Journal of Applied Mathematics and Physics, 2014, 2, 50-54
Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp
http://dx.doi.org/10.4236/jamp.2014.25007
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
Relativity
Walter Petry
Mathematical Institute of the University Duesseldorf, Duesseldorf, Germany
Email: wpetr y @med use.d e, petryw@uni-duesseldorf.de
Received January 2014
Abstract
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.
Keywords
Gravi tati on , Flat Space-Ti me, Cosmology, Big Bounce, No Big Bang, Flat Space
1. Introduction
A previously studied covariant theory of gravitation in flat space-time is stated [1]. 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 [2]. 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
W. Petry
51
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 [1] is shortly summarized. The metric of flat space-time is
( )
2ij
ij
dsdx dx
η
= −
(2.1)
whe r e
ij
η
is a symmetric tensor. In the special case where
()
123
,,
xxx
are the Cartesian coordinates,
4
x ct=
and
( )
( )
1,1,1, 1
ij
η
= −
(2.2)
the space-time metric is the pseudo-Euclidean geometry. We put
( )
det .
ij
ηη
=
(2.3)
The gravitational field is described by a symmetric tensor
. Let
( )
ij
g
be defined by
kj j
ik i
gg
δ
=
(2.4)
and put analogously to (2.3)
( )
det
ij
Gg=
(2 .5)
Then, the proper-time is defined similarly to (2.1) by the quadratic form
( )
2
.
ij
ij
dcgdx dx
τ
= −
(2.6)
General relativity uses Equa ti o n (2.6) as metric.
The Lagrangian of the gravitational field is given by
( )
12
// //
1
2
mnik jlij kl
ijklmn mn
G
LGggg gggg
η


=−−
 


(2.7)
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).
Put
4
4k
c
π
κ
=
. (2.8)
Then, the energy-momentum of the gravitational field is
( )
12ln
/ ///
1 11
() 8 22
iirkmkl mni
jkl mnjrjrj
G
TGg ggggggLG
δ
κη




= −+
 




(2.9)
which is a tensor for this theory.
The energy-momentum of general relativity is not a tensor.
The energy-momentum tensor of matter is
( )
2
() .
iki i
jjk j
TMpgu upc
ρδ
=++
(2.10)
Here,
ρ
,
p
and
( )
i
u
denote the density, the pressure and the four-velocity
i
dx
d
τ



of matter. It holds by
virtue of Equation (2.6)
2
.
ij
ij
c guu= −
(2.11)
W. Petry
52
We define the covariant differential operator of order two in divergence form
12
/
/
imn ki
jjk n
m
G
Dgg g
η



=




. (2.12)
Then, the Lagrangian gives the field equations
14
2
i iki
j jkj
D DT
δκ
−=
, (2.13)
with the relation
( )()
i ii
j jj
T TMTG= +
(2.14)
i.e.
i
j
T
is the total energy-momentum tensor inclusive that of the gravitational field.
The equations of motion of matter are
//
1
() ()
2
k kl
ik kli
TMgTM=
(2.15)
whe r e
() ()
ij jki
k
TMg TM=
(2.16)
is the symmetric energy-momentum tensor.
In addition, we have the conservation law of the total energy-mom entum
/
0.
k
ik
T=
(2.17)
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
()
,
i iii
j jjj
DR TTM==
(2.18)
where
i
j
R
is the Ricci tensor and
i
j
T
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
()
01,2, 3
i
ui= =
(3.1)
and
,
mr mr
pp p
ρρ ρ
=+=+
(3.2)
where the indices m and r denote matter and radiation.
The equations of state are
1
3.0,
m rr
pp
ρ
= =
(3.3)
The gravitational field has by virtue of Equation (3.1), the homogeneity and the isotropy the form
W. Petry
53
() ()
()()
()
2
,1, 2, 3
1
4
,
0,
ij
gat i j
ij
ht
ij
== =
=−==
= ≠
(3.4)
The four-velocity is given by
( )
( )
0,0,0, .
i
u ch=
(3 .5)
The initial conditions at present time
00t=
are
( )()( )()
' ''
00
001,0,0,a haHhh= ===
( )( )
00
0 0.,
m mrr
ρρρ ρ
= =
(3.6)
Here, the prime denotes the
t
derivative,
0
H
is the Hubble constant and
0
'h
doesn’t appear by the use of
general relativity because
( )
1.ht =
The condition
( )
1.ht =
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
() ( )
2223
00
2
42 0
'ΩΩ Ω
21
m rm
H
aKa a
actt
κλ ϕ

=−++

 ++
(3.7)
whe r e
2
c
λ
is the constant of the conservation of the total energy,
Ω
m
and
Ω
r
are the density parameters of
matter and radiation, and
2
4
00
000 2
00
0
'
1 18
31 ,
Ω.
6 12
m
hc
HK
HH
H
ϕ
κλ
ϕ

 

=+=−

 
 


(3.8)
Furthermore, it holds
342 0
2 1.ah ctt
κλ ϕ
= ++
(3. 9)
It easily follows that non-singular solutions exist under the condition
0
Ω0.
m
K>
(3.10)
The inequality (3.10) implies
42 0
210for all.
ct tt
κλ ϕ
+ +>∈
(3.11)
Relation (3.7) gives at present time
00t=
0
Ω Ω1Ω.
rm m
K+=+
(3.12)
Let us furthermore assume
0
Ω1.
mK<<
(3.13)
Relation (3.7) implies the existence of a constant
11
1 0a witha<<<
and
23
11 0
ΩΩΩ.
rmm
aaK+=
(3.14)
Hence, there exists a time
10
0tt<=
such that
( )
1/2
23
00
42 0
'ΩΩ Ω
21
m rm
H
aKa a
actt
κλ ϕ
=− ++
++
(3.15)
Here, the upper sign holds for
1
tt
and implies a contraction of the space till the time
1
t
with
( )
11
at a=
and the lower sign holds for
1
tt
yielding an expansion of space.
Let us introduce the time
ζ
by
( )
1
d dt
ht
ζ
=
. (3.16)
W. Petry
54
Then, Equatio n (3.15) for
1
tt
together with Equation (3.9) can be rewritten
1/2
0
06 43
Ω Ω
Ω
1.
mm
r
K
da H
ad a aa
ζ

=+ −++


(3.17)
Equation (3.17) is under the assumptions (3.13) and
( )
1
at a>
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
PG [3].
References
[1] Pet ry, W. (1981) Cosmological Models without Singularities. General Relativity Gravitation, 13, 1057-1071.
http://dx.doi.org/10.1007/BF00756365
[2] 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
[3] Pet ry, W. (2014) A Theory of Gravitation in Flat Space-Time. Science PG.