Einstein (27 page)

By means of other experiments we can then investigate whether these rods actually have all the properties that geometry postulates about “straight lines.” For instance, we can measure whether such a rod is really the shortest line connecting two points. Of course in order to be able to carry out this measurement we must also describe a physical operation of measuring the length of a curved line. It may be found that when a triangle is formed by joining these points by lines that form the shortest
distances between these points, the sum of the angles of this triangle does
not
equal two right angles. We are then faced with a dilemma. If we say that the lines forming this triangle are straight lines, we retain the property of the straight line to be the shortest distance between any two points, but then the theorem of the sum of angles is no longer valid. On the other hand, if we want the theorem to be valid, the property to be the shortest distance has to be rejected. We are free to decide which property we retain for the lines we call “straight,” but we cannot have them both as in Euclidean geometry.

Einstein’s fundamental assumption can now be re-expressed in this form: In a space where masses that exert gravitational forces are present, Euclidean geometry ceases to be valid. In this theory curves which are the shortest distances between any two points have special significance, and the angles of a triangle formed by these lines do not add up to two right angles where a gravitational field exists.

This distinction between Euclidean space and the “curved” space of Einstein can be illustrated by considering a similar distinction between a plane surface and a curved surface. For all triangles on a plane surface, all of Euclid’s theorems hold true; but what happens for triangles on a curved surface? Take for example the surface of the earth. If we are restricted to only those points which actually lie on the surface and cannot consider any point lying above or below it, there are no “straight lines” in the usual sense. But the curves which form the shortest distance between two points on the earth’s surface are important in navigation and geodesy; they are called geodesic lines. For the surface of the sphere, the geodesic lines are arcs of great circles, and consequently all the meridians defining longitude and the equator are geodesic. If we consider a triangle formed by the North Pole and two points on the equator it is bounded by geodesic lines. The equator cuts all the meridians perpendicularly so that the two angles at the base of the triangle are both right angles, and hence the
sum of the angles is greater than two right angles
by just the value of the polar angle. A similar situation always holds for any curved surface, and, conversely, if the sum of the angles of a triangle formed by geodesic lines on a surface does not equal exactly two right angles, then the surface is curved.

This notion of curvature of a surface is extended to space. Geodesic lines are defined as curves forming the shortest distances between any two points in space, and the space is called “curved” if the angles of a triangle formed by three geodesic
lines do not add up to two right angles. According to Einstein’s theory, the presence of material bodies produces certain curvatures in space, and the path of a particle moving in a gravitational field is determined by this curvature of space. Einstein found that such paths can be described most simply by considering the geometry of this curved space rather than by ascribing its deviation from a straight line to the existence of forces as Newton had done. Furthermore, Einstein found that not only the paths of material particles, but also those of light rays in a gravitational field can be described simply in terms of geodesic lines in this curved space; and, conversely, that the curvature of space can be inferred from observations on the path of moving bodies and light rays.

We shall see later that many people, even some physicists, considered it absurd to say that any conclusion about the curvature of space can be drawn from the form of light rays. Some even considered it completely nonsensical to say that a space is “curved.” To them, a surface or a line may be
curved in space
, but to say that space itself is “curved” seemed preposterous and absurd. This opinion, however, is based on ignorance of the geometrical mode of expression. As we have seen above, a “curved space” simply means a space in which the sum of the angles of a triangle formed by geodesic lines does not equal two right angles, and this terminology is used because of the analogous distinction between flat and curved surfaces. It is futile to try to picture what a curved space “looks” like, except by describing the measurement of triangles.

If we wish to describe the motion of a certain particle completely, it is not sufficient to give the shape of its trajectory, but it is necessary to add how the position of the particle on this trajectory varies with the time. For instance, to say that the motion of a particle uninfluenced by any force in the Newtonian sense is rectilinear is not complete; we must add that its motion takes place with constant velocity.

The complete motion can, however, be presented in a geometrical form by adding a dimension to the number necessary to describe the trajectory. For example, in the simplest case of a rectilinear motion, the trajectory is a straight line, and the position
of the particle describing it can be specified by giving the distance that the particle is from a certain definite point on the straight line. We now take a sheet of paper and plot these distances along one direction, and for each point plot in a direction perpendicular to the distance the time corresponding to each position. Then the curve drawn through these points gives the complete geometrical presentation of the motion. If the motion takes place with constant velocity as well as being rectilinear, the curve will be a straight line. Thus motion along a straight line, or one-dimensional motion, to express it technically, can be represented
completely
on a plane — that is, in two-dimensional space. Now, the space of our experience has three dimensions; to specify the position of a ball in a room, we must give three numbers, the distances from the two walls and its height above the floor. Hence we need three dimensions to describe the trajectory of a general motion, and four dimensions to give a
complete
presentation of the motion. The motion of a particle is specified completely by a curve in a four-dimensional space.

This notion of four-dimensional space, simple as it is, has given rise to a great deal of confusion and misunderstanding. Some writers have maintained that these curves in four-dimensional space are “only aids for mathematical presentation” and “do not really exist.” The statement “do not really exist,” however, is a pure truism, since the statement “really existing” is used in daily life to describe only directly observable objects in our three-dimensional space. In contrast to this, many authors, especially philosophers and philosophically tinged physicists, have taken the point of view that
only the events in four-dimensional space are real
, and a representation in three-dimensional space is only a subjective picture of reality. We can readily see that such a position is equally justified except that the word “real” is used in a different sense. To clear up this disagreement we have to use a little
semantics
.

In his special theory of relativity developed at Bern, Einstein had shown that when mechanical and optical phenomena are described by means of clocks and measuring rods, the description depends on the motion of the laboratory in which these instruments are used. And he had been able to state the mathematical relations that correlate the various descriptions of the same physical event. In 1908 Hermann Minkowski, Einstein’s former professor of mathematics at Zurich, showed that this relationship between different descriptions of the same phenomena
can be represented mathematically in a very simple manner. He pointed out that these different descriptions of a motion represented by a curve in four-dimensional space are mathematically what are known as “projections of this four-dimensional curve on different three-dimensional spaces.” Minkowski therefore took the view that only the four-dimensional curve “really” exists, and the different descriptions are merely different pictures of the same reality. This concept is analogous to saying that a fixed object in three-dimensional space, say a house, “really exists,” but that photographs of this house taken from various directions — that is, two-dimensional projections of the three-dimensional house — never represent reality itself, but only descriptions of it from different points of view.

Obviously the word “real” is not used in the same sense here as when we say that only the three-dimensional body is “real” and that the four-dimensional presentation is simply an invented mathematical schema. In Minkowski’s speech “real” means the “simplest theoretical presentation of our experiences,” while in the other sense it means “our experience expressed as directly as possible in ordinary, everyday language.”

Einstein’s theory of gravitation started out from this representation of motion as a curve in four-dimensional space. Motion, if neither gravity nor any other force is acting, is represented by the simplest curve, the straight line, in flat, four-dimensional space. If only gravity but no other force is acting, Einstein assumed that the space becomes curved, but the motion is still represented by the simplest curve in such a space. Since there are no straight lines in curved space, he took for the simplest curve in space the curve with the shortest length between any two points — that is, the geodesic line. Hence motion of a particle under gravity is represented by a geodesic curve in four-dimensional curved space, and this curvature of space is determined by the distribution of matter which produces the gravitational field.

Thus Einstein’s general theory of relativity consists of two groups of laws:

First: The field laws which state how the masses present produce the curvature in space.

Second: The laws of motion both for material particles and for light rays which state how the geodesic lines can be found for a space whose curvature is known.

This new theory of Einstein was a fulfillment of the program of Ernst Mach. From the material bodies present in space it
enables one to calculate the curvature of space, and from this the motion of bodies. According to Einstein, the inertia of bodies is not due, as Newton has assumed, to their efforts to maintain their direction of motion in absolute space, but rather to the influence of the masses about them — the fixed stars, as Mach had suggested.

Einstein’s new theory, which so boldly and fundamentally changed the tested and successful Newtonian theory, was originally based on arguments of logical simplicity and generality. The question naturally arose whether new phenomena could be deduced from this theory which differed from those derived from the old, and which could be used as experimental tests between the two theories. Otherwise Einstein’s theory remained only a mathematical-philosophical construction, which provided a certain degree of mental stimulation and pleasure but contributed nothing about physical reality. Einstein himself always recognized a new theory only if it uncovered a new field of the physical world.

Einstein showed mathematically that in “weak” gravitational fields his theory predicted the same results as Newton’s. Here the curvature of our three-dimensional space is negligible, and the only difference comes from the new mathematical approach in the addition of the fourth dimension. The calculation of motion — for example, that of the earth around the sun — gives exactly the same result as that obtained from Newton’s law of force and his theory of gravitation. It is only when the velocity of a body is comparable to that of light that any difference between the two theories can be detected.

In order to find phenomena where spatial curvature plays possibly a role, Einstein searched among the observations of celestial bodies for motions that were inconsistent with the predictions of Newtonian mechanics. He found one case. It had long been known that Mercury, a planet close to the sun and finis strongly exposed to its gravitational field, did not move exactly as predicted by Newton’s theory. According to the old theory, all planets should perform elliptical orbits whose position in space are fixed in relation to the stars, but, it had been observed that the elliptical orbit of Mercury rotates around the sun at the very
small rate of 43.5 seconds of an arc per century. This discrepancy had never been given a satisfactory explanation. When Einstein calculated the motion of Mercury according to his theory, he found that the orbit should actually rotate as observed. From the very beginning this achievement has been a strong argument in favor of Einstein’s theory.

The effect of the curvature of space on the path of light rays is more impressive. While still at Prague Einstein had pointed out the possibility of bending rays of light as they passed close to the surface of the sun. He had calculated, on the basis of Newton’s law of force and his own theory of gravitation of 1911, that the deflection should be 0.87 seconds of an arc. According to his new theory of curved space, Einstein found the deflection is 1.75 seconds, actually twice as great as his former result.

The third prediction that Einstein made was on the change in wave length of light emitted by a star. His calculation showed that light, in leaving the star where it is emitted, has to pass through its gravitational field, and this passage shifts the wave length toward the red. Even for the sun the effect turned out to be hardly observable, but in the case of the very dense companion star to Sirius it seemed to be of observable magnitude.

It is important to note that of these three phenomena predicted by the theory, only one of them, the motion of Mercury, was actually known at the time Einstein developed his theory. The other two were entirely new phenomena, which had never been observed or even suspected. Both of these predictions received unqualified verification some years later, and so gave conclusive evidence of the correctness of the theory. It is very remarkable and a great tribute to Einstein that he was able to develop a theory which started from few fundamental principles and using the criterion of logical simplicity and generality led to amazing results.