The foundations of Einstein's theory of gravitation

§ 6

THE VERIFICATION OF THE NEW THEORY BY ACTUAL EXPERIENCE

According to Einstein's theory, a planet, at the distance of Mercury for instance, moves, under the action of the sun's attraction, along the "straightest path," according to the equation: The 's can be derived from the differential equations, which were given for them above, and which result from the assumed sole presence of the sun and the planet being regarded as a mass concentrated at a point. Einstein's developments give the ellipse of Kepler too as a first approximation for the path of the planet: at a higher degree of approximation, however, it is found that the radius vector from the sun to the planet, between two consecutive passages through perihelion and aphelion, sweeps out an angle, which is about 0.05" greater than 180°; so that, for each complete revolution of the planet in its path, the major axis of the path—i.e. the straight line connecting perihelion with aphelion—turns through about 0.1" in the sense in which the path is described. Therefore, in 100 years—Mercury completes a revolution in 88 days—the major axis will have turned through 43". The new theory, therefore, actually explains the hitherto inexplicable amount, 43 seconds per 100 years, in the motion of Mercury's perihelion, merely from the effect of the sun's gravitation. (The deviations due to such disturbances would only differ very inappreciably from the values obtained by Newton's theory in the case of the remaining planets.) The only arbitrary constant which enters into these calculations is the gravitational constant which figures in the differential equations for the gravitational potentials as has already been mentioned on page 50. This achievement of the new theory can scarcely be estimated too highly.

The reason that a measurable deviation from the results according to Newton's theory occurs in the case of Mercury, the planet nearest to the sun, but not in the case of the planets more distant from the sun, is that this deviation decreases rapidly with increasing distance of the planet from the sun, so that it already becomes imperceptible at the distance of the earth. In the case of Venus, the eccentricity of the path is, unfortunately, so small, that it scarcely differs from a circle: and the position of the perihelion can, therefore, only be determined with great uncertainty.

Of the other two possibilities of verifying the theory, one arises from the influence of gravitation upon the time an event takes to pass. How such an influence can come about, will be evident from the following example: According to the new theory, an observer cannot immediately distinguish whether a change, which he observes during the passage of a certain event, is due to a gravitational field or to a corresponding acceleration of his place of observation (his system of reference). Let us assume ah invariable gravitational field, denoted by parallel lines of force in the negative direction of the -axis, and having a constant value for the acceleration with which all bodies in the field fall (i.e. characterized by conditions which approximately exist on the surface of the earth). According to Einstein's theory, any event will take place in this field in just the same way as it appears to occur when referred to a co-ordinate system which has an acceleration in the positive direction of the -axis. Now if a ray of light, the time of oscillation of which is travels from a point —which is to be conveniently supposed at rest relatively to the corresponding co-ordinate system at the moment of departure of the ray—in the direction of the -axis for a distance to a point , then an observer at will, owing to his own acceleration, , have attained a velocity at the instant the ray of light reaches him (c denotes the velocity of light). According to the usual Doppler Principle, he will assign a time of oscillation to the ray of light as a first approximation, instead of . If we transfer the same event to the equivalent gravitational field, this result assumes the following form: The time of oscillation of a ray of light at a place , the gravitational potential of which differs from that of a place by the amount , is connected with the time of oscillation there observed by the relation: according to the principle of equivalence of Einstein's theory of gravitation.

This special case shows how the duration of an event is to be understood as being dependent upon the gravitational condition.

Moreover, one can regard every vibrating system (which emits a spectral line) as a clock, the motion of which, according to the investigation made just above, depends upon the gravitational potentials of the place where it is stationed. This same "clock" will have a different time of oscillation at another place in the field according to the gravitational potential, i.e. it will go at a different rate. Consequently, a particular line in the spectrum of the light which comes from the sun, e.g. an Fe-line (iron), must appear to be shifted in comparison with the corresponding line as produced by a source of light (arc-lamp) on the earth; the gravitational potential at the surface of the sun has, corresponding to the latter's great mass, a different value from that at the surface of the earth, and a definite time of oscillation (colour) is characterized in the spectrum by a definite position (Fraunhofer line). It has not yet been possible to observe this effect, which amounts to about 0.008[15] for a wave-length of 400 with certainty.

For the conditions of emission of the light from the sun's surface have not yet been sufficiently investigated, and the systematic errors in the wave-lengths in the light from the source used for comparison on the earth, the arc-lamp, are not yet sufficiently known to allow the negative results of observation hitherto obtained to be regarded as giving binding decisions. This is the more true inasmuch as in the case of the fixed stars there are, doubtless, signs of the presence of a gravitational shift of the spectral lines (vide the closing essay The Third Test of this book). It is a particularly important task of astronomy to establish this effect with certainty, for this gravitational displacement of the spectral lines is a direct consequence of the hypothesis of equivalence, and does not assume the other hypotheses of the theory such as, for example, the differential equations of the gravitational field.

The third and particularly important inference from Einstein's theory is the dependence of the velocity of light upon the gravitational potential, and the resultant curvature (based upon Huygens' principle) of a ray of light in passing through a gravitational field. The theory asserts that a ray of light, coming e.g. from a fixed star, and which passes in close proximity to the sun, has a curved path. As a consequence of this curvature, the star must appear displaced from its true position in the heavens by an amount which attains the value 1.7" at the edge of the sun's disc, and decreases in proportion to the distance from the centre of the sun. But since a ray of light which comes from a fixed star and passes by the sun can be caught only when the light of the sun, which overpowers all else by its brilliancy, is intercepted before its entrance into our atmosphere, only the rare moments of a total eclipse come into account for this observation and for the solution of the problem. The solar eclipse of 29th May, 1919, during which photographs were taken at two widely-separated stations, for the purpose of this test, has, as far as the results of measurement allow us to pass definite judgment, decided in favour of the general theory of relativity.[16]

The experimental verification of Einstein's theory of gravitation has thus not reached completion. But if, in spite of this, the theory can, even at this early stage, justly claim general attention, the reason is to be found in the unusual unity and logical structure of the ideas underlying it. In truth, it solves, at one stroke, all the riddles, concerning the motions of bodies, which have presented themselves since the time of Newton, as the result of the conventional view about the meaning of space and time in the physical description of natural phenomena.