Chapter 60

[109]The difference in the Newtonian potential between two pointsandis defined as the work necessary to drag a unit mass fromtoagainst the action of the Newtonian gravitational field. In this case the potential atis 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 theterm, 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 theterm 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 pressuretakes the place ofin our former equations. Einstein then shows, as before, that for the universe to be cylindrical, hence stable, a connection must exist between this mysterious pressureand 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 (seeChapter XXXVI). Furthermore, the connection between the density of matterand the internal pressureis easier to understand than that betweenand the curvatureof the universe; although, of course,andare 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 ofconstitutes the function of action. At all events, inasmuch as the superfluous part ofdisappears 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 becausein 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 refersnot 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 hisAllgemeine 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 outquantitativelyby 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 incomplete quantitative agreementwith 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 beifthe 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. 173ff.[153]In defence of an absolute space-time, flateverywhereandeverywhen, 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 integrateround 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 somea prioripreference 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 hisa prioribelief 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-timea priorito 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 hisa prioriconvictions. 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,everywhereandeverywhen, 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 hypothesesad 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.Ifis expressed in metres andin 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 separationof the two events as referred to the train will then be given by(since). Substituting our numerical values, we findwhich 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 ofthat 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.

[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 theterm, 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 theterm 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 pressuretakes the place ofin our former equations. Einstein then shows, as before, that for the universe to be cylindrical, hence stable, a connection must exist between this mysterious pressureand 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 (seeChapter XXXVI). Furthermore, the connection between the density of matterand the internal pressureis easier to understand than that betweenand the curvatureof the universe; although, of course,andare 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 ofconstitutes the function of action. At all events, inasmuch as the superfluous part ofdisappears 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 becausein 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.

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 refersnot 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 hisAllgemeine 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.

Thus, in hisAllgemeine 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 outquantitativelyby 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 incomplete quantitative agreementwith 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 beifthe 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.

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 incomplete quantitative agreementwith 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 beifthe 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. 173ff.

[153]In defence of an absolute space-time, flateverywhereandeverywhen, 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.

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 integrateround 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 somea prioripreference 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 hisa prioribelief 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-timea priorito 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 hisa prioriconvictions. 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,everywhereandeverywhen, 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 hypothesesad 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.

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,everywhereandeverywhen, 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 hypothesesad 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.Ifis expressed in metres andin 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 separationof the two events as referred to the train will then be given by(since). Substituting our numerical values, we findwhich 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 ofthat 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.

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