A theory of gravitation in flat space-time is applied to homogeneous, isotropic cosmological models. There are non-singular cosmological models. A natural interpretation is a non-expanding universe. The redshift is an intrinsic effect and not a Doppler effect. The universe contains only energy in the beginning, i.e. no matter exists. In the course of time matter and radiation are created from energy where the whole energy is conserved. Matter increases with time but a certain time after the beginning of the universe the creation of matter is finished and the universe appears like a static one. A modified Hubble law is considered which may explain the high redshifts of objects in the universe without the assumption of dark energy.
Einstein’s general theory of relativity yields a singularity in the beginning of a homogeneous, isotropic universe. Hence, space and time do not exist before and are created in the beginning. The redshift is interpreted as Doppler effect which yields the expansion of space. This is the mainstream interpretation of cosmological models studied by general relativity. It is implied by the singularity of the models.
A theory of gravitation in flat space-time is studied by the author in several articles, e.g. in article [
In the beginning of the universe only energy exists, i.e. there is no matter. In the curse of time matter and radiation are created at coasts of energy where the total energy is conserved. Non-singular cosmological models exist, i.e. a big bang doesn’t exist. The creation of matter is finished a certain time after the beginning of the universe, i.e. the creation of the universe is finished. It is now like a stationary universe. A natural interpretation is a non-stationary, non-expanding universe, the creation of which is finished. The redshift is an intrinsic gravitational effect and not a Doppler effect. A non-expanding universe is also experimentally indicated by several authors, e.g. the articles [
Chapter 2 contains the theory of gravitation in flat space-time which is a field theory. The metric is a flat space-time, e.g. the pseudo-Euclidean metric. The gravitational field satisfies covariant differential equations of order two with the total energy-momentum as source. Conservation of the total energy-momentum and the equations of motion for matter are stated (see e.g. article [
In Chapter 3 a homogeneous, isotropic universe is studied. The pseudo-Euclidean metric is used implying a flat space as indicated by measurements. The matter consists of dust and radiation. The universe is described by two functions. There is no singularity, i.e. no big bang. The absolute time is introduced. There is no age problem. A redshift formula is derived.
A modified Hubble law is described in Chapter 4 by the assumption that everybody is surrounded by a medium similar to that in electrodynamics. Absolute time is used and a modified Hubble law is derived. Quasars and galaxies with nearly the same redshift can have quite different distances as already experimentally indicated by Arp. Redshifts can give smaller distances of the objects than by the standard Hubble law. It may be that dark energy in the universe is not needed because the high redshifts are explained by the modified Hubble law.
The theory of gravitation in flat space-time and the general theory of relativity are compared with one another from the theoretical point of view in article [
A theory of gravitation in flat space-time is studied in several articles ( see e.g. article [
Let
A special case is the pseudo-Euclidean geometry where
Put
Define the tensor
Then, the Lagrangian for the gravitational potentials
where the bar/denotes the covariant derivative relative to the metric (3.1). Let k be the gravitational constant and put
Then, the energy-momentum
This is a tensor whereas the energy-momentum of general relativity is not a tensor.
The energy-momentum tensor of matter is
Here,
Define the differential operator of order two
Then, the field equations for the potentials from the Lagrangian (3.3) are
where
The equations of motion are
where
is the symmetric energy-momentum tensor of matter. In addition to the field Equations (3.8) we have the conservation of the total energy-momentum tensor, i.e.
The field Equations (3.8) and the equations of motion (3.10) imply the conservation law (3.12) of the total energy-momentum. In addition, the field Equations (3.8) and the conservation law (3.12) of the total energy- momentum give the equations of motion (3.10).
The field Equations (3.8) are formally similar to those of general relativity with quite different meanings of
The theory of gravitation in flat space-time and that of general relativity are compared with one another from the theoretical point of view in article [
Flat space-time theory of gravitation is applied to a homogeneous, isotropic universe. The results can be found in several articles, especially in [
The metric is the pseudo-Euclidean geometry, i.e.
The potentials are
which implies the proper-time
Assume that matter is at rest, i.e.
which gives by (3.2) the four-velocity
The matter tensor has the form
Here,
The pressure of matter is that of dust, i.e.
The energy-momentum tensor of gravitation is given by
with
The prime denotes the time-derivative.
The equations for the gravitational field have the form
The conservation of the total energy gives
where
where
The initial conditions for the differential Equations (4.8) are the values at present time
where
Let
and
The it follows from (4.9) for
Longer calculations imply the relations
The condition
gives by the use of (4.12b) that the denominator of (4.14b) is positive for all
then, there exists a time
and for all
We get for
Put
and assume
then, the solutions are approximated by
In the beginning of the universe, i.e.
This means that the universe starts with a small positive value
In addition to the time t there exist the proper time
and the absolute time
The proper time
i.e. the light-velocity in the universe is at every time equal to the vacuum light-velocity c.
The universe happens in the pseudo-Euclidean geometry with the time transformation (4.21b). Hence, space is not expanding.
Two light rays are emitted at distance r at time
The last two relation yield
Therefore, the time-intervals are at any time always identical.
The Equation (4.14b) is by the use of (4.14a) and the use of the absolute time
The frequency
where
Hence the frequency (energy) is not changed when light moves in the universe. Therefore, the measured redshift at the observer at present time is
The redshift formula (4.26) implies by the use of (4.24), differentiation of (4.24) and the constant velocity of light c in the universe
The age of the universe measured with absolute time
The age of the universe from the beginning till
Hence, the age of the universe measured with absolute time from the beginning till
There is no age problem for the universe by the use of the absolute time which is more natural than the proper time
There exist several articles which require a universe with non-expanding space (see e.g. [
A theoretical comparison of the theory of gravitation in flat space-time with general relativity is given in the article [
It is worth to mention that flat space-time theory of gravitation gives in the matter dominated universe a fast increase of the inhomogeneity of matter (see [
The interpretation of an expanding space gives with recently observed high redshift quasars an acceleration of the expansion of the space. This is partly explained by the introduction of a cosmological constant in the general theory of relativity. This is also possible for flat space-time theory of gravitation.
Here we will consider another possibility. Let us assume that a body is surrounded by a medium analogous to that of electrodynamics with electric permittivity
We follow along the lines of article [
with
Then, the proper-time is given by
Let
The electric and the magnetic field are defined by
and the derived fields are
Then, it follows from (5.5) with (5.2) and (5.3)
and with the abbreviation
Let us now consider a medium in the universe described by the absolute time
The metric is given by
and the proper-time
Assume that an atom at rest emits a light ray at time
Hence, the emitted frequency is
Assume that the light-ray moves only a small part of his way to the observer in this medium. In the exterior of the medium relation (4.22) holds, i.e. the frequency is not changed during the motion. Hence, the redshift at the observer is
which gives analogous to Section 4
Hence, we get an intrinsic redshift by the medium, i.e. the standard Hubble law doesn’t give a real distance indicator. We must use the modified Hubble law (5.14).
We can assume
Then, we conclude:
(1) The linear modified Hubble law overestimates the Hubble constant.
(2) For a fixed redshift the resulting distance of the object is smaller than by the standard Hubble law.
(3) Quasars and galaxies with nearly the same distance can have quite different redshifts depending on the medium in which light is emitted (see e.g. Arp [
(4) Quasars may be nearer than by the standard Hubble law. Hence, the measured energy emitted from these quasars is smaller than generally assumed.
(5) It is possible that the assumption of dark energy in the universe is not necessary because the higher redshift implying the acceleration of the universe may be explained by the surrounding medium of the objects, as e.g. of galaxies and quasars.
All these results can be found in the article [
The theory of gravitation in flat space-time gives a non-expanding, non-singular universe, i.e. there exists no big bang. The redshift is an intrinsic effect and not a Doppler effect. Space is not expanding. In the beginning of the universe only energy exists and no matter. In the course of time matter (dust) and radiation are created whereas the total energy is conserved. Matter increases with time and a certain time after the beginning the creation of matter is finished, i.e. the total universe appears static. A modified Hubble law is given which may explain the high redshifts of objects in the universe without the assumption of dark energy.
WalterPetry, (2015) Creation of a Non-Expanding, Non-Singular Universe. Journal of Modern Physics,06,1085-1094. doi: 10.4236/jmp.2015.68113