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General Relativity implies an expanding Universe from a singularity, the so-called Big Bang. The rate of expansion is the Hubble constant. There are two major ways of measuring the expansion of the Universe: through the cosmic distance ladder and through looking at the signals originated from the beginning of the Universe. These two methods give quite different results for the Hubble constant. Hence, the Universe doesn’t expand. The solution to this problem is the theory of gravitation in flat space-time where space isn’t expanding. All the results of gravitation for weak fields of this theory agree with those of General Relativity to measurable accuracy whereas at the beginning of the Universe the results of both theories are quite different, i.e. no singularity by gravitation in flat space-time and non-expanding universe, and a Big Bang (singularity) by General Relativity.

General Relativity (GR) implies an expanding universe where the expansion rate is the Hubble constant. There are two different methods to measure the Hubble constant. The results of these two methods are two different values for the Hubble constant (see e.g. [

The theory of GFST is shortly summarized. The metric is flat space-time given by

( d s ) 2 = − η i j d x i d x j (1)

where ( η i j ) is a symmetric tensor. Especially, pseudo-Euclidean geometry has the form

( η i j ) = ( 1 , 1 , 1 , − 1 ) . (2)

Here, ( x i ) = ( x 1 , x 2 , x 3 ) are the Cartesian coordinates and x 4 = c t . Let

η = det ( η i j ) . (3)

The gravitational field is described by a symmetric tensor ( g i j ) . Let ( g i j ) be defined by

g i k g k j = δ i j (4)

and put similar to (3)

G = det ( g i j ) . (5)

The proper time τ is defined by

( c d τ ) 2 = − g i j d x i d x j . (6)

The Lagrangian of the gravitational field is given by

L ( G ) = − ( − G − η ) 1 / 2 g i j g k l g m n ( g / m i k g / n j l − 1 2 g / m i j g / n k l ) (7)

where the bar “/” denotes the covariant derivative relative to the flat space-time metric (1). The Lagrangian of dark energy (given by the cosmological constant Λ ) has the form

L ( Λ ) = − 8 Λ ( − G − η ) 1 / 2 . (8)

Let

κ = 4 π k / c 4 (9)

where k is the gravitational constant. Then, the mixed energy-momentum tensor of gravitation, of dark energy and of matter of a perfect fluid is

T ( G ) J i = 1 8 κ [ ( − G − η ) 1 / 2 g k l g m n g i r ( g / j k m g / r ln − 1 2 g / j k l g / r m n ) + 1 2 δ j i L ( G ) ] (10a)

T ( Λ ) j i = 1 16 κ δ j i L ( Λ ) (10b)

T ( M ) j i = ( ρ + p ) g j k u k u i + δ j i p c 2 . (10c)

Here, ρ , p and u i denote density, pressure and four-velocity of matter. it holds by (6)

c 2 = − g i j u i u j . (11)

Define the covariant differential operator

D j i = [ ( − G − η ) 1 / 2 g k l g j m g / l m i ] / k (12)

of order two. Then, the field equations for the gravitational potentials ( g i j ) have the form

D j i − 1 2 δ j i D k k = 4 κ T j i (13)

where

T j i = T ( G ) j i + T ( M ) j i + T ( Λ ) j i . (14)

Define the energy-momentum tensor

T ( M ) i j = g i k T ( M ) k j . (15)

Then, the equations of motion in covariant form are

T ( M ) i / k k = 1 2 g k l / i T ( M ) k l . (16)

In addition to the field Equation (13) and the equations of motion (16) the conservation law of the total energy-momentum holds, i.e.

T i / k k = 0 . (17)

The results of this chapter may be found in the book [

GFST is defined in flat space-time metric, e.g. in the pseudo-Euclidean geometry which is used in the following to study homogeneous, isotropic, cosmological models. The matter tensor is given by a perfect fluid with velocity equal to zero. The total matter is given by the sum of density of matter ρ m and of radiation

ρ r with the corresponding pressure density of matter p m = 0 and of radiation p r = 1 3 ρ r . It holds for homogeneous, isotropic, cosmological models

g i j = a ( t ) 2 ( i = j = 1 , 2 , 3 )

g i j = − 1 / h ( t ) ( i = j = 4 )

g t j = 0 ( i ≠ j ) .

The initial conditions at present time t 0 = 0 are

a ( 0 ) = h ( 0 ) = 1 , a ˙ ( 0 ) = H 0 , h ˙ ( 0 ) = h ˙ 0 , ρ m ( 0 ) = ρ m 0 , ρ r ( 0 ) = ρ r 0

where H 0 is the Hubble constant and h ˙ 0 is an additional constant not appearing in GR. Relation (16) for i = 4 implies under the assumption that matter and radiation do not interact

ρ m = ρ m 0 / h 1 / 2 , ρ r = ρ r 0 / ( a h 1 / 2 ) (18)

It follows by the use of the field Equation (13)

d d t ( a 3 h a ˙ a ) = 2 κ c 4 ( 1 2 ρ m + 1 3 ρ r + Λ 2 κ c 4 a 3 h ) (19a)

d d t ( a 3 h h ˙ h ) = 4 κ c 4 ( 1 2 ρ m + ρ r + 1 8 κ c 4 L ( G ) − Λ 2 κ c 2 a 3 h ) (19b)

where

L ( G ) = 1 c 2 a 3 h ( − 6 ( a ˙ a ) 2 + 6 a ˙ a h ˙ h + 1 2 ( h ˙ h ) 2 )

The expression 1 16 κ L ( G ) is the density of gravitation field. The conservation law of the total energy is

( ρ m + ρ r ) c 2 + 1 16 κ L ( G ) + Λ 2 κ a 3 h = λ c 2 (20)

where λ is a constant of integration. Define the quantity

φ 0 = 3 H 0 ( 1 + 1 6 h ˙ H 0 ) .

The field Equation (19) imply by the use of the conservation law (20) and the initial conditions the relation

a 3 h = 2 κ c 4 λ t 2 + φ 0 t + 1 . (21)

It follows from (20) with the present time

the differential equation

Here,

Relation (20) with

The assumption

implies that the solution of (23) is non-singular for all

It follows from (23a)

The time

Therefore,

The differential Equation (23a) can by the use of (21) be rewritten

This differential equation is for not too small functions

Then, the conditions (25) and (27) give

i.e.

There are two methods of measuring the Hubble constant of the universe: the cosmic distance ladder and looking at the signals originated from the beginning of the universe. Two different results for the Hubble constant are received. Therefore, the universe doesn’t expand because the methods use the expansion of the universe. It is worth to mention that GR implies expansion because the universe starts from a point singularity and the observed universe is very big. Furthermore, the universe must be inflationary expanding because the observed universe is flat. Summarizing, it follows that GR doesn’t correctly describe gravitation if two Hubble constants are measured.

A theory of gravitation in pseudo-Euclidean geometry has been given in article [

The author declares no conflicts of interest regarding the publication of this paper.

Petry, W. (2020) The Hubble Constant Problem and the Solution by Gravitation in Flat Space-Time. Journal of Modern Physics, 11, 214-220. https://doi.org/10.4236/jmp.2020.112013