de Sitter SchwarzLochCosmos

In some discussions with Albert Einstein, Willem de Sitter came to the conclusion that the energy-momentum tensor has no influence on the structure of the universe. To develop a model of the universe, he removed this tensor from the field equations, as well as the Ricci scalar, since its function of adjusting between the Ricci tensor and the energy-momentum tensor was no longer necessary. Because it is initially left open whether the cosmological constant Λ is positive or negative, the sign of Λ can be chosen arbitrarily.

In 1917, de Sitter determined his model of the universe from the modified field equations

By analogy to the Schwarzschild line element, we have

(1)

(2)

In spherical coordinates, the calculation is less cumbersome than in the coordinate systems used by Schwarzschild in his groundbreaking work.

From equation (2) we have

The Christoffel symbols Γ_μν^α and Γ_μα^α are

Γ₀₀¹ and Γ₁₀⁰ are used to calculate R₀₀

R₀₀ is equated to Λ⋅g₀₀.

The simple differential equation for f(r) has the general solution

The result allows for different de-Sitter universes.

A special solution with a=0, b=1 and positive Λ=1/R² is

If Λ is also replaced by 1/R² in the equations (1), the field equations

lead to the same line element.

(7)

(8)

For r=R the line element is singular, there is no geodate from the interior with r<R to the exterior. This de-Sitter world has a cosmological horizon like the interior of a black hole, from which nothing, neither mass nor energy, can penetrate the event horizon to the outside.

A (stellar) black hole is formed when a very massive star, in whose center the last possible nuclear fusion has ceased, explodes in a supernova and leaves a stellar remnant in the center, which still has more than two and a half solar masses (m₀=5·10³⁰kg). This remnant star then collapses to form a black hole. A black hole with less mass than m₀ cannot be formed in this way.

The physical processes that take place in a core-collapse supernova are theoretically well calculated and the results of the calculations agree with corresponding observations, for example, of the supernova SN1987A. Dark matter and dark energy are not taken into account in the calculations, The same applies to the following considerations.

If one believes the theoretical models, then all the mass of a black hole disappears in a point-like (Schwarzschild metric) or ring-like (Kerr metric) singularity. Since both singularities have no volume, the mass density there is infinite. Given such an unphysical result, one can doubt whether the theory is applicable to the interior of a black hole.

Every black hole is spherical in shape, its radius R is the Schwarzschild radius. Consequently, the de Sitter universe is spherical in shape, its radius R is the Schwarzschild radius of a black hole.

The equations for the mass M, the volume V and the average density ρ follow.

Thus, the mass M and the average mass density ρ of the de Sitter universe (7) are uniquely determined by the Schwarzschild radius R, just as in the case of a black hole.

It applies

with the constant

(9)

(10)

(11)

As long as the product ρ·M² remains smaller than the constant L, the mass M and its density ρ can assume arbitrary values independently of each other. However, if in a fixed volume the mass increases so much that ρ·M² becomes equal to L, then a black hole is created, and due to the relationship ρ·M²=L,
ρ and M are now inseparably connected. If the mass M of the black hole increases, the volume V becomes larger and the density ρ decreases according to equation (8c). Because no mass can escape from the black hole and, according to equation (8c), ρ is inversely proportional to M², the density ρ cannot become larger. The constant L is an upper limit for the product ρ·M². It follows:

1. The maximum possible density ρ of a mass M is L/M².
2. The maximum possible mass M with the density ρ is

So a black hole with the average mass density ρ has the mass

therefore

(12)

Some examples

ρ_KG is the density of a quark-gluon plasma.

ρ_HH is the density in the photosphere of the Sun.

ρ_K is the average density of the cosmos.

(13)

(14)

(15)

The Schwarzschild radii are

(16)

(17)

M_KG and R_KG correspond to the values of the smallest stellar black hole discovered so far. A significantly smaller one is hardly possible.
At 5700K, the H-He plasma in the photosphere of the Sun has the density ρ_HH. When the recombination era of the cosmos begins the density is nearly ρ_HH.
The mass M_K differs hardly at all from the estimated (visible) mass of the universe (Wikipedia: ≈ 10⁵³ kg).
For the radius of the universe, AI gives 46 billion light-years. That is 4.4·10²⁶ m, i.e. about twice R_K.

That the estimated data for the mass M, the average density ρ, and the radius R of the universe fit the picture of a black hole could be a strange coincidence. Nevertheless, it is a strong indication for the black-hole universe. Instead of believing in a Big Bang, in which all mass and energy appeared out of nothing in 10^-4s, contrary to all conservation laws of physics, it is assumed here that the origin of our universe was a stellar black hole (sBH) within a larger universe. This black hole could have had a mass of about M_KG (20% more than m₀), thus being at the lower limit of the mass and the upper limit of the density of an sBH.

After its formation, the black-hole cosmos continuously absorbs more or less mass from the surrounding universe. The temporal evolution of the cosmos proceeds from a quark-gluon phase through the same states that are also assumed in the Big Bang theory, but it is not determined by an extrapolated Hubble time; rather, it depends on the irregular absorption of masses from the encompassing universe. According to equations (8), this is associated with an increase in volume and a decrease in density. The cosmos expands, the waves of electromagnetic radiation, which in the early cosmos constitute the main part of the energy and mass, are stretched apart, so that the radiation loses energy. Since radiation and mass are still in thermal equilibrium, the temperature of the cosmos decreases. At 5700 K, the mass of the cosmos has grown to half a trillion solar masses (Eq. 14) and the radius to one tenth of a light-year (Eq. 17b). The cosmos absorbs more mass, the density becomes lower, the expansion continues, and the temperature falls until, at about 3000 K, matter and radiation decouple. The mass of the cosmos now consists of neutral atoms, and the radiation survives as background radiation, undisturbed but constantly becoming longer-wavelength, for billions of years. The measurement of the background radiation provides the earliest experimental result of all, while for the time before it there are only purely theoretical considerations. With z=1100, its redshift is greater than that of any other event in the cosmos.

The cosmos as a black hole in the universe has angular momentum. When it absorbs new mass, the event horizon expands, and the centrifugal force pushes mass into the space between the old and the new horizon. The cosmos becomes more inflated at a lower density, and the distances between the galaxies become greater. Because the surface area also grows with each increase in mass, the probability of absorbing new mass becomes greater, causing the expansion to accelerate.

With one hundred billion solar masses already at the beginning of the recombination era (Eq. 14), the cosmos, as a black hole, is by far large enough to absorb black holes with several tens of thousands of solar masses from the surrounding universe. In a hydrogen-rich environment, they then grow into the supermassive black holes that appear as very early AGNs of early galaxies.

Comparison between the Big Bang theory and the de-Sitter_Cosmos..

Problem

Big Bang Theory

de-Sitter-Cosmos

Origin of mass and energy	from nothing	from the surrounding universe
Reason for the absence of antimatter	???	already absent in the surrounding universe
Mass ratio of light nuclei	Primordial nucleosynthesis (PNS)	PNS and the same ratio already present in the surrounding universe
Reason for the expansion	Dark Energy	Angular momentum of the BH cosmos
Reason for the acceleration of the expansion	Dark Energy	A larger surface area of the BH cosmos increases mass uptake.
Formation of very early AGNs	until now mysterious	Black holes are taken into the BH cosmos at a very early stage.

Münster, 2 September 2026
JHM