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Reference: Einstein’s 1920 Book

Section XXIX (Part 2)
The Solution of the Problem of Gravitation on the Basis of the General Principle of Relativity

Please see Section XXIX at the link above.

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Summary

In the Galilean domain of the special theory of relativity there is no gravitational field, and “isolated” material points move uniformly and in straight lines.

In the random Gaussian domain of the general theory of relativity, the influence of gravitational fields on the behavior of measuring-rods, clocks and freely-moving material points can be studied simply by mathematical transformations. With this mathematical device, we investigate the space-time behavior of the gravitational field, which was derived from the Galilean special case, and formulate that behavior into a law.

We then generalize this law for all gravitational fields by taking into consideration the conservation laws and the general postulate of relativity. We then determine the influence of the gravitational field on the course of all those processes which take place according to known laws when a gravitational field is absent. The theory of gravitation derived in this way from the general postulate of relativity has already explained a result of observation in astronomy, against which classical mechanics is powerless.

The Newtonian Theory can be derived as a special case from this general theory of gravitation. The deviations from Newtonian theory can also be calculated, which, otherwise, are difficult to observe because of their smallness. For example, the general theory of relativity accurately predicts the delicate rotation of the ellipse of mercury as observed. But it also predicts the rotation of the ellipse of all other planets, which is too small to observe.

The general theory of relativity has been confirmed further, by phenomenon, such as, the curvature of light rays by the gravitational field of the sun.

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Final Comments

In an abstract sense, Time reflects endurance of Space. The Gaussian coordinates express time as a variable at each location of space. If the endurance of a location is infinite it has no motion. It is totally rigid. If a location has no endurance it is all motion. It is totally flexible. In short, the lesser is the endurance the greater is the motion. 

In a practical sense, time introduces a gradient of motion among locations, which introduces acceleration and force. This is sensed as if space has substance. Thus, space-time-substance become covariants.

With no time, there is no gradient of motion, no acceleration, no force, and nothing is sensed. As time comes about, so does the sense of substance. With greater variations in time, there is greater acceleration and force built up in space. When the gradient of variation in time is small, the space is sensed as the electromagnetic radiation, such as, light. When the gradient of variation in time is very large, the space is sensed as rigid matter.

The gradient of time comes about in terms of vibrations. The gradient is small when vibrations are small. The gradient is large when the vibrations are large. Matter has much higher vibrations than electromagnetic radiation or light.

The Gaussian coordinates thus produce space that can be sensed. This is the basis of a gravitational field.

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Reference: Einstein’s 1920 Book

Section XXVII (Part 2)
The Space-Time Continuum of the General Theory of Relativity Is not a Euclidean Continuum

Please see Section XXVII at the link above.

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Summary

The space time continuum of the special theory can be regarded as four-dimensional Cartesian co-ordinates if we consider the velocity of light to be constant. For a rotating body of reference, a gravitational field is present; and, therefore, the velocity of light will depend on the coordinates. We cannot consider four-dimensional Cartesian co-ordinates in the general theory of relativity.

We refer the four-dimensional space-time continuum in an arbitrary manner to Gauss co-ordinates. In themselves these co-ordinates have no significance. A momentary existence of a material point at a location without duration will be represented by a single coordinate point. A permanent existence of a material point with motion shall then be represented by an infinitely large number of coordinate points as in a continuous line. Encounters among material points shall be represented by coincidences among coordinate points.

The motion of a material point relative to a body of reference shall be represented by encounters of that material point with particular points of the reference-body. This will take into account all kind of variations in both space and time in terms of flexibility, inertia or consistency.

All physical descriptions of space-time relationships can then be represented by coincidences among Gaussian coordinates of a body with a reference body. The Gaussian coordinates do not suffer the limitation of absolute rigidity of the Cartesian coordinates.

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Final Comments

A body in uniform motion may not have acceleration, but it has a constant velocity. This constant velocity differs from body to body due to differences in their inherent structure. This inherent structure appears as the mass, inertia, rigidity or consistency of the body.

Light has near zero consistency and near infinite velocity; whereas, matter has near infinite consistency and extremely low range of velocities. By extrapolating between these data points, the special theory of relativity manages to come up with an approximate method to calculate the relative velocity in uniform motion for matter.

The general theory of relativity accounts for acceleration by relating instantaneous changes in consistency to changes in velocity throughout a continuum. Thus, it accounts for acceleration that manifests in the form of gravitational field.

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