Cosmology Should Directly Use the Doppler's Formula to Calculate The

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International Journal of Astronomy and Astrophysics, 2013, 3, 303-317 http://dx.doi.org/10.4236/ijaa.2013.33036 Published Online September 2013 (http://www.scirp.org/journal/ijaa)

Cosmology Should Directly Use the Doppler’s Formula to Calculate the Red Shift of Ia Supernova —The Proofs That Metric Red Shift Is Inapplicable and Dark Energy Does Not Exist

Xiaochun Mei, Ping Yu Institute of Innovative Physics, Fuzhou, China Email: [email protected], [email protected]

Received March 28, 2013; revised April 29, 2013; accepted May 7, 2013

Copyright © 2013 Xiaochun Mei, Ping Yu. This is an open access article distributed under the Creative Commons Attribution Li- cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT The Doppler formula should be used directly to calculate red shift of Cosmology. The first is gravity, the second is the Doppler’s effect and the third is the Compton scattering. The red shift of cosmology is considered to be caused by the receding motions of celestial bodies, of which essence is the Doppler’s effect. However, the basic formula used to calculate the relationship between red shift and distance for Ia supernova in cosmology is zRtRt1 01 which is based on the R-W metric and related to the scalar factor Rt  . This is different from the Doppler formula which is related to speed factor Rt . Because the R-W metric is only a mathematical structure of space, the metric red shift is not an independent law of physics, this inconsistence is not allowed in physics. It is proved strictly in this paper that the formula of metric red shift is only the result of the first order approximation. If higher order approximations are considered, we can obtain a restrict condition Rt  constant . It indicates that if the formula of metric red shift holds, it can only be suitable to describe the spatial uniform expansion, unsuitable for the practical universal process with acceleration. The further study reveals that the R-W metric violates the invariability principle of light’s speed in vacuum. The time delay caused by relative velocity between light’s source and observer is neglected. So the R-W metric violates the special relativity and is not suitable to be used as the basic space-time frame in cosmology. The formula used to calculate the relationship between red shift and distance of Ia supernova is wrong and subsequently the deduced results about dark energy and the accelerating universe are incredible. Cosmology should use directly the Doppler formula to calculate red shift. It is proved that based on the Doppler’s formula and by the method of numerical calculation, the relation of red shift and distance of Ia supernova can be explained well. The hypotheses of dark energy and the accelerating expansion of the universe are completely unnecessary in cosmology.

Keywords: Cosmology; Doppler Formula; R-W Metric; Hubble Law; Supernova; Dark Energy; Dark Material

1. Introduction gists declared that about 26% material in the universe was dark material and about 70% was dark energy. The As we known that there are mainly three mechanisms universe seems to do accelerating expansion [1,2]. leading to the red shift of spectrum. The first is gravity The red shift of cosmology is considered to be caused which relates to mass and gravity constant. The second is by the receding motions of celestial bodies, so its essence Doppler’s effect which relates to velocity. The third is is the Doppler’s effect related to speed. We should use the Compton scattering which is related to the energy Doppler’s formula to calculate the red shift. However, it transformation of photon. According to the Hubble law, is strange that the basic formula used to calculate the the spectrum red shift of extragalactic nebula was proportional to the distance between observer and luminous relation of red shift and distance of Ia supernova is com- celestial body. The red shift of cosmology is considered pletely different from the Doppler formula. The basic to be the Doppler’s effect. In 1998, cosmic observers formula used in cosmology is [3] found that the red shift of Ia supernova deviated from the v Rt linear relation of the Hubble law. By fitting the observa- z 1 1  0 (1) tion values with standard theory of cosmology, cosmolo- vRt01

The formula is based on the R-W metric in which 1 2. Unavailability of Metric Red Shift is the frequency of emitted light at time t1 ,  0 is the Formula frequency of received light at time t . The formula is 0 2.1. The Restriction Condition Rt    Constant related to scalar factor Rt. But the Doppler’s formula for the Metric Red Shift Formula is related to speed factor Rt . It is proved in this paper that they can not be consistent. This inconsistence is not Standard cosmology uses the R-W metric to describe the allowed in physics. universal space-time with form It is proved strictly that Formula (1) is only the result ddsctRt2222  of first order approximation. When higher order ap- 2 (2) proximations are taken into account, the restriction con- dr 22 2 2 2 2 rrdsind   dition Rt  constan will be obtained. Therefore, For- 1r mula (1) is only suitable to describe the process of spatial Here Rt  is scalar factor and  is considered as uniform expansion; it’s unsuitable for the practical proc- the factor of spatial curvature. The co-moving coordinate ess of the universal expansion with acceleration. More is used in cosmology. The space is considered flat when strictly, if we do not consider approximation, both For-   0 . When (2) is used to describe the expansive uni- mula (1) and the restriction relations do not exist. That is verse, luminous celestial bodies are considered to be to say, we have no metric red shift actually. fixed on the coordinate r ,,  which do not changed In fact, the W-R metric is only a mathematical struc- with time. Let’s repeat the process to obtain Formula (1) ture of space-time, rather than an independent law of based on the R-W metric. Suppose that light moves along physics. It is introduced to simplify the Einstein’s equa- the radius direction     0 with d0s  , we get tion of gravity field to obtain the Friedmann equation of from (2) cosmology. It is not absolutely necessary to use the R-W ctdd r metric in cosmology. Using normal coordinates directly,  (3) Rt 2 we can also establish the equation of cosmology based on  1r the Einstein’s equation of gravity field. In this case, there Light source is fixed at point r and r does not is no the metric red shift formula (1). In this sense, the change with time. But to light’s motion, r changes red shift of metric is not independent in physics. If it is with time. Suppose that photon’s coordinate is r1 at correct, it cannot be contradicted with the Doppler’s and moment t1 and photon arrives at the original point gravity red shifts. r0  0 at moment t0 . The integral of (3) is [3] The further analysis indicates that the R-W metric vio- t 0 0 ctdd r lates the invariability principle of light’s speed in vac-  sin nr1 Rt 2 uum when it is used to describe the uniform motion of tr111 r (4) light’s source. The time contraction between observer sinrk1  1 and moving light’s source is also neglected when the  rk1 0 R-W metric is use to describe the spatial expansion. So sinhrk 1 the R-W metric is not one of special relativity and is un-  1 suitable to be used as the basic space-time frame in cos- The negative sign indicates that light moves along the mology. Especially, it is not suitable to be used to de- direction to decrease r . Suppose that a light wave is scribe the high red shift of supernova in which the high emitted during the period of time from t1 to t11 speed expansion of the universe is involved. with period 1  c  1 . Observer receives the light during In summary, the formula used to calculate the relation the period of time from t0 to t00 with period of red shift and distance of Ia supernova in cosmology  00 c  . According to the current understanding, be- has essential mistake. Therefore, dark energy and the cause t00 and t11 are decided by the same r1 , accelerating universe are incredible. The Doppler for- we have mula should be used directly to calculate red shift of tt00ddtt0  (5) Cosmology. It is proved by the method of numerical Rt Rt calculation that if the Doppler’s formula and the (revised) tt111

Newtonian formula of gravity are used, the high red shift Because  0 and 1 are small, according to present of Ia supernova can be explained well. We do not need the theory, we obtain from (5) hypothesis of dark energy and accelerating universe again.   0  1 (6) Let’s discuss the problem in the metric red shift for- Rt Rt mula. The inconsistence between the metric red shift and 01 the Doppler’s red shift will be discussed in appendix. Based on (6) and the definition of red shift, we obtain

(1). By considering (1) and the Friedmann equation of     3  Rt 0000 66 Rt  Rt R t  4 cosmology, as described in Section 3, we can obtain (1). 23  4 0  Rt00 Rt Rt 0 We now prove that (6) is the result of the first order ap- (18) proximation. If higher order approximations are consid-  Rt 66 Rt  Rt   R  3  t  1111    4 ered, we can obtain R()tRt () or R()t  constant.  23 4 1 01  Rt Rt Rt So the formula (1) is only suitable to the uniformly ex-  11 1 pansive universe, unsuitable to describe the universal Because of Rt    0 , we have Rt 0 . By consid- process with acceleration. Let ering (15), we still obtain (6). It can be known that no 1 matter what high orders are considered, we always obtain ft (7) Rt these two relations. So the metric can only be written as and take the integral of (5), we have Rt   a bt (19) That is to say, if (6) holds, it can only be used to de- ft10 ft ft 1100 ft (8) scribe the uniform expansion of space. Because gravity By developing (8) into the Taylor series, we obtain exists in the universe, Formula (1) can not be used to 1 describe the practical process of the universal expansion ft ft ft f  t  f t 2 01 0 00002! with acceleration. More strictly, we only have Formula (9). Formulas (10)- 1 3  ft 00 ft 1  ft  11 (9) (13) do not exist in mathematics actually. It means that it 3! is impossible for us to have a new red shift mechanism 1122 ft 11 ft  11 based on the R-W metric, which is different from the 2! 3! Doppler’s and gravity red shifts. The reason is that the R-W metric is only a mathematical structure of space- Because  0 and 1 are very small, if let the items with same orders are equal to each other, we have time, rather an independent physical law. f tft   (10) 00 11 3.2. The Problem of the R-W Metric 22 ft00  ft   11 (11) Firstly, constant  in (2) is considered as curvature factor and the metric is considered to describe flat space- ft 33 ft   (12) 00 11 time when   0 . This idea has been proved wrong that 44 when scalar factor Rt is related to time [4]. That is to ft ft    (13)   00 11 say that the R-W metric has no constant curvature in By considering (7), (10) is just (6). So the metric red general situations. By using the formula of the Rieman- shift formula (1) is actually the approximation of first nian geometry and making strict calculation, the space- order. (11) can be written as time curvatures of the R-W metric actually are Rt  Rt   R 0  22 1  (14) KKK 2201 01 02 03 R Rt01 Rt (20)   R 2 Substituting (6) in (14), we obtain KKK 12 13 23 R2  Rt01 Rt (15) Here K0 j is the curvature of space-time closing part

Because t0 and t1 are arbitrary, (15) indicates that and Kij is the curvature of pure spatial part. This result Rt  constant , i.e., the expansion speed does not de- is completely different from the current understanding.    pend on time and we have Rt01 Rt Rt  0 . Therefore, constant  is not the factor of spatial curva- From (12), we have ture. Instead, it is a certain adjustable parameter. The R-W metric does not represent the flat space-time when Rt2 R  2 t Rt2 R  2 t 00 3311   0 . 23 01  23 (16) Rt00 Rt  Rt 11 Rt In fact, we can use more simple method to prove the result above. As we know in geometry, the principle to Substituting (6) and (15) in (16), we get judge an arbitrary metric to be flat or not is that if we can  3  3 find a coordinate to transform the metric to one with flat 0  1 (17) 33 space-time, the original metric is flat essentially. If we Rt01 Rt cannot, the original metric is curved in essence. By us- It is still (6). From (13), we get ing common coordinate system, the four dimensional

metric of flat space-time is point r0 and emits a bundle of light at time t  0 . This

22222222 2 light arrives at observer located at position rt  at time ddddsct rr r sind  (21) t . Suppose that space expands in a uniform speed which corresponds to the uniform motion of light’s source. Let By using the co-moving coordinate or the coordinate Rt   a bt and take the integral of (24), we obtain transformation rRtr  in (21), we obtain cb 22 rt rln 1  t (26) Rtr  0  22 2  ba d1s c2 d2ddt  RtRtrrt  c (22) 8 Let r0  1 , a  1 and b 310, we get the speed 2222222 of light’s source Vt br 3108 m/s which is near Rtdd r r r sind   0 light’s speed. Suppose that observer is at the original However, let   0 in (2), we obtain point of coordinate system with rt 0 , according to 8 7 2222 22222 2 (26), we have 1ln1310t and t 5.73 10 s . ddsctRtrr dd   r sind  (23)   This result is obviously incorrect. According to special (22) is different from (23) unless Rt  0 . Because we relativity, light’s speed does not depend on the speed of can obtain (21) by using rRtr  in (22), so the light’s source. Because the initial distance between ob- space-time described by (22) is flat in essential. However, server and light’s source is 1m, the time that light arrives 7 it is easy to see that when Rt   constant we can not at observer should be 3.33 10 s . The error is as great find a coordinate transform to change (23) into (21), so as 58%. In fact, charged particles move at near light’s (23) can not be the metric of flat space-time. speed in high energy accelerators. And experiments show We can prove generally that if (23) described light’s that radiation lights can not propagate in this way. motion in flat space, it would violate the invariability Suppose that the acceleration of spatial expansion is a 2 principle of light’s speed in vacuum. For the light source constant and let Rt   a btgt  2 . Take rr 0 fixed on the reference fame, its coordinate r does not when t  0 . The integral of (24) is change with time. But for the motion of light, the coor- c rr0 dinate r changes with time. Suppose that light moves 2 along the direction of radius, we have d0s  and bag 2 (27) dd0. According to (23), we have  gtbbagbbag22 22    ln ln  drc 22  (24)  gtbbagbbag 22    dtRt Taking a  1 , b  1 and g  0.3 , we have The velocity of light relative to the observer rested at 21/2 the original point of coordinate system is R01  and (2)0.633bag. If t  8.16 , we have R 18.9 2.83 . The speed of light’s source is drt ddr   Vt r Rt  Rt  c (25) ddtt d t Vbgtr  00 1.45 r. Meanwhile, we have Rtr  c Vt c gt b b2 20 ag and bb2 21.63 ag . (25) indicates that light’s velocity is related to the velocity of special expansion. Suppose that space expands The first item in (27) becomes minus infinite and the uniformly, let Rt a bt, according to (25), we have second item is meaningless, so (27) can not yet describe the motion of light in this case. Therefore, in general, the Vtc  brc. At the moment when light is emitted out, (25) is just the Galileo’s addition rule of light’s velocity. R-W metric is unsuitable to describe light’s motion. When light moves towards observer, minus sign is taken This problem does not exist in (22). Suppose that light and light’s speed is less than its speed in vacuum. When moves along the direction of radius, we have the light moves apart from observer, plus sign is taken Rtr 22 and light’s speed is great than its speed in vacuum. Espe- c221d2dd t RtRtrrt 2   cially, because r increases with time, enough long c (28) time later, light’s speed may greatly exceed its speed in Rtr22d0 vacuum. This is not allowed in physics, so the R-W metric is not the metric of special relativity. drcRtr     (29) We can use two simple examples to show that if (23) dtRtRt  describe the metric of flat space, it can not describe light’s motion. Suppose that light’s source is located at By considering (29), light’s velocity is

ddrrdRtr formulas similar to (1). VR  tr Rtc  (30) c ddtt dt The R-W metric violates the invariability principle of light’s speed in vacuum and neglects the time delay be- It indicates that light’s speed is unchanged in vacuum, tween observers and moving light’s source caused by so (22) is the metric of special relativity. relative motion. So the red shift formula of metric based We have g00  1 in (23), so it is considered to have a on the R-W metric can not be correct. It is improper to uniform time for whole space. However, because of spa- use the metric as the basic frame of space-time in cos- tial expansion, the clock fixed at position r has a speed mology. We should use the Doppler’s formula to calcu- relative to observer who is rested at the original point of late the red shift directly. coordinate system. According to special relativity, there is time delay between them, but (23) can not describe this 2.3. Other Problems in the Formula for relation. According to (22), we have Calculating the Red Shift of Ia Supernova Rtr 22 V 2 Based on (1) in cosmology, the following formula is ob- dd1 tt  d1 (31) cc22 tained to calculate the relation of red shift and distance of Ia supernova [5] This is just the delay of special relativity, so (22) is the cz1  metric of relativity in flat space-time, but (23) is not. Hdsin n Similarly, it is proved in mathematics that the four di- 00Lk k 0  mensional metric in which three dimensional space has (36)  constant curvature  is z dz   2  2  dr 0 222 2222 2 11zzzzm0  2   ddsct2  r d  r sind  (32)  1r Here dL is luminosity distance. In light of the R-W

By using co-moving coordinate in (32), we obtain metric, we have dzrzRrL 11  0 . By fitting with practical observations, cosmologists deduced that 22 22Rtr 22dd RtRtrtr  dark energy was about 70% and non-baryon dark mate- ddsc22 t  22 11Rtr Rtr rial was about 26% of the universal material. An accel- (33) erating universe is assumed. dr 2 Rt222222 rdsind r We now discuss the problems existing in (36). The 22 1 Rtr Friedmann equation of cosmology is 2 When   0 , (33) becomes (22), so (33) is correct RG  8π 2 m  (37) metric which satisfies special relativity with special cur- R R 3 vature  . We have g00  1 , g01  0 and g11  Rt  in (33). Its form is much more complicated than the R-W Here m is the density of normal material,  is metric (2). When light moves along the direction of ra- the density corresponding to cosmic constant. At present dius with d0s  , we have time t0 , we have 00 t and H00 Ht  , the Friedmann equation of cosmology can be written as Rtr22 2dd RtRtrtr  ct22d 3H 2 3 22 22 0  (38) 11Rtr Rtr m0  2 (34) 8πG 8πGR0 22 Rtrd 2 Rt 0 Defining critical density c as 1 Rtr22 3H 2   0 (39) We obtain c 8πG   22 dr Rtr cRtr1  Let mmc00   ,   c , 00m  ,   (35) we write (38) as dtRtRt   If we use (29) and (35) to calculate the red shift of 10  (40) RH22 cosmology, i.e., also so called metric red shift, the results 00 are certainly related to velocity. However, we can not Let at   Rt  Rt 0  , we have at0  1 at pre- separate variables in (29) and (35), so we can not write sent time. Because  is a constant, we can write the them in the simple form of (3) and can not get the same Friedmann equation as

228πGG 28π  1 at m  a H00 (41) dR1sin z n 33R2 L 0   0  RH00 (50) By considering z  dz   33 3  2  RR 00 or a  0 (42) 0 11zz zz  2 m0   (41) can be written as When space is flat, we have   0 , m0  1, 2 and sin nr r . So (50) becomes da 22m0 Ha001 m   (43) z dta dz HRr00  (51)  2 0 11 zzzz  2   We have ddat a or ddtaa  , so (4) can be writ- m0  ten as Comparing with (49), (51) has an additional item con- 1 1da taining  in the radical sign of integrated function. sin nr  (44) Raa  The reason is that when   0 , the right side of (43) has 01 1 z only two items with 2 The upper limit of the integral is at0  1 and lower da 22m0 limit is at1 11  z . By intruding so-called energy H0  a (52) effective curvature density [5] dta 3 But (36) is calculated without considering this factor, k  2 (45) so that additional three items are added. If space is 8πGR curved for   1, we still have m0  1 and can We have kkc  and 00kkc  . We write obtain (40) as   z    1dz  dRL 0 1sin z (53) 100  k (46)   3  22 RH000 1z RH00  m0   According to Document [5], (46) is written as Similarly, for   1, we have 1 R  (47)  z  0  1dz  H  dRL 0 1sinh z (54) 00k   3  RH000 1z m0  By introducing transformation az11 in (44)   we obtained (36). However, (47) is obviously wrong. (46) (53) and (54) are also different from (36), so (36) is contains constant  , but (47) does not. According to wrong. Note that this is a mistake of mathematics, and

(46), when   0 we have k 0  0 , so R0 is limited. has nothing to do with physics. So we can not use it to But according to (47), when k 0  0 we have R0   . calculate the relation of red shift and distance of Ia su-  It means that R0 is infinite at present time in flat space. pernova. More essential, if Rt constant , we have But this is completely impossible. According to (46), at    constant . Substituting it in (44), we can not ob- correct result should be tain (36). In this case, let aAR  / 0 , substitute it in (44), for   0 , we obtain Arz ln(1 ) without practical  k R0  (48) significance. H0 It is proved below that if the Doppler’s formula is used In fact, it is unnecessary for us to introduce relation for calculation, the hypotheses of dark energy and the 22 accelerated universe will become unnecessary. k  aR . Whether or not space is flat depends on sin nr r , sinnr sin r or sinnr sinh r . For flat space, by taking   0 in (41) and substituting it in 3. Using the Doppler’s Formula to Calculate the Red Shift of Ia Supernova (44), and by considering m0  1 and sin nr r , we get 3.1. The Friedmann Equation Needs Relativity z dz Revision HRr00  (49)  2 Standard cosmology is based on the principle of cos- 0 11zz m0 mology. The principle declares that the distribution of Because (47) can not hold, (36) can only be written as universal material is uniform and isotropous. So the ma-

Copyright © 2013 SciRes. IJAA X. C. MEI, P. YU 309 terial density is only related to time and unrelated to spa- d322rrL tial coordinates with the form  t . To match with the mGMm0001  (55)  d 2223cr r principle, the R-W metric is used in cosmology which is  considered with biggest symmetry of space-time. In (55) all quantities are defined in flat space. We have Standard cosmology uses the Friedmann equation as V 2 basic equation. The Friedmann equation is based on the d1d  t (56) Einstein’s equation of gravity field. But two simplified c2 conditions are used. One is the R-W metric and another This is just the time delay formula of special relativity, is static energy momentum tensors. However, British so (55) can be considered as the relativity revision for- physicist E. A. Milne proved in 1943 that the Friedmann mula of the Newtonian gravity. The space-time singular- equation could be deduced simply based on the Newto- ity in the Einstein’s theory of gravity becomes the origin- nian theory of gravity when cosmic constant is not con- nal point r  0 in the Newtonian formula of gravity. sidered [6]. This result indicates that the Friedmann The singularity problem of general relativity in curved equation is actually equivalent to the Newtonian theory space-time is eliminated thoroughly. The theory of grav- of gravity which is only suitable to describe the universal ity returns to the traditional form of dynamic description. processes with low expansion speed, unsuitable for the When the formula is used to describe the universe ex- universal processes with high expansion speed. So we pansion, the revised Friedmann equation can be obtained. have to consider the rationality of these two simplified Based on it, the high red-shift of Ia supernova can be condition. explained well. We do not need the hypotheses of the In fact, as discussed before, the R-W metric violates accelerating universe and dark energy. It is also unnec- the invariability principle of light’s speed in vacuum and essary for us to assume that non-baryon dark material is is not the metric of relativity. Static energy momentum 5 - 6 times more than normal baryon material in the uni- tensor means that the velocity and momentum of celestial verse if they really exist. The problem of the universal body which exist practically in the expansion process of age can also be solved well. the universe are neglected. So the Friedmann equation is Further more, we prove below that by the method of one which is improperly simplified. In the early period of numerical calculation and the Doppler’s formula pro- cosmology, what observed were the expansion processes posed in [7], even based on the Newtonian theory of with low speed nearing the Milky Way galaxy, the gravity, we can also explain well the relation of red shift Friedmann equation can handle it. While cosmology de- and distance of Ia supernova. velops to the current period, what observed are the expansion processes with high speed such as the red shift of 3.2. Using the Doppler’s Formula to Calculate Ia supernova, the Friedmann equation becomes unsuitable the Red Shift of Ia Supernova and need relativity revision [7]. If we use metric (33) which satisfies the constrain of As we know that the solution of differential equation is special relativity in the Einstein’s equation of gravity, determined by initial condition. However, according to the Big Bang theory, the universe blew up from a singu- because metric tensors g00  1 and g01  0 contain speed factor Rtr  , the obtained cosmic equation is larity with infinite density. That is to say, all material in much more complicated than the Friedmann equation. If the universe has a same initial position. However, infinite dynamic energy momentum tensor means are used si- density is unimaginable and singularity can not exist in multaneously, speed factor Rtr   is also introduced. In the real world. The practical situation may be that some this case, even though we consider the principle of cos- unknown interactions can avoid material to be com- mology to take material density as  t , the equation of pressed into infinite density by gravity. cosmology will becomes very complex so that it is im- According to [7], we assume that there exist a certain possible to solve [7]. This may be the real reason why the mechanism so that a uniform material sphere with mass pioneers of cosmology had to use the R-W metric and M 0 can only be compress into a finite radius r0 . The static energy momentum tensor, for they had no another motion equation of universe expansion can be written as choice. mr0  F r Fn r (57) By considering this difficulty, we have to look for Here F r is the Newtonian gravity and F r is other more proper method to study cosmology. In fact,   n   we have proved that by transforming the geodesic equa- the sum of all non-gravities. For convenience of calculation of the Schwarzschild solution of the Einstein’s equation, we assume m tion of gravity field to flat space-time description, the 0 Fn rArrr   0 (58) following revised Newtonian formula of gravity can be 2 obtained [7] Here Ar is an unknown function. Fn r corre-

sponds to an infinite barrier at position r0 . When a ma- In principle, we can write (64) as terial sphere with radius r is compressed into a sphere rfrKrfrKr100 ,,   10 (65) with radius r0 , it can not be compressed further. For the spheres with different radii r , their r are different. 0 From (65), we can get rgrK01 , in principle. Suppose that the material distribution of the universe What observer see at present time is the light emitted by is uniform with  t . The static mass contained in  the celestial body located at position rr10 and time the spherical surface with radius r is M . According 0 tt10 . At present time t0 , the celestial body has moved to (55), the revised Newtonian gravity is related to veloc- to position r0 . ity. Using it to calculate the universe expansion, under In the formulas above, 0 , r1 and z are known the condition Vc 1 , the speed of a particle located on through observations, but r0 and K r0 are unknown. the spherical surface is By connecting (60), (61) and (65), we can determinate V r and K r . The integral of (64) is difficult but can Qr Kr 0  0  c 10 be calculated by the numerical method of computer. Take (59)  b 1027 Kg/m 3 , ry1026 m , 2 0 00 2GM 0 33 26  ry1110 m , we have 2 1     K r0 cr 20rr 56  3  by0 2 x 0.25 (66) Here   2GM0 c . K r0 is a constant which de- ry scribes initial conditions. Let  r  0 in (59), we get the result of the Newtonian theory of gravity We use x as basic variable to calculate y0 and K r0 . In the calculation, we take b , z and y1 as VGM28 Gr 2 Kr Kr (60) input parameters. With this method, we actually traced c cr22003 c back to the initial situations of the universe expansion We take an expansive sphere as the expansive universe based on the present observations of red shift and dis- and use the Doppler’s formula to describe red shift with tances. In other words, as long as the initial conditions of the universe expansion are known, we can know its cur- 1Vt c rent situations. z  (61) 1Vt c 3.3. The Red Shift of Ia Supernova Suppose that luminous body moves along the direction of radius and observer is located at the origin point of flat According to photometry measurement, the density of 28 reference frame. The distance between observer and ce- luminous material in the university is about 0 210 3 lestial body is rt at moment t . The distance be- kg/m at present time. Because there exists a great mount of non-luminous material, we assume that practical ma- tween observer and celestial body is r0 at present moment t . In the expanding process of the universe, celes- terial is 10 times more than luminous material and let 0 27 3 0 210 kg/m . In Figure 1, we take mB = 5.5 + tial body moves from r1 to r0 with rr01 , while the light travels from r to observer along opposite direc- 5log dL in which dL is luminosity distance with unit 1 622 tion. Suppose that light’s speed is invariable in the proc- length 10pc  3.09 10 m . Because our discussion is ess (having nothing to do with the speed of light’s source based on flat space-time and the Doppler’s effect without or the expensive speed of space, as shown in (30)), we considering the gravity red shift in this paper, light’s fre- have following relation quency does not change in the process of propagation. It r tr00dr is unnecessary for us to use the concept of luminosity tt1 d  (62) distance (See appendix for detail). So we need to trans- cV tr11 form dL back to practical distance r . By astronomical observations, we know the universe The curved line in Figure 2 shows the relations between the red-shifts, distances and initial condition pa- material density 0 at present time t0 , but we do not 33rameters of Ia supernova based on the unrevised Newto- know  t at past time t . By the relation 00rr  , we have nian formula of gravity. The vertical coordinate is the 2 3 values of K r0 . The bottom horizontal coordinate is 8πGr 8πGr00 22 (63) the value of red-shift. On the upside, under the line of 33ccr horizontal coordinate, the numbers are the values of dis- Substituting (60) in (62) and using (63), we obtain tance r . Above the line of horizontal coordinate, the rr 00ddrr numbers are the values of r0 . For z  1 and mB  25 , r  (64) 26 1  we get r1 1.2 3 10 m . By numerical calculation, we Vc 32 26 2 rr1183Gr crKr  00 0 obtain r0 1.83 10 m and Kr0 2.60  10 . For

z  0.5 and mB  23.1 corresponding to Newtonian formula of gravity, we have Kr0  0 . But 26 26 r1 0.67 10 m , we obtain r0 0.91 10 and for the revised Newtonian formula of gravity, we have 2 Kr0 2.51  10 . For z  0.1 and mB  19.1 Kr 0  0 when z  0.7, and Kr 0  0 when z  0.7 . 26 which corresponds to r1 0.15 10 m , we obtain When z is very small, we have Kr0  0 for both si- 26 3 r0 0.16 10 and Kr0 5.30  10 . tuations. We see that by directly using the Doppler’s formula and the Newtonian formula of gravity, we can explain 3.4. The Age of the Universe the high red shift of Ia supernova well. The hypotheses of dark energy and the accelerating expansion of the uni- We consider the universe as a material sphere with radius  11 verse become unnecessary. The universe began its ex- r0 1.510 m at initial moment, which is about the pansion from a finite volume, rather than a singularity. distance between the sun and the earth. Long enough The difficulty of singularity in cosmology is eliminated. later, at time t , an observer located at the original point If the revised Newtonian formula (55) is used, accord- of reference frame receives the light emitted from a ce- ing to reference [7], the result is shown in Figure 3. lestial body on the spherical surface with radius r = 1.23 26 Comparing Figures 2 and 3, the difference is that for the × 10 m and red shift z  1 at time t0 . Suppose that

(Ωm0, Ωλ) 26 (0.0, 1.0) (0.3, 0.7) (0.5, 0.5) 24 (1.0, 0.0) Supernova (1.5, -0.5) Cosmology 22 Project

B m 20 Calan/Tololo (Hamuy et al, effective effective 18 A.J. 1996)

14 0.02 0.05 0.1 0.2 0.5 1.0

redshift z Figure 1. The Hubble diagram for red shift and distance of Ia supernova.

26 r0 0.16 0.35 0.54 0.72 0.91 1.13 1.33 1.49 1.68 1.83 × 10 m K 0.15 r 0.15 0.30 0.43 0.55 0.67 0.81 0.93 1.03 1.14 1.23 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 -0.25 -0.30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Z Figure 2. The relations between red-shifts, distances and initial parameters of Ia supernova by using the Doppler’s formula and the Newtonian gravity.

26 r0 0.16 0.35 0.54 0.73 0.91 1.14 1.35 1.54 1.73 1.90× 10 m K 0.08 r 0.15 0.30 0.43 0.55 0.67 0.81 0.93 1.03 1.14 1.23 0.06 0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Z Figure 3. The relations between red-shifts, distances and initial parameters of Ia supernova by using the Doppler’s formula and the revised Newtonian gravity.

27 the material density of the universe is 0 210 of galaxies. The problem does not exist by using the 3 17 kg/m at present time, the initial density is 0 5.9 10 Doppler’s formula to calculate the red shift of cosmology, kg/m3, to be equal to the density of neutron star. Accord- no mater we use the revised Newtonian formula or the ing to the calculation before, the real distance of celestial unrevised Newtonian formula of gravity. 26 body is r0 1.83 10 m at present time. We consider it as the radius of observable universe and take 4. Conclusions Kr0 0.026 for the following formula to calculate the time during which the universe expanded from radius There are many difficulties in the current cosmology. 11 26 They are essential and can not be solved by some small r 1.5 10 m to r0 = 1.83 × 10 m 0 revising. We need to suspect whether or not there are tr00ddrr r 0 ttd   (67) problems in the foundation of cosmology. In the process   32 trV of deducing the Friedmann equation of cosmology based 11r1 cGr800  3 cr 0.026 on the Einstein’s equation of gravity, two simplification The result is t 35 billion years. But this value is conditions are used. One is the R-W metric, in which not sensitive to r1 when it is not very large. Taking metric tensors are only related to time, not related to 20 r1  10 m , which is about the radius of the Milky Way space coordinates. Another is static energy momentum 11 galaxy, the result is almost the same with r0 1.5 10 tensors, in which celestial body’s velocities are neglected. m. It means that the age of the universe mainly depends We need to discuss whether or not these two conditions on the later expansive process. Using (67) to calculates are proper, for they are really over simplified. the time during which the radius of sphere expanses from Cosmic red shift is considered to be caused by the re- 1.23 1026 m to 1.83 1026 m , the result is 15.4 billion ceding motions of celestial bodies which are actually the years, so the time during which the radius of sphere ex- Doppler’s effect. However, in the current cosmology, 26 26 panses from 1.23 10 m to 1.83 10 m is 19.5 bil- what used to calculate the relation of red shift and dis- lion years. tance of Ia supernova is the formula of metric red shift By using the revised Newtonian formula of gravity, for based on the R-W metric, rather than the formula of the the same red shift z  1, the result in reference [8] is that Doppler’s red shift. They are completely different and the time is 30.8 billion years for a sphere’s radius ex- the inconsistence can not be allowed in physics. We 11 26 pands from r1 1.5 10 m to r0 1.95 10 m and should consider why the metric red shift is used rather 13 billion years for radius expands from 1.23 1026 m than the Doppler’s red shift in cosmology. to 1.95 1026 m . So the sphere’s radius expands from The reason is that the Doppler’s effect is related to 1.5 1011 m to 1.23 1026 m is 17.8 billion years. speed factor Rtr   . If the Doppler’s effect is used di- Therefore, for the same red shift, by using the revised rectly, we should use dynamic energy momentum tensor Newtonian gravity, the age of the universe is smaller in the Einstein’s equation of gravity too. In this case, then using the unrevised Newtonian gravity. According because speed factor R()tr is contained in energy to the current cosmology, the universe age is estimated to momentum tensor, the obtained equation of cosmology be about 10 - 15 billion years, too short to the formation would become too complex to be solved. If the metric

Copyright © 2013 SciRes. IJAA X. C. MEI, P. YU 313 red shift formula is used, it is equivalent to use static Therefore, the current formula used to calculates the energy momentum tensor, the trouble can be avoided. relation of red shift and distance of Ia supernova in cos- However, the problem is that there exist relative motion mology is wrong. We should directly use the Doppler’s speeds practically between observers at rest on the earth formula to calculate the red shift of cosmology. By the and celestial bodies. So it is very irrational to use static method of numerical calculation, based on the (revised) energy momentum tensors in cosmology. Newtonian theory of gravity and the Doppler’s formula, In order to avoid using dynamic metric, co-moving it is proved that the red shift of Ia supernova can be ex- coordinate system is used by expansion cosmology. Be- plained well. Therefore the hypotheses of dark energy cause observer moves with celestial body in co-moving and the accelerating universe are completely unnecessary. coordinates system, the speed of celestial body can be The problems of the universe age and the singularity of eliminated. However, there are three problems here. The the Big Bang in the early universe can also be solved first is that astronomers at rested on the earth observe red simultaneously. In fact, the so-called accelerating expan- shift without moving with celestial body. The second is sion of the universe is not the result of direct observation. that even though the observer moves with a celestial body, It depends on the fitting between theoretical modal and because the speed of expansion is related to the distance the observation of Ia supernova. Other proofs of the ac- between celestial bodies, the relative speeds between the celerating universe such as the cosmic background radia- observer and other celestial bodies still exist. The third is tion anisotropy and the X ray properties of galaxy clus- that if observer moves with celestial body, it means that ters and so on are not direct ones, or can be explained by observer falls freely in gravitational field. According to the other methods shown in this paper. They remain to be principle of general relativity, gravity is canceled locally. In discussed further. The hypothesis of accelerating uni- this case, the metric of space-time is flat and all calculations verse is completely unnecessary. about cosmic red shift become ineffective. The procedure we developed and used in this paper On the other hand, the formula of metric red shift and its source code are open to researchers. Demanders, based on the R-W metric is impossible. The reason is the please send us e-mail to ask for it. R-W metric is only a mathematical structure of space. It is not absolutely necessary to use the R-W metric in REFERENCES cosmology. In fact, we can also use common coordinate [1] S. Perlmutter, et al., “Measurements of Ω and Λ from 42 system to establish cosmic equation based on the Ein- High-Redshift Supernovae,” APJ, Vol. 517, No. 2, 1999, stein’s equation of gravity. So the origin of metric red shift p. 565. doi:10.1086/307221 is suspicious. At least, it is not an independent effect caused [2] B. Leibundgut, et al., “Observational Evidence from Su- by independent physical mechanism. But gravity red shift pernovae for an Accelerating Universe and a Cosmologi- and the Doppler’s red shift are caused by independent cal Constant,” The Astronomical Journal, Vol. 116, No. 3, physical laws. If we can obtain red shift from the R-W met- 1998, p. 1009. doi:10.1086/300499 ric, it should not be contradicted with gravity red shift or the [3] E. W. Kolb and M. S. Turner, “The Early Universe,” Ad- Doppler’s red shift. Otherwise, it is unacceptable. dison-Wesley Publishing Company, Boston, 1990. It is proved strictly that Formula (1) is only the result [4] X. Mei, “The R-W Metric Has No Constant Curvature of first order approximation. When higher order appro- When Scalar Factor R(t) Changes with Time,” Interna- ximations are taken into account, the constriction condi- tional Journal of Astronomy and Astrophysics, Vol. 1, No. tion Rt  constant can be obtained. Therefore, For- 4, 2011, pp. 177-182. mula (1) is only suitable to describe the process of spatial [5] S. M. Carroll and W. H. Press, “The Cosmology Con- uniform expansion; it’s unsuitable for the practical proc- stant,” Annual Review of Astronomy and Astrophysics, Vol. 30, 1992, pp. 499-542. ess of the universal expensive with acceleration. More doi:10.1146/annurev.aa.30.090192.002435 strictly, if we do not consider approximation, both rela- [6] E. A. Milne, “A Newtonian Expanding Universe, General tions the formula (6) and (15) do not exist. That is to say, Relativity and Gravitation,” Springer Netherlands, Berlin, we have no metric red shift actually. 2000. Besides, the R-W metric has many other problems. It [7] X. Mei and P. Yu, “Revised Newtonian Formula of Grav- is proved that the R-W metric violates the principle of ity and Equation of Cosmology in Flat Space-time Trans- light’s speed invariable. When Rt is related to time, formed from Schwarzschild Solution,” International the R-W metric has no constant curvature. The time de- Journal of Astronomy and Astrophysics, Vol. 2, No. 1, lay caused by relativity velocity between light’s source 2012, pp. 6-18. and observer is neglected according to the R-W metric. [8] L. Liao and Z. Zheng, “General Relativity,” 2nd Edition, So the R-W metric is not a relativity one and unsuitable Higher Education Publishing Company, Beijing, 2004. to be used as the basic space-time frame in cosmology. [9] S. Weinberg, “Gravitation and Cosmology,” John Wiley, The red shift based on it can not be correct. New Jersey, 1972.

Appendix: Inconsistence between the R-W with 10 when it propagates in space, luminosity Metric and the Doppler’s Red Shifts distance is equal to real distance. For example, as shown in this paper, light’s source moves in flat space and we 1. Proper Distance and Expansive Speed Based only consider the Doppler’s shift. In this case, the light’s on the R-W Metric frequency is unchanged and we can take drRrL 0 . The standard cosmology uses the R-W metric (2) to describe the universal space-time. According to the R-W 3. The Doppler’s Red Shift metric, at a certain moment t , the proper distance be- Suppose that luminous celestial body is at rest in refer- tween a celestial body and the original point of coordi- ence frame K , the emitted light’s proper frequency and nate system is [9] wave length are  and  . Reference frame K0 moves r dr in a uniform speed V relative to K . Observing in K , rt Rt  Rtlr 0  2 the received light’s frequency and wave length become 0 1r (68)  and  . The definition of frequency shift is r dr 0 0 lr    2 0 (71) 0 1r z  11  0 If space is flat with   0 , we get lr r and If K moves apart from K0 along the radial direc- rt Rtr  (69) tion with speed Vc , according to the Doppler’s formula, the red shift is Suppose that observer is at the original point of coordinates system and space is flat, due to spatial expansion, 1Vt c Vt z  (72) the speed of celestial body relative to observer is 1Vt c c Vr  rt  Rtr  (70) The Doppler’s frequency shift is only related to speed, It is a wrong understanding in cosmology that celestial having noting to do with distance. As along as light is bodies have no velocities when co-moving coordinates emitted, no matter how far the observers are, received are used. In fact, if Rt  0 , there is certainly a relative frequencies are the same. In this sense, the metric red velocity according to (70). shift is completely different fundamentally from the The speed shown in (70) is proportional to r . For a Doppler’s red shift. certain Rt , when r is large enough, we could have Vc . This result will cause a great trouble for cosmol- 4. The Hubble Red Shift ogy. Many problems in the current cosmology are related to it. In the early period of cosmology, because the Suppose that there is no gravitational field in space or speeds of observed celestial bodies are low with Vc , space is flat, light’s frequency does not change in its (70) is tenable approximately. In fact, the Hubble for- propagation process, so luminosity distance is equal to mula is related to (70). However, if the distance is large practical distance. By introducing coordinate transforma- enough, the inadaptability of (70) will emerge. tion rt   Rtr  which is so called space expansion and define the Hubble constant H tRtRt     , the 2. Luminosity Distance and Proper Distance Hubble red shift is The distances of celestial bodies can not be measured H tRtRtrVt      zrtRtr   (73) directly in general. The indirect methods are needed and ccRtcc the concept of luminosity distance is used. The energy that celestial body emitted in unit time is called as lumi- Comparing (73) with (72), it is obvious that the Hub- nosity L . What measured in astronomy is its brightness ble formula is the approximation of the Doppler’s formula under the condition Vc . It seems that the Hub- B . Let dL represent luminosity distance, their relation ble red shift is proportional to distance rt. However, is BL 4 dL . Suppose that a celestial body emitted  because the Hubble constant contains speed factor Rt N photons at time t1 with frequeNcy 1 , we have   actually, the formula is related to velocity essentially. In LNht11/ . During the time t0 ~ tt00 , these photon arrive at the spherical surface with radius fact, the Hubble red shift is considered to be caused by rRr 0 and the frequency becomes  0 , we have the recession velocity of celestial body. In this way, the 22 universal expansion is deduced. The red shift observed in BNh0004 Rrt. So we get drL (1 zRrz )0 (1  ) . cosmology is also related to gravity. Strictly speaking, If space-time is flat with tt01, we have we should consider both the Doppler’s and gravitational 1/2 dRrzL (1 ) . If light’s frequency does not change red shifts simultaneously in cosmology.

5. The Red Shift Caused by the R-W Metric relativity, gravity will cause red shift. In this case, besides the Doppler’s red shift, we should consider gravity As shown in Figure 4, suppose that a luminous celestial red shift too. Because the intensities of gravity are dif- body is located at position rt  Rt r at moment 11 ferent in different places,  is related to the position t . The frequency of emitted light is  . The observer at 1 1 1 where light’s source is located. So  is not proper fre- original point O receives the light at moment t . The 1 0 quency again. frequency of received light is  0 . Suppose that light source emits another light in time interval t  with However, in practical observation, we can not measure 11 the intensity of gravity field in distance place, so we do period   c  . If the light is received by observer 11 not know  actually. We only compare observed fre- located at original point O at moment t  , the fre- 1 00 quency  with proper frequency to calculate red shift. quency of received light is  . 0 0 So we can not used (1) to describe gravity red shift. In According to Figure 4 and based on the R-W metric, fact, if the universe is static, we have Rt Rt we can obtain (1) which is related to scalar Rt and 10    and z  0 , even thought there exist gravity field and not related to the speed factor Rt of spatial expan-   gravity red shift. sion. But the Doppler’s red shift is related to Rt ,   So metric red shift described by (1) is different from having nothing to do with Rt  . Both are completely the Doppler’s shift and gravity red shift. It contradicts different. with the Doppler’s red shift. We take some concrete Let’s discuss the real meaning of Formula (1) now. In examples to prove this point further below. order to determine red shift, we have to know  0 and 1 . Because we determine  0 by measurement, the key 6. The Red Shift in the Uniform Expansion is to determine 1 . In practice, 1 is considered as Process of Space light’s proper frequency. Suppose that there is no gravity field in space without gravity red shift. Suppose that The R-W metric can be used to describe both the uniform and accelerating expansion of space. The uniform there is an observer located at point rt1 who is at rest with light’s source. According to this observer, expansion is equal to the uniform motion of light’s source. In this case,  is the proper frequency of light light’s frequency is  , i.e., proper frequency. When 1 1 for the observer located at K . For the observers at rest light travels from rt to original point rt  0 , 1 00  in K , light’s proper frequency is also  , but received frequency changes from  to  . That is to say, the 0 1 1 0 frequency is  . Let Rt   a bt, a and bc are change of frequency is caused by the distance between 0 constants. The speed of spatial expansion is light’s source and observer, rather than the speed of Vt   Rtr   br. According to the Doppler’s and light’s source. This is just the essence of metric red Hubble’s formula, we have the same red shift shift. However, this idea will cause serious problem, incon- 1Vt c Vt br z  (74) sistent with the Doppler’s red shift. According to ob- 1Vt c c c server at original point rt00 0 , the celestial body at point has a moving speed. The frequency of light emitted The red shift is proportional to coordinate r (dis- by the celestial body is not proper frequency with the tance), having nothing to do with the time that light was Doppler’s red shift. The change of frequency is related to emitted. The red shift is the same no matter what time speed and not related to distance. In fact, if frequency is observer observes. However, according to the metric red shift formula (1), we have related to distance, when a static observer measure the frequency of light emitted by a static light’s source in abt bt t z 0 1 01 (75) distance, red shift would be founded. Obviously, this abt abt result violates experiments and basic physical laws and is 11 completely impossible. So the red shift caused by the z represents the red shift that observer observes at mo- spatial expansion can only be the Doppler’s red shift ment t0 . The Hubble constant at moment t1 is related to speed, rather than the metric red shift related to Rt   b distance. Ht1 (76) 1 Rt a bt If there is gravity field in space, according to general 11 The uniform expansion of space corresponds to the rt 11  Rt  r uniform motion of light’s source, and light’s speed is t0 v0 O V unrelated to the motion of light’s source, or light’s speed K K is invariable in vacuum. Let dtL 101 ctt, in this 0 c v1 case, dtL  1  is just the distance between observer at Figure 4. The red shift caused by the R-W metric. original point and luminous celestial body at moment t1 .

Therefore, we can write (75) in the form of the Hubble’s Because acceleration effect has been omitted, light’s red shift frequency is unchanged in the process of light’s propaga-

Ht1 tion. The red shift observed by observer at original point zHtt101 t dtL 1 (77) c at time t0 is also represented by (80). The Hubble constant at time t is It seems that (77) is the same as the Hubble’s formula. 1  But it is different from (74) essentially. According to Rt 1  bgt 1 Ht1  2 (81) (74), red shift is proportional to r and is not a function Rt1 abtgt112 of time. We observe at present time t0 , and red shift is z  0 . According to (77), red shift is related to So the Hubble’s red shift is  dt ctt. When tt , the distance between Ht 11 Rt L 101 01 zrtRtr light’s source and observer is zero, red shift is also zero 11 ccRt1 even though the relative speed between them is non-zero. (82) Rt  r b gtr Speaking simply, according to the Doppler’s formula, 11 red shift is related to the velocity of light source, having cc nothing to do with distance. When the distance is zero It is the same as the Doppler’s red shift (80). If we and light source’s speed is light’s speed, red shift infinite. consider (1), the red shift becomes According to the formula based on R-W metric, red shift 2 is related to the distance, and unrelated to the velocity of abtgt002 z  2 1 (83) light source. When the distance is zero and light source’s abtgt112 speed is light’s speed, red shift is still zero. The differ- By using (81) and let dtL 101 ctt, we can write ence is so great and essential so that they can not be con- (82) as sistent at all. Ht  qt H2  t zdt111 dt2  (84) 7. The Red Shift in the Uniform Accelerating c LL112 c2 Expansion Process of Space This is considered as the revised Hubble formula in

If light’s source is accelerated in vacuum, according to cosmology. In which dtL  1  may not be the real dis- general relativity, non-inertial motion is equal to gravita- tance of light’s source, we called it as apparent distance. tional field and gravity red shift will be produced. qt 1  is the so-called decelerated factor with Therefore, in this case, when luminous body moves in 2 ga  bt11 gt 2 gravitational field, gravity red shift should also be con- qt1  2 (85) sidered. Because the red shifts are different in different bgt 1 places with different intensities of gravity, the frequency (84) is completely different from (80) and (82), and they  of emitted light is different in different places. So  1 1 are incompatible. According to (80) and (82), red shift is shown in (1) is not proper frequency again. The observer still proportional to coordinate r . At present moment, located at original point O does not know the initial we have zbgtr 0  . But according to (84), the rela- value 1 . He can only compare received frequency  0 tion between red shift and dtL 1 is not linear again. At with proper frequency. So in this case, (1) has nothing to present moment, we have z  0 , even though the rela- do with practical measurement. That is to say, if gravity tive speed between them may not be zero. It is obvious or acceleration effects are considered, Formula (1) is that the Doppler’s red shift and the Hubble’s red shift are unavailable. consistent, but they are completely different from the If gravity and acceleration effects are omitted, we metric red shift. discuss the differences between the Doppler’s and Hubble’s red shifts and the metric red shift. Suppose 8. The Red Shift of Metric in General Situations that space is flat, light’s source is uniformly accelerated with If the acceleration of spatial expansion is arbitrary, according to standard theory, we develop Rt into the 1 2 Rt a bt  gt (78) Taylor series in the neighborhood of t with [8] 2 0 Vt Rtr b gtr (79)  2     Rt000 Rt t t Rt Rt001 t t  When Vt c, the Doppler’s red shift at moment t1 is Rt Rt 2  00 (86) 1bgtrc bgtr  z 11 (80)  1 2 2   Rt0000001 Ht  t t  qH t t t   1bgtrc1 c  2 

Here the decelerated factor is t0 and t1 actually. When their difference is great, we  can only develop Rt 0  into the Taylor series in the Rt 00 Rt qt00 (87) neighborhood of t0 and develop Rt1 into the Taylor Rt 2 2 0 series in the neighborhood of t1 . The results should be If the items in the bracket of (86) are small enough, let  1 22 dt ctt and substitute it in (1), we can obtain Rt00 Rt 1 Ht  00000 t qH t  t  (89) L 101  2  2 Ht000 qt H t 2  1 22 zdtLL11 dt  (88) c 2 2 Rt11 Rt 1 Ht  11111 t qH t t  (90) c  2  (88) is considered as the revised Hubble formula, al- Here t0  1 and t1  1 . So (1) should be though dtL 1 may not be the real distance of light’s 1 source. However, the formula should satisfy the condi- 1 Ht t  qH22 t  t  002 0 00 tion that tt01 is small enough, otherwise the series is z  1 1 divergent. Besides, it should note that H t0  and 22 1Ht11 t qH 1 t 11 t qt00 in (86) is the quantities at present time. But in 2 (82) and (84), H t and qt  are the quantities at 1 1 1  H ttHtt  qHtt22  (91) past time. 00 112 0 00 However, for the problems of cosmology, tt may 01 1 22 be very large. For example, for the supernova of high red qH111 t  t  2 shift, tt01 may be billion years. Because the high or- n It is completely different from (88). That is to say, we der items in (86) are proportional to tt01 , when n , the formula is infinite, so it is unsuitable in gen- can not obtain the revised Hubble’s formula (88) from the metric red shift formula (1) in general situations. eral. In fact, as shown in (1), the positions of times t0 and t1 are equal. We have two independent variables