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Reference: Einstein’s 1920 Book

Appendix I
Simple Derivation of the Lorentz Transformation

Please see Appendix 1 at the link above. This appendix is supplementary to Section XI.

The Fig. 2 above provides the relative orientation of the co-ordinate systems K and K’. K is relatively at “rest” while K’ is moving at a velocity v.

The coordinates of K and K’ are different for the same event, but they are related by the fact that the velocity of light c is the same in both coordinate systems.

The coordinates for an event in K shall be related by constants to the corresponding coordinates in K’ because v is constant. This is true for motion in reverse direction for straight line motion.

We can relate the uniform velocity v to the constants involved in the correspondence of the coordinates of K and K’.

According to the principle of relativity the unit of lengths at rest in K or K’ will appear to be the same when viewed from the other system (K’ or K). From this we can determine the value of the constants involved in terms of v and c.

This gives us the following relationships between the coordinates of K and K’.

In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light of arbitrary direction, both for the system K and for the system K’. We have thus derived the Lorentz transformation.

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Reference: Einstein’s 1920 Book

Section XXXI (Part 3)
The Possibility of a “Finite” and Yet “Unbounded” Universe

Please see Section XXXI at the link above.

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Summary

A universe can be without bounds (infinite) from the perspective of its dimensions, but from a higher dimension it can be seen to be finite.

If the ratio of the circumference of a circle to its diameter is determined very accurately, and it is less than π, then that circle is drawn on a spherical surface.

Riemann discovered a three-dimensional analogy of a two-dimensional spherical surface. That means an object moving in a straight line in any direction will ultimately reach its starting point. Such a space is finite but it has no bounds.

Thus, closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent. This is the approach taken by the general theory of relativity.

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Final Comments

Our space has the property of consistency beyond just having a location. Because of this consistency the space has a curvature. The smaller is the consistency, the larger is the curvature. Even light does not travel in straight lines but has a curvature on a cosmic scale; therefore, it will never escape this universe. This universe can therefore be viewed as unbound but finite in nature.

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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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