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Based on general relativity, J. R. Oppenheimer proved that massive celestial bodies may collapse into singular black holes with infinite densities. By analyzing the original paper of Oppenheimer, this paper reveals that the calculations had a series and serious of mistakes. The basic problem is that the calculation supposes that the density of celestial body does not change with space-time coordinates. The density is firstly assumed invariable with space coordinates and then it is assumed invariable with time. But at last, the conclusion that the density of a celestial body becomes infinity is deduced. The premise contradicts with conclusion. In fact, there is no restriction on the initial density and radius for celestial body in the calculation. According to the calculation results of Oppenheimer, a cloud of thin gas may also collapse into singular black hole under the action of gravity. The calculations neglect great rotating speeds of massive and high density celestial bodies which would make them falling apart rather than collapsing into singularities. Because we do not know the function relations that material densities depend on space-time coordinates in advance, there exists the rationality problem of procedure using the Einstein’s equation of gravity field to calculate material collapse. Besides these physical problems, the calculation of Oppenheimer also has some obvious mistakes in mathematics. Another improved method to calculate massive celestial body’s collapse also has similar problems. The results are also unreliable. The conclusion of this paper is that up to now general relativity actually has not proved that massive celestial bodies may collapse into singularity black holes.

S. Chandrasekhar proved in 1935 that the mass of white dwarf has an upper limit, i.e., so-called the Chandrasekhar limit [

After that, J. R. Oppenheimer proved that the upper limit of neutron star is about 0.75 times of solar mass [

In 1939, Oppenheimer proved further that when the masses are great enough, the celestial bodies would collapse into singular black holes with infinite density [

The calculation of Oppenheimer about celestial bodies collapsing into singularly black holes is abstruse and hard to understand so that common text books do not introduce it. Most physicists only accept Oppenheimer’s statement but do not understand the details. The author examines the original paper of Oppenheimer recently and finds many mistakes in it. The conclusion of Oppenheimer about celestial bodies collapsing into singularly black holes does not hold at all. The main problems are as follows.

1) The calculation of Oppenheimer hidden a premise, i.e., the density of celestial body was a constant

2) There are no any restriction on the initial density and radius for celestial body in the calculation. That is to say, according to Oppenheimer’s calculation, a cloud of thin gas may also collapse into singular black hole under the action of gravity. This is completely impossible.

3) Suppose that

4) There are other obvious mathematical and physical mistakes in Oppenheimer’s calculations.

5) More generally, the calculations of general relativity in this kind of problems have the mistake of procedure. In order to solve the Einstein’s equation of gravity for material collapse, we should know dynamic energy momentum tensor in advance including the forms of material density changing with space-time coordinates. Otherwise the equation of gravity field cannot be solved. Conversely, if we have known the forms of material density changing with space-time coordinates, we can judge directly whether or not massive celestial body collapses. In this case, we do not need to solve the Einstein’s equation of gravity at all. However, in the practical problems of celestial body collapse, we cannot know dynamic energy momentum tensor in advance. So the Einstein’s equation of gravity is actually incapable for this problem.

6) In order to escape this predicament, physicists introduce co-moving coordinates. The observers are assumed to move with material along the radial direction so that the speed of material seems to be canceled. However, observer moving with material is equivalent to falling freely in gravity field. According to the principle of general relativity, gravity field would be eliminated locally and space-time would become flat. In this situation, there is no problem of matter collapsing again.

7) Most celestial bodies in nature have rotating speeds. In the evolution processes of celestial bodies, angle momentums are conservative. For massive celestial bodies, rotating speeds are much greater than radial speeds. Because co-moving coordinate frame moves along the direction of radius, it is impossible using it to describe the real motion of celestial bodies. Due to very high rotating speeds, the final destiny of massive celestial bodies is falling apart, instead of collapsing into singularities. However, the calculations of general relativity completely neglect the rotation speeds of celestial bodies.

8) Another improved method to calculate star’s collapse also has similar problems in which even arbitrary coordinate transformations are used to simplify the motion equations. So the results are also incredible.

In this paper, we first introduce the improved and simplified calculation adopted in Weinberg’s book “Gravity and Cosmology” [

Weinberg’s book discusses the simplest collapsing process of a material sphere in which pressure is neglected. The solutions are divided into both external and inner parts. The metric of gravitational field with spherical symmetry in the external space of sphere is

Here

Suppose that the space-time coordinates inner sphere are

If the pressure of fluid is neglected, energy momentum tensor is

The Einstein’s equation of gravity is

By considering (2) and (3), the non-zero components are obtained with

Substituting (5) in the Einstein’s equation of gravity, we get four formulas below

Suppose that density has nothing to do with space coordinates with

We get

The formula (11) is supposed tenable without proof in the book. Substituting (11) in (10) and cancelling the sign of ripple, the following formulas are obtained

Comparing (12) with (10), we know that (11) is equivalent to let

By comparing both formulas above, we get

Here

By substituting (12) and (14) in (2), we get at last

The formula is actually the Robertson-Walker metric with constant curvature

By considering (12), (15) and (18), the Equations (8) and (14) become

By cancelling

Let

The solution of (22) can be represented by the parameter equations of cycloid curve

When

It indicates that a fluid sphere without pressure with initial density

The vacuum solution (1) outside sphere is different from the solution (17) inside the sphere. In order to make the metrics continuous, same transformations have to be introduced so that they have the same forms on the boundary of sphere. However, as Weinberg said in the book, the transformations have something confuse. So we only provide the result here. By introducing following transformations in (17) to set

Here constant

in which

By using co-moving coordinates, the radius of celestial body is

Comparing (27) and (29) with (1), we take

In this way, we can make the metric to be continuous on the boundary of sphere. Let

In which

1) In the calculation, it is assumed that the density of start has nothing to do with space coordinate with form

2) The formula (11) has no any mathematical and physical reason. The simplified formula (12) is only an artificial patchwork, not a rational deduction.

3) According to (24), the time that a celestial body collapses from static state into the state of infinite density is only related to initial density

4) (27) describes the metric inside sphere, so (25) represents the time inside sphere with

5) Substituting (15) in (13) and considering (8), we get

By cancelling

However, E. A. Milne proved in 1943 that the Freidman equation can be deduced directly from the Newtonian formula of gravity [

6) Because co-moving coordinates are used, the velocities of material are neglected. However, even though observers are at rest in the co-moving coordinate frame, because

7) In the co-moving coordinate frame, observers fall freely with material in gravity field. According to the principle of equivalence in general relativity, the local space-time is flat. So it is impossible for us to use (2) and (17) to describe the metric for an observer who is rest at the co-moving coordinate frame.

8) In fact, most celestial bodies in nature have rotating speeds. Especially, for massive celestial bodies, the rotating speeds are much greater than the speeds along the direction of radius. The co-moving coordinate frame moving along the direction of radius cannot describe the real rotations of celestial bodies. For example, for a neutron star with 10 times of solar mass and radius

9) General relativity uses the Schwarzschild and Kerr solutions to describe static and rotating black holes. Within the horizon, to ensure the square of length being a positive quantity, space coordinate

J. R. Oppenheimer considered the collapse process of a sphere with uniform density (Taking

In the calculation of Oppenheimer, the metric inner sphere is [

Based on this metric, the Einstein’s equation of gravity field becomes

In the formulas, the non-zero components

The Tolman solution is considered to use the co-moving coordinates too in which the non-zero component of energy momentum tensor is

In (39),

The solution of (43) is

In which

The integral of (45) is

Here

Oppenheimer chosen

It will be pointed out later that (48) is untenable in general situations. After obtaining (48), Oppenheimer thought further that when

Here

Here

Let

Comparing (51) with (39), we obtain

By considering (44), we get from (52)

By considering (46) and (52), we have

From (54), (55) and (56) , we get

For the situation with

Here

Here

In the region

By using condition

Substitute (61) in (60), the transformations between

For a great enough time

When

It can be obtained from (63)

According to Oppenheimer, (64) represents the condition that the observer who moves with material cannot send out a light sign from a star, or light corn is closed so that light sign cannot escape from star. Oppenheimer estimated that for a star with mass

Oppenheimer discussed the asymptotic behaviors of (65) and (66). Because the problems revealed below, the discussions become meaningless, so we do not introduce them again.

1) It is assumed at first in (48) that density

2) At a certain moment to have

Therefore, both (49) and (50) do not hold in general situations when

3) The constant

4) According to (50) and (46), in the space outside sphere with

However, the metric is related to time

5) In order to know whether or not a star collapse into a singularity in finite time really, we should take the integral of (47), then discuss the relation between

6) In (64),

7) In the calculations of Oppenheimer, there are no any restriction on the initial values of celestial body’s mass, density and radius. The formulas (62) and (64) are tenable generally even for a cloud of thin gas. That is to say, according to Oppenheimer’s calculation, a cloud of thin gas can also collapse into singular black hole. Regardless of

8) There are other mistakes in Oppenheimer’s paper. For example, it is

9) The Tolman solution is used practically in the calculation of Oppenheimer. The Tolman solution is considered to be based on co-moving coordinate. However, the real situation is that in the original equation (39),

It can be concluded that the calculations of general relativity about celestial bodies collapsing into singularity black holes are wrong. Based on general relativity, physicists have never successfully proved that material can collapse into singularity up to now. The theories of so-called singular black holes in the astrophysics have no any scientific foundation actually. Before this manuscript, the author has published a series of papers to prove that singular black holes cannot exist in nature [

In fact, The Einstein’s theory of gravity has never been used successfully in the problems that the material of field source has moving velocity. The reason is that in this case the Einstein’s equation of gravity cannot be solved. In order to solve the equation, physicists have to introduce co-moving coordinates. The purport is to cancel the influence of velocity on the equation of gravity. But it cannot represent the real process of physics. This fact reveals that the Einstein’s equation of gravity is practically invalid when the material of field source has moving velocities.

Therefore, the so-called singular black holes, white holes and wormholes in cosmology and astrophysics have nothing to do with real world. The space-time singularities in general relativity are caused by the description method of curve space-time, rather than massive mass. According to the current understanding, black holes may exist in the centers 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 as MECO (Massive Eternally Collapsing Object), not a singularity black hole [

In the modern cosmology based on the Einstein’s theory of gravity, the same problems also exist. By introducing both simplified conditions the co-moving coordinates and the R-W metric, the Freidman equation of cosmology is obtained and used to describe the expansive universe. However, as proved by British Physicist E. A. Milne, the Freidman equation can also be deduced directly from the Newtonian equation of gravity. No any revised item is contained in the Freidman equation. The result indicates that the current cosmology is based on the Newtonian theory of gravity actually. The Freidman equation is only suitable for the universal processes with low expansive speed, unsuitable for the universal processes with high expansive speed just as the high red shift of Ia supernovas. This is the reason why cosmology needs the hypotheses of dark energy, so much non-baryon dark material and accelerating expansion.

On the other hand, 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 [

All these tell us that the geometric describing form of gravity is impossible and should be given up. The description of gravity should retune to the normal form of dynamics in flat space-time through the revision of the Newtonian theory.