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Reference: Einstein’s 1920 Book

Appendix II
Minkowski’s Four-Dimensional Space (“World”)

Please see Appendix 2 at the link above. This appendix is supplementary to Section XVII.

We can characterize the Lorentz transformation still more simply if we introduce the imaginary √(—1).ct in place of t, as time-variable.

According to the theory of relativity, the “time” enters into natural laws in the same form as the space coordinates. A four-dimensional continuum was described by Minkowski as the “world” instead of “space.” In this continuum, a “location” was defined as an “event in the world.”

This four-dimensional “world” bears a close similarity to the three-dimensional “space” of (Euclidean) analytical geometry. We can regard Minkowski’s “world” in a formal manner as a four-dimensional Euclidean space (with imaginary time co-ordinate); the Lorentz transformation corresponds to a “rotation” of the co-ordinate system in the four-dimensional “world.”

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Comment

In the four-dimensional world continuum, the first three coordinates define the extents of the substance, and the fourth coordinated defines the consistency of the substance.

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Reference: Einstein’s 1920 Book

Section XXXII (Part 3)
The Structure of Space According to the General Theory of Relativity

Please see Section XXXII at the link above.

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Summary

The geometrical properties of space are not independent of matter. The matter is almost at rest; whereas, light has a tremendous velocity of transmission. They both contribute to the geometrical property of space.

The presence of gravitational fields around matter tells us that the geometry of the universe is not exactly Euclidean. But it is conceivable that our universe differs only slightly from a Euclidean one, because matter is only a very small part of it.

The detailed distribution of matter is not uniform in this universe. The real universe will deviate in individual parts from the spherical, but it will be necessarily finite.

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Final Comments

The space has substantiality, which gives it a measure of consistency. When the consistency is extremely small, the space appears as fluid energy that has a very high velocity. When the consistency is extremely large, the space appears as rigid matter that has a very low velocity. In between, the space appears as the gravitational field of variable consistency and velocity. The velocity has an inverse relationship with consistency. It is the balance of inherent motion of matter floating in a sea of energy that is perceived as the phenomenon of Gravity.

The spectrum of energy/matter based on the property of consistency suggests a vortex type pattern, which is seen repeated in nature at all scales. For example, the atoms display this pattern where the electrons form a vortex, at the center of which there is an extremely dense and small spinning nucleus. The “gravitational field” at this level appears as charge.

At the level of the solar system, planets revolve, as if they are caught up in a vortex of gravity, at the center which is a massive and spinning sun. In their turn, the spinning planets form the center of smaller vortices of gravity in which their moons are caught up.

On a much larger cosmic scale, we have solar or star systems that are caught up in a vortex of gravity which appears as a galaxy. At the center of the galaxy is an extremely dense and small spinning black hole.

All these vortices at different scales seem to be overlapping and producing a very complex pattern in which the inherent motions of the heavenly bodies balance each other in a cosmic dance. We may thus visualize the universe having a “solid” spinning center made up uncountable number of galaxies with a great periphery of curving light far away of unimaginable proportions.

This “vortex universe” is devoid of solid masses in about 99% of its volume; but that volume is filled with palpable energy. The universe may be considered to be finite yet unbounded because it seems to curve upon itself.

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Reference: Einstein’s 1920 Book

Section XXX (Part 3)
Cosmological Difficulties of Newton’s Theory

Please see Section XXX at the link above.

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Summary

The universe appears to be homogenous on a cosmic scale. But this view is not in harmony with the theory of Newton, according to which the stellar universe ought to be a finite island in the infinite ocean of space.

It is thought that light would be escaping this island universe without ever returning and this shall violate the law of conservation of energy; but this is not certain.

As a result a modification is suggested to Newton’s law of gravitation that would limit the size of gravitational fields, and make universe appear not to be centered. But we do not know if the universe is centered or not. The real problem is that we have no law of gravity that has a firm theoretical foundation.

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Final Comments

Einstein is looking at the problem of not having a strong empirical or theoretical foundation for Newton’s Law of Gravity.

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