Chapter 6

XXX.

COSMOLOGICAL DIFFICULTIES OF NEWTON’S THEORY

Part from the difficulty discussed in Section XXI, there is a second fundamental difficulty attending classical celestial mechanics, which, to the best of my knowledge, was first discussed in detail by the astronomer Seeliger. If we ponder over the question as to how the universe, considered as a whole, is to be regarded, the first answer that suggests itself to us is surely this: As regards space (and time) the universe is infinite. There are stars everywhere, so that the density of matter, although very variable in detail, is nevertheless on the average everywhere the same. In other words: However far we might travel through space, we should find everywhere an attenuated swarm of fixed stars of approrimately the same kind and density.

This view is not in harmony with the theory of Newton. The latter theory rather requires that the universe should have a kind of centre in which the density of the stars is a maximum, and that as we proceed outwards from this centre the group-density of the stars should diminish, until finally, at great distances, it is succeeded by an infinite region of emptiness. The stellar universe ought to be a finite island in the infinite ocean of space.[24]

[24] _Proof_—According to the theory of Newton, the number of “lines of force” which come from infinity and terminate in a mass m is proportional to the mass _m_. If, on the average, the mass density ρ0 is constant throughout the universe, then a sphere of volume _V_ will enclose the average mass ρ0_V_. Thus the number of lines of force passing through the surface _F_ of the sphere into its interior is proportional to ρ0_V_. For unit area of the surface of the sphere the number of lines of force which enters the sphere is thus proportional to ρ0_V/F_ or to ρ0_R_. Hence the intensity of the field at the surface would ultimately become infinite with increasing radius _R_ of the sphere, which is impossible.

This conception is in itself not very satisfactory. It is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space, never to return, and without ever again coming into interaction with other objects of nature. Such a finite material universe would be destined to become gradually but systematically impoverished.

In order to escape this dilemma, Seeliger suggested a modification of Newton’s law, in which he assumes that for great distances the force of attraction between two masses diminishes more rapidly than would result from the inverse square law. In this way it is possible for the mean density of matter to be constant everywhere, even to infinity, without infinitely large gravitational fields being produced. We thus free ourselves from the distasteful conception that the material universe ought to possess something of the nature of a centre. Of course we purchase our emancipation from the fundamental difficulties mentioned, at the cost of a modification and complication of Newton’s law which has neither empirical nor theoretical foundation. We can imagine innumerable laws which would serve the same purpose, without our being able to state a reason why one of them is to be preferred to the others; for any one of these laws would be founded just as little on more general theoretical principles as is the law of Newton.

XXXI.

THE POSSIBILITY OF A “FINITE” AND YET “UNBOUNDED” UNIVERSE

But speculations on the structure of the universe also move in quite another direction. The development of non-Euclidean geometry led to the recognition of the fact, that we can cast doubt on the _infiniteness_ of our space without coming into conflict with the laws of thought or with experience (Riemann, Helmholtz). These questions have already been treated in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them briefly here.

In the first place, we imagine an existence in two dimensional space. Flat beings with flat implements, and in particular flat rigid measuring-rods, are free to move in a _plane_. For them nothing exists outside of this plane: that which they observe to happen to themselves and to their flat “things” is the all-inclusive reality of their plane. In particular, the constructions of plane Euclidean geometry can be carried out by means of the rods _e.g._ the lattice construction, considered in Section XXIV. In contrast to ours, the universe of these beings is two-dimensional; but, like ours, it extends to infinity. In their universe there is room for an infinite number of identical squares made up of rods, _i.e._ its volume (surface) is infinite. If these beings say their universe is “plane,” there is sense in the statement, because they mean that they can perform the constructions of plane Euclidean geometry with their rods. In this connection the individual rods always represent the same distance, independently of their position.

Let us consider now a second two-dimensional existence, but this time on a spherical surface instead of on a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods withal as the realisation of “distance”? They cannot do this. For if they attempt to realise a straight line, they will obtain a curve, which we “three-dimensional beings” designate as a great circle, _i.e._ a self-contained line of definite finite length, which can be measured up by means of a measuring-rod. Similarly, this universe has a finite area that can be compared with the area, of a square constructed with rods. The great charm resulting from this consideration lies in the recognition of the fact that _the universe of these beings is finite and yet has no limits._

But the spherical-surface beings do not need to go on a world-tour in order to perceive that they are not living in a Euclidean universe. They can convince themselves of this on every part of their “world,” provided they do not use too small a piece of it. Starting from a point, they draw “straight lines” (arcs of circles as judged in three dimensional space) of equal length in all directions. They will call the line joining the free ends of these lines a “circle.” For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value π, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

image036

_i.e._ a smaller value than π, the difference being the more considerable, the greater is the radius of the circle in comparison with the radius _R_ of the “world-sphere.” By means of this relation the spherical beings can determine the radius of their universe (“world”), even when only a relatively small part of their worldsphere is available for their measurements. But if this part is very small indeed, they will no longer be able to demonstrate that they are on a spherical “world” and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.

Thus if the spherical surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they have no means of determining whether they are living in a finite or in an infinite universe, because the “piece of universe” to which they have access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that for our sphere-beings the circumference of a circle first increases with the radius until the “circumference of the universe” is reached, and that it thenceforward gradually decreases to zero for still further increasing values of the radius. During this process the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole “world-sphere.”

Perhaps the reader will wonder why we have placed our “beings” on a sphere rather than on another closed surface. But this choice has its justification in the fact that, of all closed surfaces, the sphere is unique in possessing the property that all points on it are equivalent. I admit that the ratio of the circumference _c_ of a circle to its radius _r_ depends on _r_, but for a given value of _r_ it is the same for all points of the “worldsphere”; in other words, the “world-sphere” is a “surface of constant curvature.”

To this two-dimensional sphere-universe there is a three-dimensional analogy, namely, the three-dimensional spherical space which was discovered by Riemann. its points are likewise all equivalent. It possesses a finite volume, which is determined by its “radius” (2π2_R_3). Is it possible to imagine a spherical space? To imagine a space means nothing else than that we imagine an epitome of our “space” experience, _i.e._ of experience that we can have in the movement of “rigid” bodies. In this sense we _can_ imagine a spherical space.

Suppose we draw lines or stretch strings in all directions from a point, and mark off from each of these the distance _r_ with a measuring-rod. All the free end-points of these lengths lie on a spherical surface. We can specially measure up the area (_F_) of this surface by means of a square made up of measuring-rods. If the universe is Euclidean, then _F_ = 4π_r_2; if it is spherical, then _F_ is always less than 4π_r_2. With increasing values of _r, F_ increases from zero up to a maximum value which is determined by the “world-radius,” but for still further increasing values of _r_, the area gradually diminishes to zero. At first, the straight lines which radiate from the starting point diverge farther and farther from one another, but later they approach each other, and finally they run together again at a “counter-point” to the starting point. Under such conditions they have traversed the whole spherical space. It is easily seen that the three-dimensional spherical space is quite analogous to the two-dimensional spherical surface. It is finite (_i.e._ of finite volume), and has no bounds.

It may be mentioned that there is yet another kind of curved space: “elliptical space.” It can be regarded as a curved space in which the two “counter-points” are identical (indistinguishable from each other). An elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry.

It follows from what has been said, that closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent. As a result of this discussion, a most interesting question arises for astronomers and physicists, and that is whether the universe in which we live is infinite, or whether it is finite in the manner of the spherical universe. Our experience is far from being sufficient to enable us to answer this question. But the general theory of relativity permits of our answering it with a moderate degree of certainty, and in this connection the difficulty mentioned in Section XXX finds its solution.

XXXII.

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

According to the general theory of relativity, the geometrical properties of space are not independent, but they are determined by matter. Thus we can draw conclusions about the geometrical structure of the universe only if we base our considerations on the state of the matter as being something that is known. We know from experience that, for a suitably chosen co-ordinate system, the velocities of the stars are small as compared with the velocity of transmission of light. We can thus as a rough approximation arrive at a conclusion as to the nature of the universe as a whole, if we treat the matter as being at rest.

We already know from our previous discussion that the behaviour of measuring-rods and clocks is influenced by gravitational fields, _i.e._ by the distribution of matter. This in itself is sufficient to exclude the possibility of the exact validity of Euclidean geometry in our universe. But it is conceivable that our universe differs only slightly from a Euclidean one, and this notion seems all the more probable, since calculations show that the metrics of surrounding space is influenced only to an exceedingly small extent by masses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves analogously to a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake. Such a universe might fittingly be called a quasi-Euclidean universe. As regards its space it would be infinite. But calculation shows that in a quasi-Euclidean universe the average density of matter would necessarily be _nil_. Thus such a universe could not be inhabited by matter everywhere; it would present to us that unsatisfactory picture which we portrayed in Section XXX.

If we are to have in the universe an average density of matter which differs from zero, however small may be that difference, then the universe cannot be quasi-Euclidean. On the contrary, the results of calculation indicate that if matter be distributed uniformly, the universe would necessarily be spherical (or elliptical). Since in reality the detailed distribution of matter is not uniform, the real universe will deviate in individual parts from the spherical, _i.e._ the universe will be quasi-spherical. But it will be necessarily finite. In fact, the theory supplies us with a simple connection[25] between the space-expanse of the universe and the average density of matter in it.

[25] For the radius _R_ of the universe we obtain the equation

image037

The use of the C.G.S. system in this equation gives 2/k = 1.08 x 1027; ρ is the average density of the matter and _k_ is a constant connected with the Newtonian constant of gravitation.

APPENDICES

APPENDIX I

SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION

(SUPPLEMENTARY TO SECTION XI)

For the relative orientation of the co-ordinate systems indicated in Fig. 2, the _x_-axes of both systems permanently coincide. In the present case we can divide the problem into parts by considering first only events which are localised on the _x_-axis. Any such event is represented with respect to the co-ordinate system _K_ by the abscissa _x_ and the time _t_, and with respect to the system _K′_ by the abscissa _x′_ and the time _t′_. We require to find _x′_ and _t′_ when _x_ and _t_ are given.

A light-signal, which is proceeding along the positive axis of _x_, is transmitted according to the equation

_x_ = _ct_

or

_x_ – _ct_ = 0 . . . . . (1).

Since the same light-signal has to be transmitted relative to _K′_ with the velocity _c_, the propagation relative to the system _K′_ will be represented by the analogous formula

_x′_ – _ct′_ = 0 . . . . . (2)

Those space-time points (events) which satisfy (1) must also satisfy (2). Obviously this will be the case when the relation

(_x′_ – _ct′_) = λ(_x_ – _ct_) . . . (3).

is fulfilled in general, where λ indicates a constant; for, according to (3), the disappearance of (_x_ – _ct_) involves the disappearance of (_x′_ – _ct′_).

If we apply quite similar considerations to light rays which are being transmitted along the negative _x_-axis, we obtain the condition

(_x′_ + _ct′_) = (_x + ct_) . . . (4).

By adding (or subtracting) equations (3) and (4), and introducing for convenience the constants _a_ and _b_ in place of the constants λ and μ where

image038

and

image039

we obtain the equations

image040

We should thus have the solution of our problem, if the constants _a_ and _b_ were known. These result from the following discussion.

For the origin of _K′_ we have permanently _x′_ = 0, and hence according to the first of the equations (5)

image041

If we call _v_ the velocity with which the origin of _K′_ is moving relative to _K_, we then have

image042

The same value _v_ can be obtained from equations (5), if we calculate the velocity of another point of _K′_ relative to _K_, or the velocity (directed towards the negative _x_-axis) of a point of _K_ with respect to _K′_. In short, we can designate _v_ as the relative velocity of the two systems.

Furthermore, the principle of relativity teaches us that, as judged from K, the length of a unit measuring-rod which is at rest with reference to _K′_ must be exactly the same as the length, as judged from _K′_, of a unit measuring-rod which is at rest relative to _K_. In order to see how the points of the _x′_-axis appear as viewed from _K_, we only require to take a “snapshot” of _K′_ from _K_; this means that we have to insert a particular value of _t_ (time of _K_), _e.g._ _t_ = 0. For this value of _t_ we then obtain from the first of the equations (5)

_x′_ = _ax_

Two points of the _x′_-axis which are separated by the distance Δ_x′_ = 1 when measured in the _K′_ system are thus separated in our instantaneous photograph by the distance

image043

But if the snapshot be taken from _K′_(_t′_ = 0), and if we eliminate _t_ from the equations (5), taking into account the expression (6), we obtain

image044

From this we conclude that two points on the _x_-axis separated by the distance 1 (relative to _K_) will be represented on our snapshot by the distance

image045

But from what has been said, the two snapshots must be identical; hence Δ_x_ in (7) must be equal to Δ_x′_ in (7_a_), so that we obtain

image046

The equations (6) and (7_b_) determine the constants _a_ and _b_. By inserting the values of these constants in (5), we obtain the first and the fourth of the equations given in Section XI.

image047

Thus we have obtained the Lorentz transformation for events on the _x_-axis. It satisfies the condition

_x′_2 – _c_2_t′_2 = _x_2 – _c_2_t_2 . . . . . . (8a).

The extension of this result, to include events which take place outside the _x_-axis, is obtained by retaining equations (8) and supplementing them by the relations

image048

In this way we satisfy the postulate of the constancy of the velocity of light _in vacuo_ for rays of light of arbitrary direction, both for the system _K_ and for the system _K′_. This may be shown in the following manner.

We suppose a light-signal sent out from the origin of _K_ at the time _t_ = 0. It will be propagated according to the equation

image049

or, if we square this equation, according to the equation

_x_2 + _y_2 + _z_2 – _c_2_t_2 = 0 . . . . . (10).

It is required by the law of propagation of light, in conjunction with the postulate of relativity, that the transmission of the signal in question should take place—as judged from _K′_—in accordance with the corresponding formula

_r′_ = _ct′_

or,

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = 0 . . . . . . (10_a_).

In order that equation (10_a_) may be a consequence of equation (10), we must have

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = σ (_x_2 + _y_2 + _z_2 – _c_2_t_2) (11).

Since equation (8_a_) must hold for points on the _x_-axis, we thus have σ = 1. It is easily seen that the Lorentz transformation really satisfies equation (11) for σ = 1; for (11) is a consequence of (8_a_) and (9), and hence also of (8) and (9). We have thus derived the Lorentz transformation.

The Lorentz transformation represented by (8) and (9) still requires to be generalised. Obviously it is immaterial whether the axes of _K′_ be chosen so that they are spatially parallel to those of _K_. It is also not essential that the velocity of translation of _K′_ with respect to _K_ should be in the direction of the _x_-axis. A simple consideration shows that we are able to construct the Lorentz transformation in this general sense from two kinds of transformations, viz. from Lorentz transformations in the special sense and from purely spatial transformations. which corresponds to the replacement of the rectangular co-ordinate system by a new system with its axes pointing in other directions.

Mathematically, we can characterise the generalised Lorentz transformation thus:

It expresses _x′, y′, x′, t′_, in terms of linear homogeneous functions of _x, y, x, t_, of such a kind that the relation

_x′_2 + _y′_2 + _z′_2 – _c_2_t′_2 = _x_2 + _y_2 + _z_2 – _c_2_t_2 (11_a_).

is satisficd identically. That is to say: If we substitute their expressions in _x, y, x, t_, in place of _x′, y′, x′, t′_, on the left-hand side, then the left-hand side of (11_a_) agrees with the right-hand side.

APPENDIX II

MINKOWSKI’S FOUR-DIMENSIONAL SPACE (“WORLD”)

(SUPPLEMENTARY TO SECTION XVII)

We can characterise the Lorentz transformation still more simply if we introduce the imaginary

image031

in place of _t_, as time-variable. If, in accordance with this, we insert

image050

and similarly for the accented system _K′_, then the condition which is identically satisfied by the transformation can be expressed thus:

_x_1′2 + _x_2′2 + _x_3′2 + _x_4′2 = _x_12 + _x_22 + _x_32 + _x_42 (12).

That is, by the afore-mentioned choice of “coordinates,” (11_a_) [see the end of Appendix II] is transformed into this equation.

We see from (12) that the imaginary time co-ordinate _x_4, enters into the condition of transformation in exactly the same way as the space co-ordinates _x_1, _x_2, _x_3. It is due to this fact that, according to the theory of relativity, the “time” _x_4, enters into natural laws in the same form as the space co ordinates _x_1, _x_2, _x_3.

A four-dimensional continuum described by the “co-ordinates” _x_1, _x_2, _x_3, _x_4, was called “world” by Minkowski, who also termed a point-event a “world-point.” From a “happening” in three-dimensional space, physics becomes, as it were, an “existence” in the four-dimensional “world.”

This four-dimensional “world” bears a close similarity to the three-dimensional “space” of (Euclidean) analytical geometry. If we introduce into the latter a new Cartesian co-ordinate system (_x′_1, _x′_2, _x′_3) with the same origin, then _x′_1, _x′_2, _x′_3, are linear homogeneous functions of _x_1, _x_2, _x_3 which identically satisfy the equation

_x_1′2 + _x_2′2 + _x_3′2 = _x_12 + _x_22 + _x_32

The analogy with (12) is a complete one. We can regard Minkowski’s “world” in a formal manner as a four-dimensional Euclidean space (with an imaginary time coordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”

APPENDIX III

THE EXPERIMENTAL CONFIRMATION OF THE GENERAL THEORY OF RELATIVITY

From a systematic theoretical point of view, we may imagine the process of evolution of an empirical science to be a continuous process of induction. Theories are evolved and are expressed in short compass as statements of a large number of individual observations in the form of empirical laws, from which the general laws can be ascertained by comparison. Regarded in this way, the development of a science bears some resemblance to the compilation of a classified catalogue. It is, as it were, a purely empirical enterprise.

But this point of view by no means embraces the whole of the actual process; for it slurs over the important part played by intuition and deductive thought in the development of an exact science. As soon as a science has emerged from its initial stages, theoretical advances are no longer achieved merely by a process of arrangement. Guided by empirical data, the investigator rather develops a system of thought which, in general, is built up logically from a small number of fundamental assumptions, the so-called axioms. We call such a system of thought a _theory_. The theory finds the justification for its existence in the fact that it correlates a large number of single observations, and it is just here that the “truth” of the theory lies.

Corresponding to the same complex of empirical data, there may be several theories, which differ from one another to a considerable extent. But as regards the deductions from the theories which are capable of being tested, the agreement between the theories may be so complete that it becomes difficult to find any deductions in which the two theories differ from each other. As an example, a case of general interest is available in the province of biology, in the Darwinian theory of the development of species by selection in the struggle for existence, and in the theory of development which is based on the hypothesis of the hereditary transmission of acquired characters.

We have another instance of far-reaching agreement between the deductions from two theories in Newtonian mechanics on the one hand, and the general theory of relativity on the other. This agreement goes so far, that up to the present we have been able to find only a few deductions from the general theory of relativity which are capable of investigation, and to which the physics of pre-relativity days does not also lead, and this despite the profound difference in the fundamental assumptions of the two theories. In what follows, we shall again consider these important deductions, and we shall also discuss the empirical evidence appertaining to them which has hitherto been obtained.

(_a_) Motion of the Perihelion of Mercury

According to Newtonian mechanics and Newton’s law of gravitation, a planet which is revolving round the sun would describe an ellipse round the latter, or, more correctly, round the common centre of gravity of the sun and the planet. In such a system, the sun, or the common centre of gravity, lies in one of the foci of the orbital ellipse in such a manner that, in the course of a planet-year, the distance sun-planet grows from a minimum to a maximum, and then decreases again to a minimum. If instead of Newton’s law we insert a somewhat different law of attraction into the calculation, we find that, according to this new law, the motion would still take place in such a manner that the distance sun-planet exhibits periodic variations; but in this case the angle described by the line joining sun and planet during such a period (from perihelion—closest proximity to the sun—to perihelion) would differ from 360°. The line of the orbit would not then be a closed one but in the course of time it would fill up an annular part of the orbital plane, viz. between the circle of least and the circle of greatest distance of the planet from the sun.

According also to the general theory of relativity, which differs of course from the theory of Newton, a small variation from the Newton-Kepler motion of a planet in its orbit should take place, and in such away, that the angle described by the radius sun-planet between one perhelion and the next should exceed that corresponding to one complete revolution by an amount given by

image051

(_N.B._—One complete revolution corresponds to the angle 2π in the absolute angular measure customary in physics, and the above expression given the amount by which the radius sun-planet exceeds this angle during the interval between one perihelion and the next.) In this expression _a_ represents the major semi-axis of the ellipse, _e_ its eccentricity, _c_ the velocity of light, and _T_ the period of revolution of the planet. Our result may also be stated as follows: According to the general theory of relativity, the major axis of the ellipse rotates round the sun in the same sense as the orbital motion of the planet. Theory requires that this rotation should amount to 43 seconds of arc per century for the planet Mercury, but for the other Planets of our solar system its magnitude should be so small that it would necessarily escape detection.[26]

[26] Especially since the next planet Venus has an orbit that is almost an exact circle, which makes it more difficult to locate the perihelion with precision.

In point of fact, astronomers have found that the theory of Newton does not suffice to calculate the observed motion of Mercury with an exactness corresponding to that of the delicacy of observation attainable at the present time. After taking account of all the disturbing influences exerted on Mercury by the remaining planets, it was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained perihelial movement of the orbit of Mercury remained over, the amount of which does not differ sensibly from the above mentioned +43 seconds of arc per century. The uncertainty of the empirical result amounts to a few seconds only.

(_b_) Deflection of Light by a Gravitational Field

image052

In Section XXII it has been already mentioned that according to the general theory of relativity, a ray of light will experience a curvature of its path when passing through a gravitational field, this curvature being similar to that experienced by the path of a body which is projected through a gravitational field. As a result of this theory, we should expect that a ray of light which is passing close to a heavenly body would be deviated towards the latter. For a ray of light which passes the sun at a distance of Δ sun-radii from its centre, the angle of deflection (α) should amount to

image053

It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification (“curvature”) of space caused by the sun.

This result admits of an experimental test by means of the photographic registration of stars during a total eclipse of the sun. The only reason why we must wait for a total eclipse is because at every other time the atmosphere is so strongly illuminated by the light from the sun that the stars situated near the sun’s disc are invisible. The predicted effect can be seen clearly from the accompanying diagram. If the sun (_S_) were not present, a star which is practically infinitely distant would be seen in the direction _D_1, as observed front the earth. But as a consequence of the deflection of light from the star by the sun, the star will be seen in the direction _D_2, _i.e._ at a somewhat greater distance from the centre of the sun than corresponds to its real position.

In practice, the question is tested in the following way. The stars in the neighbourhood of the sun are photographed during a solar eclipse.

In addition, a second photograph of the same stars is taken when the sun is situated at another position in the sky, _i.e._ a few months earlier or later. As compared with the standard photograph, the positions of the stars on the eclipse-photograph ought to appear displaced radially outwards (away from the centre of the sun) by an amount corresponding to the angle _a_.

We are indebted to the [British] Royal Society and to the Royal Astronomical Society for the investigation of this important deduction. Undaunted by the [first world] war and by difficulties of both a material and a psychological nature aroused by the war, these societies equipped two expeditions—to Sobral (Brazil), and to the island of Principe (West Africa)—and sent several of Britain’s most celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson), in order to obtain photographs of the solar eclipse of 29th May, 1919. The relative discrepancies to be expected between the stellar photographs obtained during the eclipse and the comparison photographs amounted to a few hundredths of a millimetre only. Thus great accuracy was necessary in making the adjustments required for the taking of the photographs, and in their subsequent measurement.

The results of the measurements confirmed the theory in a thoroughly satisfactory manner. The rectangular components of the observed and of the calculated deviations of the stars (in seconds of arc) are set forth in the following table of results:

image054

(_c_) Displacement of Spectral Lines Towards the Red

In Section XXIII it has been shown that in a system _K′_ which is in rotation with regard to a Galileian system _K_, clocks of identical construction, and which are considered at rest with respect to the rotating reference-body, go at rates which are dependent on the positions of the clocks. We shall now examine this dependence quantitatively. A clock, which is situated at a distance r from the centre of the disc, has a velocity relative to _K_ which is given by

_v_ = ω_r_,

where ω represents the angular velocity of rotation of the disc _K′_ with respect to _K_. If _v_0, represents the number of ticks of the clock per unit time (“rate” of the clock) relative to _K_ when the clock is at rest, then the “rate” of the clock (_v_) when it is moving relative to _K_ with a velocity _v_, but at rest with respect to the disc, will, in accordance with Section XII, be given by

image055

or with sufficient accuracy by

image056

This expression may also be stated in the following form:

image057

If we represent the difference of potential of the centrifugal force between the position of the clock and the centre of the disc by φ, _i.e._ the work, considered negatively, which must be performed on the unit of mass against the centrifugal force in order to transport it from the position of the clock on the rotating disc to the centre of the disc, then we have

image058

From this it follows that

image059

In the first place, we see from this expression that two clocks of identical construction will go at different rates when situated at different distances from the centre of the disc. This result is also valid from the standpoint of an observer who is rotating with the disc.

Now, as judged from the disc, the latter is in a gravitational field of potential φ, hence the result we have obtained will hold quite generally for gravitational fields. Furthermore, we can regard an atom which is emitting spectral lines as a clock, so that the following statement will hold:

_An atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated._

The frequency of an atom situated on the surface of a heavenly body will be somewhat less than the frequency of an atom of the same element which is situated in free space (or on the surface of a smaller celestial body).

Now φ = – _K (M/r)_, where _K_ is Newton’s constant of gravitation, and _M_ is the mass of the heavenly body. Thus a displacement towards the red ought to take place for spectral lines produced at the surface of stars as compared with the spectral lines of the same element produced at the surface of the earth, the amount of this displacement being

image060

For the sun, the displacement towards the red predicted by theory amounts to about two millionths of the wave-length. A trustworthy calculation is not possible in the case of the stars, because in general neither the mass _M_ nor the radius _r_ are known.

It is an open question whether or not this effect exists, and at the present time (1920) astronomers are working with great zeal towards the solution. Owing to the smallness of the effect in the case of the sun, it is difficult to form an opinion as to its existence. Whereas Grebe and Bachem (Bonn), as a result of their own measurements and those of Evershed and Schwarzschild on the cyanogen bands, have placed the existence of the effect almost beyond doubt, while other investigators, particularly St. John, have been led to the opposite opinion in consequence of their measurements.

Mean displacements of lines towards the less refrangible end of the spectrum are certainly revealed by statistical investigations of the fixed stars; but up to the present the examination of the available data does not allow of any definite decision being arrived at, as to whether or not these displacements are to be referred in reality to the effect of gravitation. The results of observation have been collected together, and discussed in detail from the standpoint of the question which has been engaging our attention here, in a paper by E. Freundlich entitled “Zur Prüfung der allgemeinen Relativitäts-Theorie” (_Die Naturwissenschaften_, 1919, No. 35, p. 520: Julius Springer, Berlin).

At all events, a definite decision will be reached during the next few years. If the displacement of spectral lines towards the red by the gravitational potential does not exist, then the general theory of relativity will be untenable. On the other hand, if the cause of the displacement of spectral lines be definitely traced to the gravitational potential, then the study of this displacement will furnish us with important information as to the mass of the heavenly bodies.[27]

[27] The displacement of spectral lines towards the red end of the spectrum was definitely established by Adams in 1924, by observations on the dense companion of Sirius, for which the effect is about thirty times greater than for the Sun. R.W.L.—translator

APPENDIX IV

THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

(SUPPLEMENTARY TO SECTION XXXII)

Since the publication of the first edition of this little book, our knowledge about the structure of space in the large (“cosmological problem”) has had an important development, which ought to be mentioned even in a popular presentation of the subject.

My original considerations on the subject were based on two hypotheses:

(1) There exists an average density of matter in the whole of space which is everywhere the same and different from zero.

(2) The magnitude (“radius”) of space is independent of time.

Both these hypotheses proved to be consistent, according to the general theory of relativity, but only after a hypothetical term was added to the field equations, a term which was not required by the theory as such nor did it seem natural from a theoretical point of view (“cosmological term of the field equations”).

Hypothesis (2) appeared unavoidable to me at the time, since I thought that one would get into bottomless speculations if one departed from it.

However, already in the ’twenties, the Russian mathematician Friedman showed that a different hypothesis was natural from a purely theoretical point of view. He realized that it was possible to preserve hypothesis (1) without introducing the less natural cosmological term into the field equations of gravitation, if one was ready to drop hypothesis (2). Namely, the original field equations admit a solution in which the “world radius” depends on time (expanding space). In that sense one can say, according to Friedman, that the theory demands an expansion of space.

A few years later Hubble showed, by a special investigation of the extra-galactic nebulae (“milky ways”), that the spectral lines emitted showed a red shift which increased regularly with the distance of the nebulae. This can be interpreted in regard to our present knowledge only in the sense of Doppler’s principle, as an expansive motion of the system of stars in the large—as required, according to Friedman, by the field equations of gravitation. Hubble’s discovery can, therefore, be considered to some extent as a confirmation of the theory.

There does arise, however, a strange difficulty. The interpretation of the galactic line-shift discovered by Hubble as an expansion (which can hardly be doubted from a theoretical point of view), leads to an origin of this expansion which lies “only” about 109 years ago, while physical astronomy makes it appear likely that the development of individual stars and systems of stars takes considerably longer. It is in no way known how this incongruity is to be overcome.

I further want to remark that the theory of expanding space, together with the empirical data of astronomy, permit no decision to be reached about the finite or infinite character of (three-dimensional) space, while the original “static” hypothesis of space yielded the closure (finiteness) of space.

_K_ = co-ordinate system

_x, y_ = two-dimensional co-ordinates

_x, y, z_ = three-dimensional co-ordinates

_x, y, z, t_ = four-dimensional co-ordinates

_t_ = time

_I_ = distance

_v_ = velocity

_F_ = force

_G_ = gravitational field