[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.[63]the direction of motion being situated along theaxis.[64]This type of geometry is sometimes calledflat-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 forby taking a new variable,wherestands 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 obtainipso factothe 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 forby writingin place of,we obtain the following result: Two Galilean systems moving with a relative velocityare represented by two space-time Cartesian co-ordinate systems differing in orientation by an imaginary angle,whereis connected withby 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 velocityas measured in the train, and if the train is moving with a speedin the same direction along the embankment, then the speed of the ball with respect to the embankment is given by,wherehas the following expression:In this formularepresents 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. Werean imaginary number, we should obtain the Riemannian type.We may note that if fororwe substitute,that is to say, if we wish to discover what would result from the addition of a velocity to the invariant velocity, we obtainThis 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. (SeeProc. 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 motionin vacuois 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 waythrough 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 tensorbeing twice covariant, and the vectorrepeated 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 fromto,whererepresents thedimensions of our continuum, we obtain thereby the various components of our tensors.Thus, in a space of two dimensions, the various components ofare,,,,which reduce to,,,owing to the identity ofand,thetensor 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,andis the mixed form. We see, then, thatshould really be written,sincerefers 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 withby 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 thenull-linesor 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 asmay 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 byFurthermore, 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 lightin vacuois given by,which is approximatelyor,whereis 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 graduallyincreasein speed till it attained its invariant speedat infinity, as though it were repelled,notattracted 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. SeeAppendix IV.[106]De Sitter’s universe is truly spherical only when we argue in terms of imaginary timeit. 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 thatwould be of the same sign as the other’s. Analogously, in the special theory, by putting it in place of,we obtainedin place of,giving usin 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 potentialat two pointsandis equal to the work which must be expended against the gravitational force for a unit mass to be moved fromto.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 thenwould have the same constant value everywhere. This would entail the vanishing of.And so Poisson’s equationwould 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,(whereis 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[63]the direction of motion being situated along theaxis.
[63]the direction of motion being situated along theaxis.
[64]This type of geometry is sometimes calledflat-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.”
[64]This type of geometry is sometimes calledflat-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 forby taking a new variable,wherestands 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 obtainipso factothe 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 forby writingin place of,we obtain the following result: Two Galilean systems moving with a relative velocityare represented by two space-time Cartesian co-ordinate systems differing in orientation by an imaginary angle,whereis connected withby the formula.
[65]Minkowski demonstrated the significance of the expression forby taking a new variable,wherestands 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 obtainipso factothe 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 forby writingin place of,we obtain the following result: Two Galilean systems moving with a relative velocityare represented by two space-time Cartesian co-ordinate systems differing in orientation by an imaginary angle,whereis connected withby 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 velocityas measured in the train, and if the train is moving with a speedin the same direction along the embankment, then the speed of the ball with respect to the embankment is given by,wherehas the following expression:In this formularepresents 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. Werean imaginary number, we should obtain the Riemannian type.We may note that if fororwe substitute,that is to say, if we wish to discover what would result from the addition of a velocity to the invariant velocity, we obtainThis 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.
[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 velocityas measured in the train, and if the train is moving with a speedin the same direction along the embankment, then the speed of the ball with respect to the embankment is given by,wherehas the following expression:In this formularepresents 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. Werean imaginary number, we should obtain the Riemannian type.
We may note that if fororwe substitute,that is to say, if we wish to discover what would result from the addition of a velocity to the invariant velocity, we obtainThis 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.”
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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. (SeeProc. Nat. Acad. of Sci., Nov., 1926.)
[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. (SeeProc. 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.
[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.
[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.
[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).
[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 motionin vacuois 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 waythrough 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.
[80]The statement that all bodies fall with the same motionin vacuois 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 waythrough 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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 tensorbeing twice covariant, and the vectorrepeated twice in the formula, being contravariant.
[90]The tensorbeing twice covariant, and the vectorrepeated 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 fromto,whererepresents thedimensions of our continuum, we obtain thereby the various components of our tensors.Thus, in a space of two dimensions, the various components ofare,,,,which reduce to,,,owing to the identity ofand,thetensor 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,andis the mixed form. We see, then, thatshould really be written,sincerefers to a contravariant vector.
[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 fromto,whererepresents thedimensions of our continuum, we obtain thereby the various components of our tensors.
Thus, in a space of two dimensions, the various components ofare,,,,which reduce to,,,owing to the identity ofand,thetensor 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,andis the mixed form. We see, then, thatshould really be written,sincerefers 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.
[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.
[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 withby the relation
[94]More precisely,is not,but is connected withby 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.
[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.
[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 thenull-linesor 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.
[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 thenull-linesor 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 asmay enter into the law of curvature in the empty space around matter; but never foreign tensors.
[98]Also constants such asmay 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.
[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.
[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 byFurthermore, 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 lightin vacuois given by,which is approximatelyor,whereis the mass of the body exciting the gravitational field.
[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 lightin vacuois given by,which is approximatelyor,whereis 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 graduallyincreasein speed till it attained its invariant speedat infinity, as though it were repelled,notattracted 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.
[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 graduallyincreasein speed till it attained its invariant speedat infinity, as though it were repelled,notattracted 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.
[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.
[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. SeeAppendix IV.
[105]Of course even from a mathematical point of view the attempt would have been impossible from the start. SeeAppendix IV.
[106]De Sitter’s universe is truly spherical only when we argue in terms of imaginary timeit. 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 thatwould be of the same sign as the other’s. Analogously, in the special theory, by putting it in place of,we obtainedin place of,giving usin place of.
[106]De Sitter’s universe is truly spherical only when we argue in terms of imaginary timeit. 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 thatwould be of the same sign as the other’s. Analogously, in the special theory, by putting it in place of,we obtainedin place of,giving usin 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.
[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 potentialat two pointsandis equal to the work which must be expended against the gravitational force for a unit mass to be moved fromto.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 thenwould have the same constant value everywhere. This would entail the vanishing of.And so Poisson’s equationwould 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,(whereis 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.
[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 potentialat two pointsandis equal to the work which must be expended against the gravitational force for a unit mass to be moved fromto.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 thenwould have the same constant value everywhere. This would entail the vanishing of.And so Poisson’s equationwould 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,(whereis 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.