Journal of Modern Physics, 2013, 4, 974-982
http://dx.doi.org/10.4236/jmp.2013.47131 Published Online July 2013 (http://www.scirp.org/journal/jmp)
The Singularities of Gravitational Fields of Static Thin
Loop and Double Spheres Reveal the Impossibility of
Singularity Black Holes
Xiaochun Mei
Institute of Innovative Physics in Fuzhou, Fuzhou, China
Email: ycwlyjs@yeah.net
Received April 21, 2013; revised May 22, 2013; accepted June 20, 2013
Copyright © 2013 Xiaochun Mei. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In the classical Newtonian mechanics, the gravity fields of static thin loop and double spheres are two simple but foun-
dational problems. However, in the Einstein’s theory of gravity, they are not simple. In fact, we do not know their solu-
tions up to now. Based on the coordinate transformations of the Kerr and the Kerr-Newman solutions of the Einstein’s
equation of gravity field with axial symmetry, the gravity fields of static thin loop and double spheres are obtained. The
results indicate that, no matter how much the mass and density are, there are singularities at the central point of thin
loop and the contact point of double spheres. What is more, the singularities are completely exposed in vacuum. Space
near the surfaces of thin loop and spheres are highly curved, although the gravity fields are very weak. These results are
inconsistent with practical experience and completely impossible. By reasonable analogy, black holes with singularity
in cosmology and astrophysics are something illusive. Caused by the mathematical description of curved space-time,
they do not exist in real world actually. If there are black holes in the universe, they can only be the types of the Newto-
nian black holes without singularities, rather than the Einstein’s singularity black holes. In order to escape the puzzle of
singularity thoroughly, the description of gravity should return to the traditional form of dynamics in flat space. The
renormalization of gravity and the unified description of four basic interactions may be possible only based on the
frame of flat space-time. Otherwise, theses problems can not be solved forever. Physicists should have a clear under-
standing about this problem.
Keywords: General Relativity; The Einstein’s Equation of Gravity Field; Axially Symmetrical Solutions; Singularity;
Kerr Metric; Kerr-Newman Metric; Gravitational Field of Static Thin Loop; Gravitational Field of Double
Spheres Black Hole; Quasar; MECO
1. Introduction
According to the Einstein’s theory of gravity, singulari-
ties exist at the centers of celestial bodies when material
densities are great enough and gravity fields are strong
enough. However, singularities are always confusing.
Real world can not be infinite. If infinite appears in our
theory, we have to argue whether the theory has some-
thing wrong. Making a general survey of scientific his-
tory, we see that physical progresses are often based on
the elimination of infinite. Facing the problem of singu-
larity in the Einstein’s theory of gravity, such as singu-
larity black holes, we should be skeptical rather than ap-
preciative.
In fact, the author has proved that the present theory of
singularity black hole is impossible by calculating the
precise inner solutions of gravity field equations of hol-
low and solid spheres [1]. To avoid space curvature infi-
nite at the center of solid sphere, we set an integral con-
stant to be zero directly at present. However, according
to the theory of differential equation, the integral con-
stant should be deter-mined by the known boundary con-
ditions of spherical surface, in stead of the metric at the
spherical center. By considering that fact that the vol-
umes of three dimensional hollow and solid spheres in
curved space are different from those in flat space, the
integral constants are proved to be nonzero. The results
indicate that no matter what the masses and densities of
hollow sphere and solid sphere are, there exist space-time
singularities at the centers of hollow sphere and solid
spheres.
Meanwhile, the intensity of pressure at the center point
C
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X. C. MEI 975
of solid sphere can not be infinite. That is to say, the ma-
terial can not collapse towards the center of so-called
black hole. At the center and its neighboring region of
solid sphere, pressure intensities become negative values.
There may be a region for hollow sphere in which pres-
sure intensities may become negative values too. The
common hollow and solid spheres in daily live can not
have such impenetrable characteristics. The results only
indicate that the singularity black holes predicated by
general relativity are caused by the descriptive method of
curved space-time actually. If black holes exist really in
the universe, they can only be the Newtonian black holes,
not the Einstein’s black holes.
According to the practical observations by Rudolf E.
Schild and Darryl J. Leiter [2], the centre of Quasar 0957
+ 561 which was considered to be a black hole is actually
a close object, called a MECO (Massive Eternally Col-
lapsing Object). Unlike an empty hole, it is surrounded
by a strong magnetic field and material. This result chal-
lenged traditional astrophysics and cosmology. It implied
that the current theory of singular black hole may be
wrong. We have reason to ask such a question. Whether
or not singularity black holes, predicted by general rela-
tivity, really exist in the universe?
In this paper, we discuss the gravitational fields of
static thin loop and double spheres. Based on the coordi-
nate transformations of the Kerr and the Kerr-Newman
solutions of the Einstein’s equation of gravity with axial
symmetry, the gravitational fields of static thin loop and
double spheres are calculated. The results indicate that,
no matter what their masses and density are, the spatial
curvatures at the central point of thin loop and the con-
tact point of two spheres are infinite. What is more, the
singularities are completely exposed in vacuum. The
spaces nearby the surfaces of loop and spheres are highly
curved, even though their masses are very small so that
the gravitational fields are very weak.
These results are completely inconsistent with practi-
cal experience. They are very absurd and completely im-
possible. The only possible explanation is that the singu-
larities are caused by the description method of curved
space-time. By logical analogy, so-called singular black
holes and white holes as well as wormholes which con-
nect both holes in the current cosmology and astrophys-
ics are something illusive. They have nothing to do with
the real world actually. If there are black holes in nature,
they can only be the type of the Newtonian black holes,
i.e., in a certain region in which light can not escape but
there is no singularity, rather than the Einstein’s singu-
larity black holes! In fact, as we know that our university
is actually a great black hole by considering its mass and
radius! However, we live in it normally. Where is singu-
larity?
In order to escape the problem of singularity thor-
oughly, we should describe gravity in flat space-time.
The author has proposed a scheme by transforming the
geodesic equation of the Schwarzschild solution of the
Einstein’s equation of gravity field to flat space-time for
description, the relativity revised Newtonian formula of
gravity can be obtained [3]. The space-time singularity in
the Einstein’s theory of gravity becomes the original
point 0r
in the Newtonian formula of gravity. It is
proved that the formula can describe the procession of
Mercury perihelion well.
When the formula is used to describe the universe ex-
pansion, the revised Friedmann equation can be obtained.
Based on it, the high red-shift of Ia supernova can be
explained well. We do not need the hypotheses of the
accelerating universe and dark energy. It is also unnec-
essary for us to assume that non-baryon dark material is
5 - 6 times more than normal baryon material in the uni-
verse if they really exist. The some puzzle problems in
cosmology such as the Hubble constant and the universal
age can also be solved well.
2. The Gravitational Field and Singularity of
Static Thin Loop
The gravitational field of static thin loop is discussed at
first. As shown in Figure 1, a thin loop with mass
M
and radius is placed on the
b
x
y plane. The center
of ring is located at the origin point of spherical coordi-
nate system. The ring is thin enough so that its cross sec-
tion can be neglected comparing with its perimeter. It
will be seen later that even though the cross section of
loop is not zero, the result is also the same essentially.
Because the mass distribution of thin loop has axial
symmetry, the metric tensor of gravitational field does
Figure 1. The gravity field of static thin loop.
Copyright © 2013 SciRes. JMP
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976
t
not depend on time and coordinate
, so the four di-
mensional linear element can be written as
Copyright © 2013 SciRes.



22
00
22
33
d,
,d
,s

2
11
22
22 2
d,d
ind
s
gr t
gr r
gr r


gr r


r
(1)
By introducing coordinate transformations

,rr,,tt



 and

,r



2
11
2
,d
ind
d
, we can
rewrite (1) as




22
00
22
22
22
33
12
d,d
,d
,s
,d
gr t
gr r
grr
gr rr







gr r




(2)
The Formulas (1) and (2) are with axial symmetry and
can be used to describe the gravitational field of thin loop.
Using these metrics in the Einstein’s equation of gravity,
we can obtain the concrete forms of metric tensor in
principle. However, it is difficult to solve the equation of
gravitational fields directly.
On the other hand, we know that there is a ready-made
solution of gravitational field’s equation with axial sym-
metry and two independent parameters, i.e., the Kerr
solution [4]. If the solution of the Einstein’s equation of
gravity with the same symmetry and parameters is
unique, we can obtain the solution of static mass distri-
bution of thin loop by means of the coordinate transfor-
mations of the Kerr solution. Besides, we seem to have
no other choice. The method is discussed below.
The Kerr solution with two free parameters is

22
222
222 2
2222
22 2
222
222
2222
222
2
d1 d
cos
cos d1cosd
2
2sin
1sind
cos
2sin
2sindd
cos
r
st
r
rrr
rrr
r
rrr
rt
r

 



 








 


 


(3)
At present, the Kerr metric is used to describe the
gravity field of a rotating sphere, in which parameters
,,1GMJ Mc


.
is considered to be the
unit angle momentum. If we use (3) to describe the
gravitational field of thin loop,
and
will have
different meanings. Because (3) contains a crossing item
ddt
which is related to time, the solution is dynamic
one, rather than static. For static mass distribution, this
item does not exist and should be canceled. We can re-
move it by the diagonalization of metric tensors. We
have

00 30
03 33
222 222
222
2222 222
22sin
1cos cos
2 sin2sin
1
cos co s
gg
gg
r
rr
rr
rr

 
 





(4)
 
From the eigen equation
00 03
30 33
0
gg
gg
(5)
w
e get





2
03300330330
1
22
222 222
222 22
222 222
222
222
222
2
200330033033
4
12 2sin 16sin
cos cos cos
2sin
cos cos
4
2
gg gg
r
rr
rr r
r
rr
rr
gg gggg


  

 








 











10
1
2
2
2
12
2
1
gg





0
1
22
222222
222 22
222 222
222
222
222
122sin 16sin
2
2cos cos cos
12 2sin
2cos cos
r
rr
rr r
r
rr
rr

  

 







 










(6)

X. C. MEI 977
The orthogonal transformations of coordinates are
 
 
30 30
22
22
03 00103002
00 1002
22
22
03 00103002
sin d
gg
gg gg
rg g
gg gg

dd
sin d
tt
r





 



 



 


 
 
 
(7)
The inverse transformation is








22
22
00 10300 103001
03 1212
22
22
00203 00103 002
03 1212
sin d
ggg gg
g
ggg gg
g

 
dd
sin d
tt
rr




ddt

 
















(8)
By the transformation, we can cancel crossing item containing
and get
2222 2
00 330312
dsind2sindddgt grgrtt222
sin dr

,tt
 (9)
Substitute (9) into (3), we can transform it into the diagonal form. For the consistency of notations, we set



again and obtain the result

 
2
1
22
222 2222
222 22222 2
222 22
222
222 2
2222
22 2
1
d2
22 sin16sin22
2cos cos
cos cos
cos d1cos d
2
12
2
s
rr
rr rr
rr rr
r
rrr
rrr

 
 






  










 

22
2
2
sin d
cos t


 
2
2
2222222
222 22222
222 222
22 2
22 sin16sin22
cos cos
cos cos
sin d
rr
rr rr
rr rrr
r
 
 
 





 







22
222
sin
cos


r
00 12g
(10)
cos ,sin,0xbybz
The Formula (10) has the form of (1), so we can use it
to describe the gravitational field of static thin loop.


. The distance between these
two points is
On the other hand, we know in general relativity that
only by comparing with the Newtonian theory in the
weak field when is great enough, the integral con-
stant of the solution of the Einstein’s equation of gravity
field can be determined. According to this principle, we
have relation

 (11)
Here
is the Newtonian gravity potential. Now let’s
discuss the concrete form of
for a thin loop. As is
shown in Figure 1, suppose that the coordinates of ob-
servation points are
00
sin cosxr y,sinrsin

 
coszr

and 0
.
The coordinates of a point on the surface of thin loop are
222
000
22
2sincos
Rxxyyzz
rb rb


 
 
0
(12)
For symmetry and simplicity, we take
so the
Newtonian potential of thin loop is
π
22
0
d2d
2sincos
GMG b
Rrb rb

 


,M
(13)
Here
and are mass, linear density and radius
of thin loop individually. Let
b
π,d d ,
 


2
coscos1 2sin2
 
 , and put them into (13),
we get
Copyright © 2013 SciRes. JMP
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
0
22
π
2d
2sin 4si
Gb
r brbrb


 
 
2
nsin 2

(14)
Then let 2
 
again, (14) can be written as
π2
22
0
π2
22 0
4d
2 sin4
4d
2 sin1
Gb
r brbrb
Gb
rb rb

2
22
sin sin
sink








 
 
 
 
(15)
We have 22
4sinkrbr2
2sinbrb


 in the
formula. Let

π2
2
0
Kk 22
d
1sink


r
(16)
(16) is just the first kind ellipse function. When
,
we have

22
22
1
2sin
3sin
1sin
12
rb rb
b
b
rr r


 
 2
1




2πbM
(18)
Substituting (16) and (18) into (15) and considering
, we get

22
2
sin



211
14
b
GM
rr
 
(19)
On the other hand, we can expand
g
into the
power series about 1r and write (10) as
2
22
2
3
222 3
2
23
2
222
2
222
22 2
23
d
22cos
1d
22sin 1248
1d
1cos d
2sin
1sind
s
t
rr
r
rrr
r
r
r
rr
 
 



 




 







 


00
(20)
By comparing (11) and the item
g
in (20) up to the
order , we get
1
r
22
11
GM
rr

(21)
Let constant GM
rr
2
r
, we get . Howeverthe
relation is only suitable for the situation when the mass
of thin loop is concentrated at the center point of the loop.
In order to obtain the more accurate gravity potential of
thin loop, we should consider higher order items. There
are no items containing
order in (19). By consider-
ing the items containing order up to , we have
3
r

22
3
22
3
22 cos
1
20.5 2.75sin
2
1
GM GM
rr
GMb
GM
rr


(22)
 
We see that the function forms on the two sides of (22)
are different. It means that the solution of the Einstein’s
equation of gravity can not asymptotically coincide with
the Newtonian theory of gravity automatically in this
case. In order to make them asymptotically consistent,
further transformation is needed. Because constant
has the dimension of length, we can take b
. Be-
cause we always have but may have
2
cos 0
2
0.5 2.75sin0
22
0.52.75sincos
, so we have

rr in general. Therefore, we have
in (22). However, we can set
so that (22)
becomes

22
22
33
0.52.75sin
1cos 1
b
b
rrrr


22 22
cos,0.52.75sinAb Bb
(23)
Let

rb
and by
considering the condition the only real number
solution of (23) is
 
 
 
1
3
2
262
3323
1
3
2
262
3323
11
33
11 i
4
22
1i
4
22
ii2cos3
BrrABr
rArA r
BrrABr
ArA r
ab abQ




 

 
(24)
Here


2
62
2
32 3
16
22
4
,
22
,tg .
rABr
Br
ab
Ar Ar
b
Qab a




(25)
,rrr
So we can write
and obtain
 
dd
dd d
dd
,d ,d
rr
rr
r
Trr Vr







,Tr
(26)
The concrete forms of functions

and
,Vr
are unimportant, so we do not write them out
here. Now we substitute (26) into (10) and obtain the
metric of gravitational equation of thin loop which has
the form of (2) with
,rrr

Copyright © 2013 SciRes. JMP
X. C. MEI
yright © 2013 SciRes. JMP
979

 

2
1
22
222 2222
22222222 2
22 222
22 2
222 222
22
22
d
122sin16 sin22s
2
2cos cos
cos cos
cos cos
,d 2
2
s
rb bbrb
rbrrb r
rr brr b
rb
rb rb
Tr r
rbr






 22
2
2
in d
cos
bt


 
















22
2
2222
2
22222
2
22
2
1
22
222222
222 222
222 222
,,dd
2
,
cos
1cos 2
d2
22sin '16sin
2cos cos cos
Tr Vrr
rbr
Vr
brrb
rrrbbr
r
rr
rbabr b
rbr rr brb



 




 













 









22
2222
22 2
22
cos
sin d
rb
rbrrr
r




2
22 2
sin
cos
b
b
0r0
r
,g
b
(27)
Cop
As is shown in Figure 1, or by the definition of coor-
dinate systems, we have both and r simul-
taneously for the original points of two coordinate sys-
tems. When , we have in (27) which leads
to 0022, and 33 . The result shows
that a singularity will appear at the centre of thin loop.
This singularity is completely exposed in vacuum, no
matter how much the mass or density of thin loop is,
even they are very small. The singularity is essential one
which can not be removed by coordinate transformation.
This result is absurd and unacceptable, for it obviously
violates common experience. It does not like the singu-
larity of the Schwarzschild solution which is considered
to hide in the center of huge mass and unobservable di-
rectly so that physicists can tolerate its existence.
0
g

0r
g
Besides, it can be proved that the space nearby the sur-
face of thin loop is also high curved. Because of
0
,
we can let
for approximation. In the nearby re-
gion of thin loop’s surface, we take π2
, so (27)
becomes





2
22 22
22
22
2
22
2222
22
22
22
2
22
dd ,π2d
,π2d
d
2,π2,π2dd
r
stTr r
rb
rV r
rr
rrrb
rb
r
rr
rTr Vrrr
rr b















0.67b
, (23) becomes Take
(28)
3
11 1
rrr

or
3
21
r
rr
(29)
By considering Equation (26), we have



42
2
2
3
,π2,,π20
1
rr
Tr Vr
r


(30)
0.67rb
Take
0.21r
0.10, 0.10gg

 
1.10g
, we have . Using these
values in (28), we obtain 1122 and
33
11 1ggg
. So the space nearby the surface of thin loop
is highly curved. The result does not agree with practical
experiences completely. On the surface of thin loop, the
gravity is very weak and space should be nearly flat with
22 33


0r
.
Because the curvature of space is a quantity which can
be measured directly, the solution (27) is improper for
the gravitational field of thin loop. In fact, according to
the result (13) of the Newtonian theory, at the center
point
, the gravitational potential of loop is a lim-
ited constant with
π
0
2d 2πGM
GG
b

 
(31)
Because
is a constant, the gravity at the center
point of loop is zero. This agrees with practical experi-
ences. The essential problem is that for such simple and
foundational material distribution, if (27) is improper,
what is the correct solution for the Einstein’s equation of
X. C. MEI
980
gravity? Can we find another solution? If can, how can
we deal with the problem of the uniqueness of theory?
On the other hand, let 22
20rb r
 in (27), we
have 2
r

 ~1KgM
2
b. By taking and
b = 1 m, we have 22
41 108
7.GM c
 b and
.
So if we take 22
b

r
r
r , would not be a real
number. Therefore, would not be a real number too.
The second singularity of (27) determined by relation
22
rb

 does not exist. In the Kerr solution,
22
rb

 
22
20rb r
 
describers a surface of elliptical sphere
which represents the event horizon of black hole. But for
the gravitational field of thin loop, because equation
has no real solution in general situa-
tions, the event horizon does not exist.
Next, we discuss the situation when the cross section
of thin loop is not zero. In this case, the gravitation field
has three independent parameters. The third is the radium
of loop’s cross section. As we have known that the Kerr-
Newman metric is one with axial symmetry and three
independent parameters [5]. At present, it is used to de-
scribe the external gravitational field of revolving
charged sphere. If the solution of the Einstein’s equation
of gravitational field with three parameters and axial
symmetry is unique, by the coordinate transformation,
we can also reach the gravitational field of loop with
cross section based on the Kerr-Newman metric. By the
same method of metric tensor’s diagonalization, we can
write the Kerr-Newman metric as

 
2
1
22
22222222 222
2
222 22222 2
222 222
222
222 2
222
22 22
d
122sin16sin 22sin
2 d
2cos cos
cos cos
cos
cos d1cos
2
s
rQrQ t
rr rr
rr rr
r
rrr
rrQ r

 
 



 

 


 


  
 




 



 
 



 

 
2
2
2
22222222222
222 22222 2
222 222
222
222
d
122 sin16sin22 sin
2
2cos cos
cos cos
cos
sin d
rQ rQ
rr rr
rr rr
r
r

 
 


 


 



  
 




 



 
(32)
According to (32), when r
r and
, we
have
2
222
00 23
22cos
1Q
grrr

  (33)
On the other hand, when the area of thin loop’s cross
section is considered, the Newtonian potential of gravity
field is very complex. We do not discuss it in detail but
can get the same conclusion by the simple estimation.
Suppose that the radius of thin loop’s cross section is h,
when , and
h ~ b, due to the axial symmetry,
we can always write the Newtonian gravity potential as
rbrh
12
23
,, ,,fbhfbh
GM
rrr

 

r 2
r
(34)
, we obtain from (11), (33) and (34) Similar to the discussion above, when , by considering terms up to order
2
1
22
,,
2
f
bh
GM QGM
rrrr

(35)
Let 1
x
rand 1r



x
, we can get from (35)
222
1
2
2GMGMQf xGMx
xQ
 
(36)
When , we have also
x
x
. That is to say,
when , we have . Substitute (36) into (32),
we can get the metric of loop with cross section. The
singularity still exists at the center point of loop which is 0r0r
Copyright © 2013 SciRes. JMP
X. C. MEI 981

also exposed in vacuum. Space nearby the surface of
loop is also highly curved. The situation is completely
the same as that when the area of cross section of thin
loop is neglected.
3. The Gravitational Field and Singularity of
Static Double Spheres
As shown in Figure 2, the masses and radius of double
spheres are
M
and . The centers of two spheres are
located at the points on the axis individually. It
is obvious that the gravity field also has axial symmetry
and two parameters and can be obtained through the co-
ordinate transformation of the Kerr solution. For this
problem, the Newtonian potential is
bb z
12
22 22
11
2 cos
GM rr
GM r bbrr b
11
2 cosbr




r rr

 


 


(37)
Here 1 and 2 are the distances between the center
of two spheres and the certain point in space,
is the
original point of coordinate system and
is the angle
between and
z axis. When , we have rbr
222
2
3cos
2
b
rr
2
1
GM b




(38)
From (11), (20) and (38) we get relationship

22
3
22cos
1
4
1
rr
GMb
GM

22
3
21
3cos
rr

 
,
b
(39)
Take 2,GM

, we have
Figure 2. The gravity field of static double spheres.
22
22
33
13cos
1cos 1
2
b
b
rr
rr


(40)
22 22
cos,1 3cos2Ab Bb


0r0r
Let in (40), we
can obtain the relations similar to (24) and (26). By con-
sidering (10), the metrics of static double spheres can be
obtained. It also agrees with the form of (27). Further
more, it is the same that we have and
simultaneously. So there is a singularity at the contact
point of double spheres with ,
00 22
,gg 
33
g 23
g and . Take 2b and π2
,
(40) becomes
3
111
rrr

or
3
21
r
rr
1KgM
(41)
Take
, i.e., the gravitational field is very
weak so that we can let 0


in (26) and get the for-
mula similar to (30) with

42
2
2
3
,,,0
1
rr
Tr Vr
r


 (42)
2r
2.67r
Take
, we get
and . Substi-
tute the values in (28), we obtain 11 ,
22
0.44T
0.15g
1.78g2.28g
and 33
11 1ggg
. It means that the space
nearby the surfaces of two spheres is also high curved.
However, this is impossible. In the weak field, we should
have 22 33

1r. More serious is that when
, becomes a negative number according to (41)
so that it is meaningless. So (26) is also unsuitable for the
gravitational field of static double spheres.
r
In fact, there are many other axial symmetry distribu-
tions of masses with two or three parameters. For exam-
ple, three spheres which are superposed one by one along
a straight line, two cones which are superposed with their
cusps meeting together, as well as the hollow column and
so on. In principle, all of their gravitational fields can be
obtained by means of the coordinate transformations
based on the Kerr solution and the Kerr-Newman solu-
tions. This method is unique actually. However, we can
imagine that same problem will occur in all cases. The
singularities would exist at some points and were ex-
posed in vacuum. The spaces nearby the surfaces of ob-
jects are highly curved under the conditions of weak
fields. All of them can not coincide with practical ex-
periences.
4. Conclusions
According to the singularity theorem proved by Stephen
Hawking, space-time singularities existed commonly and
unavoidably in the Einstein’s theory of gravity [6]. It is
now believed that black holes are created through the
collapse of material. Because black holes are considered
to be hidden at the centers of super-massive mass with
Copyright © 2013 SciRes. JMP
X. C. MEI
Copyright © 2013 SciRes. JMP
982
very high density, for example, at the centers of quasars
and galaxies so that they can not be observed directly,
physicists can tolerate their existence at present. How-
ever, if a singularity is exposed in vacuum, the problem
will become very serious.
The calculation in this paper proves that the singulari-
ties would appear at the center of a thin loop and the
contact point of two spheres according to the Einstein’s
theory of gravity. The singularities would be exposed in
vacuum completely. The space nearby the surfaces of
thin circle and double spheres were high curved. Theses
are impossible completely. If they were true, we could
held a black hole in our hand by bending a fine wire into
a circle or griping two spheres together. The ruler would
be bended when it was placed in the center region of a
thin loop. Light would bend and the effect of gravita-
tional lenses would be seen when it passed through the
central region of finger circle. These results are obvi-
ously unimaginable and absorbed.
So the singularity in the Einstein’s theory of gravity
can only be caused by the description method of curved
space-time. By the rational analogy, the so-called singu-
lar black holes, white holes and wormholes which con-
nect both holes in the current cosmology and astrophys-
ics are something illusive. They can not exist in the real
world. The true world excludes infinites. If there are
black holes in nature, they can only be the types of the
Newtonian black holes without singularity.
In fact, a correct theory of physics can not tolerate the
existence of infinites. It is well known that the history of
physics is the one to overcome infinites. Modern physics
grows up in the process to surmount infinites. Physicists
and cosmologists should take cautious and incredulous
attitude on the problems of singularity. We should think
in deep, whether or not our basic theory of gravity has
something wrong when we enjoy its so-called beauty and
symmetry.
In order to escape the puzzle of singularity thoroughly,
the description of gravity should return to traditional
form of dynamics in flat space. Only in this way, the re-
normalization of gravity can be possible. The unified
description of four basic interactions can be possible only
based on the frame of flat space-time. Otherwise, based
on curved space-time, theses problems can not be solved
further. Physicists should understand this situation clear-
ly.
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