Journal of Modern Physics, 2013, 4, 20-25
http://dx.doi.org/10.4236/jmp.2013.47A1003 Published Online July 2013 (http://www.scirp.org/journal/jmp)
Cosmology with Bounce by Flat Space-Time Theory of
Gravitation and a New Interpretation
Walter Petry
Mathematisches Institut der Universitaet Duesseldorf, Duesseldorf, Germany
Received April 15, 2013; revised May 20, 2013; accepted June 25, 2013
Copyright © 2013 Walter Petry. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
General relativity predicts a singularity in the beginning of the universe being called big bang. Recent developments in
loop quantum cosmology avoid the singularity and the big bang is replaced by a big bounce. A classical theory of
gravitation in flat space-time also avoids the singularity under natural conditions on the density parameters. The uni-
verse contracts to a positiv e minimum and then it expands during all times. It is not symmetric with regard to its mini-
mum implying a finite age measured with proper time of the universe. The space of the universe is flat and the total
energy is conserved. Under the assumptio n that the sum of the density parameters is a little bit bigger than one the uni-
verse is very hot in early times. Later on, the cosmological model agrees with the one of general relativity. A new inter-
pretation of a non-expanding universe may be given by virtue of flat space -t i me theory of gravitat i on.
Keywords: Gravitation; Cosmology; Big Bounce; Flat Space; No Big Bang; Non-Expanding Universe
1. Introduction
Einstein’s theory of general relativity is generally ac-
cepted as the most powerful theory of gravitation by vir-
tue of its well-known predictions. It gives a singularity in
the beginning of the universe being called big bang and
which has been accepted for long times. But recent de-
velopments of loop quantum cosmology avoid the singu-
larity and it is replaced by a big bounce. There are many
authors who have studied the big bounce by the use of
loop quantum cosmology, see e.g. [1-5] and the extensive
references therein. One compares also the popular book
[6] on this subject. A big bounce in the beginning of the
universe has already been studied by Priester (see e.g.
[7]). Observational hints on a big bounce can be found in
[8].
In 1981 the auth or [9] h as studied a cov ar iant th eory of
gravitation in flat space-time. There exists an extensive
study of this theory since that time. The energy-mome-
tum of gravitation is a covariant tensor and the total en-
ergy-momentum of all kinds of matter and fields includ-
ing that of gravitation is the source of the gravitational
field. The total energy-momentum is conserved. The the-
ory gives the same results as general relativity to the ac-
curacy demanded by the experiments for: gravitational
redshift, light deflection, perihelion precession, radar
time delay, post-Newtonian approximation, gravitational
radiation, and the precession of the spin axis of a gyro-
scope in the orbit of a rotating body. But there exist also
differences to the results of general relativity, these are:
the theorem of Birkhoff doesn’t hold and the theory gives
non-singular cosmological models (no big bang). A sum-
mary of flat space-time theory of gravitation with the
mentioned results can be found in [10] where also refer-
ences to the detailed studies are given. Non-singular cos-
mological models studied by the use of flat spacetime
theory of gravitation can be found e.g. in the papers [11-
15].
Subsequently, we follow along the lines of the above
mentioned articles. Let us assume a hmogeneous, iso-
tropic universe consisting of matter, radiation and dark
energy, given by a cosmological constant. The theory of
gravitation in flat space-time implies a flat space. Under
the assumption that the sum of all the de nsity parameters
is bigger than one, the solution describing the universe is
non-singular, i.e. all the energies are finite. The universe
contracts to a minimum and then it expand s for all times.
The sum of all the energies of matter, radiation, dark
energy, and the gravitational energy is conserved. As-
suming that the sum of all the density parameters is a
little bit bigger than one then the universe becomes very
hot in early times. The time where the contracting uni-
verse enters into the expanding one corresponds to the
C
opyright © 2013 SciRes. JMP

W. PETRY 21
big bang of Einstein’s theory. Some time after this point
of contraction to expansion the solution agrees with the
result of Einstein’s general theory of relativity. There is
no need of inflation because the space of the universe is
always flat. It is worth to mention that the theory of
gravitation indicates that an other interpretation as con-
tracting and expand ing universe is possible. The univ erse
is non-stationary and the time d ependence follows by the
transformation of the different kinds of energy into one
another whereas the total energy is conserved. This trans-
formation of the energies is also the reason for the ob-
served redshifts at distant galaxies. This interpretation
also solves the problem of velocities hig her than the light
velocity at very distant galaxies.
12
8G L


 
 


It is worth to mention that this article appeared in
arXiv (see reference [16]).
2. Gravitation in Flat Space-Time
In this section the subsequently used covariant theory of
gravitation in flat space-time [9] is shortly summarized.
The line-element of flat space-time is

2
ds ij
ijdx dx
 

ij
(1)
where

is a symmetric tensor. In the special case
where

123
,,
x
xx are the Cartesian coordinates, x4 = ct
and



1,1,1, 1ag
ij di

(2)
the space-time metric (1) is the pseudo-Euclidean ge-
ometry. Put


det ij



. (3)
The gravitational field is desribed by a symmetrric
tensor

ij
g
. Let

ij
g
be defined by
,ik i
kj j
gg
kj j
ik i
gg

  (4)
and put analogously to (3)


det ij Gg. (5)
The proper time

is defined similarly to (1) by the
quadratic form

2
2
cd  . (6)
ij
ij
gdxdx
The Lagrangian of the gravitational field is given by
12
//
mn ikjl
Gijkl
G
Lgggg

 
 


 //
1
2
ij kl
mnmn
ggg



(7)
where the bar / denotes the covariant derivative relative
to the flat space-time metric (1). The Lagrangian of the
dark energy (given by the cosmological constant
. (8)
Put
4
4kc  
k
(9)
where denotes the gravitational constant, then the
mixed energy-momentum tensors of the gravitational
field, of dark energy and of matter of a perfect fluid are
given by the following expres sions
12
/// /
11
82
1
2
Giirkmlnklmn
j
kl mnjrjr
i
jG
G
Tggggggg
L




  






(10a)

) has
the form
1
16
ii
jj
TL 





2.
Miiki
jjkj
Tpguupc   
,p
(10b)
(10c)


i
u denote density, pressure and

and Here,
four-velocity d
d
i
x




2. ij
ij
cguu
of matter. It holds by virtue of (6)
(11)
Define the covariant differential operator
12
/
/
iklmi
jjml
k
G
Rggg











 


 (12)
of order two in divergence form, then the field equations
for the potentials


ij
g
can be written in the covariant
form
14
2
iik i
j
jk j
RRT

 

GM
iiii
(13)
where
j
jjj
TTTT
 (14)


is the total energy-momentum tensor. The equations of
motion are given in covariant form by
M
//
1
2
M
kkl
ik kl i
TgT (15)
where
M
M
ijjk i
j
TgT
addition to the field Equations (13) and the equations of
(16)
is the symmetric energy-momentum tensor of matter. In

W. PETRY 22
motion (15) the conservation law
0
k
T
/ ik (17)
of the total energy-momentum tensor holds. All the
Equations (13), (15) and (17) are generally covariant and
the energy-momentum (10a) of the gravitational field is a
tensor in contrast to that of general relativity. The field
Equation (13) are formally similar to the equations of
Einstein’s general relativity theory but i
j
R
 is not the
Ricci tensor and the source of the gravitatnal field in-
cludes the energy-momentum tensor of gravitation. It is
worth to mention that the field Equation (13) together
with the Equation (15), respectively (17) imply the Equa-
tion (17), respec t ively (15).
io
3. Isotropic Cosmological Model
f gravitation is
(18)
and
mr
In this section the flat space-time theory o
applied to homogeneous, isotropic cosmological models.
The pseudo-Euclidean geometry (1) with (2) is assumed.
The matter tensor is given by (10c) with

01,2,3
i
ui
,
mr
pp p

  (19)
where inds m and rdenote matt
qu o

e thiceer and radiation
respectively. The eations f state for matter (dust) and
radiation are
0, 3.
mrr
pp

 (20)
By virtue of (18) the potentials are given by
 
 

1, 4
0,
ij
gh tij
ij


 




(21)
where all the functions dend only on t by virtue of
l
2,1,2,3ati j

pe
the homogeneity of the model. The four-veocity (11) has
by the use of (18) and (21) the form

12
0,0,0, .
i
uch (22)
For thme 00t the fo
di

00
,0,
r
Hhh



(23)
where the dot denotes the tierivative, 0
e present tillowing initial con-
tions are assumed:
 
00ah

 
00
1,0
0,0
mmr
a

 


me-d
H
is the
tanwell-known Hu bble constan t and 0
h
 is a const which
doesn’t appear by Einstein’s theo The constants 0m ry.

and 0r

denote the present densities of matter and ra-
diatioIt is worth to mention that general relativity im-
plies

1ht  which is not possible in flat space-time
theoryitation. This will be important to avoid the
singularity.
Under the
n.
of grav
assumption that matter and radiation do not
interact the equations of motion (15) can be solved by the
use of (18) to (22). It follows

12,0,hp
   12
00
3 .
mmmrrr
p ah   (24)
The field Equations (13) with (10) to (14) im
(1 ply by 8) to (22) the two non-linear differential equ ations:
3
d11aa
31
2 4212
2
d2
32
mr
ah c
ta ch



 







 (25a)

3
31242212
d11
4
d282
mr G
ha
ah cL
thc ch











 
.
(25b)
Here, it holds
2
2
312
2
11
66
2
Gaahh
Lah aahhc








 


 (26)
where the gravitational energy is given by 1
16 G
L

.


The proper time is:



222 2
23
1
cdadxdxdxct
h
  . (27)
The conservation of the to tal energy has the form:

2
22
1
3
1a
22
12
16 2
mr G
cL c
h



  
(28)
where


is a constant of integration. The Equations (25)
and (26) give by the use of (28) and the initial conditions
(23):
40
42 0
4
62
21
ct ha
ha ctt
 
 

 
 (29)
with
0
00 0
1
31
6
h
Hh





. (30)
Integration of relation (29) yields:
3124 2
21ahc tt  
0.

 (31)
Equation (28) gives at present time 0
t
of0 by the use (23) and (24)

2
42 2
0000
84 .
338
mr c
ck H
k
 
18


 
 






 (32)
It follows from (2 8) by the use of (26), (29), (31) , (24)
and (31) the formula

W. PETRY
© 2013 SciRes. JMP
23
Copyright

2
a

22
00 0
2
42 0
2632
00
18
38
21
88
.
33 3
mr
mr
a
c
kH
k
ct t
cakaka









  




  







(33)
Let us introduce the density parameters:
2
2
0
,,
3
c
00
22
00
88
33
mr
mr
kk
H
HH


 (34)
and define

 
 
0mr
K 1m  (35)
then the differential Equation (33) can be rewritten in the
form:

2
2
00
2
42 0
21
mr
a
a
H236
m
K
aaa




(36a)
with the initial cond ition

01.
ct t 




 


a

(36b)
Hence, a solution of (36) togeth
a
4. Cosmology with a Bounce
utions of (36) with
e rewritten by the use of the density
pa
er with (31) describes
homogeneous, isotropic cosmological model in flat
space-time theory of gravitation.
In this section we will study the sol
(31) and show that non-singular cosmological models
with a bounce exist.
Relation (32) can b
rameters (34) and the definition (35)
2
400
20
0
812 .
m
c
K
HH
 



 (37)
A necessary condition to avoid a singular solution of
(3
00K. (38)
Inequality (38) is by the use of (37
42 0
210ct t   
tR
(39)
for all

. Then, the differential Equation (36) has a
positive solutio n


at and relation (31) gives a positive
function


12 .ht
1
t
Therefore, condition (38) is necessary
and sufficient for non-singular cosmological models by
virtue of (24). Then, there exists a time such that


10.at
 (40)


Put

1
aat
1
tt
1, then the differential Equation (36a)
implies at

:
236
111 0rm m
aaaK


tR
(41)
and for all



10at a

101aa
01K
0
1,
rm m
. (42)
The assumption
(43)
gives by virtue of (41)
. (44)
Hence, it follows by (35)
K
(45)


  
i.e. the sum of all the density parameters is a little bit
bigger than one. The cond ition (43) is also iumportan t for
a very hot universe in the beginning because the tem-
perature is given by


6) is :
) equivalent to
0
TtT at
T
1
tt
(46)
where 0 is the present temperature of radiation. For

the universe contracts and then it expands as
Hence, there exists a bounce in the early universe.
1
The differential Equation (36) is written in the case
.tt
1
tt

(expanding universe) in the form:

12
236
00
42 0
21
mrm
H aKa aa
actt  






01a
1
tt
(47)
and for

(contracting universe) in the form:

1
236
00
42 0
21
mrm
H aKa aa
actt
 

 

2


11
at a
0
. (48)
In the special case r


 
an analytic solution of the
expanding univers e (47) can be gi v en (see e.g. [ 12]):






 
12
300 0
2112cos3 2sin3
m
at KKtKt   (49a)

W. PETRY 24
where
 
12
00
arctg3 m0
1
12 Htt



 




. (49b)
For the subsequent considerations compare
The time 1
t is given byof 0
r
 )
tK 
[14].
(in the ca se

12

0
0
1
3
1
1mK Ht



 

(
00
2HA


50a)
with


12
0
1
tgarctg 2
3
4.0338 .
K
A
OK
 

12
0
0
1
m
K







 
 



(50b)
Relation (49) gives two different kinds of solutions
(see [14]):
ying
1) The denominator of (4 9) is positive and vanishes as
t impl

0
13
2H

0
1
2
m
 (51)
where expressintaining 0 ons co
K
are om
of (44). )
itted by virtue
2) The denominator of (49is always positive as
t implying

0
0
13
2H

1.
2
m
 (52)
In both cases te function h


at is in
time 1 t. Int case

at converges to infinity as
tn the
a
creasing after the
the firs
goes to infinity whereas i second case the func-
tion

t converges to a e value as t goes to in-
finity. Subsequently, we will only consider the interest-
ing case 1). Condition (51) is by the use of(30) a condi-
tion on the initial value 0 h
 of (23).
Let us define the time 1 t
 by
finit
2
0
01 1100
00
3
2
2m
H
t



 
K
HH


 




(53)
en, for 1
tt
 the solution (49a) can be app
Equation (54a) below and by the use of (31) the full


th
by roximated
solution is
 

2.
m
12 00
1
31ht HtHt




For 1 tt

(54b)

the function

at of the differentia
tion (48) starts at l Equa-






13
2a
11
1 cos31.8161aa (55)
and decreases as 1 tt to 1 a.
The proper time

from the beginning of t

he uni-
verse is given by

12
t1dthtt


. (56)
The proper time


1
t  , i.e. from t
universe till 1 t, is finite by virtue of (31), (55) and (43).
Th
he beginning of the

e proper time

t
rt
 of the universe is by (56) in-
creasing with increasing t and goes to infinity as t
goes to infinity by viue of (54b). It follows for 1
tt


the proper time
 

12
11
ln
tht  
00
3H

(57a)
with a suitable con stant 0. Therefore, it holds for


t 

sufficiently large
  


12 exp 3htH t
00

  . ()
57b
Hence, under the condition (51), the function


at
ositive value starts by virtue of (55) from a small p


1
a

a

 and decreases to 1 a as 1 tt. Then,

at
t. It ih

t imet
um
Let ime (5
e
increases for all times 1 tt and goes to infinity as
s worth to mention tat as not sym-
ric with regard to its minim at 1
t.
us now introduce the proper t6) into the dif-
ferential Equation (36). It follows by thuse of (31)
220
0
2643
1d
dmm
r
K aH
aaaa 

 
 
 
 
 


01.a (58)
Hence, this differential equa
and (38) for 1 aa identical
tio e. The
’s th
tion is by the use of (44)
with the differential equa-
n of general relativity describing a homogeneous, iso-
tropic universrefore, all the results of general rela-
tivity are valid. But for su fficiently s mall

at, th e solu -
tion is qu ite differen t fro m that of g eneral relativity and it
has no singularity in contrast to Einsteineory, i.e.,
there is no b i g b ang.
 

22
3001 001
331
21
mm
a tHt HtHt Ht





2










(54a)

W. PETRY 25
5. A New Interpretation
In this section a new interpretetion of the results of sec-
on 4 is given. All the fo rmulae and results of the previ-
nterpretation of a bounce,
ng universe. This is pos-
whereas the total energy is conse
La
ti
ous chapter are valid ex ept the i
i.e. of a collapsing and expandi
sible by virtue of the conservation of the total energy (28)
with (24) and (26) in flat space-ti me theory of gravitation.
The universe can be interpreted as non-expanding where
the redshift follows by the transformation of the different
kinds of energy into one another (see [14,15]). The for-
mula for the redshift is identical with the one of the ex-
panding universe. The derivation of this result can be
found in [13-15].
In the beginning of the non-expanding universe, no
matter, no radiation and no dark energy exist by virtue of
(31) and (55). In the course of time radiation, matter and
dark energy arise rved.
ter on, matter and radiation decrease (this corresponds
to the expanding universe). Formula (24) for matter with
(57b) implies that matter is exponentially decaying for
sufficienly large proper time

in the non-expanding
universe analogously to the radioactive decay whereas
dark energy increases to a finite value as t, respctively

goes to infinity. It is worth to mention that this result
depends on the assumed dark eergy given by a cosmo-
logical constant. For 0 , again no matter and no ra-
diation exist in th e beginning of the universe. Matter and
diation arise in the non-expanding universe and matter
increases to a finite value whereas radiation goes to zero
in the course of time. Ttal energy is again conserved
(see, e.g. [11,12]).
It seems that a non-expanding space is a more natural
interpretation of the universe implied by the use of gra-
vitation in flat space-time. The problem of velocities of
galaxies higher th
n
ra
he to
an light velocity doesn’t arise and
so
models. The u
lar, i.e., in the beginning of the
r on it expands for all tim
pretation by the use of flat space-time theory of gravita-
tion than an expanding space. Here, th
axies is explained by the change of the different kinds of
energies in the course of time where the total energy is
wald, Physical Review Letters, Vol. 86, 2001, pp.
5227-5230. doi:10.1103/PhysRevLett.86.5227
mething like inflation is superfluous because in the
beginning of the universe, no matter, no radiation and no
dark energy exist, i.e. all the energy is in form of gravita-
tion and the space is flat for all times.
6. Conclusion
The theory of gravitation in flat sace-time is applied to
homogeneous, isotropic, cosmologocalni-
verse is non-singu
verse space contracts and lateuni- es.
A non-expanding universe seems a more natural inter-
conserved.
REFERENCES
[1] M. Bojo
e redshift of gal-
[2] A. Ashtekar, wski, Ad-
vances Theoresics, Vol. 7, 2003,
l. 8, 2005,
0.1007/s10714-009-0763-4
M. Bojowald and J. Lewando
tical Mathemaical Phy
pp. 233-268.
[3] M. Bojowald, Living Reviews in Relativity, Vo
p. 11.
[4] A. Ashtekar, General Relativity Gravitation, 2009, pp.
707-741. doi:1
005.5491.
Bounce to the -Dominated Uni-
nd M. Szyd-
, Vol. 13, 1981,
[5] A. Ashtekar, “The Big Bang and the Quantum,” 2010,
arXiv:1
[6] M. Bojowald, “Zurück vor den Urknall, Die ganze Ges-
chichte des Universums,” S. Fischer, 2009.
[7] J. Overduin, H.-J. Bloome and J. Hoell, “Wolfgang Pries-
ter: From the Big
verse,” 2011, arXiv:astro-ph/0608644.
[8] J. Mielczarek, M. Kamionka, A. Kurek a
lowski, “Observational Hints on the Big Bounce,” 2010,
arXiv:1005.0814.
[9] W. Petry, General Relativity Gravitation
pp. 865-872. doi:10.1007/BF00764272
[10] W. Petry, “A Theory of Gravitation in Flat Space-Time,
with Applications,” In: M. C. Duffy and M. Wegener,
Eds., Recent Advances in Relativity Theory, Vol. II, Had-
ronic Press, 2002, p. 196.
[11] W. Petry, General Relativity Gravitation, Vol. 13, 1981,
pp. 1057-1071. doi:10.1007/BF00756365
[12] W. Petry, General Relativity Gravitation, Vol. 22, 1990,
pp. 1045-1065. doi:10.1007/BF00757815
[13] W. Petry, Astrophysics & Space Science, Vol. 222, 1994,
pp. 127-145. doi:10.1007/BF00627088
[14] W. Petry, Astrophysics & Space Science, Vol. 254, 1997,
pp. 305-317. doi:10.1023/A:1000938931517
ion,” 2011,
[15] W. Petry, “Is the Universe Really Expanding?” 2011,
arXiv:0705.4359.
[16] W. Petry, “Cosmology with Bounce by Flat Space-Time
Theory of Gravitation and New Interpretat
arXiv:1102.1063.