9 6 4 A P P E N D I X F Moreover, we must first ask if, on our first approach, both the equations of motion and those of gravitation are reduced to those of classical mechanics, as this—he repeated— gives a very approximate idea of reality. A first approach, as we well know, consists of choosing the g values so that the determi- nant they make up differs very little from the value –1, which means, as classical science dictates, that space is Euclidean. It also implies calculating the forces as if the bodies were at rest, which means assuming, as really is the case, that the speed of movement is very small with respect to that of light. With these assumptions he showed that the equations of motion in a gravitational field are reduced to those of classical mechanics the movement depends entirely on the potential , which, as he pointed out, is connected with the grav- itational potential of Newtonian mechanics through a very simple relationship. He went on to demonstrate that by leaving pressures and impulses aside and considering only the mass, the gravitational equations in (2) are reduced to Poisson’s equation. (This had been announced earlier.) Three Results of the General Theory of Relativity We have yet to consider—Dr. Einstein said—empirical proofs of the general theory of relativity. In order to do so it is necessary to find phenomena for which the results predicted by this theory deviate from those of classical science and in particular those of Newtonian mechanics. Those deviations generally are so small that only three effects have been found that allow us to compare theory and experiment, he added. (Those three effects, which ap- pear to be the only ones, were found by Einstein.) In order to calculate them, he went on, it is necessary to work with a greater degree of accuracy than previously indicated. The first effect is the very slow motion of Mercury’s orbit (perihelion motion), which classical mechanics was unable to explain. Using equations of motion provided by the gen- eral theory of relativity, we obtain a value for the precession of the orbit that coincides very satisfactorily with the value provided by observation. (The theory predicts a precession of 42”9 per century, so that the agreement is truly extraordinary.) The second effect is the bending of a ray of light passing through a gravitational field. (This phenomenon is easily understandable by using the principle of equivalence between mass and energy. According to this principle, a ray of light—an electromagnetic beam—is, in a sense, the same as a projectile launched with the speed of light if that is its nature, it is clear that it will be diverted on passing through a gravitational field.) Following this, he developed the equation of motion, obtaining the g values from the ex- pression of the interval, which, in this case, was canceled out. The concordance of the theory’s predictions with the deviations obtained is perfect, he concluded. (The theory gives the deviation of a ray passing close to the Sun a value of 1″75, and observations obtained during the Sobral and Príncipe eclipses gave 1″98 and 1″61, respectively.) The last effect refers to the functioning of clocks in gravitational fields of different in- tensities, that is, of different potential. Einstein then concluded that the times registered by g44 ds2

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L E C T U R E S A T U N I V E R S I T Y O F B U E N O S A I R E S 9 6 3 While in Newton’s theory, he added, there is just one gravitational potential, in the gen- eral theory there are ten. There are also ten equations, therefore, that define them. The latter, and thus the former, are reduced to six—he said—through four equations that connect them (for reasons given above)…. Furthermore, Dr. Einstein said, since as we have seen in the theory of the electromagnet- ic field, instead of the density of matter, we have the energetic tensor made up of the six components of the pressures and tensions that are stimulated in matter because of de- formations: the three quantities of movement and the density of energy or of mass. Since they are identified in relativity, it seems plausible to write the following as gravitational equations: (1) The second term—Dr. Einstein said—is reduced to the product of the gravitational con- stant κ times the density of the matter when the first term is reduced to Laplace’s equa- tion from ordinary mechanics. This must be required of all equations that seek to define the gravitational field, he added, since Newton’s equations describe reality in a first and subtle attempt. The reduction of (1) to Poisson’s equation, which we have just finished discussing, is produced when there is no movement within the matter, since in that case the tensions and quantities of movement disappear.) The equations in (1) cannot be taken as gravitational equations—Dr. Einstein said— even though they satisfy the conditions stipulated, in that they do not conform to some of the invariance characters demanded by the principle of relativity, for which reason it is nec- essary to replace them with others. He defined such others as follows: (2) where R is the sum of the products . Those equations, he said, possess the proper- ties of (1) and of other invariant equations, since the divergence of the first term is zero. Those are, he said, the most general and only equations that can be calculated in an ex- clusively mathematical way and that satisfy the required conditions we have mentioned. If they do not represent reality, there are no others. Determinants of g Values The problem of determining gravitational potentials, Dr. Einstein continued, is a daunt- ing one. They are defined, he said, by ten equations, each of which contains an infinitude of terms, some of them made up. However, the solution becomes much simpler, since we can give ourselves four of the equations for reasons provided earlier. By choosing those four equations arbitrarily, we are merely establishing at will a certain system of Gaussian coordinates, which we have every right to do because of the principle of general relativity itself. Tμν Rμν gTμν = Rμν Rμν 1 2 --gμνR - – kTμν = gμνRμν

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L E C T U R E S A T U N I V E R S I T Y O F B U E N O S A I R E S 9 6 5 two identical clocks, one located in field of potential U and the other in a field of potential , are related by the expression divided by 1 – U. A clock located on the Sun, he continued, would run more slowly than an identical one located on Earth. The optical lines emitted by the same substances on the Sun and on the Earth originate in natural mechanisms that, in a sense, can be thought of as identical clocks. Those coming from the Sun, therefore, must have a lower frequency than the terrestrial lines originating from the same matter. That is to say, the former must be shifted to the red. The experiment, he said, suggests that they are redshifted, but there are so many causes of disturbances in an atmosphere as complicated as the solar atmosphere that we cannot be absolutely sure if this effect is really the one predicted by the theory. Some experimenters think so. But really, we cannot affirm such a thing categorically. Future Development of the Theory We will conclude, Dr. Einstein said, by making some statements and predictions related to the future development of the theory. We are familiar with two kinds of forces, he said: electrical and gravitational that is, two fields, the electromagnetic field and the gravitational field. Each one of them, he added, has its own equations. These forces are essentially of the same nature, he said gravitational forces may also be electrical. The future development of the theory, therefore, cannot help but consist of establishing equations that will encompass both gravitational and electromagnetic phenomena. Weyl and Eddington have made attempts in this direction. I believe—the sage concluded—that none of the paths chosen is the true one, but I am certain that it will not be long until it is discovered. U′ 1 U′ –

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