Einstein 1920 (App III) The Experimental Confirmation of the General Theory of Relativity

Reference: Einstein’s 1920 Book

Appendix III
The Experimental Confirmation of the General Theory of Relativity

Please see Appendix 3 at the link above.

The development of a science a purely empirical enterprise. The theory correlates a large number of single observations. Although there are different theories, but they may agree completely in terms of deductions from them, which are capable of being tested.

General theory of relativity differs fundamentally from Newtonian mechanics, but the deductions from them are very much in agreement except in few testable cases.

(a) Motion of the Perihelion of Mercury
According to the general theory of relativity, the major axis of the elliptical orbit rotates round the sun in the same sense as the orbital motion of the planet. Theory requires that this rotation should amount to 43 seconds of arc per century for the planet Mercury. This is conformed by actual observations. The theory of Newton does not suffice to calculate the observed motion of Mercury with an
exactness as observed.

(b) Deflection of Light by a Gravitational Field
As a result of this theory, it was expected that a ray of light which is passing close to a heavenly body would be deviated towards the latter. This was tested during the total solar eclipse of 29th May, 1919. The stars in the neighborhood of the sun were photographed during the solar eclipse. The photograph was then compared for the position of the stars when sun was not present. The positions of the stars on the eclipse-photograph appeared displaced radially outwards (away from the centre of the sun) as predicted. The results of the measurements confirmed the theory in a thoroughly satisfactory manner.

(c) Displacement of Spectral Lines towards the Red.
An atom absorbs or emits light of a frequency which is dependent on the potential of the gravitational field in which it is situated. The frequency of an atom situated on the surface of a heavenly body will be somewhat less than the frequency of an atom of the same element which is situated in free space (or on the surface of a smaller celestial body). Thus a displacement towards the red ought to take place for spectral lines produced at the surface of stars as compared with the spectral lines of the same element produced at the surface of the earth. For the sun, the displacement towards the red predicted by theory amounts to about two millionths of the wave-length. A trustworthy calculation is not possible in the case of the stars, because in general neither the mass M nor the radius r is known. Up to the present the examination of the available data does not allow of any definite decision being arrived at, as to whether or not these displacements are to be referred in reality to the effect of gravitation.

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June 10, 2019 – 5:29 PM Categories: Science | Post a comment

Einstein 1920 (App II) Minkowski’s Four-Dimensional “World”

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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June 9, 2019 – 9:47 AM Categories: Einstein | Post a comment

Einstein 1920 (App I) Simple Derivation of the Lorentz Transformation

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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June 9, 2019 – 8:15 AM Categories: Einstein | Post a comment

Einstein 1920 (XXXII) The Structure of Space

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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June 8, 2019 – 8:01 PM Categories: Einstein | Post a comment

Einstein 1920 (XXXI) The Possibility of a “Finite” and Yet “Unbounded” Universe

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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June 8, 2019 – 4:59 PM Categories: Einstein | Post a comment