Relativity - The Special and General Theory Part 9

In the first place, we imagine an existence in two dimensional s.p.a.ce.

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 24. 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 s.p.a.ce) 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 circ.u.mference 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 p, which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value

eq. 27: file eq27.gif

i.e. a smaller value than p, 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 circ.u.mference of a circle first increases with the radius until the " circ.u.mference 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 circ.u.mference 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 a.n.a.logy, namely, the three-dimensional spherical s.p.a.ce which was discovered by Riemann. its points are likewise all equivalent. It possesses a finite volume, which is determined by its "radius"

(2p2R3). Is it possible to imagine a spherical s.p.a.ce? To imagine a s.p.a.ce means nothing else than that we imagine an epitome of our "

s.p.a.ce " experience, i.e. of experience that we can have in the movement of " rigid " bodies. In this sense we can imagine a spherical s.p.a.ce.

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 = 4pR2 ; if it is spherical, then F is always less than 4pR2. 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 s.p.a.ce. It is easily seen that the three-dimensional spherical s.p.a.ce is quite a.n.a.logous 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 s.p.a.ce: "

elliptical s.p.a.ce." It can be regarded as a curved s.p.a.ce 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 s.p.a.ces without limits are conceivable. From amongst these, the spherical s.p.a.ce (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 moduate degree of certainty, and in this connection the difficulty mentioned in Section 30 finds its solution.

THE STRUCTURE OF s.p.a.cE ACCORDING TO THE GENERAL THEORY OF RELATIVITY

According to the general theory of relativity, the geometrical properties of s.p.a.ce 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 s.p.a.ce is influenced only to an exceedingly small extent by ma.s.ses even of the magnitude of our sun. We might imagine that, as regards geometry, our universe behaves a.n.a.logously 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 s.p.a.ce 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 connection * between the s.p.a.ce-expanse of the universe and the average density of matter in it.

Notes

*) For the radius R of the universe we obtain the equation

eq. 28: file eq28.gif

The use of the C.G.S. system in this equation gives 2/k = 1^.08.10^27; p is the average density of the matter and k is a constant connected with the Newtonian constant of gravitation.

APPENDIX I

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 pernumently 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 K1 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 K1 with the velocity c, the propagation relative to the system K1 will be represented by the a.n.a.logous formula

x' - ct' = O . . . (2)

Those s.p.a.ce-time points (events) which satisfy (x) must also satisfy (2). Obviously this will be the case when the relation

(x' - ct') = l (x - ct) . . . (3).

is fulfilled in general, where l 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 l and , where

eq. 29: file eq29.gif

and

eq. 30: file eq30.gif

we obtain the equations

eq. 31: file eq31.gif

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 K1 we have permanently x' = 0, and hence according to the first of the equations (5)