Hittjo Hummelen

The einstein equation as published in the General Theory of Relativity in 1916 reads:

$R_{\mu \nu }-{\tfrac {1}{2}}g_{\mu \nu }R={\frac {8\pi G}{c^{4}}}T_{\mu \nu }$

Left of the equal sign is the metric field, which describes the extent to which spacetime is curved or deformed at a certain spot (μ, ν) in the earlier world. The part to the right of the equal sign describes the quantity of matter and energy (T) at the corresponding spot (μ, ν) in the later world with a correction factor consisting of a few constants. In addition to the number 8 and the well-known symbol π, they are G, the Cavendish constant or the gravitational constant, and c, the speed of light. These constants are the direct result of all differential calculi that Einstein serves up to us, and are only needed for the conversion of the different units of the four variables that make up the formula. Those variables are:

R_μν describes to what extent spacetime at (μ, ν) differs from Euclidean space and is known as the Ricci tensor
g_μν is called the metric tensor and is a description of the local geometrical structure at (μ,ν)
R is a scalar function that describes the extent to which a surface differs from a flat space (2D)
T_μν is the stress–energy tensor. T denotes the density and flux of energy and stress at location (μ, ν).

Note 40 Density is an intensive proporty that expresses how much mass is present in a certain volume. The term flux (Latin: fluere, "flow") denotes the flow of a quantity through a surface. With a closed surface you can speak of the flux of a vector field through the surface, from the inside to the outside. In astronomy, flux therefore represents the amount of energy emitted by a star per square meter and per second in a wavelength interval of 1 nanometer. Other applications of the vector field include the electric field and the strength of the gravitational field. In electrostatics the divergence of the electric field is proportional to the charge per unit volume, and the divergence of the gravitational field is proportional to aforenamed density.

The first three variables together form the einsteintensor and is therefore directly proportional to the stress–energy tensor T_μν.

An event (T_μν) in the later world therefore is the direct source for the local curvature of spacetime in the earlier world. It is possible to deduce from the degree of curvature in the earlier world how much gravity the event generates in the later world. Einsteins equation therefore is a 'back and forth between two worlds'. Since the later world in Einstein's field equation is formed by the galaxies and the earlier world by timelike gravitational spacetime, that he called the metric field, they have different coordinate systems, '3 + 1' and '2 + 2' respectively (see Note 39). Every object one wants to describe, therefore, has to be transformed in the right way, and that is precisely the property that makes tensors so special. Three of the four variables in the field equation of the General Theory of Relativity are tensors. This ensures that the dynamics that follow from the equation, are independent of the coordinate system in which one is working.

There are different versions of the einstein equation, each of which provides the description of a different universe. In 1915 it was not yet known that the universe is expanding, so Einstein logically assumed the then generally accepted principle of a static universe. Einstein's first solution was therefore a cylindrical universe (see image below).

The cylinder-metric of Einstein (RU Leiden)

Translation figure:
Einstein’s cosmological model (cylindrical universe) and the cosmological constant:
Question: is the metric field g_μν a solution to the equations of Einstein’s cylindrical universe of 1915?
FORMULA….
[Read: curvature (xx) = material resources (xxx)]
Answer: NO! This metric field is a solution only when (a) an extra term is added – the infamous cosmological constant λ:
FORMULA……
and (b) the relation [FORMULA] is satisfied.
R = the radius of the universe
ρ = the density of mass
κ = Einstein’s gravitational constant
The cylindrical universe therefore has to be crammed with matter

However, this solution has a few disadvantages. The cylindrical universe would be full of matter, which is not the case in our universe. In addition, this universe would collapse under itself and implode. To prevent this, Einstein introduced a new element in the equation in 1917, the notorious cosmological constant Λ. This provides a repulsive force that should prevent a collapse of the universe.

A fellow physicist from the Netherlands - Willem de Sitter (1872-1934) - came up with another solution soon after that, a hyperboloid universe: The hyperboloid universe of De Sitter (RU in Leiden). Only parts of the metric change.

Translation figure:
Not even a week later (the 20th of march, 1917): De Sitter proposes his own cosmological model, the ‘hyperboloid universe’

Question: is the metric field g_μν a solution to the field equations with a cosmological constant?
FORMULA ….
Answer: YES, as long as the relations λ = 3 / R² and ρ = 0 are satisfied. Opposite to Einstein’s cylindrical universe, the hyperboloid universe of De Sitter is completely void!
De Sitter reports to Einstein on March the 20th, 1917, and submits the article with his results to the Akademie van Wetenschapen on March the 26th, 1917.

Note 41 Both images are borrowed from a pdf. on the correspondence between Einstein and De Sitter in 1917, on resp. pages 23 and 29).

The De Sitter universe seemed to have a number of advantages, but it turned out to be completely empty. Not so good either, although it seems to approach the status of our universe in the distant future, and also in its very beginning, during the cosmic inflation.

In 1929 Hubble discovered that the universe is expanding and that it is not static at all. That changed things quite a bit and Einstein deleted the cosmological constant he had introduced earlier, just as easily from his theory.

Overvieuw Einstein equation 1917 with or without *Λg_μν*.

Since the 90s of the last century, however, the cosmological constant Λ is back again. The following section explains how this equation should be interpreted today, and how this relates to the shell structure of the Sphere Observer.

Continue to: 5.4.2. Is gravitational spacetime compatible with the Sphere Observer’s spacetime?

The Oldest Theory Of Everything

5.4.1. Einstein equation with its variants