International Journal of Astronomy and Astrophysics, 2012, 2, 183-193
http://dx.doi.org/10.4236/ijaa.2012.23023 Published Online September 2012 (http://www.SciRP.org/journal/ijaa)
The Formula to Calculate the Red Shift-Distance Relation
of Ia Supernova in Cosmology Has Essential Mistakes
The Calculation of Cosmological Red Shift Should Use the Doppler Formula Directly
Xiaochun Mei, Ping Yu
Institute of Innovative Physics in Fuzhou, Department of Physics, Fuzhou University, Fuzhou, China
Email: ycwlyjs@yeah.net, Yupingpingyu@yahoo.com
Received March 21, 2012; revised April 25, 2012; accepted May 8, 2012
ABSTRACT
There are three main mechanisms to cause the red shift of spectrum in physics. The first is gravity which is related to
mass. The second is the Compton scattering which is related to the energy transformation of photon. The third is the
Doppler’s effect which is related to velocity. The basic formula used to calculate the relation of red shift and distance of
Ia supernova in cosmology is

0
1zRR t which is related to the scalar factor
t of the R-W metric. It is com-
pletely different from the Doppler formula of red shift which is related to velocity factor
Rt
. This kind of inconsis-
tency is not allowed in physics. Because of
0
Rt R
, when
Rt became larger and larger with time increase, z
became smaller and smaller, means that space expansion leads to red shift becoming smaller. At present time, we have
and , means that there is no red shift for the light emitted from distance celestial bodies at present.
The results obviously violate the Hubble law! It is proved strictly in mathematics that the formula

0
Rt R00z
0
1zRR t
is
untenable unless constant and

Rt 0
R0z
. The further study reveals that the essential reason of the mistake is
that the R-W metric violates the principle of light’s speed invariable. The time delay caused by relativity velocity be-
tween light’s source and observer is neglected. Besides, there exists the problem of time misalignment between theo-
retical calculation and practical observations in the original documents of Ia supernova projects. So the formula used to
calculate the relation between red shift and distance of Ia supernova is wrong and the deduced conclusion about dark
energy and the accelerating expansion of the universe are incredible. It is proved in this paper that based on the Dop-
pler’s formula and 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 unneces-
sary in cosmology.
Keywords: Cosmology; Doppler Formula; Hubble Law; Supernova; Dark Energy; R-W Metric
1. Introduction
As we know that there are three main mechanisms to
cause the red shift of spectrum. One is gravity which is
related to mass and another is the Doppler’s effect which
is related to velocity. According to the Hubble law, the
spectrum red shift of extragalactic nebula was propor-
tional to the distance between observer and luminous
celestial body. The red shift of cosmology is considered
to be the Doppler’s effect. In 1998, cosmic observations
found that the high red shift of Ia supernova deviated
from the linear relation of Hubble law. By fitting the ob-
servation values with standard theory of cosmology,
cosmologists concluded that more than 70% of the uni-
verse material was dark energy. The universe seems be
doing accelerating expansion [1,2].
Now that the red shift of cosmology is considered as
the Doppler’s effect, we should use the Doppler’s for-
mula to do calculation. However, it is strange that the
basic formula used to calculate the relation of red shift
and distance of Ia supernova is completely different from
the Doppler formula. The formula is

 
00
1Rt R
zRt Rt
 (1)
Here
Rt is the scalar factor of R-W metric. At pre-
sent moment 0, we take . It is well
known that the Doppler’s formula is related to the velo-
city factor
t

00
1Rt R
Rt
of spatial expansion. But (1) is only
C
opyright © 2012 SciRes. IJAA
X. C. MEI, P. YU
184
relative to , which has nothing to do with

Rt
Rt
.
The difference is so big that they are completely incom-
patible. This inconsistence is not allowed in physics.
More serious is that according to (1), at initial moment
when the distance between observer and light’s source is
zero with , we have . Such initial red
shift is strange. Meanwhile, at past time, we had
. With time increased, became greater
and greater, z became smaller and smaller, means that
space expansion leads to red shift becoming smaller. At
present time, we have and , means
that there is no red shift for the light emitted from dis-
tance celestial bodies at present. The result obviously
violates the Hubble law!

0Rtz

Rt
00
R

0
Rt R

Rt 0z
The further analysis indicates that the problem origin-
nates from the R-W metric. It is proved that the R-W
metric violates the principle of light’s speed invariable so
it is not the metric of relativity. The time contraction
between observer and moving light’s source is neglected
when we us the R-W metric to describe the spatial ex-
pansion. So the R-W metric is unsuitable to be used as
the basic space-time frame. Especially, it is unsuitable to
be used to describe the high red shift of supernova in
which the high speed expansion of the universe is in-
volved.
Besides, there exist a problem of time misalignment
between the result of theoretical calculation and practical
observation. Because light’s speed is finite, it needs time
for light to propagate from luminous celestial bodies to
observers on the earth. The light observed by the obser-
vers on the earth was emitted by Ia supernova billions of
years ago. The positions and the red shift values of Ia
supernovas observed by observers on the earth were that
were emitted billions of years ago. At present moment,
the real positions and red shift values of these supernovas
are completely different from that we observe now.
However, in the original documents [1] and [2] of Ia
supernova observations, the problem of time alignment
was neglected. The observed values which presented the
situations of past time were fitted directly with the cal-
culating values of theory which presents the situations of
present time. Then the dark energy and the accelerating
expansion of the universe were deduced. The error is
very great so that the result can not be tenable.
In sum, the formula used to calculate the relation of
red and distance of supernova has essential mistake.
Based on this formula, the concepts of dark energy and
the accelerating expansion of the universe are incredible.
It is proved that if the Doppler’s formula is used directly
to calculate the relation of red shift and distance of Ia
supernova, we do not need the hypothesis of dark energy
and the accelerating expansion of the universe again.
2. Inconsistency of Two Formulas to
Calculate the Relation of Red Shift and
Distance
In order to use (1) to describe the red shift of cosmology,
we should fix
Rt and let change with time.
When the universe expands, z increases with

0
Rt
0
Rt
bt
increasing. However, in this case, (1) also lead irrational
result. For simplicity, we assume that space expands
linearly with time increasing and let . The
distance between observer and light’s source is

Rt a
rt
Rtr abtr. The speed of spatial expansion
Vt br is a constant. Suppose initial time 10t
, we
have
1
rt ar
0
rta bt r
10
10
0
. Take present time
, let
10
010tab
 and 1r. According to (1),
we have
10
00
0
1
11
abt abt
zt
abt a


10 (2)
If (2) is used to describe the special expansion between
two luminous atoms, it means that distance between them
is
10
110 mrt
at beginning. Then they separated
with each other in a uniform speed 10
10m sV
. After
0 (about 317 years), their distance becomes 1 m
but the red shift reaches . The bigger problem is that
the red shift increases with time’s increasing, though
atoms move in uniform speeds. The result is very absurd.
If using Doppler’s formula, when
10
10 st
10
10
1Vc, we have
10
19
8
10 3.310
310
Vbr
zcc
 
(3)
The difference between (2) and (3) is times!
Because the Doppler’s formula is verified by many ex-
periments, (3) should be correct and (2) is certainly
wrong. The origin of mistake is that in the deduction of
(1), following relation is used
28
310


01
01
Rt Rt
(4)
In which 1
is the period of emitted light and 0
is
the period of received light. We will prove strictly below
that (4) is untenable unless constant or


01
Rt Rt
0z
.
At first, we discuss how to use the Doppler’s formula
to calculate the red shift in cosmology. Suppose that ob-
server is rest at the original point of reference frame. The
light’s source moves in velocity relative to observer.
The proper frequency and period of light observed by
observer who is at rest with light’s source are 1
V
and
1
. The frequency and period which observer receives at
the original point of reference frame are 0
and 0
.
According to the Doppler’s formula, we have relations
01
10
22 22
1cos 1cos
,
11
Vc Vc
Vc Vc





(5)
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X. C. MEI, P. YU 185
If light’s source leaves observer, we have cos 1
.
According to the definition of red shift, we have
1
0
1
11
1
Vc
zVc
 
or
2
2
2
22
Vzz
czz

(6)
The co-moving coordinate

rRtr is used in
cosmology. The Friedmann equation of cosmology is
2
2
8π
3m
RG
RR

 


(7)
Here m
is the density of normal material and
is
the density that cosmic constant
corresponds to. Let
m
be the total density of the universe material,
suppose that the universe expands along the direction of
radius, according to (7), we have
2
8π
3
GR
VRrr
 
(8)
We define
 

Rt
Ht Rt
(9)
The observation of WMAP shows that our universe is
approximately flat with curvature factor , so at
present moment , the Hubble formula is
0
0
t
8π
3
G
VRrr HRr
 
(10)
(6) becomes
11
1
H
Rr c
zHRrc
(11)
The relation between red shift and distance is not lin-
ear. If 1Vc, we obtain the Hubble law
VHRr
zcc
 (12)
In this case, the relation between red shift and distance
becomes linear.
However, the problem is that in the current cosmology,
we do not use (6) to calculate the red shift of Ia super-
nova. In stead of, we use following formula [3]



00
0
2
00
1
dsin
d
11 2
Lk
k
z
m
z
Hn
z
zzzz



(13)
In which
L
d is the luminosity distance
 
0
11
L
dzrzR r (14)
For flat space,
L
d is common distance, i.e., L
dr
.
By introducing so-called effective energy density of cur-
vature [3]
2
3
8π
kGR
 (15)
we define
k
kc
 (16)
Here c
is critical density. At present time , we have
0
t
00mmc
, 00kkc
, c

 . Cos-
mologists use (13) to calculate the red shift of Ia super-
nova and deduces that about 70% of the universe mate-
rial is dark energy, about 25% is dark material. Based on
these, the conclusion is that our universe is doing accel-
erating expansion now. However, (13) is completely dif-
ferent from the Doppler’s formula (6). Which one is cor-
rect? We prove blow that (13) is certainly incorrect. We
should directly use the Doppler’s formula (6) to calculate
the red shift of Ia supernova.
3. The Mistake in the Deduction of Ia
Supernova’s Red Shift-Distance Formula
3.1. The Relation

τRt τRt
0011
is
Untenable
The formula (13) is based on the R-W metric

2222
2
222 22
2
dd
ddsind
1
sctRt
rrr
r





(17)
For light’s motion, we have and get from (17)
d0s

2
22 2
2
d
d1
r
ct Rtr
(18)
Because light’s source is fixed at point r, r does
not change with time for light’s source, but for light’s
motion, r changes with time. Suppose that photon’s
coordinate is 1
r at moment 1 and photon arrives at the
original point
t
00r
at moment . The integral of (18)
is [4]
0
t

0
11
0
1
2
1
1
1
dd
sin
1
sin 1
0
sinh 1
t
tr
ctr nr
Rt r
rk
rk
rk
 


(19)
The negative sign indicates that light moves along the
direction of decreasing r. Suppose that a light wave is
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186
emitted during the period of time from 1 to 11
t t
with period 11
c
. Observer receives the light during
the period of time from 0
t to 00
t
with period
00
c
0
. According to the current understanding on
(19), because 0
t
and 1
t1
are decided by the
same 1
r, we have
 
00
11
d
tt
tt
0
1
d
Rt Rt

t
t
(20)
Because 0
and 1
are small, we obtain from (20)


0
Rt
1
01
Rt
(21)
Based on (21) and the definition of red shift, we have


0
0
1
01
Rt
zRt

 
1
1 (22)
By considering (22) and the Freidamann equation of
cosmology, as described in Section 3, we can obtain (13).
We now prove that (21) is untenable unless
0
Rt
and 01

1
Rt
. Let

1
f
tR
t
and take the
integral of (20), we have

10
ft


110
ftft ft
0
 (23)
By developing (23) into the Taylor series, we obtain
 



 
 
01 0 00
3
0011 1
22
11 11
1
2!
1
3!
11
2! 3!
ftftftf tft
ftft ft
ft ft
2
00



 
 
 
 

(24)
Because 0
and 1
are very small, the items with
same orders should be equal to each other, so we have


00 11
f
tft

(25)


2
00 11
ftft 2
 
(26)


3
00 11
ft ft3
 
(27)


4
00 11
ft ft4
 

(28)
Because of
 
1
f
tRt
, (25) is actually the same
with (21). (26) can be written as




01
2
0
22
01
Rt Rt
Rt Rt
2
1
(29)
Substituting (21) in (29), we obtain


0
Rt Rt

1
(30)
Because 0 and 1 are arbitrary, (30) indicates t t
Rt
=
constant, so we have . From (27), we have

0Rt









22
00 11
33
01
23 23
00 11
22
RtR tRtR t
RtRtRtRt


 






  
(31)
Substituting (21) and (30) in (31), we get

00 11
Rt Rt
 (32)
From (28), we get

 



 



3
000 0
4
0
234
00 0
3
111 1
4
1
23 4
11 1
66
()
66
RtRt RtR t
Rt RtRt
RtRtRtRt
Rt RtRt
 
 
 
 (33)
We have
0Rt
 too. So (33) becomes

00 11
Rt Rt

(34)
By considering (30), we have 01
. Substituting it
in (21), we have at last

0
Rt Rt1
(35)
It means that
Rt = constant. So only for stationary
space, (21) can be tenable. According to (22), we have
0z
. No red shift can be observed.
We take two simple examples. Suppose space expands
with time increase, let
2
Rt at and take the integral
of (20), we have
10110 0
11 1111
at tatt

 




0
(36)
Because time coordinates 0 and are arbitrary, if t1
t
0
t
and 1
t1
, we obtain
2
2
00
11
23 23
11 00
tt tt


   (37)
Substitute
2
0
Rt at01
and in (37), we
get

2
1
Rt at
 
 
32 2
32 2
00
11
32 32
10
10
aa
aa
Rt Rt
Rt Rt


  (38)
Let the items with same orders are equal to each other,
we obtain


0
1
10
Rt Rt
(39)


2
2
0
1
32 32
10
Rt Rt
(40)
(39) is just (21). Substitute (39) in (40), we get
12 12
10
Rt Rt or

1
Rt Rt0
. So we get 10
and 0z
. Therefore, (39) can not be used to describe
red shift. We should solve (36) directly and obtain
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X. C. MEI, P. YU 187






2
01
02
1011
01
101
t
ttt
Rt a
Rt aRtaRt a
1


(41)





0
1
0
101
1
1
z
Rt a
Rt aRtaRt a1



(42)
The result is completely different from (22). If space
contracts with time, we take

Rt at. By taking the
integral of (20), we obtain
2
000111
22tt
2


(43)
or


2
01
0
01
22
aa
Rt Rt
2
1
  (44)
It (21) holds, we get 01
and

01
Rt Rt
from (44). We can not observe reds shift too. In order to
reach useful result, we consider (43) directly and obtain
 

22
000111
2
2
1
1
2
01
0
2
2
ttt
a
aa
Rt Rt
Rt

 
 (45)
 

0
1
2
22
10 11
10
1
211
z
aaa
Rt Rt
Rt



(46)
So (21) and (22) can not hold in general situation. It
should be emphasized that red shift is only related to
ratio 01
. Very small changes would cause great
change of red shift, so we should do strict calculation.
However, though (21) can not hold in general, if (20)
is tenable, we can obtain the strict relation between 0
and 1
from (23). Based on definition 01
1z

and (42) or (46), the red shift is still related to
Rt and
unrelated to . The result is still inconsistent with
the Doppler’s formula. Where is the trouble? We discuss
this problem below.

Rt
3.2. The Problem of the R-W Metric
In the R-W metric, we have 00 . It means that we
have universal time in whole space. However, due to the
expansion of the universe, there is a relative speed be-
tween observer who is rest at the original point of coor-
dinate system and the luminous celestial body which is
fixed at a certain point with
1g
0r
. According to special
relativity, there is time delay between them. The R-W
metric can not describe this relation, so it can not be the
metric of relativity. In fact, by using common coordinate
system, the four dimensional metric of flat space-time is
22222222 2
dddd sindsct rrr
  (47)
By using co-moving coordinate

rRtr in (47), we
obtain
  


22
222
2
2222222
d1d2 d
dd sind
Rtr d
s
ctRtR
c
Rt rrr


 




trrt
(48)
It is completely different from the R-W metric (17) when
0
. The metric (48) seems to be curved but is flat
essentially. According to the principle of the Riemannian
geometry, if we can find a transformation to turn a
curved metric into flat, the original one is flat essentially.
If we can not find such a transformation, the original
metric is a curved one in essence. It is obvious that we
can not find a transformation to turn (17) into (47) when
Rt 0
, so the spatial part of (17) can not be flat! Ac-
cording to (48), the time delay of special relativity is

22 2
22
dd1d1
Rtr V
tt
cc
 
(49)
So (48) is the metric of relativity in flat space-time.
Conversely, we can prove that the R-W metric violates
the principle of invariance of light’s velocity. For the
light’s source fixed on the reference fame, coordinate r
does not change with time. But for the light’s motion,
coordinate r changes with time. Suppose that light
moves along the direction of radius with dds
d0
, according to (17), when , we have 0

d
d
rc
tRt
 (50)
The velocity of light relative is

 
 
ddd
dd
c
rt r
VtrRtRt
tt
Rtrc Vtc
 

dt
(51)
(51) indicates that light’s velocity is related to the velo-
city of special expansion. At the moment when light is
emitted out, (51) is just the Galileo’s addition rule of
light’s velocity. When light moves towards observer,
minus sign is taken in (48) so light’s speed is less than its
speed in vacuum. When the light moves apart from ob-
server, plus sign is taken so the light’s speed is great than
its speed in vacuum. Especially, because r increases
with time, enough long time later, light’s speed may
greatly exceed its speed in vacuum.
So the R-W metric violates the principle of invariance
Copyright © 2012 SciRes. IJAA
X. C. MEI, P. YU
188
of light’s velocity. This is not allowed in physics. As we
know that the watershed between classical physics and
modern physics is just on the invariance principle of
light’s speed. Because the R-W metric violates this prin-
ciple, it can not be used as the space-time frame for
modern cosmology which is considered as the theory of
relativity. Especially when the expansion speed of the
universe is very high, huge error will be caused.
Therefore, we should use (48) to describe light’s mo-
tion in flat space-time. Suppose that light moves along
the direction of radius, we have dd ds0

 and
obtain

 
22
22
2
22
1d
2ddd
Rtr
ct
c
RtRtrrtR tr





0
(52)

 
d
d
Rtr
r
tRtR
 
c
t
(53)
By considering (53), light’s velocity is
  
d
dd
dd d
c
Rtr
rr
VRtrRt
tt t
 
c
(54)
The result indicates that light’s speed is invariable.
Similarly, the four dimensional metric in which three
dimensional space has constant curvature is
2
2222222 2
2
d
ddd sind
1
r
sctr r
r

(55)
By using co-moving coordinate in (55), we obtain


 

 
2
22 2
22 22
2
2222
22
2dd
dd
11
ddsind
1
R tRtRttr
sc t
Rtr Rtr
r
Rtr r
Rtr

22

 








(56)
When light moves along the direction of radius, we
have


 



22
22
22
22
2222
d
1
2dd d
0
11
Rtr
ct
Rtr
RtRttrRt r
Rtr Rtr







(57)




22
1
d
d
cRtr
Rtr
r
tRt Rt
 
(58)
If we use (53) and (58) to calculate the red shift of
cosmology, the results are related to velocity. However,
we can not separate variables in (53) and (58), so we can
not write them in the simple form of (19). It is more
convenient for us to use the Doppler’s formula directly to
calculate the red shift of supernova in cosmology.
On the other hand, as proved in document [5], when
scalar factor
Rt is related to time, the R-W metric
has no constant curvature. By using the formula of the
Riemannian geometry to do strict calculation, the space-
time curvatures of the R-W metric actually are [5]
01 0203
2
12 13 232
R
KKK R
R
KKKR



(59)
Here 0
j
K
is the curvature of space-time crossing parts
and ij
K
is the curvature of pure spatial parts. This result
is completely different from the current understanding.
Therefore, constant
is not the factor of spatial curva-
ture. It is a certain adjustable parameter. The R-W metric
does not represent flat space-time when . It does
not represent the metric of curved space with constant
curvature too. It is improper for us to use the R-W metric
to describe the expansion universe with zero or constant
spatial curvature.
0
3.3. Time Misalignment Problem of Theoretical
Calculation and Practical Observations
Because light’s speed is finite, it needs time for light to
propagate from luminous celestial bodies to observers on
the earth. If the celestial body is far ways from the earth,
the light may take billions of years to arrive at the ob-
server. That is to say, light observed by observer on the
earth was emitted billions of years ago. So the positions
and the red shift values of Ia supernovas observed by
observers at present on the earth were that of billions of
years ago. At present moment, the real positions and the
real red shift values of these supernovas are completely
different from that we observe now.
In the formulas (13), 0
H
,
L
d, 0m and 0k
are
the values at present moment 0, therefore, (13) describe
the relation between red shift and distance. However,
Figure 1 describe the observed relation of red shift and
distance of Ia supernova at past moments. Moreover, for
the different points of curves in Figure 1, the times are
different. In order to match times, we should transform
all observation values at past moments to the value at
present moment. Then fit them with the result of theory.
Only in this way, the discussion can be meaningful.
t
However, in the original documents of supernova cos-
mology projects, we have not found this kind of trans-
formations. In stead, the observation values which repre-
sent the red shifts and distances of supernova at past
times are compared directly with theoretical values at
present time. Then the conclusions of and
00.3
m

0.7
are deduced out.
It is obvious that there is a problem of time match be-
Copyright © 2012 SciRes. IJAA
X. C. MEI, P. YU 189
Figure 1. Hubble diagram for red shift and distance of Ia
supernova.
tween theoretical calculation and practical observations.
This problem exists in cosmology commonly, not only
for supernova. The values of theory are present ones, but
the observed values were past ones. At the early period
of cosmology while Hubble deduced the Hubble law, the
observed red shift were small. Because celestial bodies
were near the earth, time for light to reach the earth was
not very great so that the differences can be neglected.
But for supernova of high red shift, great error would be
caused. With this point alone, the result of 00.3
m
and is unbelievable.
0.7

3.4. The Problem of Constant
k
According to definition (9), at present time 0, we have
0
and 0
t

0
t


0
H
Ht, the Friedmann equation
of cosmology can be written as

2
0
02
0
33
8π8π
m
H
GGR

 (60)
Defining critical density c
as
2
0
3
8π
c
H
G
(61)
Because we define 00mmc

 , c


 and
00m
 , (60) can be written as
022
00
1RH
  (62)
Let

0
atRtRt, we have at pre-
sent time. Because is a constant, we can write the
Friedamman equation as

01at


22
2
00
2
0
8π
3
8π
3
m
G
at a
G
H
R


 
(63)
Because we have
33
00
mm
RR

or 3
0
m
am
(64)
so (63) can be written as
2
22
0
00
d1
d
m
m
aHa
ta


 
 
 
(65)
We have ddat a
and ddta
a, so (19) can be
written as
1
01/(1 )
1d
sin
z
a
nr Ra
a
(66)
The upper limit of the integral is and lower
limit is

01at
111at z
. Meanwhile, according to (16),
we can write (62) as
0
22
00
1k
RH
0
 (67)
According to reference [4], from (67) we can get
0
00
1
k
RH
(68)
By introducing transformations
11az
in (66)
and considering (14) and (68), the formula (13) is ob-
tained. However, (68) is obviously wrong. (67) contains
constant
, but (68) does not. According to (67), when
0
we have 00
k
0
k
, 0 is limited. But according
to (68), when 0
R
we have 0. That is to say,
0 is infinite in flat space. This is completely impossi-
ble. According to (67), correct result should be
R
R
0
0
k
RH

(69)
In fact, it is unnecessary to introduce relation
22
kaR

 . Whether or not space is flat depends
on sin nr r
, sin sinnrr
or sin sinhnr r. For
flat space, to take 0
in (63) and substitute it in (66),
by considering 0m1
 , sinrr and L
dr
,
we get


00 2
00
d
11
z
m
z
HRr zz


(70)
Because (68) can not hold, (13) can only be written as




0
00
2
00
1
1sin
d
11 2
L
z
m
dRz n
RH
z
zz zz

 
 
(71)
When space is flat, we have , ,
0
01
m

sin rr
and L
dr
. So (71) becomes
Copyright © 2012 SciRes. IJAA
X. C. MEI, P. YU
190



00 2
00
d
11 2
z
m
z
HRr zzzz


(72)
The reason is that when , the right side of (65) has
only two items with
0
2
2
0
0
d
d
m
a2
H
a
ta



 
(73)
But we still use (65) to deduce (13). If space curved, for
, we still have and obtain
1
01
m

 
03
000 0
1d
1sin
1
z
L
m
z
dR zRH z
 


(74)
Similarly, for , we also have and
1
 01
m

 
03
000 0
1d
1sinh
1
z
L
m
z
dR zRHz

 


(75)
(74) and (75) are also different from (13). It means that
(13) is wrong. This is a mistake of mathematics, having
nothing to do with physics.
Therefore, the formula used to calculate the relation of
red shift and distance of Ia supernova in current cosmo-
logy is wrong. We should use the Doppler formula di-
rectly. The result shows that we do not need the hy-
potheses of dark energy and the accelerating expansion
of the universe in cosmology.
4. Using the Doppler’s Formula to Calculate
the Red Shift of Ia Supernova
4.1. The Friedmaann Equation Needs Relativity
Revision
Standard cosmology uses the Friedmann equation as ba-
sic equation. However, British physicist E. A. Milne
proved in 1943 that the Friedmann equation could be
deduced simply based on the Newtonian theory of gra-
vity. Although the Friedmann equation is described in
curved space-time, the Newtonian theory of gravity is
described in flat space-time, the results are actually the
same when they are used to calculate practical problems,
especially when we take curvature constant 0
.
However, as we know, Newtonian theory is only suitable
for the motions of low speeds. For the high speed expan-
sion of the universe, it is unsuitable. The Friedmann
equation needs relativity revision due to this fact.
The reason leading to this result is that two simplifica-
tions and improper conditions are used in the deduction
process of the Friedmann equation. One is the R-W met-
ric and another is static energy momentum tensor. The
problem of the R-W metric has been discussed above. If
we use static energy momentum tensor in the equation of
cosmology, it means that the velocity and momentum of
material are neglected in the expansion process of the
universe. So the Friedmann equation is the one to be im-
properly simplified and needs relativity revision [5].
However, if dynamic energy momentum tensor is used in
the Einstein’s equation of gravity, the equation of cos-
mology would become very complex to be solved [5].
We have to looking for other more proper method to
study cosmology.
We have proved that by transforming the geodesic
equation of the Schwarzschild solution of the Einstein’s
equation of gravity field to flat space-time for description,
the revised Newtonian formula of gravity is obtained [6]
22
000
2
d
1
d
L
mGMm
cr r

 


rr
2
23
3
(77)
In (76) all quantities are defined in flat space. We have
2
2
d1
Vt
c
d (78)
This is just the time delay formula of special relativity,
so (77) can be considered as the revised formula of rela-
tivity of the Newtonian’ gravity. The space-time singu-
larity in the Einstein’s theory of gravity becomes the
original point r = 0 in the Newtonian formula of gravity.
The singularity problem of gravity theory in curved
space-time is eliminated thoroughly. The theory of gravity
returns to the traditional form of dynamic description.
When the formula is used to describe the universe ex-
pansion, the revised Friedmann equation can be obtained.
Based on the revised theory of gravity, the high red-shift
of Ia supernova can be explained well. We do not need
the hypotheses of the accelerating expansion of the uni-
verse and dark energy. It is also unnecessary for us to
assume that non-baryon dark material is 5 ~ 6 times more
than normal baryon material in the universe if they really
exist. The problem of the universal age can also be
solved well.
We prove below that by using the method of numerical
calculation and the Doppler’s formula proposed in [6],
even based on the Newtonian theory of gravity, we can
also explain the relation of red shift and distance of Ia
supernova well. The hypotheses of dark energy and the
accelerating expansion of the universe become unneces-
sary.
4.2. Using the Doppler’s Formula to Calculate
the Red Shift of Ia Supernova
As we know that the solution of differential equation is
determined by initial condition. However, according to
the big bang cosmology, the universe blew up from a
singularity with infinite density. That is to say, all mate-
rial in the universe has a same initial position. However,
Copyright © 2012 SciRes. IJAA
X. C. MEI, P. YU 191
infinite density is imaginable and singularity can not ex-
ist in the real world. The practical situation should be that
at initial time, strong, weak and electromagnetic interact-
tion can not be neglected. Meanwhile, unknown interac-
tion may exist, so that material can be compressed into
infinite density by gravity.
According to the discussion in [6], we assume that
there exist a certain mechanism so that a uniform mate-
rial sphere with mass 0
M
can only be compress into a
finite radius 0. The motion equation of universe expan-
sion can be written as
r
 
0n
mrFrFr
 (79)
Here

F
r is the Newtonian gravity and
n
F
r is the
sum of all non-gravities. For convenience of calculation,
we suppose
 
0
0
2
n
m
F
rArr
r
(80)
Here

A
r is an unknown function.

n
F
r
corre-
sponds to an infinite barrier at position 0. When a ma-
terial sphere with radius is contracted into a sphere
with radius 0, it can not be contracted again. For the
sphere with different radius , is different.
r
r
r
r0
Suppose that the material distribution of the universe
is uniform with . The static mass contained in
the spherical surface with radius is 0
r

t

r
M
. According
to the revised Newtonian theory, gravity is related to
velocity. Using it to calculate the universe expansion,
under the condition 1Vc, the speed of a particle
located on the spherical surface is



10
2
0
0
2
233
120 56
VQrKr
c
GM
K
r
rr
cr









(81)
Here 2
0
2GM c
.

0
K
ris a constant which de-
scribes initial conditions. Let 0r
in bracket, we
get the result of the Newtonian theory of gravity
 
2
00
22
8
2
3
Gr
VGM
K
rK
ccr c
 r
(82)
We consider an expansion sphere as expansion uni-
verse and use Doppler’s formula (6) to describe red shift.
Suppose that luminous bodies move along the directions
of radius and observer is located at the origin point of flat
reference frame. The distance between observer and ce-
lestial body is at moment . The real distance
between observer and celestial body is 0 at present
moment 0. In the expanding process of the universe,
celestial body moves from 1 to 0
r with 01
, while
the light travels from 1 to observer along opposite di-
rection. Suppose light’s speed is invariable in the process,
we have following relation

rt tr
trrr
r
00
11
1d
d
tr
tr
rr
tt
cV
 
(83)
By astronomical observations, we know the universe
material density 0
at present time 0, but do not
know
t
t
at past time . By relation t33
00
rr
,
we have
3
2
00
2
8π
8π
33
Gr
Gr
cc
2
r
(84)
According to (82) and (84), we can obtain from (83)


00
11
122
00
dd
83
rr
rr
rr
rVc Grcr Kr

 (85)
In principle, we can write (85) as

100 10
,,rfrKrfrKr
 (86)
From (86), we can get in principle. In
other word, what observer see now is the light that celes-
tial body emitted at position 10
and at time 10
01
,rgrK
rr
tt
.
But at present time 0, the celestial body has moved to
position 0. In the formulas above, 0
t
r
, 1 and are
known through observations, but 0 and
r z
r
0
K
r
are
unknown. By connecting (6) and (86), we can determi-
nate 0 and r
0
K
r
. (84) can only be calculated by nu-
merical method through computer. We take
27 Kg m
3
0, ,
and have
10
b 26
00
10 mry 26
10 m
11
ry
3
0
0.25by
xry
 (87)
We use
x
as basic variable to calculate 0 and y
0
K
r
. In the calculation, we take b, and 1 as
input parameters. Therefore, according to this method,
we actually deduce the initial situations of the universe
expansion based on the present observations of red shift
and distances. In other words, as long as the initial condi-
tions of the universe expansion are known, we can know
its current situations.
z y
4.3. The Red Shift of Ia Supernova
In Figure 1, the curved line with 0 and 0.3
m

0.7
t
represents practical relation between the red
shift and distance of Ia supernova at the early period of
time . According to photometry measurement, the den-
sity of luminous material in the university is about
3
28
021
0 kgm
at present day. Because there exist
a great mount of non-luminous material, we suppose that
practical material is 10 times more than luminous mate-
rial and let 3
27
0210kgm
 . In Figure 1, we take
5.5 5log
B
L
md
in which
L
d is luminosity distance
Copyright © 2012 SciRes. IJAA
X. C. MEI, P. YU
192
with unit length . But the concept
of luminosity distance is unnecessary in this paper, be-
cause our discussion is based on flat space-time. So we
need to transform
6
103.0910mpc 
22
L
d
into practical distance r.
The curved line in Figure 2 shows the relations be-
tween the red-shifts, distances and initial condition pa-
rameters of Ia supernova. The vertical coordinate is the
values of 0
K
r. The bottom horizontal coordinate is
the value of red-shift. On the upside, under the line of
horizontal coordinate, are the values of distance r, above
the line is the values of 0. For and r1z25
B
m
, we
get . By numerical calculation, we ob-
tain and 0
26 m
26
10 m

Kr
11.r
0
r23 10
1.83 2
2.60 10

23.1
00.91r
20.1z
26
0.15 10m

3
05.30 10Kr


00Kr

00Kr
0.7zz
1z
26
.1
. For
and corresponding to
, we obtain and
0. For and corre-
sponding to , we obtain
and .
0.5
0.67

0.16
z
1
r
Kr
0
r
B
m
26
10 m
2.51 10
1
r
26
10
10
B
m19
0
We see that by directly using the Doppler’s formula
and the Newtonian formula of gravity, we can explain
the high red shift of Ia supernova well. The hypotheses of
dark energy and the accelerating expansion of the uni-
verse become unnecessary. The universe began its ex-
pansion from a finite volume, rather than a singularity.
The difficulty of singularity in cosmology is eliminated.
If we use the revised Newtonian formula, according to
reference [6], the result is shown in Figure 3. Comparing
Figures 2 and 3, the difference is that for the Newtonian
gravity, we have . But for the revised Newto-
nian gravity, we have when , and
when . When is very small, we
have for both situations.
0.7z

00Kr
0
Kr

0
5. The Age of the Universe
We consider the universe as a material sphere with radius
at initial moment, which is about the
distance between the sun and the earth. Long enough
later, at time , an observer located at the original point
of reference frame receives the light emitted from a ce-
lestial body on the spherical surface with radius r =
and red shift at time 0. Suppose that
the material density of the universe is
11
01.5 10 mr
t
26
1.23 10t
27
210

3
kg m at present, the initial density inside the sphere is
3
17 k

0
Kr
5
0, to be equal to the density of neu-
tron star. According to the calculation before, the real
distance of celestial body is 0 at present
time. We consider it as the radius of observable universe
and take for following formula to cal-
culate the time during which the universe expands from
radius to r
5.9 1

0
r
0 g
1.1
m
0.
11
0 m
26
10
26
0 m
1.83
1.83 1
m
r
6
0
02

00 0
11
1
32
00
dd
d
830.02
r
6
r
r
tt cGr cr
 
 
trr
tr
V (88)
r
0
0.16 0.35 0.54
0.72 0.91
1.13 1.33 1.49
1.681.83×10
26
m
r 0.15 0.30 0.43
0.55
0.67 0.81
0.93 1.03
1.14
1.23
K
0.15
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.
r
0
0.16
0.35 0.54
0.73 0.91
1.14 1.35 1.54
1.73 1.90×10
26
m
r
0.15 0.30
0.43
0.55
0.67
0.81
0.93 1.03
1.14
1.23
K
0.08
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.
The result is 35t
r
1.8310
26
1.23 10
billion years. But this value is
not sensitive to 1 when it is not very large. Taking
, which is about the radius of the Milky Way
galaxy, the result is almost the same with
. It means that the age of the universe mainly depends
on the later expansive process. Using (88) to calculates
the time during which the radius of sphere expanses from
to , the result is 15.4 billion
years, so the time during which the radius of sphere ex-
panses from to is 19.5
billion years.
20
110mr
m
26
1.23 10m
11
01.5 10r
26 m
26 m
m1.83 10
By using the revised Newtonian formula of gravity, for
the same red shift 1z
, the result is that the time is 30.8
billion years for a sphere’s radius expands from 1
r
to and 13 billion years
for radius expands to . So
the sphere’s radius expands from to
is 17.8 billion years.
11
1.5 10m
26
1.23 10m
26
010 mr
26
10 m
1.95
1.23 26
1.95 10m
11
1.5 10m
Therefore, for the same red shift, by using the revised
Newtonian gravity, the age of the universe is smaller
then using the unrevised Newtonian gravity. The reason
is that gravity becomes small after the revision of relati-
vity. Material needs more time moving to the same posi-
tion. According to the current cosmology, the universe
age is estimated to be about 10 - 15 billion years, too
Copyright © 2012 SciRes. IJAA
X. C. MEI, P. YU
Copyright © 2012 SciRes. IJAA
193
short to the formation of galaxies. The problem does not
exist by using the Doppler’s formula to calculate the red
shift of cosmology, no mater we use the revised Newto-
nian formula or the unrevised Newtonian formula of
gravity.
6. Conclusions
The red shift of cosmology is considered as the Dop-
pler’s effect. However, the basic formula used to calcu-
late the high red shift of Ia supernova in current cosmo-
logy is related to scalar factor rather than velocity
factor . There exists inconsistency which is not
allowed in physics. It is proved that the current formula
used to calculates the relation of red shift and distance of
Ia supernova is wrong in cosmology. We should directly
use the Doppler’s formula to calculate the red shift of
cosmology. By the method of numerical calculation,
based on the Newtonian gravity and the Doppler’s for-
mula, it is proved that the red shift of Ia supernova can be
explained well. The hypotheses of dark energy and the
accelerating expansion of the universe are completely
unnecessary. The problem of the universe age can be
solved well.

Rt

Rt
The procedure we developed and used in this paper
tilled “Using Revised Newtonian Gravity and Doppler’s
Formula to Calculate Cosmological Red Shift” and its
source code are open to researchers. Demanders please
send us e-mail to obtain it.
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