

The suggestion is particularly attractive because the Principle of Least Action now becomes the Principle of Greatest Probability. The law of nature is that the actual state of the world is that which is statistically most probable.

Weyl's theory also shows that the mass of a portion of matter is necessarily positive; on the original theory no adequate reason is given why negative matter should not exist. It is further claimed that the theory shows to some extent why the world is four-dimensional. To the mathematician it seems so easy to generalise geometry to $n$ dimensions, that we naturally expect a world of four dimensions to have an analogue in five dimensions. Apparently this is not the case, and there are some essential properties, without which it could scarcely be a world, which exist only for four dimensions. Perhaps this may be compared with the well-known difficulty of generalising the idea of a knot; a knot can exist in space of any odd number of dimensions, but not in space of an even number.

Finally the theory suggests a mode of attacking the problem of how the electric charge of an electron is held together; at least it gives an explanation of why the gravitational force is so extremely weak compared with the electric force. It will be remembered that associated with the mass of the sun is a certain length, called the gravitational mass, which is equal to 1.5 kilometres. In the same way the gravitational mass or radius of an electron is 7.10^-56 cms. Its electrical properties are similarly associated with a length 2.10^-13 cms., which is called the electrical radius. The latter is generally supposed to correspond to the electron's actual dimensions. The theory suggests that the ratio of the gravitational to the electrical radius, 3.10⁴², ought to be of the same order as the ratio of the latter to the radius of curvature of the world. This would require the radius of space to be of the order 6.10²⁹ cms., or 2.10¹¹ parsecs., which though somewhat larger than the provisional estimates made by de Sitter, is within the realm of possibility.

1. We refuse to contemplate the idea that when the metre rod changes its length to two metres, each centimetre of it changes to three centimetres.
2. It might be thought that if the observer preserved mentally the original direction in three-dimensional space, and obtained the direction at any point in his two-dimensional space by projecting it, there would be no ambiguity. But the three-dimensional space in which a curved two-dimensional space is conceived to exist is quite arbitrary. A two-dimensional observer cannot ascertain by any observation whether he is on a plane or a cylinder, a sphere or any other convex surface of the same total curvature.
3. Appendix, Note 15.
4. I do not think that any numerical estimate has been made.
5. Appendix, Note 16.