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Classical Reasons of General Relativity

It is here the point where we can locate one of the reasons of the existence of the Theory of General Relativity, the outstanding creation of Einstein (see Einstein, 1920). The Relativity, both Special and General appeared first as necessary due to the apparently unphysical aspect of the classical Newtonian theory marked by the existence of a privileged reference frame. This was discussed by Einstein in the Chapter XXI of the reference just cited above. Here we reproduce the short Chapter XXX of that work, which refers specifically to the distribution of matter in the Universe around us:

“Apart from the difficulty discussed in Section XXI, there is a second fundamental difficulty attending classical celestial mechanics, which, to the best of my knowledge, was first discussed in detail by the astronomer Seeliger. If we ponder over the question as to how the universe, considered as a whole, is to be regarded, the first answer that suggests itself to us is surely this: As regards space (and time) the universe is infinite. There are stars everywhere, so that the density of matter, although very variable in detail, is nevertheless on the average everywhere the same. In other words: However far we might travel through space, we should find everywhere an attenuated swarm of fixed stars of approximately the same kind and density.

This view is not in harmony with the theory of Newton. The latter theory rather requires that the universe should have a kind of centre in which the density of the stars is a maximum, and that as we proceed outwards from this centre the group-density of the stars should diminish, until finally, at great distances, it is succeeded by an infinite region of emptiness. The stellar universe ought to be a finite island in the infinite ocean of space…

This conception is in itself not very satisfactory. It is still less satisfactory because it leads to the result that the light emitted by the stars and also individual stars of the stellar system are perpetually passing out into infinite space, never to return, and without ever again coming into interaction with other objects of nature. Such a finite material universe would be destined to become gradually but systematically impoverished.

In order to escape this dilemma, Seeliger suggested a modification of Newton’s law, in which he assumes that for great distances the force of attraction between two masses diminishes more rapidly than would result from the inverse square law. In this way it is possible for the mean density of matter to be constant everywhere, even to infinity, without infinitely large gravitational fields being produced. We thus free ourselves from the distasteful conception that the material universe ought to possess something of the nature of a centre. Of course we purchase our emancipation from the fundamental difficulties mentioned, at the cost of a modification and complication of Newton’s law, which has neither empirical nor theoretical foundation. We can imagine innumerable laws which would serve the same purpose, without our being able to state a reason why one of them is to be preferred to the other; for any one of these laws would be founded just as little on more general theoretical principles as is the law of Newton” (Our Italics)

When Einstein talks of “empirical and theoretical foundation” what he understands by that is the Poisson equation (12). This equation, which is the basis of General Relativity, was always taken as defining the potential by assuming the density of matter as known. It is just natural then to conclude that the Newton’s law (of inverse square force) lacks empirical and theoretical foundation, inasmuch as we are sure that the density in the Universe is not zero while the inverse square force corresponds, in Einstein’s conditions above, to a zero density.

General Relativity originates basically from the classical idea that the gravity is omnipresent. If this is true and if the gravity is characterized as a force, then what classically is taken as a free motion is not actually a free motion, but the motion of a particle under forces that may or may not be in equilibrium. Classical Mechanics has here a contradiction in terms. Thus, there should be a mathematical description of the situation that includes the Classical Mechanics as a particular case. The only indication we have is the motion of particles. In Classical Mechanics this motion is described by Newtonian equations of motion (3). As the trajectory is geometrically a line in space, one wonders what line in a space would correspond to a general second order differential equation. If we assume that we are lucky enough to live in the case of a space with affine connection, it turns out that this is the geodesic line - the line of shortest length between two points - generalizing the line of shortest distance - straight line - from the Euclidean case. Then the dynamical roots of General Relativity can be sketched in the following fashion (Misner & Al, 1977, Ch. 12).

Start from the observation that the trajectory of motion of a particle is a geodesic (generalization of straight line - the trajectory of the free motion). Then the time as, affine parameter of geodesics, is defined up to a linear transformation. Thus the Newtonian equations of motion in a field of potential U:

can be written in the form

These four equations represent the equations of geodesics in four dimensions

in case the components of connection are defined as

any other component being zero. In view of this, the classical Poisson equation for potential

translates into an equation for determination of the Ricci curvature tensor

It is here the point where Einstein makes the bold step of assuming that there is a metric of the 4D space-time generalizing the Minkowski metric so that the Ricci tensor gets primacy over Newtonian potential. The last equation then indicates that matter is the source of curvature.

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