The R-W Metric Has No Constant Curvature When Scalar Factor R(t) Changes with Time

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1 Intenational Jounal of Astonomy and Astophysics,,, 77-8 doi:.436/ijaa..43 Published Online Decembe ( The -W Metic Has No Constant Cuvatue When Scala Facto (t) Changes with Time Influence of Space Cuvatue on the Densities of Dak Mateial and Dak Enegy Abstact Xiaochun Mei Institute of Innovative Physics in Fuzhou, Depatment of Physics, Fuzhou Univesity, Fuzhou, China eceived Septembe 3, ; evised Octobe 6, ; accepted Novembe 3, The tue meaning of the constant in the obetson-walke metic is discussed when the scala facto () t changes with time. By stict calculation based on the iemannian geomety, it is poved that the spatial cuvatue of the -W metic is K. The esult indicates that the -W metic has no constant cuvatue when t () and is not spatial cuvatue facto. We can only conside as an adjustable paamete with in geneal situations. The esult is completely diffeent fom the cuent undestanding which is based on the pecondition that the scala facto () t is fixed. Due to this esult, many conclusions in the cuent cosmology such as the densities of dak mateial and dak enegy should be e-estimated. In this way, we may ovecome the cuent puzzling situation of cosmology thooughly. Keywods: Cosmology, Geneal elativity, -W Metic, iemannian Geomety, Space-Time Cuvatue, Dak Mateial, Dak Enegy, Hubble Constant. The -W Metic has No Constant Cuvatue when Scala Facto () t is elated to Time Standad cosmology takes the obetson-walke metic as the basic famewok of space-time. Accoding to the pinciple of cosmology, ou univese is unifom and isotopic. It is poved that the space with homogeneity and isotopy is the one with constant cuvatue []. The -W metic which is consideed to be the one with maximum symmety is d ds c dt t d sin d () Accoding to the cuent undestanding, the constant in () is consideed to be a spatial cuvatue facto. That is to say, the -W metic is with constant cuvatues. When space is flat, we have and the metic becomes ds c d t ( t) d d sin d () Accoding to the cuent undestanding, the tem of thee dimensional spaces in backet in () is flat. By multiplying a scala facto t () which does not depend on spatial coodinates, the spatial tem can still be consideed as flat. Howeve, this tuned out to be not tue. We will pove below that when t (), the -W metic has no constant spatial cuvatue. The spatial cuvatue is elated to t (), although it still does not depend on spatial coodinates. To explain this esult clealy, let s fist epeat the deduction pocedue of the -W metic. As we know in geomety that the lowe dimensional cuved space can be embedded in the highe dimensional flat space. The thee dimensional space with a constant cuvatue can be consideed as a supe-cuved suface embedded in the fou dimensional flat space. The flat space-time metic of fou dimensions can be witten as ds dxdx dxidxi dx4 (3) The condition of thee dimensional supe spheical suface in the fou dimensional space-time is xx x x x3 x4 G (4) Copyight Scies.

2 78 X. C. MEI Hee G is the adius of supe-spheical suface and constant is the cuvatue of supe-spheical suface. Fom the fomula above, we get dx dx 4 4 xidxi - x 4 xidxi / x x Substituting (5) in (3) and intoducing coodinate tansfomation x sincos, x sinsin, and x3 cos, we get thee dimensional supe-cuved suface metic with a constant cuvatue d ds d sin d When co-moving coodinate () t is used and time is fixed with t t and t constant, (6) can be witten as d ds t d sin d ( t) (7) d t d sin d Hee t is consideed as new space cuvatue. When time is not fixed and t () constant, accoding the cuent undestanding, we can let and extend (7) into d ds t d sin d (8) On the othe hand, the metic of fou dimensional space-time in which the thee dimensional space has a constant is d ds c dt d sin d We think that it possible to intoduce united cosmic time in the expanding univese with homogeneity and isotopy. So at last, the fou dimensional metic of spacetime can be witten as ds c dt ds c dt t d () d sin d This is the standad pocedue to deive the -W metic. It is obvious that the deduction is not stict. We only pove that the metic (6) has a constant cuvatue. When t ()constant, we have not poved that the metics (8) and (9) still have constant cuvatues. In fact, when t () constant, by using () t in (6), we i i (5) (6) (9) get tdt ttdtd ds ( t) ( t) () d t d sin d ( t) Similaly, substituting () t in (9), we get t tt dt d ds c d t ( t) ( t) () d t d sin d ( t) We can not get (8) and () fom Equation (6) and (9). That is to say, when t () o t () constant, we have not poved that the constant is still the spatial cuvatue facto of the -W metic! Howeve, is geneally consideed as the cuvatue constant of the -W metic and the spatial pat of the metic is consideed being flat when at pesent. This is because that the spatial pat of thee dimension metic in the backet of () is flat. When it multiplies a scala facto which does not depend on spatial coodinates, the space can be still consideed flat. On the othe hand, when t () constant, we have metic tenso g constant, () is obviously not the metic of flat space. In fact, the metic of the fou dimensional flat space-time is ds c dt d d sin d (3) By using co-moving coodinate t t ds c dt t t d c t d d sin d t, we get dt (4) This fom is completely diffeent fom the -W metic () when. The space-time of metic (4) seems to be cuved, but it is flat essentially. Accoding to the pinciple of the iemannian geomety, if we can find a tansfomation to tun a cuved space into flat, then the oiginal space is flat essentially. If we can not find such a tansfomation, the oiginal space is a eally cuved space. Now, because we can find a tansfomation to tun (4) into (3), the metic (4) is flat essentially. Because we can not find a tansfomation to tun () into (3) when t (), the spatial pat of () can not be flat! On the othe hand, in ode to make the spatial distance of the -W metic be positive, we should have ()d t (5) Copyight Scies.

3 X. C. MEI 79 So we have, i.e., o. 6 Fo the obsevable univese today, we have m. 5 So fo, we should have. Theefoe, if is a positive numbe, it should be vey small. If the univese is infinite, we should have. In the cuent cosmology, we take and as well as to epesent the diffeent univeses with negative and positive cuvatues as well as flat individually. Howeve, is impope to descibe the univese, fo we have to take to ensue the -W metic meaningful. Unfotunately, this poblem is completely neglected in the cuent theoy.. The Spatial Cuvatue of the -W Metic Now let s pecisely calculate the cuvatues of () unde the condition t () constant. As we know that cuvatue has concete definition in mathematics. We should judge the flatness of space by stict calculation, not by appaent estimation. In the iemannian geomety, the iemannian cuvatue at a cetain point of N dimensional space is defined as [] K p q p p g g g g p q p p (6) K is elated to the selections of diection vectos p and q at each point in space. We define the covaiant tenso g in which and ae calculated by following fomulas g g g g (7) x x x g g g x x x, g x x g g g x x x x g,, (8) (9) () If cuvatues ae the same at all points in space, we have K constant with K g g g g () Futhe, if space is flat, the iemannian-chistoffel tensos becomes zeo eveywhee with o. Now we calculate the space-time cuvatues of the -W metic. The non-zeo Chistoffel signs ae 33 sin k 3 3 ( ) ( ) () 3 33 ( ) sin sincos 3 ctg The non-zeo cuvatue tensos of the -W metic ae 33 sin (3) sin 33 4 sin 33 The coesponding cuvatues ae K K K3 (4) K K3 K3 Hee K i ae the cuvatue of space-time cossing pat, and K kl ( kl, ) ae the cuvatues of pue space. Fo Ki, we take o constant. In this case, Kkl becomes a constant. This is just the esult that physicists think of the -W metic at pesent. In ode to make puely spatial cuvatue Kkl, we have two chooses. The fist is to set and simultaneously, epesenting static space-time. Anothe is to set t at cetain moment t t. In this case, we have Kkl but still have K i. When but t (), we have Kkl. The esult is completely diffeent fom the cuent undestanding. Now let s estimate the cuvatue s magnitudes of the -W metic. Taking in (4), we get K K3 K3 (5) In cosmology, we define the Hubble constant H () t () t () t. At pesent moment t, we have 8 36 H t H s. So we have K kl 4, i.e., the spatial cuvatue of the expansive univese is vey small. On the othe hand, fom the equation of cosmology, 36 we have t t 4πG t 3 H s, 36 so we have Ki. The cuvatue of cossing pat of space and tine is also vey small. Copyight Scies.

4 8 X. C. MEI Next, how do we ecognize the spatial cuvatue of the metic ()? In the iemannian geomety, the intuitive pictue is that when a vecto moves along a loop and then etuns to its stating point, if it can supeimpose with oiginal vecto, the space is consideed flat. If it can not, the space is cuved. Theefoe, if the space expands with time in the manne of the metic (), a vecto will not supeimpose with the oiginal vecto when it moves along a loop and etuns to the oiginal point. We should ecognize the space cuvatue of the metic () in this way. Howeve, the esult of WMAP showed that the space of ou univese seems nealy flat [3,4]. Accoding to (4), the diffeence between constant and t should be vey mall at pesent moment. If we think that the constant has same magnitude with t, the spatial 36 cuvatue of the expansive univese may be about. 3 As we know that the pecision of WMAP is about, so the expeiment is unable to find such small cuvatue. 3. Influence on the Densities of Dak Mateial and Dak Enegy The esult above may cause geat influence on cosmology. By taking into account of cosmic constant, the Fiedmann equation of cosmology is 8πG (6) 3 By defining the Hubble constant H t () t () and the effective mateial density of the univese () t e () t 3 8π G, we wite (6) as 3 ( t) 3 3 H ( t) 3 e() t (7) 8π G () t 8π G () t 8πG 8π G () t, H t Let e t H, t and define citical density c 3H 8πG, (7) becomes 3 c (8) 8πG The fomula is used to estimate the mateial density of the univese. If epesents space cuvatue facto, we have fo the flat univese so that the cuent mateial density is equal to the citical density. Defining c, we have c fo the univese today. Howeve, obsevat ions indicate that we only have.4 fo nomal mateial, geatly less than. Theefoe, non-bayon dak mateial and dak enegy have to be intoduced to fill the univese. Howeve, if is adjustable paamete, in stead of cuvatue facto, fo the nealy flat univese and pactically obseved, we can chose pope to satisfy (8). We take the Hubble constant H 65Km s Mpc 7 3 and obtain c 7.9 kg m at pesent. By pac- 8 3 tical measuements, we have kg m fo luminous mateial [5-7]. Suppose that pactical density of bayon mateial in the univese is about times moe than that of luminous mateial, i.e., suppose kg m, by taking 3.3, we can make (8) to be satisfied. In this way, it becomes unnecessay fo us to assume that the non-bayon mateial is about 6 times moe than bayon mateial if they eally exist. As fo dak enegy, it becomes suplus. We can make theoy and obsevations consistent by adjusting the value of constant without the hypotheses of dak enegy. At last, if, fom (7) we have 8π Ge() t H () t (9) 3 ( t) It means that the Hubble constant is not only elated to mateial density, but also elative to scala facto. At pesent moment, we have t. Accoding to (8), we have H 8πG 8πGc (3) 3 3 By the stict calculation based on the iemannian ge- omety, the pape poves that when the scala facto t () is elated to time, the spatial cuvatue of the -W metic is K, in stead of. That is to say, the -W metic has no constant space cuvatue in geneal situations. Theefoe, if the Fiedmann equation is used to descibe the expansionay univese, even space is flat, we can only conside as an adjustable and non-zeo paamete. The esult may geatly impact cosmology. We need to e-estimate the values of dak mateial and dak enegy densities. Accoding to the cuent estimation in cosmology, nomal bayon mateial only makes up 4%, non-bayon dak mateial makes up 6% and dak enegy makes up 7% of the univesal mateial. Howeve, nonbayon dak mateial with such huge amount of quantity can not been found up to pesent days, though physicists have stuggled to find it fo decades. The situation is even moe puzzling fo dak enegy. In fact, the concept of dak enegy is simila to the concept of ethe in the nineteen centuy. Classical physics Theefoe, the Hubble constant is the same. In fact, if is an adjustable paamete in stead of a cuvatue facto, many conclusions of the cuent cosmology should be e-evaluated. These poblems will be discussed late. 4. Conclusions Copyight Scies.

5 X. C. MEI 8 could not explain the light s popagation in the univesal vacuum so that the concept of ethe as a medium had to be intoduced. One pupose that Einstein putted fowad special elativity was to eliminate the concept of ethe fom physics. Time goes into the twenty-fist centuy, the histoy is epeating. Because physicists can not explain the high ed-shift of supenovae, the concept of dak enegy has to be intoduced. By the coect undestanding of the constant in the -W metic and the Fiedmann equation, we no longe need the concept of dak enegy again. Meanwhile, we may not need to assume that non-bayon dak mateial is 6 times moe than nomal bayon mateial, if non-bayon dak mateial exists actually. In this way, we may ovecome the cuent puzzling situation of cosmology thooughly. 5. efeence [] S. Weinbeg, Gavitation and Cosmology, Science Pub- lishing Company, 984, p. 44. [] S. Zhiming, Tenso in Physics, Beijing Nomal Univesity Publishing Company, Beijing, 985, p.8. [3] P. de Benadis, et al., A High Spatial esolution Analysis of the MAXIMA- Cosmic Micowave Backgound Anisotopy Data, Natue, Vol. 44,, pp doi:.38/3535 [4] C. L. Meiment, et al., Wilkinson Micowave Anisotopy Pobe (WMAP), Obsevations: Tempeatue-Polaization Coelation, Astophysical Jounal Supplement Seies, Vol. 48, No., 3, pp [5] J. H. Oot, In La Stucctue et Evolution de Unives, Institute Intenational de Physique Solvay,. Stoops, Bussles, 958, p. 58. [6]. G. Calbeg, et al., Galaxy Cluste Viial Masses and Omega, Astophysical Jounal, 996, Vol. 46, pp doi:.86/775 [7] D. N. Schamm, M. S. Fune, Big-Bang Nucleosynthesis Entes the Pecision Ea, eviews of Moden Physics, Vol. 7, No., 998, pp doi:.3/evmodphys.7.33 Appendix: The Cuvatues of Two Dimensional Cuved Sufaces in the Thee Dimensional Flat Space Because the geometical figue of thee dimensional suppe spheical suface is abstact and not visualized easily, in ode to have a diect visualization, we discuss two dimensional cuved sufaces in thee dimensional flat spaces futhe in this section. Simila to (3) and (4), the metic of thee-dimensional flat space and the condition of spheical suface ae individually d dx dx dx3 (3) xxxxx3x3 G Hee constant is the cuvatue and G is the adius of two dimensional spheical suface. By using column coodinates x sin, x cos and x 3 z, (3) becomes d dz d d z G (3) Hee x x is the adius of a cicle on the plane z constant. Accoding to the same calculation poced ue, we have dz d z - dz d d d (d) d (33) (34) The fom of (34) is simila to (6). We see that the metic of (3) is flat, but the metic of (34) becomes cuved. The eason is that the tansfomation (33) is non-linea. That is to say, non-linea tansfomations may change spatial cuvatues. Similaly, by intoducing nonlinea tansfomation ( z) and let ( z) d( z) dz in (34), we obtain ( z) d z ( z) ( z) dzd d ( z) ( z) (35) d ( z) d ( z) Let ds dz d dx dx dz and substitute (35) in (36), we get ( z) ( z) ( z) dzd ds d z ( z) ( z) d ( z) d ( z) (36) (37) Copyight Scies.

6 8 X. C. MEI The fom of (37) is simila to that of (). The equations coesponding to (8) and (9) ae d d ( ) d z d ds d z ( z) d (38) (39) It is obvious that the cuvatue of (39) can not be constant when z constant. In fact, by consideing ( z), the spheical suface equation (3) becomes ( z) z (4) Because new vaiables ae and z, (4) is not the equation of spheical suface and constant is no longe cuvatue. When z z constant, accoding to (4), we have z z K (4) Only in this case, constant K epesent the cuvatue of cicle with adius on the plane z constant. Similaly, by using () t in suppe spheical suface equation (4), we obtain () t c t (4) Because new vaiables ae and t, (4) is not the equation of thee dimensional suppe spheical suface and the constant is no longe cuvatue again. When t t constant, we have c t t K (43) Only in this case, constant K can be consideed as the cuvatue of suppe spheical suface with adius. We calculate the cuvatue tenso of the metic (38). The non-zeo metic tensos of (38) ae g g (44) g g The non-zeo Chistoffel signs ae k (45) Accoding to (), the only non-zeo cuvatue tenso is (46) So accoding to the fomula (6), the iemannian cuvatue of (38) is K p q p q gg gg p q p q gg (47) That is to say, the cuvatue of (38) is ( z), athe then. In fact when z ( ) constant, let ( z) in (38), we get (34). Next, we calculate the cuvatue of (39). We conside z as x. The metic tensos of (39) ae g g g g g g The non-zeo Chistoffel signs ae k ( ) The non-zeo cuvatue tensos ae Accoding to (6), we have K K gg gg K g g (48) (49) (5) (5) Because K contains, the esult is diffeent fom (47). It means that the spatial cuvatues of (38) and (39) ae diffeent. It implies that the dimensions of space can change cuvatue, despite that the metic fom (39) of two dimensional cuved suface is completely the same as (38). In fact, as we know that the metic ds d d sin d is flat with zeo cuvatue, but d d sin d is cuved with cuvatue. elated to the adius of spheical suface, the cuvatue is not a constant. Copyight Scies.