Journal of Applied Mathematics and Physics, 2014, 2, 50-54
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
How to cite this paper: Petry, W. (2014) Gravitation in Flat Space-Time and General Relativity. Journal of Applied
Mathematics and Physics, 2, 50-54. http://dx.doi.org/10.4236/jamp.2014.25007
Gravitation in Flat Space-Time and General
Mathematical Institute of the University Duesseldorf, Duesseldorf, Germany
Email: wpetr y @med use.d e, firstname.lastname@example.org
Received January 2014
A covariant theory of gravitation in flat space-time is stated and compared with general relativity.
The results of the theory of gravitation in flat space-time and of general relativity agree for weak
gravitational fields to low approximations. For strong fields the results of the two theories deviate
from one another. Flat space-time theory of gravitation gives under some natural assumptions
non-singular cosmological models with a flat space. The universe contracts to a positive minimum
and then it expands for all times. Shortly, after the minimum is reached, the cosmological models
of two theories approximately agree with one another if models in general relativity with z ero
curvature are considered. A flat space is proved by experiments.
Gravi tati on , Flat Space-Ti me, Cosmology, Big Bounce, No Big Bang, Flat Space
A previously studied covariant theory of gravitation in flat space-time is stated . The energy-momentum of
the gravitational field is a tensor. The source of the gravitational field is the total energy-momentum of all the
fields inclusive that of gravitation. This is quite different from general relativity for which the energy-mo men-
tum of gravitation is not a tensor. Hence, the energy-momentum of the gravitational field cannot explicitly ap-
pear as source by virtue of the covariance of general relativity. Therefore, the Ricci tensor is used as differential
operator yielding a non-Euclidean geometry. An extensive study exists of flat space-time theory of gra vitation. It
follows that the results of the two theories agree with one another for weak field approximations but there are
differences if the gravitational fields are strong. Therefore, the theory of flat space-time theory is applied to ho-
mogeneous, isotropic cosmological models where only matter and radiation are considered. A cosmological
constant could also be included . The universe is non-singular under the assumption that the sum of the den-
sity parameters is a little bit greater than one. In the beginning of the universe there is no matter and no radiation.
The universe contacts to a small minimum creating matter and radiation with very high temperature. All the
densities of matter and radiation are always finite. After the minimum is reached the universe expands for all
times. Shortly after the time when the minimum is reached the results of the two theories approximately agree if
a vanishing curvature of general relativity is assumed. The space of flat space-time theory of gravitation is flat,
i.e. there is no necessity of inflation in the beginning of the universe in contrast to general relativity where strong
curvature exists in the neighbourhood of the singularity which corresponds to the minimum of the universe by
the use of gravitation in flat space-time .
2. Gravitation in Flat Space-Time
The covariant theory of gravitation in flat space-time  is shortly summarized. The metric of flat space-time is
whe r e
is a symmetric tensor. In the special case where
are the Cartesian coordinates,
the space-time metric is the pseudo-Euclidean geometry. We put
The gravitational field is described by a symmetric tensor
be defined by
and put analogously to (2.3)
Then, the proper-time is defined similarly to (2.1) by the quadratic form
General relativity uses Equa ti o n (2.6) as metric.
The Lagrangian of the gravitational field is given by
mnik jlij kl
where the bar / denotes the covariant derivative relative to the metric (2.1).
We mention that a Lagrangian of the form (2.7) for general relativity doesn’t exist because the metric is given
by Equation (2.6).
Then, the energy-momentum of the gravitational field is
() 8 22
which is a tensor for this theory.
The energy-momentum of general relativity is not a tensor.
The energy-momentum tensor of matter is
denote the density, the pressure and the four-velocity
of matter. It holds by
virtue of Equation (2.6)
c guu= −
We define the covariant differential operator of order two in divergence form
Then, the Lagrangian gives the field equations
with the relation
T TMTG= +
is the total energy-momentum tensor inclusive that of the gravitational field.
The equations of motion of matter are
whe r e
is the symmetric energy-momentum tensor.
In addition, we have the conservation law of the total energy-mom entum
The field Equations (2.13) with Equation (2.14) and the equations of motion (2.15) imply the conservation
law of the total energy-momentum Equation (2.17). Conversely, the field Equations (2.13) with Equat io n (2.14)
and the conservation law of the total energy-momentum Equation (2.17) yield the equations of motion Equation
(2.15). All the stated equations are covariant.
General relativity is formally similar to the Equation (2.13) but it replaces
is the Ricci tensor and
contains only the matter tensor. Hence, the equations of general relativity
are also covariant relative to the metric given by Equation (2.6) but we get a non-Euclidean geometry. In addi-
tion, the condition of Einstein that any sort of energy is equal to matter is not fulfilled because gravitational en-
ergy is not contained as source.
It is worth mentio ning that the theory of Maxwell is analogous to flat space-time theory of gravitation because
the source of the electro-magnetic potentials
is the electrical four-current and the differential operator for
the potentials is in divergence form of order two.
3. Homogeneous, Isotropic, Cosmological Model
Let us use the pseudo-Euclidean metric Equation (2.1) with Equation (2.2).
The matter tensor Equation (2.10) is given with
where the indices m and r denote matter and radiation.
The equations of state are
The gravitational field has by virtue of Equation (3.1), the homogeneity and the isotropy the form
,1, 2, 3
gat i j
The four-velocity is given by
The initial conditions at present time
( )()( )()
001,0,0,a haHhh= ===
( )( )
Here, the prime denotes the
is the Hubble constant and
doesn’t appear by the use of
general relativity because
is not possible by flat space-time theory of gravi-
tation. Then, the field equations and the conservation of the total energy give after longer calculations
() ( )
whe r e
is the constant of the conservation of the total energy,
are the density parameters of
matter and radiation, and
Furthermore, it holds
2 1.ah ctt
It easily follows that non-singular solutions exist under the condition
The inequality (3.10) implies
Relation (3.7) gives at present time
Let us furthermore assume
Relation (3.7) implies the existence of a constant
1 0a witha<<<
Hence, there exists a time
Here, the upper sign holds for
and implies a contraction of the space till the time
and the lower sign holds for
yielding an expansion of space.
Let us introduce the time
Then, Equatio n (3.15) for
together with Equation (3.9) can be rewritten
ad a aa
Equation (3.17) is under the assumptions (3.13) and
the differential equation of general relativity
for a universe with zero curvature.
Therefore, flat space-time theory of gravitation and general relativity give approximately the same result for
the expanding, flat space. But in the beginning of the universe the results of both cosmological models are quite
different, i.e. we have a bounce and not a big bang.
It is worth mentioning that a cosmological constant could also be included without changing the statements.
Furthermore, the cosmological models of gravitation in flat space-time also permit the interpretation of a
non-expanding space. For this case the redshift is explained by the transformation of the different sorts of energy
into one another whereas the conservation of the total energy is valid.
It is well-known that general relativity is only experimentally verified for weak fields.
More details of the theory of gravitation in flat space-time and the received results can be found in several ar-
ticles of the author and in the book “A theory of gravitation in flat space-time” which appears soon in Science
 Pet ry, W. (1981) Cosmological Models without Singularities. General Relativity Gravitation, 13, 1057-1071.
 Pet ry, W. (2013) Cosmology with Bounce by Flat Space-Time Theory of Gravitation and a New Interpretation. Jou r-
nal Modern Physics, 4, 20-25. http://dx.doi.org/10.4236/jmp.2013.47A1003
 Pet ry, W. (2014) A Theory of Gravitation in Flat Space-Time. Science PG.