Cosmology Should Directly Use the Doppler s Formula to Calculate the Red Shift of Ia Supernova

Transcription

1 Inernaional Journal of Asronomy and Asrophysis,,, -7 hp://dx.doi.org/.46/ijaa..6 Published Online Sepember (hp:// Cosmology Should Direly Use he Doppler s Formula o Calulae he Red Shif of Ia Supernova The Proofs Tha Meri Red Shif Is Inappliable and Dark Energy Does No Exis Xiaohun Mei, Ping Yu Insiue of Innovaive Physis, Fuzhou, China ywlyjs@yeah.ne, Yupingpingyu@yahoo.om Reeived Marh 8, ; revised April 9, ; aeped May 7, Copyrigh Xiaohun Mei, Ping Yu. This is an open aess arile disribued under he Creaive Commons Aribuion Liense, whih permis unresried use, disribuion, and reproduion in any medium, provided he original work is properly ied. ABSTRACT The Doppler formula should be used direly o alulae red shif of Cosmology. The firs is graviy, he seond is he Doppler s effe and he hird is he Compon saering. The red shif of osmology is onsidered o be aused by he reeding moions of elesial bodies, of whih essene is he Doppler s effe. However, he basi formula used o alulae he relaionship beween red shif and disane for Ia supernova in osmology is z R R whih is based on he R-W meri and relaed o he salar faor R. This is differen from he Doppler formula whih is relaed o speed faor R. Beause he R-W meri is only a mahemaial sruure of spae, he meri red shif is no an independen law of physis, his inonsisene is no allowed in physis. I is proved srily in his paper ha he formula of meri red shif is only he resul of he firs order approximaion. If higher order approximaions are onsidered, we an obain a resri ondiion R onsan. I indiaes ha if he formula of meri red shif holds, i an only be suiable o desribe he spaial uniform expansion, unsuiable for he praial universal proess wih aeleraion. The furher sudy reveals ha he R-W meri violaes he invariabiliy priniple of ligh s speed in vauum. The ime delay aused by relaive veloiy beween ligh s soure and observer is negleed. So he R-W meri violaes he speial relaiviy and is no suiable o be used as he basi spae-ime frame in osmology. The formula used o alulae he relaionship beween red shif and disane of Ia supernova is wrong and subsequenly he dedued resuls abou dark energy and he aeleraing universe are inredible. Cosmology should use direly he Doppler formula o alulae red shif. I is proved ha based on he Doppler s formula and by he mehod of numerial alulaion, he relaion of red shif and disane of Ia supernova an be explained well. The hypoheses of dark energy and he aeleraing expansion of he universe are ompleely unneessary in osmology. Keywords: Cosmology; Doppler Formula; R-W Meri; Hubble Law; Supernova; Dark Energy; Dark Maerial. Inroduion As we known ha here are mainly hree mehanisms leading o he red shif of sperum. The firs is graviy whih relaes o mass and graviy onsan. The seond is Doppler s effe whih relaes o veloiy. The hird is he Compon saering whih is relaed o he energy ransformaion of phoon. Aording o he Hubble law, he sperum red shif of exragalai nebula was proporional o he disane beween observer and luminous elesial body. The red shif of osmology is onsidered o be he Doppler s effe. In 998, osmi observers found ha he red shif of Ia supernova deviaed from he linear relaion of he Hubble law. By fiing he observaion values wih sandard heory of osmology, osmolo- giss delared ha abou 6% maerial in he universe was dark maerial and abou 7% was dark energy. The universe seems o do aeleraing expansion [,]. The red shif of osmology is onsidered o be aused by he reeding moions of elesial bodies, so is essene is he Doppler s effe relaed o speed. We should use Doppler s formula o alulae he red shif. However, i is srange ha he basi formula used o alulae he relaion of red shif and disane of Ia supernova is ompleely differen from he Doppler formula. The basi formula used in osmology is [] z v R v R () Copyrigh SiRes.

2 4 X. C. MEI, P. YU The formula is based on he R-W meri in whih is he frequeny of emied ligh a ime, is he frequeny of reeived ligh a ime. The formula is relaed o salar faor R. Bu he Doppler s formula is relaed o speed faor R. I is proved in his paper ha hey an no be onsisen. This inonsisene is no allowed in physis. I is proved srily ha Formula () is only he resul of firs order approximaion. When higher order approximaions are aken ino aoun, he resriion ondiion R onsan will be obained. Therefore, Formula () is only suiable o desribe he proess of spaial uniform expansion; i s unsuiable for he praial proess of he universal expansion wih aeleraion. More srily, if we do no onsider approximaion, boh Formula () and he resriion relaions do no exis. Tha is o say, we have no meri red shif aually. In fa, he W-R meri is only a mahemaial sruure of spae-ime, raher han an independen law of physis. I is inrodued o simplify he Einsein s equaion of graviy field o obain he Friedmann equaion of osmology. I is no absoluely neessary o use he R-W meri in osmology. Using normal oordinaes direly, we an also esablish he equaion of osmology based on he Einsein s equaion of graviy field. In his ase, here is no he meri red shif formula (). In his sense, he red shif of meri is no independen in physis. If i is orre, i anno be onradied wih he Doppler s and graviy red shifs. The furher analysis indiaes ha he R-W meri violaes he invariabiliy priniple of ligh s speed in vauum when i is used o desribe he uniform moion of ligh s soure. The ime onraion beween observer and moving ligh s soure is also negleed when he R-W meri is use o desribe he spaial expansion. So he R-W meri is no one of speial relaiviy and is unsuiable o be used as he basi spae-ime frame in osmology. Espeially, i is no suiable o be used o desribe he high red shif of supernova in whih he high speed expansion of he universe is involved. In summary, he formula used o alulae he relaion of red shif and disane of Ia supernova in osmology has essenial misake. Therefore, dark energy and he aeleraing universe are inredible. The Doppler formula should be used direly o alulae red shif of Cosmology. I is proved by he mehod of numerial alulaion ha if he Doppler s formula and he (revised) Newonian formula of graviy are used, he high red shif of Ia supernova an be explained well. We do no need he hypohesis of dark energy and aeleraing universe again. Le s disuss he problem in he meri red shif formula. The inonsisene beween he meri red shif and he Doppler s red shif will be disussed in appendix.. Unavailabiliy of Meri Red Shif Formula.. The Resriion Condiion R Consan for he Meri Red Shif Formula Sandard osmology uses he R-W meri o desribe he universal spae-ime wih form ds d R () r d r sin d r Here R is salar faor and is onsidered as he faor of spaial urvaure. The o-moving oordinae is used in osmology. The spae is onsidered fla when. When () is used o desribe he expansive universe, luminous elesial bodies are onsidered o be fixed on he oordinae r,, whih do no hanged wih ime. Le s repea he proess o obain Formula () based on he R-W meri. Suppose ha ligh moves along he radius direion wih ds, we ge from () d () R r Ligh soure is fixed a poin r and r does no hange wih ime. Bu o ligh s moion, r hanges wih ime. Suppose ha phoon s oordinae is r a momen and phoon arrives a he original poin r a momen. The inegral of () is [] d sin nr R r r sin r k (4) r k sinh r k The negaive sign indiaes ha ligh moves along he direion o derease r. Suppose ha a ligh wave is emied during he period of ime from o wih period. Observer reeives he ligh during he period of ime from o wih period. Aording o he urren undersanding, beause and are deided by he same r, we have d R (5) R d Beause and are small, aording o presen heory, we obain from (5) (6) R R Based on (6) and he definiion of red shif, we obain Copyrigh SiRes.

3 X. C. MEI, P. YU 5 (). By onsidering () and he Friedmann equaion of osmology, as desribed in Seion, we an obain (). We now prove ha (6) is he resul of he firs order approximaion. If higher order approximaions are onsidered, we an obain R ( ) R ( ) or R () onsan. So he formula () is only suiable o he uniformly expansive universe, unsuiable o desribe he universal proess wih aeleraion. Le f (7) R and ake he inegral of (5), we have f f f f (8) By developing (8) ino he Taylor series, we obain f f f f f! f f f (9)! f f!! Beause and are very small, if le he iems wih same orders are equal o eah oher, we have f f () f f () f f () f f () 4 4 By onsidering (7), () is jus (6). So he meri red shif formula () is aually he approximaion of firs order. () an be wrien as R R (4) R R Subsiuing (6) in (4), we obain R R (5) Beause and are arbirary, (5) indiaes ha R onsan, i.e., he expansion speed does no depend on ime and we have R R R. From (), we have R R R R (6) R R R R Subsiuing (6) and (5) in (6), we ge (7) R R I is sill (6). From (), we ge R 6R R 6R 4 4 R R R (8) R 6R R 6R 4 4 R R R Beause of R, we have R. By onsidering (5), we sill obain (6). I an be known ha no maer wha high orders are onsidered, we always obain hese wo relaions. So he meri an only be wrien as R a b (9) Tha is o say, if (6) holds, i an only be used o desribe he uniform expansion of spae. Beause graviy exiss in he universe, Formula () an no be used o desribe he praial proess of he universal expansion wih aeleraion. More srily, we only have Formula (9). Formulas ()- () do no exis in mahemais aually. I means ha i is impossible for us o have a new red shif mehanism based on he R-W meri, whih is differen from he Doppler s and graviy red shifs. The reason is ha he R-W meri is only a mahemaial sruure of spaeime, raher an independen physial law... The Problem of he R-W Meri Firsly, onsan in () is onsidered as urvaure faor and he meri is onsidered o desribe fla spaeime when. This idea has been proved wrong ha when salar faor R is relaed o ime [4]. Tha is o say ha he R-W meri has no onsan urvaure in general siuaions. By using he formula of he Riemannian geomery and making sri alulaion, he spaeime urvaures of he R-W meri aually are R K K K R () R K K K R Here K j is he urvaure of spae-ime losing par and K ij is he urvaure of pure spaial par. This resul is ompleely differen from he urren undersanding. Therefore, onsan is no he faor of spaial urvaure. Insead, i is a erain adjusable parameer. The R-W meri does no represen he fla spae-ime when. In fa, we an use more simple mehod o prove he resul above. As we know in geomery, he priniple o judge an arbirary meri o be fla or no is ha if we an find a oordinae o ransform he meri o one wih fla spae-ime, he original meri is fla essenially. If we anno, he original meri is urved in essene. By using ommon oordinae sysem, he four dimensional Copyrigh SiRes.

4 6 X. C. MEI, P. YU meri of fla spae-ime is ds d r d r sin d () By using he o-moving oordinae or he oordinae ransformaion r R r in (), we obain R r ds d R R rd R r d r sin d However, le in (), we obain () ds d R r d r sin d () () is differen from () unless R. Beause we an obain () by using r R r in (), so he spae-ime desribed by () is fla in essenial. However, i is easy o see ha when R onsan we an no find a oordinae ransform o hange () ino (), so () an no be he meri of fla spae-ime. We an prove generally ha if () desribed ligh s moion in fla spae, i would violae he invariabiliy priniple of ligh s speed in vauum. For he ligh soure fixed on he referene fame, is oordinae r does no hange wih ime. Bu for he moion of ligh, he oordinae r hanges wih ime. Suppose ha ligh moves along he direion of radius, we have ds and d d. Aording o (), we have (4) d R The veloiy of ligh relaive o he observer resed a he original poin of oordinae sysem is d V r R R d d d R r V (5) (5) indiaes ha ligh s veloiy is relaed o he veloiy of speial expansion. Suppose ha spae expands uniformly, le R a b, aording o (5), we have V br. A he momen when ligh is emied ou, (5) is jus he Galileo s addiion rule of ligh s veloiy. When ligh moves owards observer, minus sign is aken and ligh s speed is less han is speed in vauum. When he ligh moves apar from observer, plus sign is aken and ligh s speed is grea han is speed in vauum. Espeially, beause r inreases wih ime, enough long ime laer, ligh s speed may grealy exeed is speed in vauum. This is no allowed in physis, so he R-W meri is no he meri of speial relaiviy. We an use wo simple examples o show ha if () desribe he meri of fla spae, i an no desribe ligh s moion. Suppose ha ligh s soure is loaed a poin r and emis a bundle of ligh a ime. This ligh arrives a observer loaed a posiion r a ime. Suppose ha spae expands in a uniform speed whih orresponds o he uniform moion of ligh s soure. Le R a b and ake he inegral of (4), we obain b r r ln (6) b a 8 Le r, a and b, we ge he speed 8 of ligh s soure V br m/s whih is near ligh s speed. Suppose ha observer is a he original poin of oordinae sysem wih r, aording o 8 7 (6), we have ln and 5.7 s. This resul is obviously inorre. Aording o speial relaiviy, ligh s speed does no depend on he speed of ligh s soure. Beause he iniial disane beween observer and ligh s soure is m, he ime ha ligh arrives 7 a observer should be. s. The error is as grea as 58%. In fa, harged pariles move a near ligh s speed in high energy aeleraors. And experimens show ha radiaion lighs an no propagae in his way. Suppose ha he aeleraion of spaial expansion is a onsan and le R ab g. Take r r when. The inegral of (4) is r r b ag (7) gb b ag b b ag ln ln gb b ag b b ag Taking a, b and g. R and, we have / ( b ag).6. If 8.6, we have R The speed of ligh s soure is.45 V bg r r. Meanwhile, we have g b b ag and b b ag.6. The firs iem in (7) beomes minus infinie and he seond iem is meaningless, so (7) an no ye desribe he moion of ligh in his ase. Therefore, in general, he R-W meri is unsuiable o desribe ligh s moion. This problem does no exis in (). Suppose ha ligh moves along he direion of radius, we have R r d R R rd R R r d R R By onsidering (9), ligh s veloiy is (8) (9) Copyrigh SiRes.

5 X. C. MEI, P. YU 7 dr r V R r R () d d d I indiaes ha ligh s speed is unhanged in vauum, so () is he meri of speial relaiviy. We have g in (), so i is onsidered o have a uniform ime for whole spae. However, beause of spaial expansion, he lok fixed a posiion r has a speed relaive o observer who is resed a he original poin of oordinae sysem. Aording o speial relaiviy, here is ime delay beween hem, bu () an no desribe his relaion. Aording o (), we have R r V d d d () This is jus he delay of speial relaiviy, so () is he meri of relaiviy in fla spae-ime, bu () is no. Similarly, i is proved in mahemais ha he four dimensional meri in whih hree dimensional spae has onsan urvaure is ds d r d r sin d r By using o-moving oordinae in (), we obain R r R R rd ds d R r R r R r d r sin d R r R R r () () When, () beomes (), so () is orre meri whih saisfies speial relaiviy wih speial urvaure. We have g, g and g R in (). Is form is muh more ompliaed han he R-W meri (). When ligh moves along he direion of radius wih ds, we have R r R R rd d R r R r (4) We obain R R r R r d R R (5) If we use (9) and (5) o alulae he red shif of osmology, i.e., also so alled meri red shif, he resuls are erainly relaed o veloiy. However, we an no separae variables in (9) and (5), so we an no wrie hem in he simple form of () and an no ge he same formulas similar o (). The R-W meri violaes he invariabiliy priniple of ligh s speed in vauum and negles he ime delay beween observers and moving ligh s soure aused by relaive moion. So he red shif formula of meri based on he R-W meri an no be orre. I is improper o use he meri as he basi frame of spae-ime in osmology. We should use he Doppler s formula o alulae he red shif direly... Oher Problems in he Formula for Calulaing he Red Shif of Ia Supernova Based on () in osmology, he following formula is obained o alulae he relaion of red shif and disane of Ia supernova [5] Hd z sin n z dz z z z z L k k m (6) Here d L is luminosiy disane. In ligh of he R-W meri, we have dl zr zrr. By fiing wih praial observaions, osmologiss dedued ha dark energy was abou 7% and non-baryon dark maerial was abou 6% of he universal maerial. An aeleraing universe is assumed. We now disuss he problems exising in (6). The Friedmann equaion of osmology is R 8πG m R R (7) Here m is he densiy of normal maerial, is he densiy orresponding o osmi onsan. A presen ime, we have and H H, he Friedmann equaion of osmology an be wrien as H (8) 8πG m 8πGR Defining riial densiy as H (9) 8πG Le m m,, m, we wrie (8) as (4) RH Le, we have a R R a a presen ime. Beause is a onsan, we an wrie he Friedmann equaion as Copyrigh SiRes.

6 8 X. C. MEI, P. YU 8πG 8πG a a H (4) m R By onsidering R (4) an be wrien as R or a (4) da m H m a d a (4) We have da d a or d da a, so (4) an be wrien as d a sin nr R aa z The upper limi of he inegral is a limi is a z (44) and lower. By inruding so-alled energy effeive urvaure densiy [5] k (45) 8πGR We have k k and k k. We wrie (4) as (46) RH k Aording o Doumen [5], (46) is wrien as R H k By inroduing ransformaion a z (47) in (44) we obained (6). However, (47) is obviously wrong. (46) onains onsan, bu (47) does no. Aording o (46), when we have k, so R is limied. Bu aording o (47), when k we have R. I means ha R is infinie a presen ime in fla spae. Bu his is ompleely impossible. Aording o (46), orre resul should be R k (48) H In fa, i is unneessary for us o inrodue relaion k ar. Wheher or no spae is fla depends on sin nr r, sin nr sin r or sin nr sinh r. For fla spae, by aking in (4) and subsiuing i in (44), and by onsidering m and sin nr r, we ge z dz HRr (49) z z m Beause (47) an no hold, (6) an only be wrien as sin dl R z n RH z dz z z m z z (5) When spae is fla, we have, m, and sin nr r. So (5) beomes z dz HRr z zzz (5) Comparing wih (49), (5) has an addiional iem onaining in he radial sign of inegraed funion. The reason is ha when, he righ side of (4) has only wo iems wih m da m H a d a (5) Bu (6) is alulaed wihou onsidering his faor, so ha addiional hree iems are added. If spae is urved for, we sill have m and an obain dl R zsin Similarly, for, we have dl R zsinh z dz (5) RH z m z dz (54) m RH z (5) and (54) are also differen from (6), so (6) is wrong. Noe ha his is a misake of mahemais, and has nohing o do wih physis. So we an no use i o alulae he relaion of red shif and disane of Ia supernova. More essenial, if R onsan, we have a onsan. Subsiuing i in (44), we an no obain (6). In his ase, le a A/ R, subsiue i in (44), for, we obain Ar ln( z) wihou praial signifiane. I is proved below ha if he Doppler s formula is used for alulaion, he hypoheses of dark energy and he aeleraed universe will beome unneessary.. Using he Doppler s Formula o Calulae he Red Shif of Ia Supernova.. The Friedmann Equaion Needs Relaiviy Revision Sandard osmology is based on he priniple of osmology. The priniple delares ha he disribuion of universal maerial is uniform and isoropous. So he ma- Copyrigh SiRes.

7 X. C. MEI, P. YU 9 erial densiy is only relaed o ime and unrelaed o spaial oordinaes wih he form. To mah wih he priniple, he R-W meri is used in osmology whih is onsidered wih bigges symmery of spae-ime. Sandard osmology uses he Friedmann equaion as basi equaion. The Friedmann equaion is based on he Einsein s equaion of graviy field. Bu wo simplified ondiions are used. One is he R-W meri and anoher is sai energy momenum ensors. However, Briish physiis E. A. Milne proved in 94 ha he Friedmann equaion ould be dedued simply based on he Newonian heory of graviy when osmi onsan is no onsidered [6]. This resul indiaes ha he Friedmann equaion is aually equivalen o he Newonian heory of graviy whih is only suiable o desribe he universal proesses wih low expansion speed, unsuiable for he universal proesses wih high expansion speed. So we have o onsider he raionaliy of hese wo simplified ondiion. In fa, as disussed before, he R-W meri violaes he invariabiliy priniple of ligh s speed in vauum and is no he meri of relaiviy. Sai energy momenum ensor means ha he veloiy and momenum of elesial body whih exis praially in he expansion proess of he universe are negleed. So he Friedmann equaion is one whih is improperly simplified. In he early period of osmology, wha observed were he expansion proesses wih low speed nearing he Milky Way galaxy, he Friedmann equaion an handle i. While osmology develops o he urren period, wha observed are he expansion proesses wih high speed suh as he red shif of Ia supernova, he Friedmann equaion beomes unsuiable and need relaiviy revision [7]. If we use meri () whih saisfies he onsrain of speial relaiviy in he Einsein s equaion of graviy, beause meri ensors g and g onain speed faor R r, he obained osmi equaion is muh more ompliaed han he Friedmann equaion. If dynami energy momenum ensor means are used simulaneously, speed faor R r is also inrodued. In his ase, even hough we onsider he priniple of osmology o ake maerial densiy as, he equaion of osmology will beomes very omplex so ha i is impossible o solve [7]. This may be he real reason why he pioneers of osmology had o use he R-W meri and sai energy momenum ensor, for hey had no anoher hoie. By onsidering his diffiuly, we have o look for oher more proper mehod o sudy osmology. In fa, we have proved ha by ransforming he geodesi equaion of he Shwarzshild soluion of he Einsein s equaion of graviy field o fla spae-ime desripion, he following revised Newonian formula of graviy an be obained [7] d r L m GM m d r r r (55) In (55) all quaniies are defined in fla spae. We have V d d (56) This is jus he ime delay formula of speial relaiviy, so (55) an be onsidered as he relaiviy revision formula of he Newonian graviy. The spae-ime singulariy in he Einsein s heory of graviy beomes he originnal poin r in he Newonian formula of graviy. The singulariy problem of general relaiviy in urved spae-ime is eliminaed horoughly. The heory of graviy reurns o he radiional form of dynami desripion. When he formula is used o desribe he universe expansion, he revised Friedmann equaion an be obained. Based on i, he high red-shif of Ia supernova an be explained well. We do no need he hypoheses of he aeleraing universe and dark energy. I is also unneessary for us o assume ha non-baryon dark maerial is 5-6 imes more han normal baryon maerial in he universe if hey really exis. The problem of he universal age an also be solved well. Furher more, we prove below ha by he mehod of numerial alulaion and he Doppler s formula proposed in [7], even based on he Newonian heory of graviy, we an also explain well he relaion of red shif and disane of Ia supernova... Using he Doppler s Formula o Calulae he Red Shif of Ia Supernova As we know ha he soluion of differenial equaion is deermined by iniial ondiion. However, aording o he Big Bang heory, he universe blew up from a singulariy wih infinie densiy. Tha is o say, all maerial in he universe has a same iniial posiion. However, infinie densiy is unimaginable and singulariy an no exis in he real world. The praial siuaion may be ha some unknown ineraions an avoid maerial o be ompressed ino infinie densiy by graviy. Aording o [7], we assume ha here exis a erain mehanism so ha a uniform maerial sphere wih mass M an only be ompress ino a finie radius r. The moion equaion of universe expansion an be wrien as mr F r F r (57) n Here F r is he Newonian graviy and Fn r is he sum of all non-graviies. For onveniene of alulaion, we assume m Fn r Ar r r (58) Here r orre- Ar is an unknown funion. Fn Copyrigh SiRes.

8 X. C. MEI, P. YU sponds o an infinie barrier a posiion r. When a maerial sphere wih radius r is ompressed ino a sphere wih radius r, i an no be ompressed furher. For he spheres wih differen radii r, heir r are differen. Suppose ha he maerial disribuion of he universe is uniform wih. The sai mass onained in he spherial surfae wih radius r is M. Aording o (55), he revised Newonian graviy is relaed o veloiy. Using i o alulae he universe expansion, under he ondiion V, he speed of a parile loaed on he spherial surfae is V QrKr (59) GM K r r r 56 r Here GM K r is a onsan whih desribes iniial ondiions. Le r in (59), we ge he resul of he Newonian heory of graviy. V GM 8Gr K r K r (6) r We ake an expansive sphere as he expansive universe and use he Doppler s formula o desribe red shif wih z V V (6) Suppose ha luminous body moves along he direion of radius and observer is loaed a he origin poin of fla referene frame. The disane beween observer and elesial body is r a momen. The disane beween observer and elesial body is r a presen momen. In he expanding proess of he universe, elesial body moves from r o r wih r r, while he ligh ravels from r o observer along opposie direion. Suppose ha ligh s speed is invariable in he proess (having nohing o do wih he speed of ligh s soure or he expensive speed of spae, as shown in ()), we have following relaion r r d (6) V r By asronomial observaions, we know he universe maerial densiy a presen ime, bu we do no know a pas ime. By he relaion r r, we have 8πGr 8πGr (6) r Subsiuing (6) in (6) and using (6), we obain r r r V 8G r r K r (64) r r In priniple, we an wrie (64) as,, r f r K r f r K r (65) From (65), we an ge r g r K, in priniple. Wha observer see a presen ime is he ligh emied by he elesial body loaed a posiion r r and ime. A presen ime, he elesial body has moved o posiion r. In he formulas above,, r and z are known hrough observaions, bu r and K r are unknown. By onneing (6), (6) and (65), we an deerminae r and K r. The inegral of (64) is diffiul bu an be alulaed by he numerial mehod of ompuer. Take 7 6 b Kg/m, r y m, 6 r y m, we have.5 by x (66) r y We use x as basi variable o alulae y and K r. In he alulaion, we ake b, z and y as inpu parameers. Wih his mehod, we aually raed bak o he iniial siuaions of he universe expansion based on he presen observaions of red shif and disanes. In oher words, as long as he iniial ondiions of he universe expansion are known, we an know is urren siuaions... The Red Shif of Ia Supernova Aording o phoomery measuremen, he densiy of 8 luminous maerial in he universiy is abou kg/m a presen ime. Beause here exiss a grea moun of non-luminous maerial, we assume ha praial maerial is imes more han luminous maerial and le 7 kg/m. In Figure, we ake m B = log d L in whih d L is luminosiy disane wih uni 6 lengh p.9 m. Beause our disussion is based on fla spae-ime and he Doppler s effe wihou onsidering he graviy red shif in his paper, ligh s frequeny does no hange in he proess of propagaion. I is unneessary for us o use he onep of luminosiy disane (See appendix for deail). So we need o ransform d L bak o praial disane r. The urved line in Figure shows he relaions beween he red-shifs, disanes and iniial ondiion parameers of Ia supernova based on he unrevised Newonian formula of graviy. The verial oordinae is he values of K r. The boom horizonal oordinae is he value of red-shif. On he upside, under he line of horizonal oordinae, he numbers are he values of disane r. Above he line of horizonal oordinae, he numbers are he values of r. For z and mb 5, 6 we ge r. m. By numerial alulaion, we 6 obain r.8 m and Kr.6. For Copyrigh SiRes.

9 X. C. MEI, P. YU z.5 and mb. orresponding o 6 6 r.67 m, we obain r.9 and Kr.5. For z. and mb 9. 6 whih orresponds o r.5 m, we obain 6 r.6 and Kr 5.. We see ha by direly using he Doppler s formula and he Newonian formula of graviy, we an explain he high red shif of Ia supernova well. The hypoheses of dark energy and he aeleraing expansion of he universe beome unneessary. The universe began is expansion from a finie volume, raher han a singulariy. The diffiuly of singulariy in osmology is eliminaed. If he revised Newonian formula (55) is used, aording o referene [7], he resul is shown in Figure. Comparing Figures and, he differene is ha for he K r. Bu for he revised Newonian formula of graviy, we have Kr when z.7, and Kr when z.7. When z is very small, we have Kr for boh siuaions. Newonian formula of graviy, we have.4. The Age of he Universe We onsider he universe as a maerial sphere wih radius r.5 m a iniial momen, whih is abou he disane beween he sun and he earh. Long enough laer, a ime, an observer loaed a he original poin of referene frame reeives he ligh emied from a elesial body on he spherial surfae wih radius r =. 6 m and red shif z a ime. Suppose ha 6 4 Supernova Cosmology Proje (Ω m, Ω λ) (.,.) (.,.7) (.5,.5) (.,.) (.5, -.5) effeive mb 8 Calan/Tololo (Hamuy e al, A.J. 996) redshif z K Figure. The Hubble diagram for red shif and disane of Ia supernova. r m r Z Figure. The relaions beween red-shifs, disanes and iniial parameers of Ia supernova by using he Doppler s formula and he Newonian graviy. Copyrigh SiRes.

10 X. C. MEI, P. YU K r m r Z Figure. The relaions beween red-shifs, disanes and iniial parameers of Ia supernova by using he Doppler s formula and he revised Newonian graviy. 7 he maerial densiy of he universe is kg/m 7 a presen ime, he iniial densiy is 5.9 kg/m, o be equal o he densiy of neuron sar. Aording o he alulaion before, he real disane of elesial 6 body is r.8 m a presen ime. We onsider i as he radius of observable universe and ake Kr.6 for he following formula o alulae he ime during whih he universe expanded from radius r.5 m o r =.8 6 m r r d V 8G r r.6 (67) r r The resul is 5 billion years. Bu his value is no sensiive o r when i is no very large. Taking r m, whih is abou he radius of he Milky Way galaxy, he resul is almos he same wih r.5 m. I means ha he age of he universe mainly depends on he laer expansive proess. Using (67) o alulaes he ime during whih he radius of sphere expanses from 6 6. m o.8 m, he resul is 5.4 billion years, so he ime during whih he radius of sphere ex- 6 6 panses from. m o.8 m is 9.5 billion years. By using he revised Newonian formula of graviy, for he same red shif z, he resul in referene [8] is ha he ime is.8 billion years for a sphere s radius expands from r.5 m o r.95 m and 6 6 billion years for radius expands from. m 6 o.95 m. So he sphere s radius expands from 6.5 m o. m is 7.8 billion years. Therefore, for he same red shif, by using he revised Newonian graviy, he age of he universe is smaller hen using he unrevised Newonian graviy. Aording o he urren osmology, he universe age is esimaed o be abou - 5 billion years, oo shor o he formaion of galaxies. The problem does no exis by using he Doppler s formula o alulae he red shif of osmology, no maer we use he revised Newonian formula or he unrevised Newonian formula of graviy. 4. Conlusions There are many diffiulies in he urren osmology. They are essenial and an no be solved by some small revising. We need o suspe wheher or no here are problems in he foundaion of osmology. In he proess of deduing he Friedmann equaion of osmology based on he Einsein s equaion of graviy, wo simplifiaion ondiions are used. One is he R-W meri, in whih meri ensors are only relaed o ime, no relaed o spae oordinaes. Anoher is sai energy momenum ensors, in whih elesial body s veloiies are negleed. We need o disuss wheher or no hese wo ondiions are proper, for hey are really over simplified. Cosmi red shif is onsidered o be aused by he reeding moions of elesial bodies whih are aually he Doppler s effe. However, in he urren osmology, wha used o alulae he relaion of red shif and disane of Ia supernova is he formula of meri red shif based on he R-W meri, raher han he formula of he Doppler s red shif. They are ompleely differen and he inonsisene an no be allowed in physis. We should onsider why he meri red shif is used raher han he Doppler s red shif in osmology. The reason is ha he Doppler s effe is relaed o speed faor Rr. If he Doppler s effe is used direly, we should use dynami energy momenum ensor in he Einsein s equaion of graviy oo. In his ase, beause speed faor R () r is onained in energy momenum ensor, he obained equaion of osmology would beome oo omplex o be solved. If he meri Copyrigh SiRes.

11 X. C. MEI, P. YU red shif formula is used, i is equivalen o use sai energy momenum ensor, he rouble an be avoided. However, he problem is ha here exis relaive moion speeds praially beween observers a res on he earh and elesial bodies. So i is very irraional o use sai energy momenum ensors in osmology. In order o avoid using dynami meri, o-moving oordinae sysem is used by expansion osmology. Beause observer moves wih elesial body in o-moving oordinaes sysem, he speed of elesial body an be eliminaed. However, here are hree problems here. The firs is ha asronomers a resed on he earh observe red shif wihou moving wih elesial body. The seond is ha even hough he observer moves wih a elesial body, beause he speed of expansion is relaed o he disane beween elesial bodies, he relaive speeds beween he observer and oher elesial bodies sill exis. The hird is ha if observer moves wih elesial body, i means ha observer falls freely in graviaional field. Aording o he priniple of general relaiviy, graviy is aneled loally. In his ase, he meri of spae-ime is fla and all alulaions abou osmi red shif beome ineffeive. On he oher hand, he formula of meri red shif based on he R-W meri is impossible. The reason is he R-W meri is only a mahemaial sruure of spae. I is no absoluely neessary o use he R-W meri in osmology. In fa, we an also use ommon oordinae sysem o esablish osmi equaion based on he Einsein s equaion of graviy. So he origin of meri red shif is suspiious. A leas, i is no an independen effe aused by independen physial mehanism. Bu graviy red shif and he Doppler s red shif are aused by independen physial laws. If we an obain red shif from he R-W meri, i should no be onradied wih graviy red shif or he Doppler s red shif. Oherwise, i is unaepable. I is proved srily ha Formula () is only he resul of firs order approximaion. When higher order approximaions are aken ino aoun, he onsriion ondiion R onsan an be obained. Therefore, Formula () is only suiable o desribe he proess of spaial uniform expansion; i s unsuiable for he praial proess of he universal expensive wih aeleraion. More srily, if we do no onsider approximaion, boh relaions he formula (6) and (5) do no exis. Tha is o say, we have no meri red shif aually. Besides, he R-W meri has many oher problems. I is proved ha he R-W meri violaes he priniple of ligh s speed invariable. When R is relaed o ime, he R-W meri has no onsan urvaure. The ime delay aused by relaiviy veloiy beween ligh s soure and observer is negleed aording o he R-W meri. So he R-W meri is no a relaiviy one and unsuiable o be used as he basi spae-ime frame in osmology. The red shif based on i an no be orre. Therefore, he urren formula used o alulaes he relaion of red shif and disane of Ia supernova in osmology is wrong. We should direly use he Doppler s formula o alulae he red shif of osmology. By he mehod of numerial alulaion, based on he (revised) Newonian heory of graviy and he Doppler s formula, i is proved ha he red shif of Ia supernova an be explained well. Therefore he hypoheses of dark energy and he aeleraing universe are ompleely unneessary. The problems of he universe age and he singulariy of he Big Bang in he early universe an also be solved simulaneously. In fa, he so-alled aeleraing expansion of he universe is no he resul of dire observaion. I depends on he fiing beween heoreial modal and he observaion of Ia supernova. Oher proofs of he aeleraing universe suh as he osmi bakground radiaion anisoropy and he X ray properies of galaxy lusers and so on are no dire ones, or an be explained by oher mehods shown in his paper. They remain o be disussed furher. The hypohesis of aeleraing universe is ompleely unneessary. The proedure we developed and used in his paper and is soure ode are open o researhers. Demanders, please send us o ask for i. REFERENCES [] S. Perlmuer, e al., Measuremens of Ω and Λ from 4 High-Redshif Supernovae, APJ, Vol. 57, No., 999, p doi:.86/7 [] B. Leibundgu, e al., Observaional Evidene from Supernovae for an Aeleraing Universe and a Cosmologial Consan, The Asronomial Journal, Vol. 6, No., 998, p. 9. doi:.86/499 [] E. W. Kolb and M. S. Turner, The Early Universe, Addison-Wesley Publishing Company, Boson, 99. [4] X. Mei, The R-W Meri Has No Consan Curvaure When Salar Faor R() Changes wih Time, Inernaional Journal of Asronomy and Asrophysis, Vol., No. 4,, pp [5] S. M. Carroll and W. H. Press, The Cosmology Consan, Annual Review of Asronomy and Asrophysis, Vol., 99, pp doi:.46/annurev.aa [6] E. A. Milne, A Newonian Expanding Universe, General Relaiviy and Graviaion, Springer Neherlands, Berlin,. [7] X. Mei and P. Yu, Revised Newonian Formula of Graviy and Equaion of Cosmology in Fla Spae-ime Transformed from Shwarzshild Soluion, Inernaional Journal of Asronomy and Asrophysis, Vol., No.,, pp [8] L. Liao and Z. Zheng, General Relaiviy, nd Ediion, Higher Eduaion Publishing Company, Beijing, 4. [9] S. Weinberg, Graviaion and Cosmology, John Wiley, New Jersey, 97. Copyrigh SiRes.

12 4 X. C. MEI, P. YU Appendix: Inonsisene beween he R-W Meri and he Doppler s Red Shifs. Proper Disane and Expansive Speed Based on he R-W Meri The sandard osmology uses he R-W meri () o desribe he universal spae-ime. Aording o he R-W meri, a a erain momen, he proper disane beween a elesial body and he original poin of oordinae sysem is [9] r R r R l r r r lr r If spae is fla wih l r, we ge R r r r and (68) (69) Suppose ha observer is a he original poin of oordinaes sysem and spae is fla, due o spaial expansion, he speed of elesial body relaive o observer is V r r R r (7) I is a wrong undersanding in osmology ha elesial bodies have no veloiies when o-moving oordinaes are used. In fa, if R, here is erainly a relaive veloiy aording o (7). The speed shown in (7) is proporional o r. For a erain R, when r is large enough, we ould have V. This resul will ause a grea rouble for osmology. Many problems in he urren osmology are relaed o i. In he early period of osmology, beause he speeds of observed elesial bodies are low wih V, (7) is enable approximaely. In fa, he Hubble formula is relaed o (7). However, if he disane is large enough, he inadapabiliy of (7) will emerge.. Luminosiy Disane and Proper Disane The disanes of elesial bodies an no be measured direly in general. The indire mehods are needed and he onep of luminosiy disane is used. The energy ha elesial body emied in uni ime is alled as luminosiy L. Wha measured in asronomy is is brighness B. Le d L represen luminosiy disane, heir relaion is B L 4 dl. Suppose ha a elesial body emied N phoons a ime wih frequeny, we have L Nh /. During he ime ~, hese phoon arrive a he spherial surfae wih radius r R r and he frequeny beomes, we have B Nh 4Rr. So we ge dl r( z) Rr( z). If spae-ime is fla wih, we have / d Rr( z). If ligh s frequeny does no hange L wih when i propagaes in spae, luminosiy disane is equal o real disane. For example, as shown in his paper, ligh s soure moves in fla spae and we only onsider he Doppler s shif. In his ase, he ligh s frequeny is unhanged and we an ake dl r Rr.. The Doppler s Red Shif Suppose ha luminous elesial body is a res in referene frame K, he emied ligh s proper frequeny and wave lengh are and. Referene frame K moves in a uniform speed V relaive o K. Observing in K, he reeived ligh s frequeny and wave lengh beome and. The definiion of frequeny shif is z (7) If K moves apar from K along he radial direion wih speed V, aording o he Doppler s formula, he red shif is z V V V (7) The Doppler s frequeny shif is only relaed o speed, having noing o do wih disane. As along as ligh is emied, no maer how far he observers are, reeived frequenies are he same. In his sense, he meri red shif is ompleely differen fundamenally from he Doppler s red shif. 4. The Hubble Red Shif Suppose ha here is no graviaional field in spae or spae is fla, ligh s frequeny does no hange in is propagaion proess, so luminosiy disane is equal o praial disane. By inroduing oordinae ransformaion r R r whih is so alled spae expansion and define he Hubble onsan H R R, he Hubble red shif is H R R r V z r R r (7) R Comparing (7) wih (7), i is obvious ha he Hubble formula is he approximaion of he Doppler s formula under he ondiion V. I seems ha he Hubble red shif is proporional o disane r. However, beause he Hubble onsan onains speed faor R aually, he formula is relaed o veloiy essenially. In fa, he Hubble red shif is onsidered o be aused by he reession veloiy of elesial body. In his way, he universal expansion is dedued. The red shif observed in osmology is also relaed o graviy. Srily speaking, we should onsider boh he Doppler s and graviaional red shifs simulaneously in osmology. Copyrigh SiRes.

13 X. C. MEI, P. YU 5 5. The Red Shif Caused by he R-W Meri As shown in Figure 4, suppose ha a luminous elesial body is loaed a posiion r Rr a momen. The frequeny of emied ligh is. The observer a original poin O reeives he ligh a momen. The frequeny of reeived ligh is. Suppose ha ligh soure emis anoher ligh in ime inerval wih period. If he ligh is reeived by observer loaed a original poin O a momen, he frequeny of reeived ligh is. Aording o Figure 4 and based on he R-W meri, we an obain () whih is relaed o salar R and no relaed o he speed faor R of spaial expansion. Bu he Doppler s red shif is relaed o R, having nohing o do wih R. Boh are ompleely differen. Le s disuss he real meaning of Formula () now. In order o deermine red shif, we have o know and. Beause we deermine by measuremen, he key is o deermine. In praie, is onsidered as ligh s proper frequeny. Suppose ha here is no graviy field in spae wihou graviy red shif. Suppose ha here is an observer loaed a poin r who is a res wih ligh s soure. Aording o his observer, ligh s frequeny is, i.e., proper frequeny. When ligh ravels from r o original poin r, frequeny hanges from o. Tha is o say, he hange of frequeny is aused by he disane beween ligh s soure and observer, raher han he speed of ligh s soure. This is jus he essene of meri red shif. However, his idea will ause serious problem, inonsisen wih he Doppler s red shif. Aording o observer a original poin r, he elesial body a poin has a moving speed. The frequeny of ligh emied by he elesial body is no proper frequeny wih he Doppler s red shif. The hange of frequeny is relaed o speed and no relaed o disane. In fa, if frequeny is relaed o disane, when a sai observer measure he frequeny of ligh emied by a sai ligh s soure in disane, red shif would be founded. Obviously, his resul violaes experimens and basi physial laws and is ompleely impossible. So he red shif aused by he spaial expansion an only be he Doppler s red shif relaed o speed, raher han he meri red shif relaed o disane. If here is graviy field in spae, aording o general O K v r R r v Figure 4. The red shif aused by he R-W meri. K V relaiviy, graviy will ause red shif. In his ase, besides he Doppler s red shif, we should onsider graviy red shif oo. Beause he inensiies of graviy are differen in differen plaes, is relaed o he posiion where ligh s soure is loaed. So is no proper frequeny again. However, in praial observaion, we an no measure he inensiy of graviy field in disane plae, so we do no know aually. We only ompare observed frequeny wih proper frequeny o alulae red shif. So we an no used () o desribe graviy red shif. In fa, if he universe is sai, we have R R and z, even hough here exis graviy field and graviy red shif. So meri red shif desribed by () is differen from he Doppler s shif and graviy red shif. I onradis wih he Doppler s red shif. We ake some onree examples o prove his poin furher below. 6. The Red Shif in he Uniform Expansion Proess of Spae The R-W meri an be used o desribe boh he uniform and aeleraing expansion of spae. The uniform expansion is equal o he uniform moion of ligh s soure. In his ase, is he proper frequeny of ligh for he observer loaed a K. For he observers a res in K, ligh s proper frequeny is also, bu reeived frequeny is. Le R a b, a and b are onsans. The speed of spaial expansion is V R r br. Aording o he Doppler s and Hubble s formula, we have he same red shif V V br z V (74) The red shif is proporional o oordinae r (disane), having nohing o do wih he ime ha ligh was emied. The red shif is he same no maer wha ime observer observes. However, aording o he meri red shif formula (), we have a b b z (75) ab ab z represens he red shif ha observer observes a momen. The Hubble onsan a momen is R b H (76) R a b The uniform expansion of spae orresponds o he uniform moion of ligh s soure, and ligh s speed is unrelaed o he moion of ligh s soure, or ligh s speed is invariable in vauum. Le dl, in his ase, dl is jus he disane beween observer a original poin and luminous elesial body a momen. Copyrigh SiRes.

14 6 X. C. MEI, P. YU Therefore, we an wrie (75) in he form of he Hubble s red shif H z H dl (77) I seems ha (77) is he same as he Hubble s formula. Bu i is differen from (74) essenially. Aording o (74), red shif is proporional o r and is no a funion of ime. We observe a presen ime, and red shif is z. Aording o (77), red shif is relaed o dl. When, he disane beween ligh s soure and observer is zero, red shif is also zero even hough he relaive speed beween hem is non-zero. Speaking simply, aording o he Doppler s formula, red shif is relaed o he veloiy of ligh soure, having nohing o do wih disane. When he disane is zero and ligh soure s speed is ligh s speed, red shif infinie. Aording o he formula based on R-W meri, red shif is relaed o he disane, and unrelaed o he veloiy of ligh soure. When he disane is zero and ligh soure s speed is ligh s speed, red shif is sill zero. The differene is so grea and essenial so ha hey an no be onsisen a all. 7. The Red Shif in he Uniform Aeleraing Expansion Proess of Spae If ligh s soure is aeleraed in vauum, aording o general relaiviy, non-inerial moion is equal o graviaional field and graviy red shif will be produed. Therefore, in his ase, when luminous body moves in graviaional field, graviy red shif should also be onsidered. Beause he red shifs are differen in differen plaes wih differen inensiies of graviy, he frequeny of emied ligh is differen in differen plaes. So shown in () is no proper frequeny again. The observer loaed a original poin O does no know he iniial value. He an only ompare reeived frequeny wih proper frequeny. So in his ase, () has nohing o do wih praial measuremen. Tha is o say, if graviy or aeleraion effes are onsidered, Formula () is unavailable. If graviy and aeleraion effes are omied, we disuss he differenes beween he Doppler s and Hubble s red shifs and he meri red shif. Suppose ha spae is fla, ligh s soure is uniformly aeleraed wih R ab g (78) V R r bgr (79) When V, he Doppler s red shif a momen is bgr bgr z (8) bg r Beause aeleraion effe has been omied, ligh s frequeny is unhanged in he proess of ligh s propagaion. The red shif observed by observer a original poin a ime is also represened by (8). The Hubble onsan a ime is R b g H (8) R a b g So he Hubble s red shif is H R z r Rr R (8) R r b gr I is he same as he Doppler s red shif (8). If we onsider (), he red shif beomes ab g z ab g By using (8) and le d (8) L, we an wrie (8) as H q H z dl d L (84) This is onsidered as he revised Hubble formula in osmology. In whih dl may no be he real disane of ligh s soure, we alled i as apparen disane. q is he so-alled deeleraed faor wih q g ab g b g (85) (84) is ompleely differen from (8) and (8), and hey are inompaible. Aording o (8) and (8), red shif is sill proporional o oordinae r. A presen momen, we have z b g r. Bu aording o (84), he relaion beween red shif and dl is no linear again. A presen momen, we have z, even hough he relaive speed beween hem may no be zero. I is obvious ha he Doppler s red shif and he Hubble s red shif are onsisen, bu hey are ompleely differen from he meri red shif. 8. The Red Shif of Meri in General Siuaions If he aeleraion of spaial expansion is arbirary, aording o sandard heory, we develop R ino he Taylor series in he neighborhood of wih [8] R R R R R R (86) R H qh Copyrigh SiRes.

15 X. C. MEI, P. YU 7 Here he deeleraed faor is q R R R (87) If he iems in he brake of (86) are small enough, le d and subsiue i in (), we an obain L H q H z dl d L (88) (88) is onsidered as he revised Hubble formula, alhough dl may no be he real disane of ligh s soure. However, he formula should saisfy he ondiion ha is small enough, oherwise he series is divergen. Besides, i should noe ha H and q in (86) is he quaniies a presen ime. Bu in (8) and (84), H and q are he quaniies a pas ime. However, for he problems of osmology, may be very large. For example, for he supernova of high red shif, may be billion years. Beause he high orn der iems in (86) are proporional o, when n, he formula is infinie, so i is unsuiable in general. In fa, as shown in (), he posiions of imes and are equal. We have wo independen variables and aually. When heir differene is grea, we an only develop R ino he Taylor series in he neighborhood of and develop R ino he Taylor series in he neighborhood of. The resuls should be R R H qh (89) R R H qh (9) Here and. So () should be H qh z H qh H H qh (9) qh I is ompleely differen from (88). Tha is o say, we an no obain he revised Hubble s formula (88) from he meri red shif formula () in general siuaions. Copyrigh SiRes.