The Precise Inner Solutions of Gravity Field Equations of
Hollow and Solid Spheres and the Theorem of Singularity
Xiaochu Mei
Institute of Innovative Physics in Fuzhou, Department of Physics, Fuzhou University, Fuzhou, China E-mail: [email protected]
Received May 26, 2011; revised June 28, 2011; accepted July 12, 2011
Abstract
The precise inner solutions of gravity field equations of hollow and solid spheres are calculated in this paper.
To avoid space curvature infinite at the center of solid sphere, we set an integral constant to be zero directly
at present. However, according to the theory of differential equation, the integral constant should be
deter-mined by the known boundary conditions of spherical surface, in stead of the metric at the spherical center.
By considering that fact that the volumes of three dimensional hollow and solid spheres in curved space are
different from that 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
singulari-ties at the centers of hollow sphere and solid spheres. Meanwhile, the intensity of pressure at the center point
of solid sphere can not be infinite. That is to say, the material 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 pressure 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
de-scriptive 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. The results revealed in the paper are consistent
with the Hawking theorem of singularity actually. They can be considered as the practical examples of the
theorem.
Keywords:
General Relativity, Inner Solutions of Hollow and Solid Spheres, Black Hole, Theorem of
Singularity
1. Introduction
We know that the static solutions of the Einstein’s equa-tion of gravity field with spherical symmetry are the Schwarzschild solutions which include inner and exter-nal ones. We consider a static and uniform sphere with radius r0 and constant density0, inner pressure
inten-sityp r
is related to coordinate but does not dependson time. By considering static energy momentum tensor of idea fluid, the Schwarzschild inner solution is [1,2].
2
2 2 2 2 2 2 2
0
1 2
2 2 2 2 2
2
3 1
d 1 1
2 2
1 d d sin d
Here 2 2 0
3 8
R c G . The metric is finite at the
center point of sphere. However, it should be pointed out that in the process of solving the Einstein’s equation of gravity field, what we obtain is actually
2 111 1 2
r A
g r
r R
(2)
It can be proved based on (2) that the space curvature is infinite at point r0. In order to avoid the infinity, we let integral constant to be zero directly in the
d
s c r R r R
r
r r
R
t
(1)
A
we can prove A0. Therefore, no matter what the mass and density of solid sphere are, the curvature infin-ity at the center of sphere is inevitable.
On the other hand, according to the current theory, the inner pressure intensity of sphere is [3].
2 2
0
2 2
0
1 1
1
r R r R
p r
r R r R
r r
2 0
3 1
c
(3)
On the spherical surface 0 we have p r
0 0. To make pressure intensity to be finite at the center of sphere, we have to introduce a constraint condition for spherical radius with2 2 0
8 9 R
r or 0
9 8
g
r
r (4)
Here 2 2
g
r GM c is the Schwarzschild radius. If
0 9 g 8
r r
0
A
1
, pressure intensity will become infinite. In this case, stable solution is impossible and material would collapse towards the center of sphere so that sin-gularity black holes appear. However, if integral constant
, pressure intensity (3) and constraint condition (4) will be changed. All calculations based on (3) and (4) about high density celestial bodies in the current astro-physics should be reconsidered.
Let’s first strictly calculate the solutions of gravity field equations of hollow and solid spheres, and then discuss the problems of singularities below.
2. The Strict Inner Solution of Gravity Field
of Hollow Sphere
Suppose that the inner radius of hollow sphere is R
and the external radius is R2, the gravity mass is M .
The region I3 with rR2 and the region I1 with
are vacuum. The region
1
rR I2 inside two spherical
shells with R1 r R2 is composed of complete liquid
with constant density 0 and pressure intensityp r
.Because material is distributed with spherical symmetry, the metric can be written as
2 2 ) 2 2 2 2
ds c ev r( dt e(r)dr2r d sin d 2 (5)
The solution of the Einstein’s equation of gravity in the region I3 is the well-known Schwarzschild metric
with
ev r 1 3 00A
g r
r
1 3
11 e 1
r A
g r
r
(6)
In order to determine integral constant A3 , we
compare (6) with the Newtonian theory under the asymptotic condition with r
00 r 1 2
g r
c
2 r
, r GM
r
(7)
Here M is the static gravity mass of hollow sphere
in the Newtonian theory. By comparing (6) with (7), we obtain A 2GM c2
3 . So we have the same result for
hollow sphere
00 2
2 1 GM
g r
c r
,
1
11
g r 1 2GM2
c r
(8)
To calculate the metric in the region I2 beneath two
spherical shells, the mixing energy momentum tensor of complete fluid is used [1]
0 2
0 0
T c , T11T22T33 p0, R1 r R2 (9)
The Einstein’s equation of gravity field is
4
2
R g R
c
8 GT
1
(10)
According to the standard procedure of calculation in general relativity, we obtain
2
G c
( ) 0
2 2 4
er
1 1 8
r r r c
(11)
( )er
2 2 4
8
1 1 Gp r
v
r r r c
(12)
2( )
4
8 er v v v v Gp r
2 4 4 2r c
(13)
The integral of (11) is
( ) 2 2 111 e 1 2
r r A
g r
r R
(14)
In which 2 2 0
3 8 G
R c and A2 is an integral
constant. We will prove 2 in the next section. (12)
minus (13) then multiplied by 0
A
2 rand considering (11),
we obtain
2
0
d
0
d 2
p r v r
c p r
r
(15)
The integral of (15) is
2 ( ) 2
2e
v r
B
0c p r (16)
Here B2 is an integral constant. By considering (11),
(12) and (16), we obtain
4( ) 2 ( ) 2
ev r er r v r c B
4
r G
r
On the other hand, by taking the differential of (14) with respect to , we get
( ) 2 2
e r 2 2 A r r R r
(18)
Substituting (14) and (18) in (17), we get
4( ) 2 2 2 2
2 3 2 2
2 1
e
4
v r A r A c B
v r
r G
R r R r
(19)
By considering the relation ev r( ) 2
v r 2dev r( ) 2 dr,
(19) can be written as
( ) 2
( ) 2
de
e
v r
v r
P r B K r
2
dr (20)
here
3 2
3 2
3r R
2
1 1
2 2
P r
r A r r R
4 2 3 2
2
8
c r K r
G A r r R
(21)
The integral of (20) is
( )d ( )
e e P r r eP r dr d
B K r
2( ) 2 2 v r r C
C 0
(22)
Here 2 is an integral constant. If let 2 in (14)
and (21), we reach the result of current theory [1]
A
2 ( )
00 e 1 2
v r r
g r B C
2 R 1 2 ( )r r
11 e 1 2
g r
R
(23)
In which constants 4 2
2 8Bc B R G , C C2
2 0
A
. If , we have
ln 1 ( )d 1 ( )d
( )d 2 2 1 2
e e e e
r
Q r r Q r r
P r r
r (24)
here
3 2
2
3 2
2 2
1 3 3
1 3 3
3 2
A r
r R
Q r
A r r R A r
r R (25)
let
0
3
3 2 2
1 r r r R 4 1 2
2 5 6 7
A r r r
(26)
The forms of constants ii
A R2,
0 are com-plex, but it is unnecessary for us to write them out. We have
0 20 2 0 1
2
2 6 7
2
5 6 7 2
7 6 5 7
2
7 6 6 5 7
2
7 6 6 5 7
d 3 1
1 3 3
1 3 2
3 ln( )
2 4
2 4
ln
2 4
Q r r r
A A r r r r
2 1 2
2
ln r
(27) we write
20 2 0 2 2 0 1 2
7
2 6 7
2
1 ( )d 2 3
(1 ) 1 3 / 3
2
1 2 3 2 2
5 6 7
1 3 /(2 ) 2
e
e
Q r r
r A A F r r r r
6 45 7
7 6 6 5 7
2
7 6 6 5 7
2 4 ln 2 4 r r (28)
dr K r
D r r
F r
0
r
(29)
let in (28), we get
2
7
0 2 2 0 1 2
2 6 7
2
6 5 7
3 (2 ) 1 3 3
1 5
1 3 (2 )
2 4 0 4 ln 0 A A
F
6 6 5 7
2 6 6 4 5 7
(30)
Therefore, D r
is finite at point r0. We canwrite (22) as
22 ( )
2 2
ev r F r C B D r
r 2 (31)
I , the metric can be written as at last
In the region
2 2 2 2 2 2 2 1 2 2 d d sin dc F r
s C B D r
r A r r r r R 2
2 2 2
2 2
1 d d
t 1 (32)
I of hollow sphere cavity, the
solution of the Einstein’s equation of gravity field is still the Schwarzschild solution
100 1
A
g r
r
,
1 1
11 1
A
g r
r
1
(33)
Let’s determine the integral constants A, A2, B2
and C2 below.
3. The Calculations of Integral Constants for
Hollow Sphere
By considering the continuity of metric tensors on the external spherical surface , according to (8), (14) and (32), we have 2
rR
3 0R
3
2 2
2 2 2 2 2
8
2 2
3
G R
GM GM
A
c R c c
(34)
22 2
2 2 2
2
2 2
2
1 GM F R C B D R
R c R
1
r R
(35)
Similarly, on the internal sphere surface with , according to (14), (33), (35) and (36), we have
3 3
01 2 2 2
8 2
3
G GM
1
A R R
c c
(36)
22 1 1
2 2 1
1 1
1 A F R C B D R
R R
(37)
Let’s prove 1 in curved space now. If space is
flat, the relation between mass 0
A
M and volume V of
hollow sphere is M 0 0V with 0
3 3
0 2 1
4 3
G
V R R (38)
It should be emphasized that M is the Newtonian
gravity mass. We introduce it by considering the asymptotic relation (7) between the Einstein’s theory and the Newtonian theory of gravity. Substitute (38) in (37), we get 1 . This is just the current calculating result
of general relativity. 0
A
However, (38) can not hold in curved space. Because there is a length contraction along the direction of radius, we should have dl dl g11
r and d d
, so
11
dVdl d dV g r
. In the curved space, the volume should be calculated by the following formula
2
1
2 2 2
11 2
d 4 d
1
R
R
V r r
V
g r r R A r
(39)The integral of (39) is difficult. If the third item in radical sign is neglected, we obtain [1].
3 3
5 5
2 1 2 2 1 2 1
4 2 , ,
5
R R R R Q R R
R
2 2
1 1
2 3 2
2 2 2
d 4
4 arcsin 1
3 1
3
R R
R R
r r R r r r
V
R R R
r R
R
(40) If we consider factor A r2 in (39), the integral
be-comes more complex. So in curved space, we have:
3 3
3 3
0 0
2 1 2 1
4 4
3 3
M R R M R R (41)
Substitute (41) in (37), we get
1 2
2G M A
c
1
(42)
R ,
Because R2 and 0 can be chosen arbitrarily,
we have M0 and 1 in general. Therefore,
from (35) and (37), we obtain 0 A
2 2
2 1
1 2
2 2
R
2
2 1
GM c R G M c
F R F R
B
D R D R (43)
2 2
2 1
2 1
2
2 2
D 1 2
1 2
R R
F R
GM c D R R G M c
F R C
D R D R
(44) Similarly, because R1, R2 and 0 are arbitrary, we have B20 and C20 in general. In this way, all integral constants are determined. In the region I1 of spherical cavity, we can write (33) as
00 2
2
1 G M
g r
c r
,
1
11 2
2
1 G M
g r
c r
(45)
According to the Newtonian theory, the material dis-tributed outside the spherical cavity with spherical sym-metry does not affect the gravity field in the cavity. But according to (45), it will have some effect for the cavity.
4. The Singularity of the Inner Metric of
Hollow Sphere
According to (45), the metric g00
r and curvature has(14) and let
2 2 2
1 r A 0
r R
2
2 0
r R A
or 3 2 (46)
r R
It seems that there is a singularity surface beneath two spherical shells. We now discuss this problem. The real number solution of (46) is
1 3
2 3
2 2 2
2 2
2 2
2 2
2 2 3
2 2 3
R A R A R
r
R A R A R
1 3 2 2 3
(47)
However, if is really a real number, the following relation should be satisfied r
2 3
2 2
2 0
2 3
R A R
(48)
Let rR2 in (46) and considering (34), we have
2 2
2 3 3
2 2 2 1 2
2GMR 2G MR
R A R R
c c
Δ
(49)
In the weak field, the item containing Mcan be ne-glected. By considering (48) and (49), we get
1 1
2 2
3
2 3 9.26
8 27
c
G
12
1
0 0
10
R (50)
We know that even for high density celestial body just as white dwarf, the difference is still very small when we do calculation based on both general relativity and the Newtonian theory. The material density of white dwarf is
11 3 0 10 Kg m
7
2.7 10 m
R
. By using this value in (50), we get
1 , similar to the size of white dwarf. For
common galaxy, we have 21 3 0 10 Kg m
22
3 10 m
R
. By using this value in (50), we have 1 , which is just
the size of galaxies. So (50) can be satisfied for common spheres and (47) becomes
1 1
3 3 3 3
1 1
2 2
2
2 2
R G M R G M
r
c c
0
M
(51)
By developing (51) into the Taylor series, if , we have 1. That is to say, there is a singularity
surface in the cavity. Because the metric of cavity is (33), in stead of (14), there is no singularity surface in the region 1 2
rR
R r R
0
M
rR
inside the hollow sphere. If , we have 1. In this case, there is a
singularity surface in the region R1 r R2.
According to (16), the pressure intensity in the hollow
sphere is
( ) 2 22e 0
v r
p r B c
2
0c22 2
B r
F r C B D r
rr
D
(52) If there is a surface with radius inside the hol-low sphere on which we have
r 02 2
C B (53)
The pressure intensity on the surface will become infinite. Therefore, if there exists black hole in hollow sphere, the black hole would be a spherical surface. Substitute (44) and (45) in (53), we get
2 2
2 1 0 1 2
1 2
2 2
1 0
1 2
2 2
2
D R R G V c D
2 2
R R GM c
F R F R
D r
R G V c R GM c
F R F R
(54) Because the radii R1 and R2 are arbitrary, from (54)
that we may find a proper r so that (68) can be
satisfied. However, on this spherical surface composed of black holes, space-time has no singularity. That is to say, the surface of space-time singularity does not overlap with the surface on which material collapses. This is incomprehensible. It should be noted that up to now we have no any restriction on the mass and density of hollow sphere. This result indicates that common hollow spheres may be unstable. They may collapse into the black hole of spherical surface!
r
Similarly, because the internal and external radii are arbitrary, let R2 or rR1, we have p R
2 0 and p R
1 0 in general. Because the hollow sphere is placed in vacuum without material outside and inside its two surfaces, this result is also incomprehensible. The singularities of hollow sphere are shown in Figure 1. It [image:5.595.311.537.562.703.2]is obvious that the results can not be true.
2
rR
5. The Singularities of Solid Sphere’s Metric
and Black Holes
According to the present calculation of general relativity, the internal metric of a common solid sphere has no singularity when the radius of sphere is greater than the Schwarzschild radius. According to the strict calculation in this paper, the situation is completely different. The solid sphere is a special situation of hollow sphere when its internal radius becomes zero. The internal metric of solid sphere is still described by (30), but the conditions of boundary are different. On the spherical surface
, we have
2
2 2
2 2
2 2
2
1 GM 1 R A
R
c R R
(55)
22 2
2 2 2
2
2 2
2
1 GM F R C B D R
R c R
(56)
In order to determine A2 in (55), we have to know
the relation between M and R2. In curved space, we
have
2 2
0 0
2 2 0
11 2
d 4 d
1
R
V r r
M
g r r R A r
(57)By substituting (57) in (55), we can decide A2 in
principle. We have in general. If suppose , we have 2
0
A
2 0
A
2 4 2
R
r dr
0 2 2
0
3 5
0 2 0 2 2
3 0 2
1
4 2
,
3 5
4 3
M
r R
2
R R O R R
0
V V V
R
R M
(58)
Because , the boundary condition (55) can not be satisfied. We estimate the magnitude of volume’s change in curved space based on (58). Let
and omitting high order items, we have 0
M
2 0 2
2 2
0
4 5
G R
V
V R c
15
~ 10
R ~ 104 2
2
3 10
R
(59)
For neutron stars, we have 0 and 2 ,
so 4
0 1.8 10
V V
27 0 ~ 10
26
2 ~ 10
R
. If considering the universe as a uniform sphere, we have and , so
2 0 1.8 10
V V
. For so-called black hole, we have
2
2 3 0
R c G according to (4) and
0 . For common spheres just as the
sum and the earth, is a very small but non-zero
quantity.
After 2 is determined, by substituting it into (28) and (29), we can determine
A
2
2 2
F R and D R .
However, we can not yet determinate C2 and B2 only
based on (56). Another condition is needed. By considering the fact that the pressure intensity on the surface of sphere should be zero with p R
2 0, from(16) and (31) we have
R2 C2
R2
2
2 2 0 2
R B c F B D (60)
From (56) and (60), we obtain
4 15 0.27
V V
0
V V
2
B20c 2
2
2 1 GM
c R
2
2 0
C 2 2 c
2 2
2
1 GM R
F R
c R D R2
(62)
Now, all integral constants are determined. The inter-nal metric of solid sphere is
2 2
2 2 2 2 2
2 2
1 dr r d sin d
r R
2 2
2 2
1 2
ds F r C B D r dt
r
A
r
(63)
Because F 0
0 and in general, we have
0 02 2
C B D
0g00 and g11
0 0 at point r0,so the infinite of space-time curvature also appears at the center of sphere.
The pressure intensity of solid sphere is also repre-sented by (52), but the integral constants should be rep-resented by (61). In general, we have F
0 0 and
2 2 0 0C B D , so we have p
0 0. That is tosay, no mater what are the mass and density of sphere, the pressure intensity at the center of sphere can not be infinite. So-called the singularity black holes in which material collapses towards its center are impossible. Meanwhile, the pressure intensity may become negative value at the center of sphere and its nearby region too. On the other hand, if there is spherical surface with ra-dius rr so that we have
2 2 0
C B D r (64)
the pressure intensity may become infinite on the surface. Substitute (61) in (63), we obtain
22 2
0 2
R D R
c F R
D r (65)
Figure 2. The singularity of solid sphere.
only take the form of a spherical surface. Such solid sphere is not stable for material will collapse to its spherical surface. But according to our common experi- ences there is no singularity of space-time curvature on the spherical surface. Such result is also uncanny.
The singularity of solid sphere is shown in Figure 2.
Notice that we did not impose any restriction for mass and density, and there are so many strange characteristics for common solid spheres. The results are completely different from the current understanding in general rela- tivity. The theory of singularity black hole in the current astrophysics and cosmology has to be reconsidered.
6. Discussions on the Theorem of Singularity
and the Rationality of Current Singularity
Black Hole Theory
S. W. Hawking etc. proved the theorem of singularity by means of the method of differential geometry [4]. The theorem was based on three prerequisite conditions. 1. General relativity was tenable. 2. The law of causality was tenable. 3. There were some points in space-time at which material densities were non-zero. The theorem claimed that if theses three conditions were satisfied, singularity inevitably existed in space-time. Hawking etc. considered singularities as the beginning and ending of time. The Big Bang theory was considered as the begin-ning of time and the black holes were regarded as the ending of time.
We note that the theorem had no restriction on mate- rial’s mass and density and did not demand that singu- larities were embodied in material. That is to say, ac- cording to the theorem, singularities may bare in vacuum. In order to avoid this embarrassing situation, Penrose proposed the so-called principle of the universe supervi- sor. The principle declares that there exists the universe supervisor who prohibit the appearance of bare singulari- ties in vacuum. In other word, due to the existence of the universe supervisors, all singularities will be wrapped in the centers of black holes with great masses and high
densities. According to the solutions of the Einstein’s equation of gravity, there exist Schwarzschild black holes with spherical symmetry and the Kerr black holes with axial symmetry and so on. The singularities were hidden in the centers of material. In this way, they can not be perceived directly, and physicists seem tolerate their existence.
The results revealed in the paper are consistent with the Hawking theorem of singularity actually. We can consider them as the practical examples of the theorem. By considering the fact that the volume of hollow and solid spheres in curved space are different from that in flat space, the strict calculation reveals that singularities can not be avoided at the centers of common hollow and solid spheres with small masses and low densities. On the other hand, because the pressure intensity can not be infinite at the center of sphere, material can not collapse towards the spherical center. Also the result shows that the pressure intensity may become negative values at the center and its nearby region.
Meanwhile, there may be curved surfaces inside the common hollow and solid spheres on which pressure intensities can become infinite so that material will col-lapse to them. But the space-time curvatures are still fi-nite on the surfaces. The surfaces of space-time singular-ity do not always overlap with the surfaces with infinite pressure intensities. All these characters can not agree with our practical experiences of common hollow and solid spheres. They are incomprehensible in physics.
According to the current understanding, the black holes exist at the center of Quasars. However, according to the observations of Rudolf E. Schild and Darryl J. Leiter, the center of Quasar 0957 + 561 is a close object, called MECO (Massive Eternally Collapsing Object) [5]. It is not a singularity black hole, and is surrounded by a strong magnetic field. The observation of Rudolf E. Schild was consistent with the calculation and analyses in this paper. That is to say, if there are black holes in the universal space, they can only be the Newtonian black holes, not the Einstein’s singularity black holes!
should think in deep, whether or not our theory has something wrong. When we enjoy the beauty and sym-metry of the Einstein’s theory of gravity, remember that we should not neglect its limitations and possible mis-take.
7. References
[1] J. R. Oppenheimer and H. Snyder, “On Continued Gravi-tational Contraction,” Physical Review, Vol. 56, No. 5,
1939, pp. 455-459.
[2] Zhang Yongli, “Introduce to Relativity,” The Publishing
Company of Yunnan People, Kunming, 1989, p. 388. [3] L. B. Feng, X. C. Liu and M. C. Li, “Genarel Relativity,”
Jilin Science Publishing Company, Jilin, 1995, p. 109. [4] S. W. Hawking and G. F. R. Eills, “The Large Scale
Structure of Space Time,” Cambridge University Press, New York, 1972.
