A Stubbornly Persistent Illusion

by Stephen Hawking

  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

  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 world-sphere 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 circle to its radius r depends on r, but for a given value of r it is the same for all points of the “world-sphere”; 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” (2π2R3). 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 γ 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 = 4πγ2; if it is spherical, then F is always less than 4πγ2. With increasing values of γ, F increases from zero up to a maximum value which is determined by the “world-radius,” but for still further increasing values of γ, 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 30 finds its solution.

  THIRTY-TWO

  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 extend 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 30.

  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 connection1 between the space-expanse of the universe and the average density of matter in it.

  1For the “radius” R of the universe we obtain the equation

  The use of the C.G.S. system is in this equation gives is the average density of the matter and x is a constant connected with the Newtonian constant of gravitational.

  APPENDIX ONE

  SIMPLE DERIVATION OF THE LORENTZ TRANSFORMATION

  [SUPPLEMENTARY TO SECTION 11]

  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

  or

  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

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

  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

  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

  and

  we obtain the equations

  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)

  If we call v the velocity with which the origin of K′ is moving relative to K, we then have

  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)

  Two points of the x′-axis which are separated by the distance Δx′ = 1 when measured in the K′ system are thus separated in our instantaneous photograph by the distance

  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

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

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

  The equations (6) and (7b) 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.

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

  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

  In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light of arbitrary directions, 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

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

  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

  or,

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

  Since equation (8a) 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 (8a) 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 characterize the generalized Lorentz transformation thus:

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

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

  APPENDIX TWO

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

  [SUPPLEMENTARY TO SECTION 17]

  We can characterise the Lorentz transformation still more simply if we introduce the imaginary . ct in place of t, as time-variable, If, in accordance with this, we insert

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

  That is, by the afore-mentioned choice of “co-ordinates,” (11a) is transformed into this equation.

  We see from (12) that the imaginary time co-ordinate x4 enters into the condition of transformation in exactly the same way as the space co-ordinates x1, x2, x3. It is due to this fact that, according to the theory of relativity, the “time” x4 enters into natural laws in the same form as the space co-ordinates x1, x2, x3.

  A four-dimensional continuum described by the “co-ordinates” x1, x2, x3, x4, 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 with the same origin, then are linear homogeneous functions of x1, x2, x3, which identically satisfy the equation

  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 imaginary time co-ordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”

  APPENDIX THREE

  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 n
o 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.