Einstein's Theories of Relativity and Gravitation / A selection of material from the essays submitted in the competition for the Eugene Higgins prize of USD5,000

THE NEW WORLD

A Universe in Which Geometry Takes the Place of Physics, and Curvature That of Force

BY GEORGE FREDERICK HEMENS, M.C., B.SC., LONDON

It is familiar knowledge that the line, the surface and ordinary Euclidean space are to be regarded as spaces of one, two and three dimensions respectively and readers of this journal are aware that a hypothetical space of four dimensions has been closely investigated. The most convenient space to study is the surface or two-space, since we can regard it as embedded in a three-space. If a surface is curved it is generally impossible to draw a straight line on it, for as we see clearly, the “straightest” line is changing its direction at every point. To describe this property accurately it is necessary to ascribe to each point a magnitude which expresses what happens to the direction of a short line in the region when displaced a short distance parallel to itself. This is called the direction-defining magnitude. Different sets of values of this magnitude relate to surfaces of different curvatures.

A second fundamental property has recently been pointed out. There is inherent in every part of a space a measure of length peculiar to that particular region and which in general varies from region to region. To describe this variation accurately it is necessary to ascribe to each point another magnitude called the length-defining magnitude, which expresses the change from each point to the next of the unit of length. These two magnitudes define the surface completely.

Similarly, a space of any number of dimensions is defined completely by a similar pair of magnitudes. A space is the “field” of such a magnitude-pair and the nature of these magnitudes defines the dimensions of the space. The four-space usually described is the Euclidean member of an infinity of four-spaces.

When we look into a mirror we see a space differing from ordinary space in that right and left are interchanged and this is described mathematically by saying that if we locate points as usual by specifying three distances upper X 1, upper X 2, upper X 3 of the point from three mutually perpendicular planes, then a point upper X 1, upper X 2, upper X 3, in actual space corresponds with a point upper X 1, upper X 2, minus upper X 3 in the mirrored space: in other words the mirrored space is derived from the real space by multiplying the upper X 3 coordinates by negative 1. If we were to multiply by StartRoot negative 1 EndRoot instead of negative 1 we should derive a different space; in this case, however, we have no mirror to show us what it looks like. Such a space is said to have one negative dimension and it has the peculiar property that in the figure derived from the right triangle of ordinary space the square of the “hypotenuse” equals the difference and not the sum of the squares of the other two sides, so that the length of a line may sometimes have to be represented by the square-root of a negative number, a “complex” number.

In considering what at first sight may appear to be fantastic statements made by this theory, it must be borne in mind that all our knowledge of the external universe comes through our sense-impressions, and our most confident statements about external things are really of the nature of inferences from these sense-impressions and, being inferences, liable to be wrong. So that if the theory says that a stone lying on the ground is not a simple three-dimensional object, and that its substance is not the same as its substance a moment before, the matter is one for due consideration and not immediate disbelief.

The idea that the universe extends in time as well as in space is not new, and fiction-writers have familiarized us with wonderful machines in which travellers journey in time and are present at various stages of the world’s history. This conception of the universe, to which the name “space-time” is usually applied, is adopted by the new theory and assigned the status of a physical reality.

The World Geometry

The fundamental creed of the new theory is that the space-time universe constitutes a true four-dimensional space of one negative dimension, this dimension being time. The variations from point to point of the direction-defining and length-defining magnitudes generate the geometrical properties of curvature, etc., and these are cognised by the human mind as physical phenomena: our sense-impressions are nothing more nor less than perceptions of the geometry of a fourspace. So instead of inferring from our sense-impressions the existence of matter, motion and the like as we are accustomed to do, we should with equal justice infer the existence of a geometrical fourspace. Thus it becomes necessary to prepare a dictionary in which the familiar things of our world are identified with those geometrical properties of the four-space which really constitute them, and in so doing parts of our geometrical knowledge assume the guise of new physical knowledge.

Through the fourspace our consciousness travels, cognising a changing three-dimensional section of it as it goes and thus giving rise to time. It becomes aware that the fourspace is pleated or folded along lines all running roughly in the same direction, and possibly because this is the easiest direction to follow, it travels along the lines. The direction of this motion is the negative dimension. Thus consciousness is always aware of the nearly constant forms of the cross-sections of the pleats along which it travels. These unvarying forms constitute matter: matter is the form of a section through a uniform pleat of the fourspace—a three-dimensional aspect of a four-dimensional curvature; so that in strict accuracy we should say that a stone is the shape or form of a changing section of a four-dimensional object, the complete object being a long fold in the fourspace. The physical interpretation of this conservation of form of the cross-section is that matter is conserved. It is thus seen that the conscious mind, by following these pleats, has so determined time that the law of the conservation of matter must hold. The mathematical treatment of the subject makes it clear that practically all other physical laws similarly follow as a direct result of this choice of time. The type of order prevailing in the physical universe, the laws of gravitation, heat, motion and the rest are not directly imposed by some external power, but are apparently chosen by mind itself.

In the neighborhood of these pleats the fourspace is still curved, but to a smaller degree. This we cognise as energy or as a field of force. Thus energy is seen to be the same kind of thing as matter and would therefore be expected to have weight. This was experimentally demonstrated in 1919 when light was in effect actually weighed. Conversely, matter consists of energy; and it is calculated that one liter of water contains sufficient energy to develop a million horsepower for about four years. It is now believed that the sun’s energy is derived from the disintegration of the matter of which it is made.

The method of establishing these identifications will be clear from the following: We already knew that matter is made up of electrons and that radiant energy is electromagnetic and before the advent of this theory it was regarded as certain that practically all observed physical phenomena except gravitation were manifestations of the electromagnetic field. The new theory has confirmed this belief. It is found that the gravitational and electromagnetic conditions of the universe are completely defined if to each point of space-time a gravitational and an electric potential are ascribed. These are magnitudes of the same nature as the direction-defining and length-defining magnitudes which must necessarily be associated with every point of space-time if it is a true “space,” and they are therefore identified with these. By performing ordinary mathematical operations on these magnitudes statements of fact clothed in mathematical form are obtained, which are to be interpreted on the one hand as physical laws and on the other as geometrical properties of the fourspace. Nearly all our physical laws are derivable mathematically in this way, so that an extensive identification is effected which has been fruitful of results.

It has been mentioned that a slight curvature is sometimes cognised as force and as this identification appeared originally as a postulate its history is interesting.

The Genesis of the Theory

An experiment by Michelson and Morley (1887), on which the whole theory is based, made it appear that if a man measures the velocity at which light passes him he will get the same result whether he is stationary, rushing to meet the light, or moving in the same direction as the light. The solution was provided by Einstein in 1905. He suggested that since we know the results of these determinations ought not to agree, something must have happened to the clocks and measuring-rods used in measuring the velocity so that the standards of length and time were not the same in the three cases, the alterations being exactly such as to make the velocity of light constant. This solution is universally accepted as true and is the fundamental postulate. Thus the length of a stick and the rate at which time passes will change as the velocity of the person observing these things changes. If a man measured the length of an aeroplane going past him at 161,000 miles per second it would measure only half the length observed when stationary. If the aeroplane were going with the velocity of light, its length would vanish though its breadth and height would be unaltered. Similarly, if of two twin brothers one were continually moving with reference to the other their ages would gradually diverge, for time would go at different rates for the two. If one moved with the velocity of light, time would stand still for him while for the other it would go on as usual. To get actually younger it would be necessary to move quicker than light which is believed to be impossible. The velocity of light is assumed to be the greatest velocity occurring in nature.

Evidently then if the distance in space and the interval in time separating two given events, such as the firing of a gun and the bursting of the shell, are measured by two observers in uniform relative motion, their estimates will not agree. Consider now the simple problem of measuring the distance between two points on an ordinary drawing-board. If we draw two perpendicular axes, we can define this distance by specifying the lengths of the projections on the two axes of the line joining the points. If we choose two different axes the projections will not be the same but will define the same length. Similarly, in a Euclidean four-space the distance between two points will be defined by the projections on the four axes, but if these axes be rotated slightly, the projections will be different, but will define the same length. Now, returning to the two observers just mentioned, it was noticed by Minkowski in 1908 that if the space measurements between the two events are split into the usual three components, and if the time measurements are multiplied by StartRoot negative 1 EndRoot, the difference between the two sets of measurements is exactly the same as would have occurred had these two events been points in a Euclidean fourspace, and two different observations made of their distance apart using two sets of axes inclined to each other. The velocity of light is made equal to 1 in this calculation by a suitable choice of units. This discovery threw a vivid light on the problem of space-time, showing that it is probably a true four-space of one negative dimension, a simple derivative of the much-discussed and now familiar Euclidean four-space.

Although this discovery gave a tremendous impetus to the progress of the theory, it is probable that it holds a deeper significance not yet revealed. It is probably a statement of the “stuff” of which the four-space is made, and perhaps also of how it is made; but the problem remains unsolved.

It thus becomes plain that our two observers are merely looking at the same thing from different viewpoints. Each has just as much right as the other to regard himself as being at rest in ordinary space (this is the postulate of the relativity of uniform motion) and to regard his time direction as a straight line in the four-space. The difference is merely that the two time axes are inclined to each other. If, however, one were moving with an acceleration with reference to the other his path in the four-space will appear curved to the other, though he himself, since he regards it as his time axis, will still assume it to be straight. If there is a body moving in what one observer sees to be a straight line, the other will, of course, in general see it as curved, and following the usual custom, since this body, without apparent reason, deviates from the straight path, will say there must be some force acting on it. Thus the curvature of his time axis, due to his accelerated motion, makes it appear that there is round him a field of force, which causes freely moving bodies to deviate from the straight path. Now if space-time is itself inherently curved it is not generally possible for any line in it to be straight any more than it is possible for any line on the surface of a sphere to be straight. Hence, all axes must be curved, and all observers, whatever their states of motion, must experience fields of force which are of the same nature as those due to motion only. The extra force experienced when a lift begins to rise is an example of force due to pure motion: gravitation is the similar force due to an inherent curvature of the four-space, and it was the postulate that these forces were similar that made possible Einstein’s solution of the general problem of gravitation.

The Time Diagram

The correlation of time with its geometrical analogue is of absorbing interest. Representing velocity by the common method of plotting a curve showing positions at various times and marking distances horizontally and times vertically, the velocity of light being 1, MM' and NN' will both represent this velocity. Since this is assumed to be the greatest velocity occurring in nature, all other possible velocities

are represented by lines falling within the upper and lower V’s. Now this diagram correctly represents two dimensions of Minkowski’s Euclidean four-space so, transmuting to real but flat four-space by multiplying times by StartRoot negative 1 EndRoot, it is seen that there is a region outside which no effect can be propagated from O since that would involve the existence of a velocity greater than that of light. This region represents the future of O. Similarly, O can only be affected by events within the region derived from the downward-opening V, which therefore represents the past of O. The region between the two represents events which may be either simultaneous with O or not, according to the velocity of the observer at O. Thus in this theory an event dictated by free-will, could affect points in its “future” region, but not in any other, which agrees with experience and shows that the theory is not essentially “determinist.” If “free-will” is really free, the future is not yet determined, and the fourspace must be in some way formed by the will as time progresses.

The trains of thought inspired by Einstein’s postulates have already carried us to a pinnacle of knowledge unprecedented in the history of man. On every hand, as we look out upon the universe from our new and lofty standpoint, unexpected and enthralling vistas open up before us, and we find ourselves confronting nature with an insight such as no man has ever before dared aspire to.

It is completely unthinkable that this theory can ever be swept aside. Apart from experimental verifications which, in point of fact, lend it the strongest support, no one could work through the theory without feeling that here, in truth, the inner workings of the universe were laid bare before him. The harmony with nature is far too complete for any doubt to arise of its truth.

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