FOOTNOTES:
[1] As exemplified in the Pythagorean discovery of the relationship between the length of a vibrating string and the pitch of its note, a discovery utilised in musical instruments. Another example is represented by Archimedes’ solution of the problem of Hieron’s gold tiara.
[2] The appellation Galilean motion does not appear to have been adopted generally. However, as it is shorter to designate “uniform translationary motion” under this name, we shall adhere to the appellation.
[3] We need not discuss here the difficult problems that relate to the connectivity of the continua. For instance, a point on a closed line does not divide the line into two parts, and yet the line remains one-dimensional. Problems of this sort pertain to one of the most difficult branches of geometry, namely, Analysis Situs, with which the names of Riemann, Betti and Poincaré are associated.
[4] The discoveries of du Bois Reymond have shown that we could proceed still further and interpose an indefinite number of additional points, but in the present state of mathematics this extension is viewed only as a mathematical curiosity. We may mention, however, that the rejection by Hilbert of the axiom of Archimedes (to be discussed in the following chapter), leading as it does to the strange non-Archimedean geometry, would be equivalent to considering the mathematical continuum as of this more general variety.
[5] That the choice of an ordering relation is all-important for the determination of dimensionality may be gathered from the following examples. Consider a number of diapasons emitting notes of various pitch. We should have no difficulty in ranging these various diapasons in order of increasing pitch. This ordering relation would be instinctive and not further analysable, since it would issue from that mysterious human capacity which enables all men to assert that one sound is shriller than another. Human appreciation is thus responsible for this definition of order in music; and we may well conceive of men whose reactions to sonorous impressions might differ from ours and who in consequence would range the diapasons in some totally different linear order. As a result, a note which we should consider as lying “between” two given notes might, in the opinion of these other beings, lie outside them. In either case, however, if we should assume that the various notes differed in pitch from one another by insensible gradations, a sensory continuum of sound pitches would have been constructed, though what we should call a continuum of pitches would appear to the other beings as a discontinuity of notes, and vice versa. In either case the respective sensory continua would be one-dimensional, since by suppressing any given note, continuity of passage would be broken. And now let us suppose that these strange beings were still more unlike us humans. Let us assume that not only would every note of their continuum appear to them as indiscernible from the one immediately preceding it and the one immediately following it, but that it would also be impossible for them to differentiate each given note from two additional notes. This assumption is by no means as arbitrary as it might appear, for we know full well that even normal human beings experience considerable difficulty in differentiating extreme heat from extreme cold and we know, too, that people afflicted with colour blindness are unable to distinguish red from green. But then, to return to our illustration, we should realise that the sound continuum, when ordered with this novel understanding of nextness or contiguity, would no longer remain one-dimensional.
In other words, in virtue of this new ordering relation, imposed by the idiosyncrasies of their perceptions, the same aggregate of notes would range itself automatically into a two-dimensional sensory continuum. This is what is meant when we claim that an aggregate of elements has of itself no particular dimensionality and that some ordering relation must be imposed from without. We must realise, therefore, that when we speak of the space of our experience as being three-dimensional in points, no intrinsic property of space can be implied by this statement. It is only when account is taken of the complex of our experiences interpreted in the light of that sensory order which appears to be imposed upon out co-ordinative faculties that the statement can acquire meaning. These conclusions are by no means vitiated by such facts as our inability to get out of a closed room or to tie a knot in a space of an even number of dimensions. Unfortunately, owing to lack of time, we cannot dwell further on these difficult questions.
[6] Here again we are neglecting, from motives of simplicity, our awareness of the focussing effort and of that of convergence. If we take these into account, we are in all truth considering not merely one private perspective, but a considerable number. Even so, our data would be incomplete.
[7] This difficulty of counting points might be obviated to a certain degree were space to be considered discrete or atomic; for in that case we might count the atoms of space separating points and thereby establish absolute comparisons between distances. But here again the procedure would be artificial, for it would be nullified unless we were to assume that the spatial voids separating the successive atoms were always the same in magnitude; furthermore (as Dr. Silberstein points out in his book, “The Theory of Relativity”), we should be in a quandary to know how a succession of atoms would have to be defined, since this definition would depend on a definition of order. At any rate, we need not concern ourselves with atomic or discrete manifolds, for Riemann assumed that mathematical space was a continuous manifold. In view of the quantum phenomena we may eventually be led to modify these views and to attribute a discrete nature to space, but this is a vague possibility which there is no advantage in discussing in the present state of our knowledge.
[8] Were it not for this restriction, comparisons of distance in space situated in different places could never be obtained by transporting rods from one place to another; for since the measurements yielded by our standard rods when they had reached their point of destination would depend essentially on the route they had followed, they could scarcely be called rigid. When the restriction is adhered to we obtain the most general type of Riemannian spaces or geometries exemplified by the three major types, the Euclidean type, the Riemannian type and the Lobatchewskian type. Weyl, however, dispenses with Riemann’s postulate and thereby obtains a more generalised type of space or geometry.
The non-mathematical reader is likely to become impatient at this rejection on the part of mathematicians of apparently self-evident postulates. But it must be remembered that a postulate which can be rejected and whose contrary leads to a perfectly consistent doctrine can certainly not be regarded as rationally self-evident; so that in a number of cases the legitimacy of so-called self-evident propositions can only be discussed a posteriori and not a priori. As a matter of fact our belief in self-evident propositions is derived in the majority of cases from crude experience and we cannot exclude the possibility that more refined observations may compel us to modify our opinions in a radical way. Einstein’s discovery that Euclid’s parallel postulate would have to be rejected in the world of reality is a case in point.
So far as Weyl’s exceedingly strange geometry is concerned, it is conceivable that it also may turn out to represent reality after all, for Weyl found it possible to account for the existence of electromagnetic phenomena in nature by assuming that the space-time of relativity was of the more general Weylian variety and not, as Einstein had assumed, of the more restricted Riemannian type.
[9] The association of a straight line with the shortest distance between two points only holds, however, provided the several dimensions of the space are of an identical nature. When we consider continua in which the several dimensions differ in nature, as in the space-time of relativity, the straight line may turn out to be the longest distance between two points.
[10] In the chapter on Weyl’s theory we shall mention Weyl’s method briefly.
[11] We are referring solely to those bodies which would yield the numerical results of Riemann’s or Lobatchewski’s geometry—both these types of geometry being compatible with the homogeneity of space.
[12] Also the behaviour of light during refraction proved that a form of Least Action known as Fermat’s principle of minimum time was involved.
[13] In this discussion we are always assuming that the observer is standing right above the coin; hence we are not considering the variations in apparent shape due to a slanting line of sight. This latter problem is of a totally different nature.
[14] The reason why Riemann refers specifically to the infinitely small is because he took it as proved that for ordinary extensions experiment had shown space to be Euclidean, and because for exceedingly great extensions he did not consider that measurements would be feasible.
[15] This metrical ether must not be confused with the classical ether of optics and electromagnetics.
[16] In all fairness to Einstein, however, it should be noted that he does not appear to have been influenced directly by Riemann.
[17] A fact by no means evident a priori.
[18] Calling
the dimensionality of the non-Euclidean space the number of dimensions of the generating Euclidean space would be
and not
, as at first sight we might be inclined to believe.
[19] An equivalent form of the criticisms discussed in the text consists in assuming that as Euclidean geometry is associated with the number zero, it must be logically antecedent to the non-Euclidean varieties since these are associated with non-vanishing numbers. But here, again, it is only because we take “curvature” as fundamental that Euclidean geometry is connected with zero. Should we choose to take “radius of curvature,” Euclidean geometry would be associated with infinity, and the non-Euclidean varieties with finite numbers. Hence, the problem would now resolve itself into determining whether “curvature” or “radius of curvature” was the more fundamental; and, in point of fact, “curvature” is a more complex conception than “radius of curvature.”
Then again, we might characterise the various geometries by means of the parallel postulate. In this case, Riemann’s geometry would be associated with the number zero, Euclidean geometry with the number one, and Lobatchewski’s geometry with infinity. Still another method of presentation would be to approach the problem through projective geometry. Then we should find that Lobatchewski’s geometry was associated with a circle, and Euclid’s with the intersection of the line at infinity with two imaginary lines, yielding the “circular points at infinity,” whereas Riemann’s geometry would be connected with a circle of imaginary radius. With this method of representation, zero would never enter into our discussions. And no one would maintain that the concept of imaginary points at infinity (the circular points) was logically antecedent to a real circle of finite radius. In short, we see that there are a large number of different methods of representing the various geometries; and, according to the method selected, the number zero may be associated with Euclideanism or with non-Euclideanism, or, again, with neither. All that we are justified in saying is that Euclidean geometry is the easiest of the geometries, but not necessarily the most fundamental.
[20] It is to be noted that had the source of light been placed at the centre of the sphere instead of at the north pole, all the great circles on the sphere would have cast straight lines for shadows on the tangent plane. Still, the shadows on the plane would, as before, have yielded Riemann’s geometry (of the elliptical variety). We see then, once again, that absolute shape, size and straightness escape us in every case, and we cannot even say that Reimann’s straight lines are curved with respect to Euclidean ones. All that is relevant is the mutual behaviour of bodies, their laws of disposition.
[21] For the sake of completeness we may mention that connectivity can be studied by the same procedure as that by which dimensionality was investigated. Thus, a space was considered to be two-dimensional when it was possible to intercept the path of continuous transfer between any two points by tracing a continuous line throughout the space. If we apply this test to the surface of a doughnut, it will be seen that a continuous line, such as a circumference enclosing part of the doughnut, like a ring, would be insufficient to divide the doughnut into two separate parts which could not be connected by a route of continuous transfer over the surface. This property would differentiate the surface of a doughnut from that of a sphere or an ellipsoid, and yet the surface of the doughnut would still be two-dimensional, but its connectivity would be different owing to the hole passing through its substance.
[22] An instant of time is of course a mere abstraction, but so is a mathematical point in space.
[23] We are not referring solely to the tides of the sea, but also to the solid tides generated in the earth’s substance.
[24] The problem is of course much more complex than would appear from the present analysis. For the slowing down of the earth’s rate of rotation, owing to tidal action, would also result in causing a retreat of the moon, producing thereby a variation in the period of its revolution round the earth.
[25] This is because in Michelson’s experiment it is not necessary to consider a sphere. The two arms of the apparatus may be of different lengths; and all that is observed is the continued coincidence of the interference-bands with markings on the instrument.
[26] More precisely
[27] When we consider the
’s in this light, we realise that the expression for
might have been anticipated directly without any regard to measurement. We might, for instance, have attempted to construct in a purely mathematical way the possible expressions containing variables such as
and
, together with corrective factors whose rôle it would be to ensure invariance. Riemann remarked that in addition to the classical expression, a number of such expressions could be constructed. But if we wish our value of
to be compatible with the existence of the Pythagorean theorem, namely (
for a right triangle), the classical expression of
must be adhered to. For this reason the type of space which is obtained under these conditions is called Pythagorean space; and in what is to follow, we shall have no occasion to consider any other variety.
[28] In a very brief way the difficulties are as follows: Differential geometry involves continuity; hence in a discrete continuum it would lose its force. But even this is not all, for there are various kinds of continuity, and continuity must be of a special type for differential geometry to remain applicable. For instance, in the foregoing exposition of the method we assumed that the expression of
would tend to a definite limit
when the points were taken closer and closer together. In particular we assumed that for infinitesimal areas of the surface the Pythagorean theorem
would hold. This was equivalent to stating that infinitesimal areas of our curved surface could be regarded as flat or Euclidean, hence as identified with the plane lying tangent to them. Inasmuch as the restriction of Euclideanism in the infinitesimal is precisely one that Riemann has imposed on space, we need have no fear of applying the differential method to the types of non-Euclidean space thus far discussed.
Nevertheless, we may conceive of spaces where
would not tend to a limit, and where, however tiny the area of our surface, we should still be faced with waves within waves ad infinitum. The situation would be similar to that presented by curves with no tangents at any point. However, such cases are only of theoretical interest.
[29] It should be mentioned that there exists another type of curvature, different from the Gaussian curvature. This other type of curvature is called the mean curvature at a point and is given by
.
In this chapter we have mentioned only the Gaussian curvature, for, as Gauss discovered, it is this type of curvature alone which characterises the geometry of the surface when explored with Euclidean measuring rods. This discovery is a direct consequence of the following considerations:
Two surfaces which have the same Gaussian curvature can be superposed on each other without being torn or stretched, whereas two surfaces which have different Gaussian curvatures can never be applied on each other unless we tear them or stretch them. Obviously the metric relations of a surface are not disturbed so long as we do not stretch the surface; and this is why surfaces which have the same Gaussian curvature and which can therefore be superposed without being stretched, must necessarily possess the same geometry.
A few illustrations may be helpful: A plane sheet of paper has a zero Gaussian curvature and a zero mean curvature at every point. A cylinder, a cone or a roll has a zero Gaussian curvature but a positive mean curvature. It is the differences in the mean curvatures that cause these surfaces to appear to us visually as differing from the plane. But owing to the fact that the plane, the cylinder, the cone and the roll have the same zero Gaussian curvature, we can wrap the plane on to the cylinder or cone without stretching it.
For this reason all these surfaces have the same Euclidean geometry (Euclidean measuring rods being used). On the other hand, a minimal surface (such as the surface defined by a soap film, stretched from wires situated in different planes) has a zero curvature just like the plane, but in contradistinction to the plane it has a negative Gaussian curvature; hence, cannot be applied to a plane, and its geometry is therefore non-Euclidean.
For the same reason a sphere, whose Gaussian curvature is a constant positive number, or that saddle-shaped surface called the pseudosphere whose Gaussian curvature is a constant negative number, can neither of them be flattened out on to a plane without being stretched. As a result their geometries differ from the Euclidean geometry of the plane. We have seen that these geometries are Riemannian and Lobatchewskian, respectively.
[30] If we were to represent non-Euclidean geometry as arising from the behaviour of our measuring rods, squirming when displaced as compared with Euclidean rods, we should see that Euclideanism in the infinitesimal implies that when our rods are of infinitesimal length and are displaced over infinitesimal distances, they behave in the same way as rigid Euclidean rods.
[31] These invariant types of curvature are given by
and
. If we consider the terms
and
separately, we obtain “tensors.” In a four-dimensional space there are 10 components for the
tensor and 20 for the
tensor. These components represent the curvatures measured in various directions.
[32] It is necessary to make this distinction for a space may be flat and yet only semi-Euclidean, as will be understood in later chapters when discussing space-time.
[33] The
’s were also the components of a tensor.
[34] We may note that the geometry, hence the nature, of a space is fully determined only when we express it in terms of the Riemann-Christoffel tensor
. Thus
at every point denotes perfectly flat or homaloidal space;
, on the other hand, comprises a whole series of generally non-homogeneous spaces of which
(the homogeneous and homaloidal variety) is only one particular illustration. Hence,
by itself does not yield us any very definite information.
Likewise, a perfectly homogeneous spherical space of four dimensions, i.e., the type of space which corresponds to Riemann’s geometry of four dimensions, is given by
where
is a constant depending on the intensity of the curvature. On the other hand,
represents a whole series of spaces, generally non-homogeneous, of which the above-mentioned truly spherical and homogeneous variety is only one particular case.
[35] Quoted from one of the standard text-books of philosophy.
[36] We are endeavouring to explain things as simply as possible, but as a matter of fact the statement we are making that acceleration remains absolute in Einstein’s special theory is not quite correct. The acceleration of a body which in Newtonian science remained the same regardless of our selection of one Galilean frame or another, varies in value in the special theory under similar circumstances. Nevertheless, inasmuch as a sharp distinction still persists between velocity and acceleration, we have felt justified for reasons of simplicity in presenting the problem as we have done.
[37] Subject to the restrictions mentioned in a note in the previous chapter.
[38] This formula
or
can also be written:
, since
. We may conclude that the velocity of the ball with respect to the embankment is equal to its velocity in the train plus the velocity of the train with respect to the embankment. This expresses the classical belief that velocities add up like numbers.
[39] In the classical theory, however, aberrational observations of very high refinement should reveal our speed through the ether, but observations of this sort are beyond our present powers; and Einstein’s theory has since proved that however precise our experiments, this velocity could never be revealed.
[40] Although no experimental results could be claimed to have justified any such assumption, Maxwell introduced it unhesitatingly into his theory. His celebrated equations of electromagnetics represented, therefore, the results of experiment, supplemented by this additional hypothetical assumption. The advisability of making this hypothesis was accentuated when it was found to ensure the law of the conservation of electricity.
[41] If two magnetic poles of equal strength, situated in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either pole is said to represent one unit of magnetic pole strength in the electromagnetic system of units. Owing to the interconnections between magnetism and electricity, we can deduce therefrom the unit of electric charge also in the electromagnetic system. Likewise, if two electric charges of equal strength, also situated in empty space at a distance of one centimetre apart, attract or repel each other with a force of one dyne, either charge is said to represent one unit of electric charge in the electrostatic system of units. From this we may derive the unit of magnetic pole strength in the electrostatic system.
[42] It is to this pressure of light that the repulsion of comets’ tails away from the sun is due.
[43] By a first-order experiment, we mean one that is refined enough to detect magnitudes of the order of
where
is the velocity of the earth through the stagnant ether and
is the velocity of light. Likewise, by a second-order experiment, we mean one capable of detecting magnitudes of the order of
. Inasmuch as
is certainly very much smaller than
,
is an extremely small quantity, and
is very much smaller still. So we see that a second-order experiment is necessarily very much more precise than a first-order one. We may also mention that no experiments have yet been successful in exceeding those of the second order in precision.
[44] The classical or Galilean transformations were
,
,
,
, where
was the velocity of the frame with respect to a frame at rest in the ether, hence with respect to the ether itself; this velocity being directed along the
axis and the two frames being assumed to have coincided at the initial instant
. Under the same conditions the Lorentz transformations were
,
,
;
where
being the velocity of light. It is easy to see that when
is small compared to
, the difference between the two types of transformations becomes negligible.
[45] As a matter of fact the appellation “FitzGerald contraction” should also be abandoned in Einstein’s theory, since FitzGerald and Lorentz had always regarded the contraction in the light of a real physical contraction in the stagnant ether. Nevertheless, if the reader realises the difference in the two conceptions of the contraction, it will simplify matters to retain the original name.
[46] It is necessary to take both of these conditions into account. If, for instance, we limited ourselves to the invariance of light’s velocity without taking into consideration the relativity of velocity, transformations such as
;
would also satisfy our requirements, and these would not be the Lorentz-Einstein transformations and would not be in accord with the relativity of Galilean motion.
[47] These results had already been anticipated by Lorentz, since as we have seen, he considered that his transformations were applicable to purely electromagnetic magnitudes. But Lorentz was always dealing with a fixed ether and with motion through the fixed ether; hence the interpretation of these discoveries was much less satisfactory.
[48] We must remember that it is only light in vacuo which is propagated with the speed
, hence which possesses an invariant velocity. Light passing through a transparent medium moves with a velocity which is less than
; hence in this case its velocity ceases to be invariant and is affected by the velocity of the medium, as also by that of the observer.
[49] The objective world of science has nothing in common with the world of things-in-themselves of the metaphysician. This metaphysical world, assuming that it has any meaning at all, is irrelevant to science.
[50] The identification of mass with energy necessitated by the Einstein theory allows us to obviate the paradoxical appearance of the following example: Consider an incandescent sphere radiating light in every direction. If placed at rest in some Galilean frame, it should remain there undisturbed, since the backward pressure exerted by the light radiation is distributed symmetrically round the sphere. But suppose now we view the incandescent sphere from some other Galilean frame moving away from the first with a velocity
. From this new frame the incandescent sphere will be moving with a velocity
, whereas the light waves radiated by it will still possess a velocity
in all directions in our frame. It will then appear to us that the light waves are speeding away from the sphere in the direction of its motion with a velocity
while they are traveling with a speed
in the opposite direction. The conditions of light pressure will no longer be symmetrical, and calculation shows that the motion of the sphere should gradually slow down under the action of this resistance. Now this is impossible; for if the sphere appeared to us to be slowing down, it could never remain attached to its evenly moving Galilean frame, and we should be faced with a contradiction. As soon, however, as we recognise that the incandescent sphere by radiating light is losing energy and hence mass, the principle of the conservation of momentum shows that this gradual loss of mass would have for effect an increase in the velocity of the sphere and would thus compensate exactly the loss of velocity generated by the unsymmetrical light pressure. In this way the paradox is explained very simply.
[51] A further verification was afforded when the mechanism of Bohr’s atom was studied by Sommerfeld. In Bohr’s atom the electrons revolve round the central nucleus in certain stable orbits. Sommerfeld, by taking into consideration the variations in mass of the electrons due to their motions as necessitated by Einstein’s theory, proved that in place of the sharp spectral lines which had always been observed it was legitimate to expect that more accurate observation would prove these sharp lines to be bundles of very fine ones closely huddled together. These anticipations were confirmed by experiment both qualitatively and quantitatively.
[52] Whenever we refer to “another Galilean frame,” we invariably mean one in motion with respect to the first. Were it not in a state of relative motion, it would constitute the same frame.
[53] We may mention that the physical existence of a finite invariant velocity is by no means impossible. It is acceptable mathematically, and the only question that we shall have to consider is whether it corresponds to physical reality.
[54] By an external event we do not necessarily mean an event occurring outside our body. We mean one that does not reduce to a mere awareness of consciousness. A sudden pain in our toe would constitute an external event in exactly the same measure as would the explosion of a barrel of gunpowder a mile distant.
[55] The relativity of simultaneity is a most revolutionary concept, as will be seen from the following illustration:
Consider two observers, one on a train moving uniformly along a straight line, the other on the embankment. At the precise instant these two observers pass each other at a point
, a flash of light is produced at the point
. The light wave produced by this instantaneous flash will present the shape of an expanding sphere. Since the invariant velocity of light holds equally for either observer, we must assume that either observer will find himself at all times situated at the centre of the expanding sphere.
Our first reaction might be to say: “What nonsense! How can different people, travelling apart, all be at the centre of the same sphere?” Our objection, however, would be unjustified.
The fact is that the spherical surface is constantly expanding, so that the points which fix its position must be determined at the same instant of time; they must be determined simultaneously. And this is where the indeterminateness arises. The same instant of time for all the points of the surface has not the same significance for the various observers; hence each observer is in reality talking of a different instantaneous surface.
[56] “The Principles of Natural Knowledge.”
[57] In the present chapter we are classing under the common name of “neo-realism” all those metaphysical doctrines which agree in considering perception, especially visual perception, a matter of direct apprehension. This classification may not be in accord with that of the metaphysicians. But as, from the standpoint of science, all doctrines that hold to the theory of the direct apprehension of reality are equally objectionable, we shall not trouble to differentiate between them.
[58] By giving the same name to these two conflicting types of simultaneity, Dr. Whitehead soon falls a victim to his own terminology. Thus, in his book, “The Theory of Relativity,” he informs us that he accepts Einstein’s discovery of the relativity of simultaneity. But in so doing he does not appear to notice that the type of simultaneity which Einstein has proved to be relative to motion is the scientific type; and it has nothing in common with the neo-realistic type. This latter species is never relative to motion, but solely to position. Hence, without any justification whatsoever, Whitehead has suddenly extended to his own understanding of simultaneity a characteristic which belongs to the type he is attacking. But owing to the same name having been given to the two types, the non-scientific reader is apt to overlook the confusion, and in this way be led to a very erroneous understanding of the nature of Einstein’s discoveries.
[59] The majority of Dr. Whitehead’s views are unfortunately couched in such loose and obscure terms that it is usually possible to place a variety of conflicting interpretations upon them. As he himself candidly admits in his book, “The Principles of Natural Knowledge”: “The whole of Part II, i.e., Chapters V to VII, suffers from a vagueness of expression due to the fact that the implications of my ideas had not shaped themselves with sufficient emphasis in my mind.”
Obviously, if the author of a book is not quite clear as to what he means, there is always some danger in settling the matter for him. Accordingly, we should have omitted any reference to Dr. Whitehead’s writings had it not been that some of his criticisms (though totally erroneous in our opinion) are often of considerable interest, leading as they do to a novel vision of things. At any rate, in any further reference to Whitehead, we shall view his criticisms in the light of type-objections, regardless of whether or not, as interpreted by us, they depict his own personal views.
[60] Italics ours.
[61] For the present we are confining ourselves to the consideration of Galilean frames; the problem becomes highly complicated when we study the peculiarities arising from a choice of accelerated frames. In this latter case, a non-Euclideanism, or curvature of space, appears with the presence of acceleration (as also of gravitation), so that we cannot even refer to a multiplicity of superposed Euclidean spaces. Also, space and time become so hopelessly confused that we are compelled to express ourselves in terms of four-dimensional space-time, where everything becomes clear once more.
[62] Conformal transformations are those which vary the shape of lines while leaving the values of their angles of intersection unaltered. They are of wide use in maps, e.g., in Mercator’s projection or in the stereographic projection.
the direction of motion being situated along the
axis.
[64] This type of geometry is sometimes called flat-hyperbolic; but as the appellation “hyperbolic” is also attributed to Lobatchewski’s geometry, it is apt to be misleading. Hence it is preferable to use the expression “semi-Euclidean.”
[65] Minkowski demonstrated the significance of the expression for
by taking a new variable
, where
stands for
. With this change,
can be written:
which is the expression of the square of a distance in a four-dimensional Euclidean space when a Cartesian co-ordinate system is taken. Since this expression is to remain unmodified in value and form in all Galilean frames, we must conclude that in a space-time representation a passage from one Galilean frame to another is given by a rotation of our four-dimensional Cartesian space-time mesh-system. Now rotation constitutes a change of mesh-system to the same extent as would a deformation of the mesh-system, and all changes of this sort entail a variation in the co-ordinates of the points of the continuum. In other words, they correspond to mathematical transformations. The transformations which accompany a rotation of a Cartesian co-ordinate system are of a particularly simple nature; they are called “orthogonal transformations.” It follows that if we write out the orthogonal transformations for Minkowski’s four-dimensional Euclidean space-time, we should obtain ipso facto the celebrated Lorentz-Einstein transformations which represent the passage from one Galilean system to another. This fact is easily verified. If, now, we recall that Minkowski had got rid of the minus sign in the expression for
by writing
in place of
, we obtain the following result: Two Galilean systems moving with a relative velocity
are represented by two space-time Cartesian co-ordinate systems differing in orientation by an imaginary angle
, where
is connected with
by the formula
.
[66] The general expression giving the composition of velocities in all three cases can be understood as follows: If a ball is moving through a train with velocity
as measured in the train, and if the train is moving with a speed
in the same direction along the embankment, then the speed of the ball with respect to the embankment is given by
, where
has the following expression:
In this formula
represents the value of the invariant velocity in that particular type of world we are considering.
In classical science,
is infinite, and the formula degenerates into
. This is the well-known classical composition where velocities lying in the same direction are added algebraically.
In Einstein’s theory,
is equal to Maxwell’s constant
. This being a real finite number, we obtain the Lobatchewskian type of composition. Were
an imaginary number, we should obtain the Riemannian type.
We may note that if for
or
we substitute
, that is to say, if we wish to discover what would result from the addition of a velocity to the invariant velocity, we obtain
This proves that the addition of a velocity to the invariant velocity still leaves us with the invariant velocity, justifying thereby the maximum and invariant nature credited to
.
[67] “The Principle of Relativity.”
[68] We may mention that the logicians have often endeavoured to introduce definitions of this sort into mathematics; but in mathematics, as in physics, their attempts have always been condemned. For instance, the definition of an inductive number, also given by Dr. Whitehead, was severely criticised by Poincaré (in “Science and Method”), who remarked that the object of a definition was to define, and that Whitehead’s definition being circular, defined nothing.
[69] In the case of Fresnel, his ignorance of non-Euclidean geometry might be claimed to lessen the force of the argument, but this contention would not hold for Lorentz or for the theoretical physicists of the latter part of the nineteenth century.
[70] It is important to understand the precise significance of the principle of entropy. Keeping in mind the fact that the principle refers to probabilities and not to absolute certainties, we may express its general formulation as follows: For all changes of a system the total entropy either is increased (irreversible changes) or else remains constant (reversible changes). At first sight it might appear as though the transformation of heat into work in the steam engine, or the withdrawal of heat from a colder body and its transfer to a warmer one, as in the case of refrigerating machines, would constitute flagrant violations of the principle. But, as a matter of fact, such is not the case. For, with the steam engine, the unnatural change exhibited by the transformation of heat into work is counterbalanced by a natural change illustrated by the fall of heat from generator to condenser. In fine, the total entropy thus remains constant (in a perfectly reversible cycle, for instance in a Carnot cycle), or else suffers an increase. Again, with the refrigerating machine, the unnatural passage of heat from the cold to the warmer body is compensated by a natural transformation; this is represented by the transformation into heat of the mechanical work performed on the apparatus. So once again the total entropy is increased, or, at best, remains constant. Still more striking cases, where the principle would appear to be at fault, are illustrated by the endothermic reactions in chemistry. Contrary to what we should expect, they absorb heat so that low-grade heat energy is transformed into higher-grade chemical energy; but here again compensating influences are at work. We cannot dwell further on these points. The reader who is interested in studying them must consult some standard work on thermodynamics.
[71] This decrease of the visual angle under the influence of relative motion would, however, apply only provided the observer were moving in a direction parallel to that of the pole, or at least yielding a component parallel to this direction.
[72] We do not wish to indulge in endless repetitions, but we must note once more that by absolute length of classical science we are referring solely to length once it has been defined by the norms of practical congruence. We do not imply absolute metaphysical spatial length, which is meaningless in science.
[73] A light year is an astronomical unit of length defined by the distance a wave of light covers in one year.
[74] We are considering only flat space-time far from matter.
[75] Recent experiments conducted by Kennedy have shown that Miller’s conclusions were erroneous, and that the null result of the original experiment was completely confirmed. (See Proc. Nat. Acad. of Sci., Nov., 1926.)
[76] It might be argued that a rotating disk is an accelerated body and that, in spite of this fact, no force acts upon it. This view would, however, be incorrect. Cohesive forces are acting between the various molecules of the disk, and were it not for the existence of these forces, the various molecules would fly off along tangents and would pursue Galilean motions; the disk would cease to exist. It is the presence of the cohesive forces which compels the molecules to describe circles, that is, to pursue accelerated or non-Galilean motions.
[77] This was in full accord with the principle of entropy.
[78] There are cases in which no determinate value for the potential can be found.
[79] Note how the theory of relativity is establishing the fusion of the mathematical (i.e., the
-distribution) with the physical (i.e., the forces of inertia).
[80] The statement that all bodies fall with the same motion in vacuo is correct, but, unless properly understood, is apt to lead to erroneous conclusions.
What the statement asserts is that any body, regardless of its constitution or mass, will fall in exactly the same way through a given gravitational field. If, however, the falling body causes a modification in the distribution of the field, then it is obvious that various falling bodies will no longer be situated in the same gravitational field; and there is no reason to assert that they will all fall in exactly the same way. For instance, if a relatively small mass—say, a billiard ball—then a relatively large one—say, the moon—were to be released in succession from some very distant point and allowed to fall towards the earth, it is undoubtedly correct to state that the duration of fall would be greater for the billiard ball than for the moon. It is easy to understand why this discrepancy would arise. In either case, owing to the mutual gravitational action, the earth would also be moving towards the falling body (moon or billiard ball), so that the point of collision would be some intermediate point, namely, the centre of gravity of the system earth-moon or earth-billiard ball. But in the case of the billiard ball, owing to its relatively insignificant mass, this centre of gravity, or point of collision, would be practically identical with the earth’s centre. This implies that the earth would scarcely move at all towards the billiard ball, whereas it would move an appreciable distance towards the moon. The motion of the earth would thus shorten considerably the distance through which the moon would have to fall, whereas the billiard ball would have to fall through the entire distance.
If we wish so to modify the conditions of the problem as to re-establish the perfect identity in the rate of fall of the moon and billiard ball, we must so arrange matters that the earth is unable to fall towards the body it is attracting. If, for example, it were possible to nail the earth to the Galilean frame in which earth, moon and billiard ball were originally at rest, and if this Galilean frame could be made to remain Galilean, i.e., unaccelerated, then the previous experiment attempted first with the moon, then with the billiard ball (the moon being removed entirely), would reveal exactly the same rate of fall for the two bodies. For now, indeed, the modifications in the distance covered, and in the nature of the field brought about by the displacement of the earth, would be non-existent.
A further case which presents a theoretical interest in Einstein’s discussions is afforded by what is known as a uniform field, that is to say, a field in which the gravitational force is the same in intensity and direction throughout space. A field of this sort would be generated by an infinitely extended sheet of matter of uniform density. Owing to the infinite mass a sheet of this sort would possess, its acceleration towards the falling body would be nil; all bodies would then fall with exactly the same constant acceleration towards the sheet. The reason uniform fields present a theoretical interest is because the field of force generated in an enclosure moving with constant acceleration is precisely of this type. When, therefore, Einstein identifies the field of force enduring in an accelerated enclosure with a gravitational field, we must remember that the distribution of matter which would be necessary to produce the same field is that of an infinitely extended sheet. Only to a first approximation can a finite mass of matter, like the earth, be deemed to generate a field of this kind.
[81] Subject to certain niceties which will be mentioned presently.
[82] To obviate any confusion with electromagnetic forces, we are considering only forces which act on uncharged bodies.
[83] Here the reader may well question our right to argue as though velocities combined in the classical way, when the whole significance of the special relativity theory has been to deny the validity of the classical transformations. In point of fact the objection would be legitimate; and in all rigour the Einstein-Lorentz transformations should be applied for each successive instantaneous velocity of the enclosure. But it so happens that when the Einstein-Lorentz transformations are applied to a transverse beam of light, it is found that for low velocities of the enclosure the bending is practically the same as it would have turned out to be had we followed the classical rule of composition. In other words, had the motion of the enclosure been uniform, the transverse ray of light as measured in the enclosure would have been inclined much as in classical science. Indeed, were it not that the relativity transformations entailed a variation in the slant of a ray of light moving transversally, the theory would be incompatible with the well-known phenomenon of astronomical aberration.
[84] We are assuming that the field does not vary with time. When this is not the case, we must specify that our observations must also be conducted over a very short duration of time.
[85] Qualitatively at least. The precise quantitative justification will be furnished later.
[86] If the spatial mesh-system we are considering is one of straight lines, a Cartesian one, for example, the statement in the text is accurate, but if we consider the more general case of a curvilinear mesh-system, we must introduce certain restrictions. In this last instance, our rod must be of infinitesimal length, for were its length finite, its orientation as referred to the curvilinear mesh-system would vary from place to place. This would introduce complications. Hence it is preferable to restrict our attention to rods of infinitesimal length and consider orientation as defined at a point. With this restriction in force, there is nothing to change in the explanation given in the text.
[87] Thus, calling
,
,
the components of the vector in the first frame of reference and
,
,
in the second, the components
,
,
will be connected with the components
,
,
in the following way:
where
,
,
,
, etc., are defined by the nature of the change to which our co-ordinate system has been subjected.
[88] Not to be confused with the tensor of a quaternion.
[89] A distinction of this sort does not apply, of course, to the equality of two invariants; for, as we have seen, a change of mesh-system can produce no effect on the value of an invariant, seeing that an invariant has no components.
[90] The tensor
being twice covariant, and the vector
repeated twice in the formula, being contravariant.
[91] It is customary to represent scalars by ordinary letters, and tensors of the first, second and third orders, and so on, by letters followed by indices equal in number to the order of the tensor. Thus,
is a tensor of the second order,
is one of the third order, and so on.
When we substitute for these indices all possible arrangements of the numbers from
to
, where
represents the
dimensions of our continuum, we obtain thereby the various components of our tensors.
Thus, in a space of two dimensions, the various components of
are
,
,
,
, which reduce to
,
,
, owing to the identity of
and
, the
tensor being symmetrical.
In order to differentiate at a glance contravariant tensors from covariant ones, the indices are placed above the letter. For instance,
is the contravariant form of
, and
is the mixed form. We see, then, that
should really be written
, since
refers to a contravariant vector.
[92] There exist types of fields for which the potential distribution is indeterminate, but we need not consider this case, as Newton’s gravitational field is of the kind which admits a definite potential distribution.
[93] In four-dimensional space-time there are sixteen of these
’s at every point, but, the tensor being symmetrical, six turn out to be mere repetitions, so we need only speak of ten separate
’s at every point.
[94] More precisely,
is not
, but is connected with
by the relation
[95] Later we shall see that this belief of classical science is not rigorously correct but it still remains true under certain special circumstances.
[96] In point of fact, it was when Einstein applied the principle of Action (to be discussed presently) that he first recognised the error in his original law. Also we may note that the law of curvature
, when it does not reduce to
, represents a non-homogeneous type of curvature, and most certainly not a homogeneous spherical curvature as certain lay writers have stated, drawing hasty philosophical conclusions therefrom.
[97] We shall see ([Appendix I]) that the geodesics of space-time are of two major varieties: the so-called time-like and the so-called space-like geodesics. The transition between the two is given by the null-lines or minimal geodesics; these correspond to the paths and motions of light rays. The space-like geodesics would correspond to the paths and motions of bodies moving with a speed greater than that of light. As such motions cannot exist, according to the theory of relativity, we see that free bodies can follow only the time-like geodesics. In future, therefore, when referring to the geodesics of space-time, we shall always have in mind the time-like geodesics. Also we may note that whereas the time-like geodesic defines the longest space-time distance between two points, the null-line or minimal geodesic has always a zero space-time length.
[98] Also constants such as
may enter into the law of curvature in the empty space around matter; but never foreign tensors.
[99] It is well to remember, however, that the laws we have considered are all in the image of Newton’s in that they contain no derivatives of the potentials to an order higher than the second. If this restriction is omitted, a number of alternative laws become possible. Inasmuch as their study presents tremendous mathematical difficulties they have not been investigated; and it is hard to say what might be the nature of their solutions.
[100] There are also the other time-potentials, i.e.,
,
,
. But as, in our mesh-system, the direction of time is perpendicular to those of space, these potentials vanish in the present case, and we are left with
.
[101] In reality the identification is a little more complicated; it is given by
Furthermore, it can be seen that this potential
, of varying value from place to place, is connected with the variable speed of light from place to place through the gravitational field. More precisely, the speed of light in vacuo is given by
, which is approximately
or
, where
is the mass of the body exciting the gravitational field.
[102] The Einstein effect is due to a veritable decrease in the frequency of vibration of the atom situated nearer the sun, and this retardation is caused by the increasing departure from unity of the potential
, tending as it does towards zero as we approach the sun. It would be totally incorrect to ascribe it to a slowing down in the motion of a ray of light travelling away from the sun in a radial direction, owing to the retarding effect of the sun’s gravitational pull. The gravitational pull has nothing to do with the Einstein effect; and as a matter of fact, calculation shows that a ray of light travelling away from the sun would gradually increase in speed till it attained its invariant speed
at infinity, as though it were repelled, not attracted by the sun. But over and above these results of calculation, it can be seen immediately that a modification in the speed of light would be incapable of explaining the existence of the Einstein effect. In all cases we are bound to receive the successive vibrations with the same frequency as they are emitted by the atom; for otherwise there would be a gradual accumulation or depletion of light waves travelling along the fixed distance separating us from the atom. Hence any verification of the Einstein effect could be ascribed only to a real modification in the frequency of the atom’s vibrations.
[103] The Einstein shift in the spectral lines as seen by a definite observer will increase in importance as the atom nears the star or as the star increases in mass. For a star of given mass, the effect will therefore increase as the volume of the star decreases, and hence as its density increases. We understand, therefore, why it is that for two stars of the same mass, the best conditions of observation will be afforded by the star which has the greater density; while for two stars having the same density, the best conditions will be afforded by the star having the greater mass.
[104] According to the special principle of relativity.
[105] Of course even from a mathematical point of view the attempt would have been impossible from the start. See [Appendix IV].
[106] De Sitter’s universe is truly spherical only when we argue in terms of imaginary time it. In this case, for any given observer, both time and space close round on themselves. When, however, we use real time t, as indeed we should, we find that de Sitter’s universe yields a three-dimensional spherical Riemann extension, for the space of a given observer, but that real time no longer curls round on itself. This universe can be represented on the surface of a hyperboloid of one sheet, open at both ends in the time direction, and there is no fear of a return of time with the past becoming the future. It is easy to see why time is differentiated in its curvature from space in de Sitter’s universe. All we have to do is to notice that in
, since the three space-
’s (
) are always positive and
, the time-
, is always negative, or vice versa (except when they all vanish), we always have a difference in sign between the curvatures
,
,
, on the one hand, and
, on the other. Were we to appeal to imaginary time,
would also be positive, so that
would be of the same sign as the other
’s. Analogously, in the special theory, by putting it in place of
, we obtained
in place of
, giving us
in place of
.
[107] Here we are viewing de Sitter’s universe as a spherical universe for reasons of simplicity. In other words, we are arguing in terms of imaginary time it.
[108] The reason why, under Newton’s law, a uniform distribution of matter to infinity is impossible can be understood as follows: The difference between the values of the Newtonian potential
at two points
and
is equal to the work which must be expended against the gravitational force for a unit mass to be moved from
to
. But if matter were distributed uniformly to infinity, no work would be necessary, since wherever the body stood, it would be at the centre of the infinite universe, hence would be subjected to equal forces in every direction. But then
would have the same constant value everywhere. This would entail the vanishing of
. And so Poisson’s equation
would become
; an impossibility, since
, density of matter, must have a non-vanishing value. In short, it would be necessary to modify the law of gravitation and consider, for example,
(where
is some constant) in place of Poisson’s equation. With this latter law instead of Newton’s, an infinitely extended universe of matter would be possible.
[109] The difference in the Newtonian potential between two points
and
is defined as the work necessary to drag a unit mass from
to
against the action of the Newtonian gravitational field. In this case the potential at
is always higher than at
.
[110] At all events, it could be departed from only momentarily. A more precise formulation of the law of equipartition would consist in saying that when the condition of statistical equilibrium is reached, the total energy of the system will be divided up among the different degrees of freedom of the constituent particles. However, this equipartition of energy will not be absolutely rigorous. Taken at any instant, the energies of the separate degrees of freedom will be some greater and some less than would be demanded by rigorous equipartition; but, on an average, equipartition will endure when large numbers are considered. These conclusions are necessary consequences of the law of entropy and are based on probability considerations. Inasmuch as, in the example of the billiard balls, we assumed these to be identical, the law of equipartition connoted that the kinetic energies, hence velocities, of the balls would eventually fluctuate round the mean velocity corresponding to rigorous equipartition; so that, broadly speaking, the velocities of all the balls would be the same. The precise fluctuations in the velocities are expressed by Maxwell’s law of the distribution of velocities, and would be found to be represented by the bell-shaped curve of Gauss, a celebrated mathematical curve which enters into the law of errors and into a number of probability problems.
[111] Here let us note a difference between our present problem and the one we discussed when investigating the curvature of space in the interior of a fluid sphere under the older law of space-time curvature. In the present case, by adjoining the
term, it is possible for cosmic matter to fill the entire space of the universe, so that space closes round on itself when matter is assumed to be distributed homogeneously and continuously. The universe becomes self-contained, and no matter or light can escape from it.
[112] We are, of course, referring solely to the sphere’s surface, and not to the centre of its volume, which stands outside the two-dimensional surface or space which we are considering.
[113] The problem might be on a different footing were space-time to be considered atomic; for, as we have mentioned, in a discrete manifold, in contradistinction to a continuous one, a metrics might be immanent in the continuum even in the total absence of matter. There is always the possibility that we may be on the wrong track when we assume space-time to be continuous, and it cannot be denied that the existence of quantum phenomena lends colour to this possibility. So long, however, as we assume the continuity of the fundamental extension, de Sitter’s universe, existing with a perfectly definite metrics independently of the presence of matter, appears extremely improbable.
[114] At least it appears to be the only type of universe which would also be in harmony with the existence of low star-velocities.
[115] The situation would be somewhat similar to that which exists in the case of electric and magnetic actions. Here, also, we know that an electron at rest develops a purely electric pull according to the Newtonian law, whereas, when it is in relative motion, a magnetic pull is superadded at right angles to the line of motion and to the electric pull.
[116] It is most important to note that the relativity of all motion, as expressed by the general principle of relativity, stands on an entirely different footing from the ultra-relativistic conception of rotation as upheld by Mach. The two forms of relativity have been muddled up so often that it appears necessary to point out their essential differences. The general principle of relativity merely states that the natural laws can be thrown into a form which remains covariant to all choices of mesh-system. In other words, all observers, whether Galilean, accelerated or rotating, will observe the laws of nature under the same tensor form; so that there is no reason to follow classical science and elevate one type of observer above another. This is what Einstein originally meant by the relativity of all motion. Thus the general principle of relativity in no wise implies that an observer would not realise that conditions had changed after the frame to which he was attached had been set into rotation.
[117] The increase of mass in a gravitational field can be anticipated most easily as follows: Consider a disk rotating in a Galilean frame. As referred to this frame, the points of the rim will be moving with a certain velocity; hence a mass fixed to the rim will increase when its value is computed in the Galilean frame. But the postulate of equivalence allows us to assert that conditions would be exactly the same were the disk to be at rest in an appropriate gravitational field. Inasmuch as in this case the gravitational force would be pulling outwards from centre to rim, just as though a massive body had been placed outside the rim, we may infer that the mass of a body increases as it approaches gravitational masses. For instance, the inertial mass of a billiard ball would be increased were the ball to be placed nearer the sun.
[118] Prior to his discovery of the cylindrical universe, Einstein had made several attempts to account for a self-contained nuclear universe in infinite space-time, in which the relativity of inertia would also be satisfied. The solution of this problem was intimately connected with the invariance of the boundary conditions, and this accounts in part for the numerous references to boundary conditions which we encounter in all the original papers. However, the low velocities of the stars proved that the relativity of inertia could not be realised with infinite space-time.
[119] Einstein’s original law in the case of feeble gravitating masses and low velocities is practically identical with that of Newton.
[120] It is of interest to note that in an alternative presentation of the cylindrical universe Einstein does not make use of the
term at all. He confines himself to studying under what conditions a cloud of cosmic dust could cause the universe to be cylindrical, hence to be stable. The novelty of the procedure consists in taking into consideration a certain pressure (not a hydrostatic pressure) which physicists have been led to discuss but which as yet remains unexplained. We refer to the internal pressure which prevents an electron from exploding under the mutual repulsions of its various parts, all of which are charged with negative electricity. This pressure has been called the Poincaré pressure; it is thought to be responsible for atomicity, but its nature is highly enigmatic. Under this new treatment of the problem, the mysterious cohesive pressure
takes the place of
in our former equations. Einstein then shows, as before, that for the universe to be cylindrical, hence stable, a connection must exist between this mysterious pressure
and the average density of matter throughout the universe, hence also between the pressure and the universal curvature. In this respect, at least, a slight advance in our understanding of the binding pressure of matter appears to have been obtained (see [Chapter XXXVI]). Furthermore, the connection between the density of matter
and the internal pressure
is easier to understand than that between
and the curvature
of the universe; although, of course,
and
are one and the same, so that, as before, it is matter that creates the spatio-temporal universe.
[121] We are assuming that a four-dimensional vector calculus would have been in existence; but this is a purely mathematical question.
[122] We are using the word “Action” to denote what Hamilton called the “Principal Function.” For a more rigorous treatment the reader must refer to standard works on Analytical Dynamics.
[123] Also called Lagrangian Function.
[124] This deification of the principle of action which is traceable to the influence of Hilbert and Weyl is resisted by Eddington and Silberstein, who point out that the principle has none but a formal significance.
[125] We are endeavouring to explain the problem in as elementary a way as possible. A rigorous exposition, however, would compel us to state that only a certain part of
constitutes the function of action. At all events, inasmuch as the superfluous part of
disappears when we calculate the stationary condition, no essential change need be made in our exposition.
[126] “Space, Time and Matter.”
[127] We cannot insist on numerous niceties such as the distinction between tensors and tensor-densities, etc. We may note, however, that whereas the in-magnitudes
,
,
differ from the tensors
,
, and from the invariant density
, yet
, the tensor of the electric and magnetic forces, is the same as in Einstein’s theory. This is because
in Einstein’s theory already happened to be an in-tensor.
[128] Eddington has shown that the discrepancies which might be expected to arise would in all probability be too small to be observed.
[129] See note, [page 354].
[130] When suitable units are chosen.
[131] J. S. Mackenzie.
[132] In Poisson’s case the argument had some weight, for there was reason to suppose that were Fresnel’s views justified, we should at some time or other have observed bright shadows in the course of our daily experience.
[133] It is scarcely necessary to add that our awareness of sensations does not presuppose any knowledge of space or of our human body as an object situated in space. When, for instance, an infant who is beginning to emerge into consciousness feels a pain in its leg, then one in its arm, it is not supposed that it succeeds in localising the two sensations in the two respective limbs. Only much later will it succeed in localising its sensations. For the present we are assuming that the infant knows nothing of space or of its body; it is merely registering sensations, and the pain in its leg will appear to it to differ in some obscure qualitative way from the pain in its arm.
[134] Recently considerable progress has been made by Schrödinger in the interpretation of quantum phenomena within the atom, by means of a wave theory of matter, known as wave mechanics.
[135] By realism we mean “common-sense realism,” and not that monstrous distortion known as “neo-realism.”
[136] Of recent years certain philosophers known as logicians, Bertrand Russell in England, Couturat in France, among others, have stressed the logical aspect of mathematics. The question is whether they have not over-stressed it. That mathematical reasoning implies clear thinking and complies with the rules of logic has never been denied; nevertheless the assertion that all mathematical reasonings are of a purely deductive nature, and are reducible to the rules of logic, is an opinion which is by no means unanimous. Some of the greatest among the modern mathematicians, notably Poincaré and Klein, have protested vigorously against this view and have pointed out numerous cases of circularity in the arguments and lack of rigour in the definitions presented by the logicians. Over and above this aspect of the matter, they have maintained that the rules of a game are not everything in its make-up. To say that mathematics and logic are one and the same would be equivalent to maintaining that poetry was nothing but grammar, syntax and rules of versification, or that music was nothing but counterpoint and harmony. It is conceivable that we might acquire as thorough a knowledge of counterpoint and harmony as Beethoven may have possessed and yet be unable to compose a work rivalling any of his symphonies. We should have no hesitancy in granting that a Beethoven must obviously have been gifted with some mysterious faculty which had been denied us; and this faculty, whatever its essence, would relate to music, would be a part of music, since were all men lacking in it, there would be no great music. Under the circumstances, in spite of what might be called our logical knowledge of music, could we truthfully claim to have as thorough a knowledge of it as a Beethoven?
And it is exactly the same with pure mathematics. We know from experience that many persons, though possessing highly logical minds, are yet refractory to advanced mathematics. Were mathematics nothing but logic, this situation would seem extraordinary. Logistics, from a failure to see in mathematics anything but a series of rules and regulations with no creative faculty behind it, has been christened “thoughtless thinking” by its adversaries. But without wishing to take sides in a controversy for which the majority of persons evince but little interest, there is a point which the unprejudiced onlooker must perceive. With the sole exception of Hilbert, who, though opposing Russell’s views, defends opinions of a somewhat similar nature, none of the logicians have contributed to the constructive side of mathematics. This again appears somewhat strange when we recall that one of the earliest boasts of this school of thinkers was that logistics would give them wings. One cannot help but suspect that the logicians are lacking in some creative faculty of which they may not be conscious, and that as a result they perhaps occupy in mathematics a position analogous to that of the professor of counterpoint and harmony in music. Under the circumstances, it is questionable whether they possess a sufficient understanding of this difficult science to contribute any information of value.
Of course a charge of this sort cannot apply to Hilbert, whose great work in the creative regions of mathematics has proved him to be gifted with the creative faculty in addition to the purely formal dissecting faculty which all mathematicians, regardless of their tendencies, must necessarily possess. Owing to Hilbert’s attitude, the problem is generally regarded as controversial.
[137] In this case, however, Laplace’s equation is of the two-dimensional variety and not of the usual three-dimensional type which defines the distribution of the Newtonian potential in the empty space around a gravitational mass of finite dimensions. However, the two-dimensional Laplace equation also gives the distribution of the potential around matter in special instances. Such is the case when we consider an attracting cylinder of uniform density, finite section and infinite length.
[138] In the particular case of the oscillating pendulum, we may restrict our attention to the real (i.e. non-imaginary) realm of the elliptical function considered. But the very existence of elliptical functions is dependent on the introduction of imaginary quantities.
[139] This is of course merely a figure of speech. It is not assumed that the molecules come into actual contact. Furthermore, it would be difficult to specify exactly how contact should be defined for molecules.
[140] Thus, Euler, when discussing absolute space and time, writes: “What is the essence of space and time is not important; but what is important is whether they are required for the statement of the law of inertia. If this law can only be fully and clearly explained by introducing the ideas of absolute space and absolute time, then the necessity for these ideas can be taken as proved.” Again, Riemann, when discussing the possible non-Euclideanism of space, maintains a similar attitude. We read: “It is conceivable that the measure relations of space in the infinitesimal are not in accordance with the assumptions of our [Euclidean] geometry, and, in fact, we should have to assume that they are not, if, by doing so, we should ever be enabled to explain phenomena in a more simple way.”
[141] Bergson, “Durée et Simultanéité.”
[142] Bertrand Russell, article on non-Euclidean geometry in the Encyclopaedia Britannica, and various other writings. We may note, however, that Russell has recently modified his views on this subject.
[143] Whitehead, “The Principle of Relativity.”
[144] Both attempts failed.
[145] Quantum phenomena would now appear to account for what Ritz ascribed to that new entity, the magneton.
[146] We might also mention the annual variation in the angle of aberration of a star, and its relationship with the star’s parallax. This relationship would appear to be utterly mysterious.
[147] Italics ours.
[148] Kant’s attitude towards Newton’s absolute space is somewhat confused. At times he defends the absoluteness of space, making extensive use of the arguments of Newton and Euler. At other times he presents his own arguments in favour of the relativity of space and motion. Finally, in his last work (Metaphysische Anfangsgründe der Naturwissenschafften), he writes: “Absolute space is, then, necessary not as a conception of a real object, but as a mere idea which is to serve as a rule for considering all motion therein as merely relative.” How motion can be relative while space is absolute is a problem that Kant fails to elucidate. At any rate, the problem of the absoluteness of space and time in classical science refers not to the essence of space and time (a problem which would degenerate into one of metaphysics, hence which would be meaningless to the scientists), but solely to a discussion of those conceptions which are demanded by the world of experience. Hence we may realise that a man ignorant of mechanics is in no position to pass an opinion one way or the other. And Kant’s knowledge of Newtonian mechanics was extremely poor, to say the least.
Thus, in his Allgemeine Naturgeschichte und Theorie des Himmels, we find him giving incorrect formulæ for the most elementary facts concerning falling bodies. Then again, basing his arguments on what he claims to be the laws of dynamics, he tells us of a nebula which would set itself into rotation owing to its outer parts falling towards the centre and rebounding sideways against the inner parts. But this hypothesis is in flagrant opposition to the principles of dynamics, and had Kant spoken of a man pulling himself up by his bootstraps he would have given expression to no greater absurdity. Whereas this latter statement would violate the principle of action and reaction, Kant’s violates the invariance of the quantity of moment of momentum in a self-contained dynamical system.
[149] The development and gradual acceptance of the kinetic theory of gases is particularly instructive in this connection. Some two centuries ago Daniel Bernoulli had suggested that the tendency of a gas to expand might be attributed to a rushing hither and thither of its molecules. But inasmuch as the idea was not worked out quantitatively, no great attention was paid it. Not till a century or so later was it investigated in a mathematical way by Maxwell and by Boltzmann. For this reason these scientists receive the credit for the kinetic theory; and Bernoulli’s name (in connection with this theory) has lost all but a historical interest. Maxwell’s and Boltzmann’s theoretical anticipations were borne out quantitatively by the experiments performed in their day, and a large number of scientists accepted the theory as sound. Even so, the doctrine still had its detractors, for subsequent experiment proved that in the matter of specific heats at low temperatures, theory and observation were in utter conflict. Further difficulties related to the problem of the equipartition of energy. The net result was that other physicists (Kelvin and Ostwald in particular) were hostile to the kinetic theory.
At this stage we must mention that as far back as 1827 an English doctor named Brown had noticed that fine particles suspended in a liquid appeared to be quivering when viewed under the microscope. Brown attributed these curious motions to the presence of living organisms; others suggested that they were due to inequalities of temperature brought about by the illumination of the microscope. On the other hand, the adherents of the kinetic theory maintained that these Brownian movements were due to the impacts of the molecules of the fluid on the suspended particles. Now the important point to understand is that this latter hypothesis could serve only as a suggestion. Before any reliance could be placed in it, it would be necessary to prove that the precise Brownian movements actually observed were in complete quantitative agreement with the theoretical demands of the kinetic theory; and a quantitative theory of this sort had never been formulated. Such was the state of affairs when Einstein, in one of his first papers, gave an exhaustive quantitative solution of the problem of Brownian movements (in the case both of translations and of rotations), stating what the precise movements would have to be if the kinetic theory of fluids corresponded to reality.
Perrin then submitted the Brownian movements to precise quantitative measurements with a view to checking up on Einstein’s anticipations. The result was a disappointment; for Perrin found a considerable discrepancy between those anticipations and experiment. And so the kinetic theory of Brownian movements appeared to be untenable, and, more generally, the whole kinetic theory of gases and fluids to be in peril. When informed of Perrin’s results, Einstein went over his calculations afresh and discovered a numerical mistake in his computations. On rectifying this error he found that the theoretical anticipations were in perfect agreement with Perrin’s measurements. About the same time, by means of mathematical calculations based on the quantum theory, Einstein succeeded in accounting for the specific-heat difficulty mentioned previously. As for the arguments directed against the kinetic theory by reason of the “equipartition-of-energy theorem,” they in turn were answered, thanks to the quantum theory, itself a product of quantitative investigation. The net result was that the kinetic theory was finally established, the principle of entropy ceased to be regarded as an absolute principle, and Ostwald surrendered.
Our purpose in giving this brief historical sketch of the problem of Brownian movements has been to show that loose guesses, unless supported by arguments of a precise quantitative nature, are of little interest to physical science. For this reason no great importance is attributed to the vague atomistic speculations of Democritus or to the relativistic speculations of Mach, even though, in the light of subsequent developments, the latter may prove to be correct. In all cases of this sort, the credit goes (and rightly so) to the theoretical investigator who has succeeded in overcoming the major difficulty, that of working out the theory along rigid mathematical lines. To be sure, in many instances the thinker who makes the guess or advances the hypothesis also follows it up mathematically. Such was the case with Einstein when he formulated the postulate of equivalence for the purpose of interpreting the significance of the equality of the two masses. When a dual contribution of this type occurs, the credit is of course twofold.
[150] “The Meaning of Relativity.”
[151] Quite recently Dr. Whitehead has endeavoured to work out the same problem afresh.
[152] A very lucid exposition of Poincaré’s attack is given in Cunningham’s “Principle of Relativity” (1914), pp. 173 ff.
[153] In defence of an absolute space-time, flat everywhere and everywhen, Dr. Whitehead argues that the variable curvature forced upon space-time by matter in Einstein’s theory “leaves the whole antecedent theory of measurement in confusion” and hence must render knowledge impossible.
We do not consider the argument sound, for the absolute magnitudes of the relativity theory never were spatial or temporal, even in the special theory with its flat space-time. It was only the space-time interval that was absolute. As for the charge that the general theory renders knowledge impossible, it is refuted by the fact that Einstein’s astronomical predictions are of so precise a nature that it has required the most perfect instruments and most competent astronomers to detect them. This would scarcely be expected of a theory which had rendered knowledge impossible.
A further argument of Dr. Whitehead’s deals with the ambiguity in the measure of rotation which is entailed by Einstein’s matter-modified space-time theory of gravitation. Thus he writes: “The Einstein theory in explaining gravitation has made rotation an entire mystery.” Following this criticism, Dr. Whitehead proceeds to confuse the general theory of gravitation with the cylindrical universe and Mach’s mechanics, neglecting to notice that the former does not necessarily entail the latter. Einstein has insisted on this point repeatedly; besides, the gravitational equations themselves make it quite clear.
At all events, it is difficult to see any merit in the criticism based on rotation. All that we have a right to demand of any theory is that it be in harmony with the existence of an inertial frame with respect to which rotation develops centrifugal forces; for these forces have been detected by experiment. But we have no right to maintain that any experiment performed to this day has ever been sufficiently precise to demonstrate the absolute fixity of the inertial frame, or to deny that the frame may not suffer from a certain measure of indeterminateness when large masses move in its neighbourhood. Dr. Whitehead’s argument, which he claims is “based entirely on the direct results of experience,” would thus appear to be scientifically unsound.
As a matter of fact, calculations have been performed by J. A. Schouten (see Eddington’s illuminating discussion of the matter in “The Mathematical Theory of Relativity,” pp. 99-100), and the results have been to show that the indeterminateness of rotation resulting from the sun’s relative motion is no more than 1".94 a century in Einstein’s theory. And, most certainly, neither Newton’s experiment of the rotating bucket of water nor even the more precise tests taken with pendulums and gyroscopes would be able to detect a negligible magnitude of this sort. Hence it would appear that the general theory, though necessitating a fluctuating space-time background, is yet in perfect harmony with those dynamical facts of rotation which drove Newton to absolute space more than two hundred years ago.
[154] (I
) refers to the boundary conditions of the cylindrical universe.
[155] Inserted in order to conform to previous terminology.
[156] A tensor expression built up of the
’s, and presenting the property of conservation, is given by
. We may identify this expression, therefore, with the presence of a permanent entity such as matter. Outside these permanent entities the space-time structure would be given by
, which is equivalent to
. If now we integrate
round a point-centre, assuming space-time to be flat at infinity, a certain undetermined constant of integration must be introduced into our solution. This is a purely mathematical necessity, having nothing to do with the physical significance of our equations. Now, the constant being undetermined, we may assign any value to it we please. According to the positive or negative value we might ascribe to this constant, we should obtain a field of force either attractive or repulsive around our particle; and the law of force distribution would be approximately that of the inverse square. Only if we attributed a vanishing value to the constant would no field of force be present. This solution, however, would be scarcely permissible. Thus, we have shown that the space-time theory, independently of any empirical study of nature, suggests the existence of permanent entities surrounded by fields of force. In actual practice, we identify the undetermined constant with the “mass” of the central particle; and since mass is always positive in the world of our experience, the constant is always a positive number, and attraction is the outcome. In this way the integrated form of Einstein’s law of gravitation is obtained.
[157] Or else so profoundly modified as to constitute, to all intents and purposes, a new theory.
[158] From the standpoint of scientific method, it is most instructive to contrast with Einstein’s theory one which fails to satisfy the demands of science. We refer to Dr. Whitehead’s attempt to interpret the phenomenon of gravitation in terms of Minkowski’s flat space-time. In the last chapter we noted that prior to Einstein’s sweeping generalisation a solution of gravitation on the basis of flat space-time had been worked out by Poincaré, Nordström and others. Dr. Whitehead wishes to revert to this type of solution. One of its advantages, he claims, is that it obviates the idea that matter influences the structure of space-time. Whether this is an advantage or not is a mere question of feeling, but in science it is a highly dangerous procedure to reject a simple co-ordination of the facts of experience in favour of some a priori preference which offers no means of being put to a test. Dr. Whitehead would probably claim that in his case this criticism does not hold, for he has given definite arguments in support of his contentions. But what are these arguments? The first consists in his writing: “The only possible structure is that of planes and straight lines.” Yet, as no mathematician would agree with this contention, it would appear that Whitehead was merely restating his a priori belief in the inevitable homogeneity of the space-time structure without justifying his views in any way. Elsewhere he writes: “Are these material bodies really the ultimate data of perception, incapable of further analysis? If they are, I at once surrender.” Now, in point of fact, it is scarcely probable that any scientist would agree with Whitehead’s hypothetical opponent in maintaining that material objects do constitute the ultimate data of perception. But what conceivable connection is there between this problem and the totally different one pertaining to a possible action of matter on the structure of space-time? Obviously, Whitehead is implicitly assuming that we know enough of space-time a priori to assert that it can never be affected by whatever is not an ultimate datum of perception. Hence his second argument, just like the first, reduces to a mere restatement of his a priori convictions. Let us proceed.
Having satisfied himself that material bodies are not what his hypothetical opponent claims them to be, he considers his point proved. Material bodies are henceforth referred to as “a certain coherence of sense-objects such as colours, sounds and touches.” And these sense-objects “at once proclaim themselves to be adjectives of events.” This, according to Dr. Whitehead, precludes matter from exerting any effect on space-time, so that Einstein’s general theory must be discarded.
It is hard to see how a solution of the problems of nature will ever be advanced by such loose arguments. At any rate, having assumed space-time to be flat, or at least homogeneous, everywhere and everywhen, Dr. Whitehead is compelled to account in terms of forces generated by matter for the gravitational effects predicted by Einstein. He explains the occurrence of the shift-effect by tracing it “to the combination of two causes, one being the change in the apparent mass due to the gravitational potential and the other being the change in the electric cohesive forces of the molecule due to the gravitational field.” We shall assume on his authority that the first of the two causes he mentions can be deduced from his gravitational equations, and shall concern ourselves solely with the second cause. Now, unless we get our information from text-books of physics, written more than thirty years ago, we shall never labour under the impression that the spectral lines have anything to do with the constitution of the molecule. Molecular vibrations may cause a broadening of the lines owing to a Doppler effect, and are recognised as giving rise to the diffuse band spectra, but modern experiment has traced the visible line spectra to occurrences within the atom. It follows that these molecular adjustments which Dr. Whitehead is seeking to introduce would appear to have no effect one way or another on the shift he proposes to account for. However, it is not from the standpoint of scientific accuracy that we wish to criticise Dr. Whitehead’s statements, but solely from that of method.
By his own admission, nothing can be predicted of the shift until we know more about the molecule (we presume he means atom). Thus, he tells us, “Accordingly, it requires some knowledge of the structure of the molecule to be certain what the shift (if any) of the spectral lines should be.” Dr. Whitehead wrote his book before the shift was observed on the companion of Sirius, and presumably thought it safer to include the parenthetical words “if any.”
The quotation just cited illustrates the weak point of this type of theory. For inasmuch as we know practically nothing about the intra-atomic occurrences responsible for radiation, we may postulate anything we please with impunity, and as a result make the theory predict any shift we wish, be it vanishing or great. But the purpose of a scientific theory is to give us the power to predict phenomena as yet unknown, and not merely to account for what we already know. It was for this reason that Newton’s law of gravitation was so great an advance over Kepler’s laws of planetary motion or Ptolemy’s crystal spheres. For Newton’s law enabled us to predict how a stray comet would move, whereas Kepler’s laws could tell us nothing.
A theory such as Dr. Whitehead’s allows us to predict anything—hence nothing. No experimental test is possible, since whatever experiment discloses can immediately be accounted for by varying our hypotheses ad hoc. Besides, one cannot help but feel that had not Einstein previously predicted the existence of the shift, Dr. Whitehead would never have deduced it from his theory. Similar conclusions would apply to the double bending of a ray of starlight.
One of the reasons why Einstein’s theory commands such respect among scientists is precisely because, by predicting definite occurrences and no others, it allows itself to be submitted to a test. As Einstein wrote in 1918, several years before the shift was finally observed on the companion of Sirius: “If the displacement of spectral fines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable.” It is theories permitting such categorical statements that scientists demand. The theories of Newton, of Maxwell and of Einstein are of this sort.
[159] The difference is, however, exceedingly minute, as we can judge by reverting to the expression for the invariant space-time distance
. If
is expressed in metres and
in seconds,
will be 300,000,000. Suppose, then, that the two signals, as measured from the embankment, are ten metres apart, and that the two light flashes are separated by an interval of one second. If these two events are viewed from a train travelling at the rate of ten metres a second, that is, at approximately 22 miles an hour, it will be possible for the observer in the train to pass before each of the two flashes just as they are produced, with the result that as referred to his train the two events will occur at the same spot. The time separation
of the two events as referred to the train will then be given by
(since
). Substituting our numerical values, we find
which is very nearly one second, as it would rigorously have been in classical science. We see, furthermore, that it is owing to the enormous value of
that the difference between the two sciences is so hard to detect in practice. It is for this reason that the fundamental continuum, though one of space-time, reduces for all practical purposes to the separate space and time of our forefathers, unless very high velocities are considered.
[160] Subject to the limitations mentioned in the previous paragraphs.
[161] See [Appendix I].
[162] We do not know whether Eddington is still prepared to defend these views.
[163] An exceedingly clear presentation of this point has been given by Dr. Silberstein in his remarkable books on the relativity theory.
[164] Although they do not appear to be perpendicular to each other on the diagram, the time and space directions
and
are perpendicular when account is taken of the peculiar geometry of space-time with which we are dealing.
[165] “Space, Time and Gravitation,” p. 141.
[166] This is true only in a general way, for further empirical data are necessary when we wish to consider problems involving friction and elasticity, etc. Moreover, in considering any particular problem, a knowledge of the masses and forces involved must be obtained, and this of course entails physical measurements.
[167] Geodesics are also the longest distances; the essential is that they constitute extreme distances. Thus, if we take two points at random on a sphere, the great circle passing through these points is a geodesic. But we may follow this geodesic from one point to the other in two different ways: either by going along the shortest path, or else by following the line that passes through the antipodes on the other side of the sphere. In this last case, the geodesic is the longest path.
[168] To confine ourselves to a three-dimensional space, if this space were spherical, stars situated at a finite distance would yet have vanishing parallaxes, so that they would appear to be at infinity. For this reason a spherical space, though finite, would manifest itself to us visually as infinite. Theoretically, also, images of stars should form at the antipodes. In an elliptical space, however, these images would coincide with the star itself.
Conversely, a Lobatchewskian space, though of infinite extent, would appear to be finite, since stars at infinity would have a non-vanishing parallax, and so would appear to be situated at a finite distance.