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

Home , Big Bounce

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)

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

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 universe contracts to a positive minimum and then it expands during all times. It is not symmetric with regard to its minimum 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 assumption that the sum of the density parameters is a little bit bigger than one the universe is very hot in early times. Later on, the cosmological model agrees with the one of general relativity. A new interpretation of a non-expanding universe may be given by virtue of flat space-time theory of gravitation.

Keywords: Gravitation; Cosmology; Big Bounce; Flat Space; No Big Bang; Non-Expanding Universe

1. Introduction radiation, and the precession of the spin axis of a gyro- scope in the orbit of a rotating body. But there exist also Einstein’s theory of general relativity is generally ac- differences to the results of general relativity, these are: cepted as the most powerful theory of gravitation by vir- the theorem of Birkhoff doesn’t hold and the theory gives tue of its well-known predictions. It gives a singularity in the beginning of the universe being called big bang and non-singular cosmological models (no big bang). A sum- which has been accepted for long times. But recent de- mary of flat space-time theory of gravitation with the velopments of loop quantum cosmology avoid the singu- mentioned results can be found in [10] where also refer- larity and it is replaced by a big bounce. There are many ences to the detailed studies are given. Non-singular cos- authors who have studied the big bounce by the use of mological models studied by the use of flat spacetime loop quantum cosmology, see e.g. [1-5] and the extensive theory of gravitation can be found e.g. in the papers [11- references therein. One compares also the popular book 15]. [6] on this subject. A big bounce in the beginning of the Subsequently, we follow along the lines of the above universe has already been studied by Priester (see e.g. mentioned articles. Let us assume a hmogeneous, iso- [7]). Observational hints on a big bounce can be found in tropic universe consisting of matter, radiation and dark [8]. energy, given by a cosmological constant. The theory of In 1981 the author [9] has studied a covariant theory of gravitation in flat space-time implies a flat space. Under gravitation in flat space-time. There exists an extensive the assumption that the sum of all the density parameters study of this theory since that time. The energy-mome- is bigger than one, the solution describing the universe is tum of gravitation is a covariant tensor and the total en- non-singular, i.e. all the energies are finite. The universe ergy-momentum of all kinds of matter and fields includ- contracts to a minimum and then it expands for all times. ing that of gravitation is the source of the gravitational The sum of all the energies of matter, radiation, dark field. The total energy-momentum is conserved. The the- energy, and the gravitational energy is conserved. As- ory gives the same results as general relativity to the ac- suming that the sum of all the density parameters is a curacy demanded by the experiments for: gravitational little bit bigger than one then the universe becomes very redshift, light deflection, perihelion precession, radar hot in early times. The time where the contracting uni- time delay, post-Newtonian approximation, gravitational verse enters into the expanding one corresponds to the

12 big bang of Einstein’s theory. Some time after this point G of contraction to expansion the solution agrees with the L 8 . (8)  result of Einstein’s general theory of relativity. There is  no need of inflation because the space of the universe is Put always flat. It is worth to mention that the theory of  4 kc4 (9) gravitation indicates that an other interpretation as contracting and expanding universe is possible. The universe where k denotes the gravitational constant, then the is non-stationary and the time dependence follows by the mixed energy-momentum tensors of the gravitational transformation of the different kinds of energy into one field, of dark energy and of matter of a perfect fluid are another whereas the total energy is conserved. This trans- given by the following expressions formation of the energies is also the reason for the ob- served redshifts at distant galaxies. This interpretation G  12 also solves the problem of velocities higher than the light iirkmlnklmn11G  Tgggggggj kl mn //j r /j / r velocity at very distant galaxies. 82  It is worth to mention that this article appeared in  arXiv (see reference [16]). 1 i    jGL  2  2. Gravitation in Flat Space-Time (10a) In this section the subsequently used covariant theory of  1 TLii  (10b) gravitation in flat space-time [9] is shortly summarized. jj16  The line-element of flat space-time is M 2 iiki2 ij Tpguupcjjkj  . (10c) ds ijdx dx (1) Here, , p and ui  denote density, pressure and where ij is a symmetric tensor. In the special case dxi where x123,,xx are the Cartesian coordinates, x4 = ct four-velocity  of matter. It holds by virtue of (6)  d and  cguu2  ij. (11) ij diag 1,1,1, 1 (2) ij the space-time metric (1) is the pseudo-Euclidean ge- Define the covariant differential operator ometry. Put  12   iklmiG Rgggjjml  /  (12)  det  ij  . (3)    /k The gravitational field is desribed by a symmetrric ij of order two in divergence form, then the field equations tensor gij . Let g be defined by for the potentials gij  can be written in the covariant kj j ik i form ggik i , ggkj j (4) iiki1 and put analogously to (3) RRTj  jk4 j (13) 2

Gg det  ij  . (5) where The proper time  is defined similarly to (1) by the GM TTTTiiii  (14) quadratic form j jjj

2 is the total energy-momentum tensor. The equations of cd2  gdxdxij. (6)  ij motion are given in covariant form by The Lagrangian of the gravitational field is given by M M kk1 l TgTik/  (15) 12 2 kl/ i G mn ik jl 1 ij kl LggggGijkl//mgggnm  //n (7)  2 where M M where the bar / denotes the covariant derivative relative ij jk i TgT (16) to the flat space-time metric (1). The Lagrangian of the j dark energy (given by the cosmological constant  ) has is the symmetric energy-momentum tensor of matter. In the form addition to the field Equations (13) and the equations of

Copyright © 2013 SciRes. JMP 22 W. PETRY motion (15) the conservation law theory of grav itation. This will be important to avoid the singularity. T k  0 (17) ik/ Under the assumption that matter and radiation do not of the total energy-momentum tensor holds. All the interact the equations of motion (15) can be solved by the Equations (13), (15) and (17) are generally covariant and use of (18) to (22). It follows the energy-momentum (10a) of the gravitational field is a hp12,0,3p ah12 . (24) tensor in contrast to that of general relativity. The field mm00 m r rr Equation (13) are formally similar to the equations of The field Equations (13) with (10) to (14) imply by  i Einstein’s general relativity theory but R j is not the (18) to (22) the two non-linear differential equations: Ricci tensor and the source of the gravitatio nal field in- 3 cludes the energy-momentum tensor of gravitation. It is d1312aa 4 1  ah 2c  mr 212 (25a) worth to mention that the field Equation (13) together d2ta 32ch with the Equation (15), respectively (17) imply the Equa-  3 tion (17), respectively (15). d11312ha 4   ah4 c mr2212 L G  . d282th  c ch 3. Isotropic Cosmological Model (25b) In this section the flat space-time theory of gravitation is Here, it holds applied to homogeneous, isotropic cosmological models. 2 The pseudo-Euclidean geometry (1) with (2) is assumed. 11aahh2  312 LahG 2 66  (26) The matter tensor is given by (10c) with c aahh2   uii 01,2,3 (18) 1 where the gravitational energy is given by L . and 16 G The proper time is: ppmr p,  mr (19) 22221 2 where th e indice s m and r denote matter and radiation cd22  a dx1  dx23  dx   ct. (27) respectively. The equ ations o f state for matter (dust) and   h radiation are The conservation of the total energy has the form: pp0,  3. (20) 3 mrr 221  a mrcLG12 c (28) By virtue of (18) the potentials are given by 162 h 2 where  is a constant of integration. The Equations (25) at,1,2,3  i j   and (26) give by the use of (28) and the initial conditions ghij  1,tij  4 (21) (23):  0, ij ha  4ct4    62  0 (29) ha21 ctt42 where all the functions depe nd only on t by virtue of  0 the homogeneity of the model. The four-vel ocity (11) has with by the use of (18) and (21) the form  i 12 1 h0 uch 0,0,0, . (22) 0031H . (30) 6 h0 For the present ti me t0  0 the following initial conditions are assumed: Integration of relation (29) yields:  312 4 2 ah00 1,0a  Hhh,0, ah 21 c t 0 t . (31) 00 (23) 0,0 mmr 00 r Equation (28) gives at present time t0  0 by the use of (23) and (24) where the dot denotes the time-d erivative, H0 is the well-known Hubble constant and h is a constan t which 2 0 1842 c 2 doesn’t appear by Einstein’s theory. The constants  84ck0000  mr    H. (32) m0 338 k  and r0 denote the present densities of matter and radiation. It is worth to mention that general relativity im- It follows from (28) by the use of (26), (29), (31), (24) plies ht1 which is not possible in flat space-time and (31) the formula

2 42 a 210ct 0 t (39)  a for all tR . Then, the differential Equation (36) has a  2 positive solution at  and relation (31) gives a positive 18c 2 12  kHmr00  0 function ht . Therefore, condition (38) is necessary 42 2  38k 21ct 0t  and sufficient for non-singular cosmological models by

virtue of (24). Then, there exists a time t1 such that c2 88 akaka632. mr00 at    0. (40) 33 3  1

(33) Put aat1  1 , then the differential Equation (36a) implies at tt : Let us introduce the density parameters: 1 236 2 rmaaa111 m K 0 (41) 88kkmr00c mr22,,   2 (34) 33H00HH30 and for all tR and define at   a1 0 . (42)

K0 mr  1 m (35) The assumption then the differential Equation (33) can be rewritten in the aa1  01 (43) form: gives by virtue of (41) 2 a  K0 1. (44) a 2 Hence, it follows by (35) H 0 K aaa23   6 2 mr0 m 42 rm  1, mK0 (45) 21ct 0t (36a) i.e. the sum of all the density parameters is a little bit bigger than one. The condition (43) is also iumportant for with the initial condition a very hot universe in the beginning because the tem- a01. (36b) perature is given by

Hence, a solution of (36) together with (31) describes Tt  T0 at (46) a homogeneous, isotropic cosmological model in flat where T is the present temperature of radiation. For space-time theory of gravitation. 0 tt 1 the universe contracts and then it expands as tt 1. Hence, there exists a bounce in the early universe. 4. Cosmology with a Bounce The differential Equation (36) is written in the case

In this section we will study the solutions of (36) with tt1  (expanding universe) in the form:

(31) and show that non-singular cosmological models 12 a H0 236 42 mrmKa0  aa with a bounce exist. actt21  Relation (32) can be rewritten by the use of the density 0 parameters (34) and the definition (35) a01  (47)

4 2 8c 0 and for tt 1 (contracting universe) in the form: 2 12m K0 . (37) HH 12 0 0 a H0 236  42 mrmKa0   aa  actt21  A necessary condition to avoid a singular solution of 0 (36) is : at 11  a. (48)

K0  0 . (38) In the special case r  0 an analytic solution of the Inequality (38) is by the use of (37) equivalent to expanding universe (47) can be given (see e.g. [12]):

at3 2112cos3 K K t  2 K 12 sin3t (49a)  00   0m  

2 where 12  ht 31 HtHt001m . (54b) 12 1 tKarctg 3 m 00Htt1 0. (49b) For tt the function at of the differential Equa- 2 1 tion (48) starts at

For the subsequent considerations compare [14]. 13 a 2 1  cos 3 a 1.8161a (55) The time t1 is given by (in the case of r 0 )    11

12 1  3 K and decreases as tt to a . Ht 1 0  m 0  ( 50a) 1 1 01 The proper time  from the beginning of the uni- 2 HA0 verse is given by with t  thtt 1d12 . (56) 12  1 K   A tg arctg 2 0 m 3 1 K The proper time  t1  , i.e. from the beginning of the 0 (50b) universe till t1 , is finite by virtue of (31), (55) and (43). 12 The proper time  t of the universe is by (56) in- 4.0338OK0 . creasing with increasing t and goes to infinity as t  Relation (49) gives two different kinds of solutions goes to infinity by virt ue of (54b). It follows for tt 1 (see [14]): the proper time 1) The denominator of (49) is positive and vanishes as 11 12 t  implying th0 ln t (57a) 3  H0 13 0 with a suitable constant  . Therefore, it holds for  t m 1  (51) 0   2H0 2 sufficiently large where expressions co ntaining K are omitted by virtue 12 0 htexp 3  H00  t   . (57b ) of (44). 2) The denominator of (49) is always positive as Hence, under the condition (51), the function at  t  implying starts by virtue of (55) from a small positive value

a a1 and decreases to a1 as tt 1 . Then, at  130 increases for all times tt and goes to infinity as m 1.  (52) 1 2H0 2 t . It is worth to mention th at at is not sym met -

ric with regard to its minimum at t1 . In both cases th e function at  is increasing after the Let us now introduce the proper time (5 6) into the dif- time t1 . In the firs t case at converges to infinity as ferential Equation (36). It follows by the use of (31) t goes to infinity whereas in the second case the func- 2 tion at converges to a finit e value as t goes to in- 1da 2 mmK0 r 2643H0   finity. Subsequently, we will only consider the interest- aad aa ing case 1). Condition (51) is by the use of (30) a condi-  a01.  (58) tion on the initial value h0 of (23).  Let us define the time t1 by Hence, this differential equation is by the use of (44)

2 and (38) for aa identical with the differential equa- 1 1   1  00 tion of general relativity describing a homogeneous, iso- H01t 3 m K0 (53) 2 HH2 00 tropic universe. The refore, all the results of general relativity are valid. But for sufficiently small at, the solu-  then, for tt1  the solution (49a) can be approximated tion is quite different from that of general relativity and it by Equation (54a) below and by the use of (31) the full has no singularity in contrast to Einstein’s th eory, i.e., solution is there is no big bang.

2 3  2 2 a3  t m Ht Ht31Ht Ht  (54a) 2 001  001m  1

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A non-expanding universe seems a more natural inter-