Cosmology with Bounce by Flat Space-Time Theory of Gravitation and a New Interpretation

doi:10.4236/jmp.2013.47A1003

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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

Email: [email protected], [email protected]

Received April 15, 2013; revised May 20, 2013; accepted June 25, 2013

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

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.









 





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



ds ij

ijdx dx







(1)

where



is a symmetric tensor. In the special case

where



123

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



. Let



be defined by

,ik i

kj j

ik i



 (4)

and put analogously to (3)





det ij

Gg. (5)

The proper time



is defined similarly to (1) by the

quadratic form





. (6)

gdxdx

The Lagrangian of the gravitational field is given by

mn ikjl

Gijkl

Lgggg







 





 //

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

4kc





(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

/// /

Giirkmlnklmn

kl mnjrjr

Tggggggg











 













(10a)



) has

the form











Miiki

jjkj

Tpguupc



 

(10b)

(10c)





u denote density, pressure and



and Here,

four-velocity d









cguu

of matter. It holds by virtue of (6)

(11)

Define the covariant differential operator

iklmi

jjml

Rggg































 (12)

of order two in divergence form, then the field equations

for the potentials





can be written in the covariant

form

iik i

jk j

RRT









iiii

(13)

where

jjj

TTTT

 (14)





is the total energy-momentum tensor. The equations of

motion are given in covariant form by

kkl

ik kl i

TgT (15)

where

ijjk i

TgT

addition to the field Equations (13) and the equations of

(16)

is the symmetric energy-momentum tensor of matter. In

W. PETRY

motion (15) the conservation law

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

 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).

3. Isotropic Cosmological Model

f gravitation is

(18)

and

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

ui

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



 (20)

By virtue of (18) the potentials are given by

 



1, 4

tij



 









(21)

where all the functions dend only on t by virtue of

2,1,2,3ati j



the homogeneity of the model. The four-veocity (11) has

by the use of (18) and (21) the form



0,0,0, .

uch (22)

For thme 00t the fo



,0,

Hhh







(23)

where the dot denotes the tierivative, 0

e present tillowing initial con-

tions are assumed:

 

00ah



 

1,0

0,0

mmr



 





me-d

is the

tanwell-known Hu bble constan t and 0

 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

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

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:

d11aa

2 4212

ah c

ta ch

























 (25a)



31242212

d11

d282

mr G

ah cL

thc ch



























.

(25b)

Here, it holds

312

Gaahh

Lah aahhc









 





 (26)

where the gravitational energy is given by 1

16 G







The proper time is:







222 2

cdadxdxdxct



 . (27)

The conservation of the to tal energy has the form:



16 2

mr G

cL c







  

(28)

where





is a constant of integration. The Equations (25)

and (26) give by the use of (28) and the initial conditions

(23):

42 0

ha ctt

 



 

 (29)

with

00 0











. (30)

Integration of relation (29) yields:

3124 2

21ahc tt

 



 (31)

Equation (28) gives at present time 0

of0 by the use

(23) and (24)



42 2

0000

84 .

338

mr c

ck H

 









 













 (32)

It follows from (2 8) by the use of (26), (29), (31) , (24)

and (31) the formula

W. PETRY





00 0

42 0

2632

33 3

ct t

cakaka



















  









 















(33)

Let us introduce the density parameters:



 (34)

and define







 

0mr

K 1m

 (35)

then the differential Equation (33) can be rewritten in the

form:



42 0

H236

aaa









(36a)

with the initial cond ition



01.

ct t











 







(36b)

Hence, a solution of (36) togeth

4. Cosmology with a Bounce

utions of (36) with

e rewritten by the use of the density

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)

400

812 .











 (37)

A necessary condition to avoid a singular solution of

00K. (38)

Inequality (38) is by the use of (37

42 0

210ct t

 



(39)

for all



. Then, the differential Equation (36) has a

positive solutio n





at and relation (31) gives a positive

function





12 .ht

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



aat

1, then the differential Equation (36a)

implies at



236

111 0rm m

aaaK





(41)

and for all







10at a



101aa

01K

rm m

. (42)

The assumption

(43)

gives by virtue of (41)

. (44)

Hence, it follows by (35)

(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

TtT at

(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.

The differential Equation (36) is written in the case

.tt



(expanding universe) in the form:



236

42 0

mrm

aKa aa

actt

 













01a

(47)

and for



(contracting universe) in the form:



236

42 0

mrm

aKa aa

actt

 



 



2





at a

. (48)

In the special case r







an analytic solution of the

expanding univers e (47) can be gi v en (see e.g. [ 12]):

















300 0

2112cos3 2sin3

at KKtKt



 (49a)

W. PETRY

where



arctg3 m0

Htt





 





. (49b)

For the subsequent considerations compare

The time 1

t is given byof 0

 )



[14].

(in the ca se





1mK







 



(

2HA





50a)

with



tgarctg 2

4.0338 .

















(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





m

 (51)

where expressintaining 0

ons co

are om

of (44). )

itted by virtue

2) The denominator of (49is always positive as

t implying





m

 (52)

In both cases te function h





at is in

time 1

t. Int case



at converges to infinity as

tn the

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

 of (23).

Let us define the time 1

 by

finit

01 1100







 

K





 







(53)

en, for 1

 the solution (49a) can be app

Equation (54a) below and by the use of (31) the full





by roximated

solution is



12 00

31ht HtHt









For 1



(54b)



the function



at of the differentia

tion (48) starts at l Equa-











2a

1 cos31.8161aa (55)

and decreases as 1

tt to 1

The proper time



from the beginning of t



he uni-

verse is given by



t1dthtt





. (56)

The proper time







, i.e. from t

universe till 1

t, is finite by virtue of (31), (55) and (43).

he beginning of the



e proper time





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





the proper time

 



tht







(57a)

with a suitable con stant 0. Therefore, it holds for







sufficiently large

 







12 exp 3htH t



 . ()

57b

Hence, under the condition (51), the function





ositive value starts by virtue of (55) from a small p









 and decreases to 1

a as 1

tt. Then,



t. It ih



t imet

Let ime (5

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

us now introduce the proper t6) into the dif-

ferential Equation (36). It follows by thuse of (31)

220

2643

dmm

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.







3001 001

331

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

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

he to

an light velocity doesn’t arise and

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.

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M. Bojowald and J. Lewando

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