Revised Newtonian Formula of Gravity and Equation of Cosmology in Flat Space-Time Transformed from Schwarzschild Solution

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1 Intenational Jounal of Astonomy and Astophysis,,, Published Online Mah ( evised Newtonian Fomula of Gavity and Equation of Cosmology in Flat Spae-Time Tansfomed fom Shwazshild Solution Xiaohun Mei, Ping Yu Institute of Innovative Physis in Fuzhou, Depatment of Physis, Fuzhou Univesity, Fuzhou, China eeived Januay, ; evised Febuay 7, ; aepted Febuay 6, ABSTACT By tansfoming the geodesi equation of the Shwazshild solution of the Einstein s equation of gavity field to flat spae-time fo desiption, the evised Newtonian fomula of gavity is obtained. The fomula an also desibe the motion of objet with mass in gavity field suh as the peihelion peession of the Meuy. The spae-time singulaity in the Einstein s theoy of gavity beomes the oiginal point = in the Newtonian fomula of gavity. The singulaity poblem of gavity in uved spae-time is eliminated thooughly. When the fomula is used to desibe the expansive univese, the evised Fiedmann equation of osmology is obtained. Based on it, the high ed-shift of Ia supenova an be explained well. We do not need the hypotheses of the univese aeleating expansion and dak enegy again. It is also unneessay fo us to assume that non-bayon dak mateial is 5-6 times moe than nomal bayon mateial in the univese if they eally exist. The poblem of the univesal age an also be solved well. The theoy of gavity etuns to the taditional fom of dynami desiption and beomes nomal one. The evised equation an be taken as the foundation of moe ational osmology. Keywods: Geneal elativity; Shwazshild Meti; Cosmology; Fiedmann Equation; Newtonian Gavity; Supenova; Hubble Constant; Dak Mateial; Dak Enegy; Univese Aeleating Expansion; Univese Age. Intodution Established on the foundation of uved spae-time, Einstein s theoy of gavity is the dominate theoy at pesent. Howeve, Einstein s theoy has some diffiulties had to be oveome suh as the poblems of nomalization, singulaity and uniqueness of gavity field s enegy and so on. In addition, it is diffiulty to solve the non-linea Einstein s equation of gavity field. It is always attative to eestablish the theoy of gavity in flat spae-time without these toubles. Sine the 94 s, many people has tied and many theoies had been poposed [,]. These theoies ae onsistent with Einstein s one unde the ondition of weak fields, but ae diffeent in stong fields. Meanwhile, these theoies also have some poblems had to be oveome. The standad theoy of osmology faes many piniple diffiulties at pesent. As is poved below, the poblems oiginate fom the Fiedmann equation whih is unsuitable to desibe the high speed expansion of the univese. The eason is that two simplified and impope onditions wee used in the dedution of the Fiedmann equation. They ae the -W meti and stati enegy momentum tenso. At pesent, the -W meti is onsideed with onstant spatial uvatue. Howeve, the autho had poved that stitly based on the uvatue fomula of the iemannian geomety, when the sala fato (t) hanges with time, the -W meti has no onstant uvatue []. The ommon undestanding about the spatial uvatue of the -W meti is wong. This idea would impose geat influene on osmology. Due to this esult, many onlusions in the osmology suh as the densities of dak mateial and dak enegy should be e-estimated. It is poved futhe in this pape that the -W meti leads to the Galileo s tansfomation of light s veloity, instead of the Einstein s tansfomation. So the -W meti is not elativity meti and unsuitable to be taken as the basi spae-time famewok of moden osmology. Meanwhile, beause elative veloities exist between mateials and obseves in the expansive univese, the equation of osmology should use dynami enegy momentum tenso, athe than stati one as ommonly used in the uent osmology. In fat, E. A. Milne pointed out in 94 that the Fiedmann equation of osmology ould be dedued based on the Newtonian fomula of gavity [4]. It means that the Copyight Sies.

2 X. C. MEI ET AL. 7 Fiedmann equation is equivalent to the Newtonian theoy atually. It is only suitable fo desibing the poess of low speed expansion of the univese, but not fo the poess of high speed expansion. Howeve, it is poved in this pape that if dynami enegy momentum tenso is used, the equation of osmology would beome vey omplex, so that it an not be solved atually. The pionee of osmology must have onsideed this poblem and had to use stati enegy momentum tenso. In the ealy stage of osmology, the Fiedmann equation seemed to be appeiable beause the expansive speed obseved was low. When osmology develops to pesent level, we obseve the high speed expansion. In this ase, the Fiedmann equation beomes unsuitable fo the poblems suh as the high ed-shift of supenova. We have to find moe peise method to desibe them. It is poved in this pape that by tansfoming the geodesi equation of the Shwazshild solution of the Einstein s equation of gavity field to flat spae-time, the evised Newtonian fomula of gavity an be obtained. The fomula an well desibe the peihelion peession of the Meuy. The spae-time singulaities in the Einstein s theoy of gavity beome the point = in the evised Newtonian fomula of gavity. We have no the touble of singulaities again. When the evised fomula is used to desibe the expansive univese, we obtain the evised Fiedmann equation. Based on it, the high ed-shift of supenova an be explained well without the hypotheses of the univesal aeleating expansion and dak enegy. Many poblems inluding the univese age to be too small an also be esolved well. In this way, we an get id of the uent puzzle situation of osmology ompletely.. evised Newtonian Fomula of Gavity Based on the Shwazshild.. evised Newtonian Fomula of Gavity Aoding to geneal elativity, the Shwazshild meti (extenal solution) is ds d sin d d () Hee GM. Let and substitute () into the equation of geodeti line, we have the integals d L ds ds () Hee and L ae onstants. By anelling ds fom the fomulas, we an obtain d L () We define d (4) In whih is eigen time, t is oodinate time. Then, let, we have fom () Then, () beomes d ds (5) L d (6) d Hee L is the angula momentum of unit mass. (6) is just the onsevation fomula of angel momentum. We only disuss the motion of patiles with mass in gavitational field. By onsideing (6), we wite () as d d d t d d d (7) By onsideing (4) and (6), the fomula above an be witten as d L L d Taking the diffeential of (8) about d, we get d L L d Note that all quantities in (9) ae defined in uved spae-time. Aoding to the theoy of the non-eulidean geomety, although we an not tansfom whole meti of uved spae-time into that of flat spae-time in geneal, we an always tansfom the geodeti line desibed in uved spae-time into that in flat spae. Let, and t epesent the spae-time oodinates of flat spae- time, due to the invaiability of ds, we have ds d d d d (8) (9) () We see that the foms of thid items on the two sides of the seond equal sign of the fomula above ae ompletely the same. So we an take, and get the elation between times t and t d () by onsideing (4), we get and fom (8) L L d Substituting it into (), we get () Copyight Sies.

3 8 X. C. MEI ET AL. L L 4 Compaing with (4), we have L L d 4 () (4) Beause we have taken, all quantities on the ight side of (4) have been defined in flat spae-time. Note that in the lassial Newtonian theoy of gavity, at the dietions of e and e in plane pola oodinates system, the patial motion equations of unit mass ae individually d d (5) d d d d Substitute (6) into (5), we get o L (6) d L (7) Compaing with (9) and let t, exept the evised item in the baket of (9), we see that the foms of (9) and (7) ae ompletely simila. So we an wite (9) as the following veto equation d L m GMm d (8) Let u and onside (6), the fomula above an be tansfomed to d u GM Lu u d L (9) This fomula is the one used to desibe the peihelion peession of the Meuy in geneal elativity. In the dedution poess above, we use the equation of geodeti line (). It means that we tansfom the equation of geodeti line into the evised fomula of the Newtonian gavity, in stead of tansfoming whole uved spae-time to flat spae-time. But it is enough fo us to desibe an objet s motion in gavity field. Now let s pove that the effet of speial elativity has been taken into aount in (8). Fom (8), () and (4), we an obtain d d d d L L L L 4 () d d d L L L 4 d L L L 4 L L 4 Compaing with (4), we get () () () d d t (4) This is just the fomula of time delay in speial elativity. The esult veifies the ationality of (8). Let t t at last, we wite (8) as dp L GMm F (5) It is the evised Newtonian fomula of gavity based on geneal elativity. In the fomula, m is the stati mass of moving patile and the ente stati mass M has spheial symmety. Angle momentum makes gavity lage but speed makes it smalle. The esult is equivalent to eplae patile s stati mass with following effetive mass in the Newtonian theoy L mm (6) We an all m as the motion mass of gavity whih is elated to objet s speed and angle momentum... The Motion of Patile in Gavitational Field with Spheial Symmety Fo simpliity, we only disuss the motion of a patile moves along the adius veto dietion with L. In this ase, by onsideing (), (5) beomes dp GMm F (7) by multiplying d on both sides of (7), the potential enegy of the patile in gavitational field is U m F df m A d (8) Copyight Sies.

4 X. C. MEI ET AL. 9 The dynami enegy of patile is dp dp d T d d p p m pd m A (9) when, we have and A m. So the law of enegy onsevation of a patile in the gavitational field an be witten as m T U m m () when and, we get the lassi law of enegy onsevation in the Newtonian theoy of gavity m GMm () In the situation of L, we alulate the poblem in the weak field with. By keeping items with the odes up to, we have U m L Fd d L m A 6 So the law of enegy onsevation is () m E m () L 6 Hee E is a onstant. Now let s disuss the motion of a patile in the gavity field. Suppose that a patile falls feely along the adium dietion of gavity field, its veloity and aeleation ae individually d a (4) (5) when, we have and a. Suppose that the patile is at point when t, by the integal of (4), we get t (6) It is obvious that evey thing is nomal within the egion. The patile is monotonously aeleated by gavitation. Thee is no any singulaity in the whole spaetime. When patile is at the oiginal point, we have lim x (7) x a lim x ( x ) x x F lim lim x x x (8) It indiates that the speed of patile tends to have light s speed in vauum at point. Aeleation is also finite. So within the egion, the motion of patile with stati mass is ontinuous. Only at point, the foe ated on patiles beomes infinite. But this kind of singulaity appeas in any theoies in whih patiles ae onsideed with infinite small size, and have nothing to do with spae-time singulaity. The singulaity of the Shwazshild solution is eliminated.. The Fiedmann Equation of Cosmology Needs elativity evision.. The Fiedmann Equation is Equivalent to the Newtonian Theoy of Gavity The Fiedmann equation of osmology is based on the Einstein s equation of gavity. Beause the equation is too omplex to solve, two simplified onditions ae used. One is the -W meti and anothe is the stati enegy momentum tenso. Using them, we obtain fom the Einstein s equation of gavity 4G p (9) 4G p Hee t () is sala fato, is uvatue onstant fato, is the univese mateial density and p is the intensity of pessue. By eliminating fom (9), we obtain the Fiedmann equation 8G (4) Cosmi onstant has not been onsideed in (9) and (4). We often eithe take it as zeo, o ombine it with effetive mateial density fo onveniene. Copyight Sies.

5 X. C. MEI ET AL. Howeve, Bitish physiist E. A. Milne poved in 94 that the Fiedmann equation ould be dedued simply based on the Newtonian theoy of gavity. Though the Fiedmann equation is desibed in uved spae-time and the Newtonian theoy of gavity is desibed in flat spae-time, the esults ae the same atually when we use them to alulate patial poblems, espeially when we take uvatue onstant. Howeve, the Newtonian theoy of gavity is only suitable fo the motions with low speeds. Fo the high speed expansion of the univese, it is unsuitable. The Fiedmann equation needs elativity evision due to this fat. We now epeat Milne s dedution below. Aoding to the piniple of osmology, the univese an be onsideed as a huge sphee with unifom and isotopi mateial distibution. Aoding to the Newtonian theoy, gavity ated on a body loated at point inside the sphee is only elated to the mass ontained in the sphee with adius, having nothing to do with the mass outside the sphee. Suppose that the mass of unifom sphee to be M 4 t t, in the dietion of sphee adius, the Newtonian equation of gavity is d GMm 4Gm m (4) Fo the expansive sphee, by onsideing o-moving oodinate t in whih has nothing to do with time, (4) beomes 4G t t t (4) (4) is the same as the fist fomula of (9) when p. Beause mass is invaiable in the expansive poess, we have t t t t, hee t is the time at pesent. We have dt dt dt dt t t d t d t (4) Substituting (4) in (4) and taking the integal, we obtain (4). In this ase, integal onstant is equivalent with uvatue onstant in the -W meti. It is obvious that (4) is the diet esult of the Newtonian theoy of gavity, fo it dose not ontain any evised item of elativity. This is why the standad theoy of osmology is effetive fo same poblems, but is ineffetive fo othe poblems suh as the high ed shift of supenova. The eason is that two simplified onditions ae used, so that the Feidmann equation beomes non-elativity theoy atually. We disuss these poblems below... The -W Meti iolates the Piniple of Invaiane of Light s eloity Aoding to the piniple of osmology, the univese is unifom and isotopy. The -W meti is onsideed with the biggest spae-time symmety. Its fom is d ds t d sin d (44) In whih is uvatue fato. When, the meti beomes flat with ds t d d sin d (45) Fo light s motion, we have ds. When light moves along adius dietion, we have d d. Aoding to (45), we obtain d (46) t Fo the light s soue fixed at point, oodinate does not hange with time. But fo the light emitted by light s soue, oodinate hanges with time as desibed in (46). The veloity of spae expansion is d t t t (47) By onsideing (46) and (47), the veloity of light elative to obseve loated at the oiginal point of efeene fame is d d t t t (48) t t The fomula indiates that light s veloity is elated to the expansion speed of spae and violates the piniple of invaiane of light s speed. In fat, at the moment when light is just emitted out, (48) is the Galileo s addition ule of light s veloity. When light moves towads obseve, minus sign is taken in (48) so light s speed is less than its speed in vauum. When the light moves apat fom obseve, plus sign is taken. In this ase, light s speed exeeds its speed in vauum. Espeially, beause ineases with time, enough long time late, light s speed may geatly exeed its speed in vauum. This is not allowed in physis. As we know that the wateshed between lassial physis and moden physis is just on the invaiane piniple of light s speed. Beause the -W meti violates this piniple, it an not be used as the spae-time fame fo moden osmology whih is onsideed as the theoy of elativity. Espeially when the expansion speed of the univese is geat, huge eo will be aused. As fo the uve spae with, let ds and d d in (44), we obtain d t Copyight Sies.

6 X. C. MEI ET AL. o d t (49) On the othe hand, as we known that oodinate has no meaning of measuement in uved spae. What is meaningful is pope distane. Suppose that an obseve stays at the oiginal point of oodinate system, the definition of pope distane fo the -W meti between obseve and light s soue is [5]. d t t tl (5) l equivalent to in the flat spae. Fo il- Hee luminant mateial moving in the expansive univese, l does not hange with time. The veloity of illumi- t t l. By nant mateial elative to obseve is onsideing (49), the veloity of light emitted by illuminant mateial moves in the uved spae is d d d d t t t d t d tl t (5) So (5) still violates the piniple of invaiane of light s speed. In fat, the fou dimensional meti of flat spae-time is ds d d sin d (5) by using o-moving oodinate t in (5), we obtain t ds t ( t) d (5) t d d sin d It is ompletely diffeent fom the -W meti (44) when. The meti (5) seems to be uved but is flat essentially. Aoding to the piniple of the iemannian geomety, if we an find a method to tun a uved spae into flat, the oiginal spae is flat essentially. If we an not find suh method, the oiginal spae is a uved spae in essene. It is obvious that we an not find a tansfomation to tun (5) into (45) when t, the spatial pat of (45) an not be flat! On the othe hand, the fou dimensional meti in whih thee dimensional spae has a onstant uvatue is ds ds d (54) d sin d by using o-moving oodinate in (54), we obtain t tt d ds t t (55) d t d sin d t Let, we eah (5) athe then (45). Theefoe, if we use o-moving oodinate to desibe the expansive univese in whih the spae is flat, we should use (5), athe than (45). If we desibe the expansive univese with onstant uvatue, we should use (55), athe than (44). Anothe esult of using the -W meti in osmology is that it leads to the united univese time. In the -W meti, g indiates that we have the same time fo any spatial point in the expansive univese. This obviously violates speial elativity. Beause thee is a elative motion speed between two objets in the expansive univese, thee exists time delay between them aoding to speial elativity. It is atually the esult of the Newtonian mehanis to use the united univese time in osmology. This is anothe eason why we say that the Fiedmann equation is equivalent to the Newtonian mehanis. Howeve, it is easy to pove that if we use flat spaetime meti (5) in the Einstein s equation of gavity, the Einstein s tenso would beome zeo with. In this way, we an not desibe the gavity field of the expansive univese. Theefoe, both the -W meti and the flat spae-time meti ae unsuitable fo osmology. We should look fo othe pope methods to desibe the expansive univese... Dynami Enegy Momentum Tenso Should Be Used in Cosmology The enegy momentum tenso of ideal liquid is used in osmology with the fom T p U U pg (56) Hee U t is the fou dimensional veloity. In the standad osmology, we take U t and Ui t with T, T i. It means that we take stati enegy momentum tenso enegy in the Einstein s equation of gavity without onsideing mateial s veloity. This is an exessively simplified appoximation. In fat, thee exist elative veloities between mateials and obseves in the expansive univese. The most basi fat fo osmology is the Hubble s ed shift, whih is explained as the kinematial effet aused by elative veloities between obseve and luminous mateial. If o-moving oodinate t t, mateial s speed is t t. In fat, on the left side of the Fiedmann equation, t. How an we take t on the we have Copyight Sies.

7 X. C. MEI ET AL. ight side of equation whih ontains enegy momentum tenso? It is absolutely unjustifiable to use stati enegy momentum tenso to desibe the expansive univese. In fat, if we use stati enegy momentum tenso to desibe the expansive univese, what do we use to desibe the stati univese? This is a poblem to make us embaassing. Aoding geneal elativity, we an use abitay efeene fame to desibe the gavity field. By using ommon spheial oodinate system, the patial veloities of an objet whih moves along the adius dietion ae t t, t t. The foth dimensional veloities ae ( ). U U U U (57) To simplify disussion below, we use the -W meti and take. Fo the expansive univese with unifom distibution of mateial, dynami enegy momentum tensos ae p T p p T p T p (58) T p T p sin T g T p (59) Substituting them in the Einstein s equation of gavity g 8GT Tg (6) We get the motion equations of osmology p 8G p 8G p (6) (6) p 8G p(6) Substitute t t in the fomulas, we get p 8G p 8G p (64) (65) p 8G p (66) Take on the ight sides but not on the left sides, we obtain the Fiedmann equation. But we an not do it in this way. Beause is a onstant, we have thee ways to make (65) tenable. The fist is to let whih desibes the stati univese. By onsideing the obsevation fat of the Hubble edshift, this is impope. The seond is to take simultaneously A and 8G p B (67) Hee A and B ae onstants. Fom (67), we obtain B A and (68) p The esult violates the Hubble law too. In addition, these veloities ae inonsistent, so (68) is impossible. The thid is to get the solution fom (65) p 8G (69) Substitute (69) in (64) and (66), we have 8G (7) 8 G (7) by anelling fom two fomulas above, we obtain at last (7) 8G The equation beomes so ompliated that it is impossible to solve atually. On the othe hand, beause the t and ight hand sides of (64)-(66) ontain, if p t ae still unelated to, we should have t, by solving the equations. The esult ontadits with the oiginal definition t. In ode to mate t unelated to, and p should be elated to. In this way, the piniple of osmology an not hold again. The esult means that we will be in dilemma if dynami enegy momentum tenso is used in osmology. Pionees of osmology must have onsideed this poblem, so they had to use stati enegy momentum to establish the equation of osmology. In the ealy stage of osmology, the obseved expansion speed of the univese was low, so the simplified motion equation ould be suitable. When osmology develops to now day s level, we obseve osmi phenomena whih take plae in the high speed expansive poesses suh as the high ed shift of supenova. The simplified Fiedmann equation beomes Copyight Sies.

8 X. C. MEI ET AL. unsuitable so that many diffiulties appea in the standad osmology at pesent. This is the main eason why we have to intodue the hypothesis of the aeleating expansion of the univese, dak enegy and non-bayon dak mateial. 4. eloity, Aeleation and Initial Conditions of the Univese Expansion 4.. eloity and Aeleation of the Univese Expansion Beause (7) an not be solved patially when dynami enegy momentum tenso is onsideed, we have to look fo othe method to desibe the expansive univese. We pove below that based on the evised fomula (5), the high ed shift of supenova an be explained well. Theefoe, we do not need the hypothesis of dak enegy and the univese aeleating expansion again. In piniple, we an take the CMB as stati efeene to desibe the univese expansion. Patially, we take the eath as stati efeene fame fo onveniene. Suppose that the univese mateial is distibuted with spheial symmety and unifom density. The stati mass of sphee with adius is M 4. Similaly we 4 with adius. Aoding to the have M Newtonian theoy, gavity ated on a small objet loated at point with mass m is [6] GM m F GM m (7) The fomulas indiate that when mass m is loated outside the sphee with, the gavity ated on it is equal to that when the spheial mass is entalized at the ente of sphee. When mass m is loated inside the sphee with, the gavity ated on it is only elated to M, having nothing to do with the mass distibuted outside the adius. Suppose that the univese expands along the dietion of adius. In the poess, angle momentum L is equal to zeo. We alulate gavity between a spheial shell with adius and an objet loated at point with stati mass m and speed along adius dietion. Suppose that satisfies () appoximately, we use () to desibe objet s effetive mass. Aoding to () and (5), we have Gm 4 df d Gm 4 d (74) Hee GM. Let 8 G,, and taking the integal of (74), we get the total gavity that the expansive sphee with adius ats on an objet loated on the spheial sufae with stati mass m and speed F 4Gm d Gm Q / ln (75) Q (76) On the othe hand, aoding to speial elativity, we have d m m F (77) / Based on (76) and (77), we get the aeleation of an objet loated on spheial sufae GQ / (78) The aeleation is just elated to the mass inside the sphee, and unelated to the mass outside the sphee. We also onside (78) as the expansion speed of spheial sufae with adius. Let and using elation d d d in (78), we obtain Let x d 4 /, we have G Q d (79) 8G GM x (8) In the expansion poess of the univese, hanges while spheial adius hanges. But spheial mass M is unhanged with onstant. We have dx d d o d dx. Suppose that initial adius is ( x x ) and initial speed is, substituting the elation into (79) and take the integal. Let Qx Q x dx (8) x we get Let Q Q (8) Copyight Sies.

9 4 X. C. MEI ET AL. we have Q C C Q Q x C (8) (84) (85) Beause (8) an not be integated dietly, we need appoximate method. When x is vey small ( ), by developing (76) into the Taylo s seies in the egion of x, we obtain 5 7 Qx x x x (86) Q x x x x (87) 56 By onsideing (8), (87) beomes 56 GM Q x Q (88) Substituting the fomulas in (78) and (85), we obtain the fomula of aeleation and speed of the univese expansion GM C Q 56 GM C 4.. Initial Condition of the Univese Expansion (89) (9) In the disussion above, we assume that mateial is only ated by gavity. Howeve, patial situation is that stong, weak and eletomagneti inteations ould not be negleted in the ealy phase of the univese duing whih mateial density was geat. Even moe, some unknown inteation may exist. Aoding to the theoy of Einstein s theoy, mateial may be ompessed into infinite density by gavity. Howeve, infinite density is unimaginable. In fat, the autho had poved that due to use the impope bounday ondition of flat spae-time in the gavity theoy of uved spae-time, the uent theoy of singulaity blak hole is wong. By stit alulation based on the Einstein s equation of gavity and uved bounday ondition, singula blak hole with infinity density do not exist [7,8]. By the same easons, the fashionable idea that the univese oiginated fom infinite small point is also impossible. In ode to avoid infinite density, we assume that thee exist a etain mehanism so that mateial sphee with mass M an only be ompessed to a finite adius. In this way, the motion equation of the univese expansion should be evised as m F F (9) n Hee F is gavity and Fn is the sum of othe foes. Fo onveniene, we simplify m Fn A (9) Hee A is undetemined funtion. It oesponds to an infinite potential baie with adius on whih the spheial sufae an not be ontated futhe. Meanwhile, by the ation of Fn at the spheial sufae with adius, the poess of ontation beome expansion and the sufae moves with a positive aeleation. When, othe foes beome zeo and only gavity ats. When, it is just the so-alled big bang of the univese fom an infinite singula point. By onsideing (9), (79) beomes d 4 / The integal of (9) is G Q A d Q xq A Let Q A K (9) (94) (95) K epesents the initial ondition of the univese expansion. Fo diffeent objets loated at diffeent position now days, thei initial positions and K ae diffeent. We will disuss how to deide K late. In this way, (94) beomes Q K (96) Unde the ondition, by onsideing (86) and (88), the fomulas of veloity and aeleation an be witten as Q K 56 GM K (97) Copyight Sies.

10 X. C. MEI ET AL. 5 / GM K (98) Hee K K. If expansive veloity is geat, we should use (78) and (96) dietly. 5. ed Shifts of Cosmology and Hubble Diagam of Supenova 5.. ed Shift of Cosmology Aoding to the Dopple s fomula, when elestial body moves along adius dietion, we have elation between speed and ed shift Z (99) Suppose that obseve is loated at the oigin point of flat efeene fame, the distane between obseve and elestial body is t at moment t. In the expanding poess of the univese, elestial body moves fom to with, while the light tavels fom to obseve along opposite dietion. Suppose light s speed is invaiable in the poess, we have following elation t d t () Aoding to (96), we have d t Q K () d The eal distane between obseve and elestial body is at pesent moment t. We know the univese mateial density at time t, but do not know its value at abitay time t. By onsideing elation, we wite (8) as 8G 8G () Using () in () and taking the integal, we an obtain the elation in piniple,, f K f K () In the fomulas above,, and Z ae known though obsevations, but K ae unknown. By t, and, we an deteminate t the elation t. By onneting () and (), we an deteminate and K. () an only be alulated by numeial method though ompute. By taking G = 6.67, = y 6 m, = y 6 m and = b 6 kg/m, we have.5 by x (4) y We use x as basi vaiable to alulate y and K in whih b, Z and y ae input paametes. Aoding to this pape, we atually dedue the initial situations of the univese expansion evesely based on the pesent obsevations of ed shift and distanes. In othe wods, as long as the initial onditions of the univese expansion ae known, we an know its uent situations. 5.. The ed Shift of Ia Supenova In Figue, the uved line with m. and.7 epesents patial elation between ed shift and distane of Ia supenova at the ealy peiod of time t. Aoding to photomety measuement, the density of luminous mateial in the univese is about 8 kg/m at pesent day. Beause thee exist a geat mount of non-luminous mateial, we suppose that patial mateial is times moe than luminous mateial and let 7 kg/m. In Figue, we take mb 5.5 5logd L in whih d L is luminosity distane with unit 6 length p.9 m. But the onept of luminosity distane is unneessay in this pape fo ou disussion is based on flat spae-time. So we need to tansfom d L to eal distane. The uved line in Figue shows the elations between ed-shifts, distanes and paametes of initial ondition of Ia supenova. The vetial oodinate is the values of K. The bottom hoizontal oodinate is the value of ed-shift. On the upside, unde the line of hoizontal oodinate ae the values of distane, above the line is the values of. Fo Z = and mb 5, we 6 get. m. By the numeial alulation, we 6 obtain.9 m and K 8.9. Fo Z =.5 and mb. oesponding to = m, we obtain.9 and K.4. Fo Z =. and mb 9. oesponding to =.5 6 m, we obtain =.6 6 and K Figue. Hubble diagam fo ed shift and distane of Ia supenova (Cited fom iss A.G. et al., 998) Copyight Sies.

11 6 X. C. MEI ET AL. Figue. elations between ed-shifts, distanes and initial paametes of Ia supenova. In this way, we an explain the high ed shift of Ia supenova well. The hypotheses of dak enegy and the aeleating expansion of the univese beome unneessay. The univese began its expansion fom a finite volume, athe than fom a singulaity. 6. evised Equation of Cosmology In ode to ompae with the equations of osmology, we now tansfom (97) and (98) to the fom of the Fiedmann equation. Suppose that the univese is a unifom sphee with density t. We define t t, in whih is a paamete unelated with time. Let t epesent today s time, we have t t, and an wite (8) as GM 8G b (5) Hee b8 G. Unde the ondition, () an be witten as / 4G b / b (6) b K Similaly, let k K, we an wite (97) as k 8 G b b 56 (7) On the othe hand, the Fiedmann equation ontaining osmi onstant ae 4G p (8) k 8G (9) Hee is onsideed as a onstant enegy density oesponding to vauum and in the uent osmology. Compaing (6) and (7) with (8) and (9), we have b b () 56 b b 5/ b p () 6 5 It is obvious that afte () and () ae used, evised equations in this pape ae with the same fom with the Feidmann equation. The diffeenes ae that is not a onstant, and p ae also elated to. In ode to be onsistent with the obsevation of Ia supenova s ed shift, the uent theoy have to assume p, so that we have to think that the univese is pushed by epulsive foe and do aeleating expansion. Aoding to this pape, we always have p, so thee is no epulsive foe and aeleating expansion again. Beause the foms of diffeential equations (6) and (7) ae vey omplex, it is moe onveniene fo us to use (97) and (98) dietly to do alulations. 7. The Hubble Constant, Dak Mateial and the Univese Age 7.. The Hubble Constant Aoding to (97), we have Z 56 GM e epesent the equivalent density of the uni- Let vese We get K () e 56 () 8Ge H Z K (4) At pesent t t and, the Hubble onstant is H e 8G K (5) We see that K and, not a eal onstant even unde ondition. This is the eason why we an not deteminate the Hubble onstant peisely up to pesent days. In fat, only taking the fist and last items in (98), we obtain the esult of the Newtonian theoy H is elated to 8G K Z Z Z Z (6) Copyight Sies.

12 X. C. MEI ET AL. 7 Taking = 6 kg/m and, we get.84 m. In osmology, we geneally take H 8 65 Km s Mp. s. We get Z.89 6 and K.57 aoding to (6). In the alulation, we onside as the pesent position of luminous elestial, without onsideing its patial position to be. Fo the situation with Z = and 6. m, aoding to (6), the esult is K 7.4. The esult indiates that even though based on the Newtonian theoy of gavity, we an also explain the high ed shift of the Ia Supenova by taking diffeent K fo diffeent objets. It is also unneessay fo us to intodue the onept of dak enegy by intoduing the effet of initial onditions. Fo Z.89 and.84 m, by using (96) fo auate numeial alulation, the esult is.85 6 and K. whih is simila to that based on (6). But fo the situations of high ed shift, the diffeenes of esults ae lage. 7.. Non-Bayon Dak Mateial Aoding to the theoy of nulei synthesis in osmology, elative density of bayon is b b h, in whih b is bayon s density and h is total density of all mateial. We have elation [9] h b.7 9 (7) Take H 65 Km s Mp whih oesponds to h.65, we have h = kg/m and b.8. Patial obsevation is m., so the theoy indiates that ou univese is mainly omposed of nonbayon mateial. Howeve, aoding to this pape, by onsideing the existene of paamete K in (5), it is enough fo us to take h 7 kg/m and get h.5 and b.. We do not need the hypothesis of non-bayon dak mateial. At least, we do not need to assume that non-bayon dak mateial is 5 ~ 6 times moe than nomal bayon mateial in the univese if non-bayon dak mateial exists atually. 7.. The Age of the Univese We onside the univese as a mateial sphee with adius.5 m at initial moment, whih is about the distane between the sun and the eath. Long enough late, an obseve loated at the oiginal point of efeene fame eeives the light omitted fom a elestial 6 body on the spheial sufae with adius. kg/m and find its ed shift is Z at time t. Suppose that the mateial density of the univese is 7 at pesent, the initial density inside the 7 sphee is 5.9 kg/m, equal to the density of neuton sta. Aoding to the alulation befoe, the 6 elestial body has moved to the position.9 m at pesent moment. We onside this distane as the adius of the obsevable univese and substitute oesponding value K.89 to following fomula to alulate the time duing whih the univese expands 6 fom adius.5 m to.9 m. t d d t t (8) Q.89 The esult is t.8 billion yeas. But this value is not sensitive to small initial adius. Taking m, equal to the adius of the Milky Way galaxy, the esult is the almost same. It means that the age of the univese mainly depends on the late expansive poess. Using (8) to alulates the time duing whih the univese adius expanses fom. 6 m to.95 6 m, the esult is billion yeas, so the time duing whih the adius of the univese expanses fom.5 m to. 6 m is 7.8 billion yeas. This is just the univese age we onside at pesent. In the pesent osmology, the univese age is estimated to be about ~ 5 billion yeas, too shot to the fomation of galaxies []. The poblem does not exist aoding to this pape. 8. Conlusions By tansfoming the geodesi equation of the Shwazshild solution of the Einstein s equation into flat spaetime to desibe, the evised Newtonian fomula of gavity and the evised equation of osmology ae obtained. The singulaity poblem in the Einstein s theoy of gavity desibed in uved spae-time is eliminated thooughly. Beause using two impope and appoximate onditions, the Feidmann equation beomes the esult of the Newtonian theoy of gavity atually. It is only suitable to desibe the low speed expansive poesses of the univese, unsuitable to desibe the high speed expansion. The equation of osmology needs elativity evision. By using the evised Newtonian fomula of gavity, the evised equation of osmology is obtained. The high ed-shift of supenova an be well explained. It is unneessay fo us to intodue the hypotheses of the univese aeleating expansion and dak enegy. It is also unneessay fo us to assume that non-bayon dak mateial is 5-6 times moe than nomal bayon dak mateial if it exists atually. Many poblems existing in osmology inluding the poblem of the univese age an be esolved well. In this way, the theoy of gavity etuns to the taditional fom of dynami desiption and beomes nomal one. The evised equation an be used as the foundation of moe ational osmology. EFEENCES [] N. osen, Geneal elativity and Flat Spae, Physial Copyight Sies.

13 8 X. C. MEI ET AL. eview, ol. 57, No., 94, pp doi:./physev [] Y. J. Wang and Z. M. Tang, Theoy and Effets of Gavitation, Hunan Siene and Tehnology Publishing Company, 99, pp [] X. C. Mei, The -W Meti Has No Constant Cuvatue When Sala Fato (t) Changes with Time, Intenational Jounal of Astonomy and Astophysis, ol., No. 4,, pp doi:.46/ijaa..4 [4] E. A. Milne, A Newtonian Expanding Univese, Geneal elativity and Gavitation, ol., No. 9,, pp doi:./a: [5] S. Weibege, Gavitation and Cosmology, John Wiley and Sons, In., New Yok, 984, p. 68. [6] C. Kittel, W. D. Knight and M. A. udeman, Mehanis, Bekeley Physis Couse, ol., MGaw-Hill, New Yok, 97. [7] X. C. Mei, The Peise Inne Solutions of Gavity Field Equations of Hollow and Solid Sphees and the Theoem of Singulaity, Intenational Jounal of Astonomy and Astophysis, ol., No.,, pp doi:.46/ijaa..6 [8] X. C. Mei, Singulaities of the Gavitational Fields of Stati Thin Loop and Double Sphees, Jounal of Cosmology, ol., Item 6,. [9] Y. Q. Yu, Physial Cosmology, Beijing Univesity Publishing Company, Beijing,, pp. 4, 84. [] M. Bolte and C. J. Hogan, Conflit Ove the Age of the Univese, Natue, ol. 76, 995, pp doi:.8/7699a Copyight Sies.