EINSTEIN AND EUCLID

Taranaki Daily News, 2 September 1933, Page 13 (Supplement)

EINSTEIN AND EUCLID

CURVED SPACE THEORY

(By

Rev. B. Dudley,

F.R.A.S.)

Among the many long-cherished beliefs which have had to be abandoned, the Euclidian geometry has suffered more than an eclipse. It was about a century | age that the first rival system was in-| vented. Up to that time it was looked upon as a necessity of thought, Euclid s well-known axioms being considered unavoidable, except perhaps for one instance, namely, that which concerned parallel lines. Even here, however, it was the form of the axiom rather than the axiom itself which was questioned. It may now be said that the number of possible geometries is legion. . Instead of regarding geometrical axioms as necessary to thought, we have come to look upon "them as the result merely of limited experience and deeply engrained ways of thinking. All sorts of geometrical axioms can be invented; and those which can be made to hang together make up a system of geometry. It is no longer necessary to take it for granted that space is Euclidian. Its geometry must be determined without any foregone conclusions regarding it; we must watch the behaviour of our measuring instruments, not expecting them to prove what we want them to prove. One outcome of the newer way of looking at these matters is that the whole Newtonian conception of the world has been undermined, or at least made liable to serious questioning.

Assuming as he did that the relations between objects in space must conform to the principles of geometry, as expounded by the ancient master, Newton postulated that force of gravitation for which his name is rightly famous. He hold that a body in motion through space would and must, unless disturbed, continue to move in a straight line, a straight line being defined as the short" est distance between two points. . But there arises in the mind of a questioner free to think independently the enquiry whether this is really so. Do straight lines drawn on the surface, of a globe, for instance, obey the Euclidian principles ? On the contrary, the interior angles of- an angle when traced on a spherical surface are not equal to two right angles. A surface of two dimensions, unless it be a plane, will not yield to Euclid’s laws. And since, there are two-dimensional spaces wherein his rules are inapplicable, doubts begin to creep in when it is suggested that three dimensional space must of necessity obey them. Theoretically, at all events,, it is possible for non-Euclidian three-dimen-sional spaces to exist (three dimensional space is space forward and backward, left and right,' up and down).

As" long as Euclid’s laws were assumed to be correct, the Newtonian doctrine that planets once launched out, as it were, into space would move forward in straight lines for ever unless some force were brought to bear upon them diverting them from their course, stood its ground fearless of challenge. But Einstein and others before him could not feel satisfied with this assumption. . If we can find a geometry of space which can explain the observed paths of the planets without adopting Newton’s force of gravitation so much is gained for simplicity; for then we are no longer obliged to believe in action at a distance. ■ Nor are we compelled to endorse those absurd doctrines of the ether that have been propounded with a view to explaining such action. Einstein therefore with great daring broke away completely from the traditional geometry. Others before him had made only partial detachment from it. This iconoclast asked himself the questions: What is space is four-dimensional, instead of three? What if, contrary to traditional belief, it is curved ? Then the planets describe - their paths about the sun not by virtue of any imagined force of gravitation acting over a distance, but because the observed “orbits” are natural to them. Assuming for a moment that space is actually curved in accordance with Einsteinian theory, how easy it would be for us to still imagine quite incorrectly that the planets are forced into curved paths by the “pull” of some other body—that is, if we knew nothing whatever about such special curvature! Suppose (to quote an illustration) that whenever I drop a marble or other spherical body to the floor of a certain room, that object invariably rolls to its centre. Then I have two possible explanations: either the floor is curved downwards, being lowest at the centre, or there is some strange force located at the middle of the floor attracting objects to it. Having acquired the constant habit of supposing the latter to be the case, it might be very difficult to wean myself away from the theory, even after serious doubts had been engendered as to its truth. Einstein Was successful in disengaging his mind from the bondage of habit and tradition; and in his effort to make all the motions usually ascribed to gravitation “natural” motions, he saw that the geometry of space could vary from one region to another. In the vicinity of the sun, for example, the planets describe ellipses in space. In the neighbourhood of a double star, on the other hand, a planet could not possibly describe that sort of curve. Its path would be determined by the kind of geodesies obtaining in that part of space. As De Sitter says, “a geodesie in curved space is exactly the same thing as a straight line in fiat space. A geodesie on a perfect plane is a straight line drawn from point to point—the shortest distance between them; whereas on the surface of a sphere the geodesie is necessarily curved: the arc of a circle.” A ray of light from a star (as experi- I mental tests have amply demonstrated) | when passing near the sun is bent, round, i and the star is thus seen in a different direction from where it would be seen if the sun had not been near to deflect it. Space cannot have a uniform geometry. Its geometry in any given part depends upon the amount of matter present there. Einstein’s theory shows that the properties of space are influenced by matter. The sun has a greater “mass” than the earth. Therefore it exerts a stronger influence. Stated in another way, owing to the “distortion” of the space they traverse—a distortion due to the sun—these raylines reach us with a direction different from that they would have if they did not pass through the markedly nonEuclidian space near the sun. I Thus ’it has become clear to modem science that the ancient system . of 1 geometry consisting of an enquiry into the properties and relationships .of volumes, areas, lines, and points involving no more than their dimensions, is based upon the capability and, at the same time, the limitation of our senses and imagination. If light travels in non-Euclidian lines in curved space, then after the lapse of, say, 100,000,000 years, it returns to its starting point. It can never reach out to an edge of boundary of the universe, but returns in the long run to its source from a direction opposite to that in which it set out. The universe is therefore without bounds and yet finite in volume.