**Abstract:
This paper deals with Hans Ertel's contribution to fundamental cosmological
problems and with the irrelevance to geophysics. Ertel's studies are related to
Einstein's relativistic physics, Eddington's large numbers in cosmology, and to
other problems.
**

The main thesis of Hans Ertel's studies based on
Einstein's, Eddington's, and Weyl's work can be expressed today with Einstein's
final sentence concluding his efforts to establish a unified field theory of
space, time, and matter. This reads as follows: “From the, quantum phenomena
it appears to follow with contents that a finite system of finite energy can be
completely described by a finite set of numbers (quantum numbers)” (Einstein
1955, p. 165).

The cosmological world models with a closed
3-dimensional (spherical) space (e = + 1
and with l > 0)
state that the universe is a system like that. The problem implied was also
called a possible "algebraification" of physics by Einstein (instead
of its geometrization according to the program of the general theory of
relativity).

The relativistic cosmology simplifies Einstein's
gravitational equations and their application to very big systems - namely to
the universe itself - by demanding a perfect "Copernicanism."
According to this idea, there are no extended inhomogeneities "en
gros" in the universe, and there never existed such ones. This makes it
possible to interpret astronomical information coming from electromagnetic waves.
Weyl and Robertson have shown that such a perfect Copernicanism leads to the
fact that the cosmological space-time-world V_{k} conforms to
Minkowski's space-time in the special theory of relativity. In this special case
the same wave optics can be derived from the Maxwellian equations of the
electromagnetic field as that in the special theory of relativity (Weyl's
cosmological principle).

**1. WORLD MODELS
**

_{4} is then the imprimitive topological
product of 3D space coordinates with time t:

V_{4}
= V_{3} x ct
(1)

This means for a conformal plane V_{4} that V_{3}
possesses again a constant curvature e = + 1,
0, - 1. This is the Robertson-Walker metric

(2)

e = + 1
means a closed (spherically curved) 3D space V_{3}. For e = - 1
and e = 0 the
cosmos is infinite in space.

Weyl's principle and the universal Einsteinian
synchronism [ Eq. (2) ] define a homogeneous and isotropic cosmic matter
according to Einstein's gravity equation that is a pure function of the cosmic
time t. In this case Friedmann's equations are valid:

(3)

and from it,

**
(4)
**

Here r(t) means
the mass density, rc^{2}
is then the energy density, and p(t) is the pressure. p and r are
connected by the dynamic equation

(5)

which simplifies for pure, pressure-free matter (p = 0)
to the conservation of mass, namely to

r_{0}R_{0}^{3}
= rR^{3}
= const

It follows from Eq. (4) that for l ≤
0 the world radius R(t) has no minimum and R < 0 is always valid. However, if
l > 0,
then there is a minimum world radius and the acceleration
of the expansion velocity
of the cosmos may also be zero.

The correct model of such a cosmos is then a spherical,
finite one, which was introduced into physics by Einstein in 1917. The mass of
this cosmos is limited; it is M = 2p^{2}rR^{3}
= const.

Eddington supposed that the initial site of the cosmos
was an unstable Einsteinian cosmos which had existed for an infinitely long
time. At that time (i.e., at the time when t ® w), it was

and
also

which means that R = l^{-1/2}.
Therefore the mass of this cosmos is always equal to the Einsteinian mass when
it collapses due to its instability:

M = M_{Einstein} = 0,5pc^{2}G^{-1}l^{-1/2}

This Eddington-Lemaitre cosmos had its maximum density
g_{0 }at the time t = - w, and at
t ® w it
transforms into the Sitter cosmos

R = l^{-1/2} exp((l/3)^{-1/2
}ct)

of a vanishingly low mass density r ~ R^{-3}
~ c^{-3} (l/3)^{-1/2}
ct.

Eddington applied his numerical relations to this world
model in which there never existed a singular state of infinitely high energy
density, and therefore there was no "big bang." On the contrary, Ertel
considered the general case of a "Friedmann-Lemaitre cosmos" with e = + 1
and of total mass

M >
0,5pc^{2}G^{-1}l^{-1/2}

This cosmos started with an infinitely high density r ~ 1/R^{3}
~ 1/t^{2} at the time t ® 0, i.e.,
with a big bang (when a radiation energy and p ~ 1/R^{4} could also
exist but later they became small in compan'son with r ~ 1/R^{3}
(see Ertel, 1935a-f, 1936, 1938a-c).

In this case, the cosmos reached a turning point R´´_{0
}= 0 at a time t = t_{0} > 0 when r_{0} = lc^{2}/4p. This
happened when in the case of a mass

the radius of the world became

The duration of this state of minimum expansion
velocity can be very long.

For
the Hubble number is

H_{0}^{2}
= lc^{2}
- c^{2}/ R_{0}^{2}

The smaller the difference l - 1/ R_{0}^{2}
is, the longer the quasi-stationary state of a very slow expansion remains. The
quantum physical eigenstates, which have existed in a primary Einsteinian cosmos
for an infinitely long time according to Eddington's cosmological scenario
(1933), came into being within a finite time in the quasi-stationary phase of
the Friedmann- Lemaitre cosmos according to Ertel (1935). Thus, the cosmological
numerical values are "frozen in quantum numbers" of this state of slow
expansion with
and
which came into being after the big
bang at t = 0.

Following this phase, which was called "atome
primitive" by Lemaitre (later in 1948), the expansion accelerated again,
and finally the Friedmann-Lemaitre cosmos goes asymptotically into a de Sitter
cosmos at t ® +w.

Thus, Ertel's scenario applies the quantum-cosmological
reasoning from the Eddingtoan-Lemaitre model with M = Mo = p/2G^{-1}l^{-1/2}c^{2}
to the Friedmann-Lemaitre cosmoses in which M > M_{0}, and considers
a quasi-stationary state with H_{0}^{2} » (l - 1/R_{0}^{2})
c^{2},
at time t ® w as the
base for the cosmological numerical relations. Thus Ertel has unified
Eddington's quantum cosmology with the idea of a primary "big bang."

**2. THEORETICAL CONSEQUENCES AND STARTING POINTS
**

Einstein's complete general-relativistic field
equa.tions contain three universal constants: the velocity of light c which
transforms time t in the special theory of relativity (STR) into a fourth
coordinate of Minkowski's space-time-world, x^{4} = ct Einstein's
gravity constant K = (8p/c^{4})
G, where G is the Cavendish-Newtonian constant in the Newtonian gravity law, and
Einstein's cosmological constant l with the
dimension of inverse surface being a measure of curvature in the space-time
world. With these constants Einstein's equations are

R_{ik} – ˝ g_{ik}
R + lg_{ik} = kT_{ik}

where R_{ik} is the once contracted tensor (the
Ricci tensor), R = R^{l}_{l} is the curvature scalar, and g_{ik}
is the metric tensor of a Riemann 4D space-time world. The tensor at the right
side holds together the remaining physics, the energy and stress distribution of
matter particles and fields as a “matter-tensor” T_{ik}. The
comparison of this matter and the existence of stable bodies and processes,
being applicable as a scale in measurements of space and time, are ensured by
the existence of Planck's effect quantum h.

With the help both of G and c, or both of h and c, a
length can be attributed to each mass M. These are on the one hand the gravity
radius GM/c^{2} and on the other hand the Compton wavelength h/Mc.
Planck discussed this problem earlier and considered h, c, and G as fundamental
constants of physics. These hypothetical particles are called today Planckions
of the mass M_{p} = (hc/G)^{˝ }and
are ideal test bodies of physics according to him. All other physical constants
could be expressed in the programs of a relativistic gravity theory as multiples
of Planck's quantities.

Planck has already noted that the elementare charge e
is of the same physical dimension as (hc)^{˝}. lt is valid that

e^{2}
= ahc

where a » 1/137 is
Sommerfeld's fine structure constant applied in the quantum theory. As early as
in 1908 Einstein demanded that the computation from h and e should be an aim of
the quantum theory of electrons and of the Maxwellian field. The linear
Maxwellian equations are, however, unable to determine the elementary charge e.

For gravity the large number w = hc/Gm^{2}
= 10^{39} is the inverse Sommerfeld number. Using Eddington's "big
cosmic number" N,

N = c^{3}/Ghl » w^{3} = 10^{117}

is valid for M:

M » c^{2}G^{-1}l^{-˝}
= N^{˝} M_{p} = N^{2/3} m

where m is the mass of a proton (generally the mass of
a stable baryon). In this case, the "total effect of the cosmos" is
defined as

Mcl^{-˝}
= c^{3}G^{-1}l^{-1} : Nhc (the
Haas relation)

Thus, the mass m of the stable elementary particles is
determined by the cosmos (this corresponds to the Einsteinian Interpretation of
Mach's principle). m=h^{2/3}G^{-1/3}l^{1/6}; thus
for l ® 0 the
rest of the mass of the elementary particles would disappear. After Weyl the
Haas relation can be written as

GM^{2} = G^{-1}c^{4}l^{-1}=Nh

This is the cosmological counterpart of Planck's
relation:

GM^{2}_{p
}= hc

Eddington supposed that these algebraic numerical
relations belong to the unstable, stationary Einsteinian cosmos. Eddington's,
Weyl's, and Haas' numerical relations should be eigenvalues of a
general-relativistic quantum theory in the closed stationary Einsteinian space,
which have existed and will exist for an "infinitely longtime" as
presumed in the Eddington-Haas model. In this case, these states are "frozen-in
quantum states."

Ertel has shown that these special models for the
Eddington, Weyl, and Haas relations between macro- and microcosmos are not
necessary. It is possible to deduce the same relations from the General
Friedmann- Lemaitre model up to a factor of the order of magnitude of one (e = 1, l > 0).
Thus, they are frozen-in quantum states from the phase of the cosmos when the
expansion of the world was very slow:

H^{2}
= lc^{2}
– c^{2}/R_{0}^{2},
with R_{0}^{2} = l^{-1}(1+ R´_{0}^{2}/c^{2})

This state was also quasi-stationary. Before it, the
world came into being through a big bang and after the stationary state the
world is asymptotically transforming into a de Sitter cosmos with

H^{2}
= lc^{2/3}

in Eddington's model. According to Ertel the
Eddington-Haas relations are principally not coupled with the primary
Einsteinian cosmos and assume a big bang, too.