The General Theory of Relativity

Special and General Principle of Relativity

The basal principle, which was the pivot of all our previous considerations, was the special principle of relativity, i.e. the principle of the physical relativity of all uniform motion. Let us once more analyse its meaning carefully. It was at all times clear that, from the point of view of the idea it conveys to us, every motion must only be considered as a relative motion. Returning to the illustration we have frequently used of the embankment and the railway carriage, we can express the fact of the motion here taking place in the following two forms, both of which are equally justifiable (Einstein 1938).

Field Equations

This is a set of 10 equations (FE) which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy. Similar to the way that electromagnetic fields are determined using charges and currents via Maxwell’s equations, the FE are used to determine the spacetime geometry resulting from the presence of mass-energy and linear momentum, that is, they determine the metric tensor of spacetime for a given arrangement of stress-energy in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the FE to be written as a set of non-linear partial differential equations when used in this way. The solutions of the FE are the components of the metric tensor. The inertial trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation. \[R_{\mu\nu} - \frac{1}{2} g_{\mu\nu}R + g_{\mu\nu}\Lambda = \frac{8 \pi G}{c^4}T_{\mu\nu}\]

The Gravitational Field

“If we pick up a stone and then let it go, why does it fall to the ground?” The usual answer to this question is: “Because it is attracted by the earth.” Modern physics formulates the answer rather differently for the following reason. As a result of the more careful study of electromagnetic phenomena, we have come to regard action at a distance as a process impossible without the intervention of some intermediary medium. If, for instance, a magnet attracts a piece of iron, we cannot be content to regard this as meaning that the magnet acts directly on the iron through the intermediate empty space, but we are constrained to imagine–after the manner of Faraday–that the magnet always calls into being something physically real in the space around it, that something being what we call a “magnetic field.” In its turn this magnetic field operates on the piece of iron, so that the latter strives to move towards the magnet (Einstein 1916).


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Euclidean and Non-Euclidean Continuum

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a “neighbouring” one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing jumps. I am sure the reader will appreciate with sufficient clearness, what I mean here by “neighbouring” and by “jumps” (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.