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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. The formula can also describe the motion of object with mass in gravity field such as the perihelion precession of the Mercury. The space-time singularity in the Einstein’s theory of gravity becomes the original point r = 0 in the Newtonian formula of gravity. The singularity problem of gravity in curved space-time is eliminated thoroughly. When the formula is used to describe the expansive universe, the revised Friedmann equation of cosmology is obtained. Based on it, the high red-shift of Ia supernova can be explained well. We do not need the hypotheses of the universe accelerating expansion and dark energy again. 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. The theory of gravity returns to the traditional form of dynamic description and becomes normal one. The revised equation can be taken as the foundation of more rational cosmology.

Established on the foundation of curved space-time, Einstein’s theory of gravity is the dominate theory at present. However, Einstein’s theory has some difficulties hard to be overcome such as the problems of normalization, singu- larity and uniqueness of gravity field’s energy and so on. In addition, it is difficulty to solve the non-linear Einstein’s equation of gravity field. It is always attractive to reestablish the theory of gravity in flat space-time without these troubles. Since the 1940’s, many people has tried and many theories had been proposed [1,2]. These theories are consistent with Einstein’s one under the condition of weak fields, but are different in strong fields. Meanwhile, these theories also have some problems hard to be overcome.

The standard theory of cosmology faces many principle difficulties at present. As is proved below, the problems originate from the Friedmann equation which is unsuitable to describe the high speed expansion of the universe. The reason is that two simplified and improper conditions were used in the deduction of the Friedmann equation. They are the R-W metric and static energy momentum tensor. At present, the R-W metric is considered with

constant spatial curvature. However, the author had proved that strictly based on the curvature formula of the Riemannian geometry, when the scalar factor R(t) changes with time, the R-W metric has no constant curvature [

It is proved further in this paper that the R-W metric leads to the Galileo’s transformation of light’s velocity, instead of the Einstein’s transformation. So the R-W metric is not relativity metric and unsuitable to be taken as the basic space-time framework of modern cosmology.

Meanwhile, because relative velocities exist between materials and observers in the expansive universe, the equation of cosmology should use dynamic energy momentum tensor, rather than static one as commonly used in the current cosmology.

In fact, E. A. Milne pointed out in 1943 that the Friedmann equation of cosmology could be deduced based on the Newtonian formula of gravity [

However, it is proved in this paper that if dynamic energy momentum tensor is used, the equation of cosmology would become very complex, so that it can not be solved actually. The pioneer of cosmology must have considered this problem and had to use static energy momentum tensor. In the early stage of cosmology, the Friedmann equation seemed to be appreciable because the expansive speed observed was low. When cosmology develops to present level, we observe the high speed expansion. In this case, the Friedmann equation becomes unsuitable for the problems such as the high red-shift of supernova. We have to find more precise method to describe them.

It is proved in this paper that by transforming the geodesic equation of the Schwarzschild solution of the Einstein’s equation of gravity field to flat space-time, the re- vised Newtonian formula of gravity can be obtained. The formula can well describe the perihelion precession of the Mercury. The space-time singularities in the Einstein’s theory of gravity become the point r = 0 in the revised New- tonian formula of gravity. We have no the trouble of singularities again.

When the revised formula is used to describe the expansive universe, we obtain the revised Friedmann equation. Based on it, the high red-shift of supernova can be explained well without the hypotheses of the universal accelerating expansion and dark energy. Many problems in- cluding the universe age to be too small can also be resolved well. In this way, we can get rid of the current puzzle situation of cosmology completely.

According to general relativity, the Schwarzschild metric (external solution) is

Here

Here

We define

In which

Then, (3) becomes

Here

We only discuss the motion of particles with mass in gravitational field. By considering (6), we write (1) as

By considering (4) and (6), the formula above can be written as

Taking the differential of (8) about

Note that all quantities in (9) are defined in curved space-time. According to the theory of the non-Euclidean geometry, although we can not transform whole metric of curved space-time into that of flat space-time in general, we can always transform the geodetic line described in curved space-time into that in flat space. Let

We see that the forms of third items on the two sides of the second equal sign of the formula above are completely the same. So we can take

by considering (4), we get and from (8)

Substituting it into (11), we get

Comparing with (4), we have

Because we have taken

Substitute (16) into (15), we get

Comparing with (9) and let

Let

This formula is the one used to describe the perihelion precession of the Mercury in general relativity. In the deduction process above, we use the equation of geodetic line (2). It means that we transform the equation of geodetic line into the revised formula of the Newtonian gravity, in stead of transforming whole curved space-time to flat space-time. But it is enough for us to describe an object’s motion in gravity field.

Now let’s prove that the effect of special relativity has been taken into account in (18). From (8), (12) and (14), we can obtain

Comparing with (14), we get

This is just the formula of time delay in special relativity. The result verifies the rationality of (18). Let

It is the revised Newtonian formula of gravity based on general relativity. In the formula,

We can call m as the motion mass of gravity which is related to object’s speed and angle momentum.

For simplicity, we only discuss the motion of a particle moves along the radius vector direction with

by multiplying

The dynamic energy of particle is

when

when

In the situation of

So the law of energy conservation is

Here E is a constant.

Now let’s discuss the motion of a particle in the gravity field. Suppose that a particle falls freely along the radium direction of gravity field, its velocity and acceleration are individually

when

It is obvious that every thing is normal within the region

It indicates that the speed of particle tends to have light’s speed in vacuum at point

The Fiedmann equation of cosmology is based on the Einstein’s equation of gravity. Because the equation is too complex to solve, two simplified conditions are used. One is the R-W metric and another is the static energy momentum tensor. Using them, we obtain from the Einstein’s equation of gravity

Here

Cosmic constant has not been considered in (39) and (40). We often either take it as zero, or combine it with effective material density for convenience.

However, British physicist E. A. Milne proved in 1943 that the Fiedmann equation could be deduced simply based on the Newtonian theory of gravity. Though the Fiedmann equation is described in curved space-time and the Newtonian theory of gravity is described in flat space-time, the results are the same actually when we use them to calculate practical problems, especially when we take curvature constant

We now repeat Milne’s deduction below. According to the principle of cosmology, the universe can be considered as a huge sphere with uniform and isotropic material distribution. According to the Newtonian theory, gravity acted on a body located at point r inside the sphere is only related to the mass contained in the sphere with radius r, having nothing to do with the mass outside the sphere. Suppose that the mass of uniform sphere to be

For the expansive sphere, by considering co-moving coordinate

(42) is the same as the first formula of (39) when

Substituting (43) in (42) and taking the integral, we obtain (40). In this case, integral constant

It is obvious that (40) is the direct result of the Newtonian theory of gravity, for it dose not contain any revised item of relativity. This is why the standard theory of cosmology is effective for same problems, but is ineffective for other problems such as the high red shift of supernova. The reason is that two simplified conditions are used, so that the Freidmann equation becomes non-relativity theory actually. We discuss these problems below.

According to the principle of cosmology, the universe is uniform and isotropy. The R-W metric is considered with the biggest space-time symmetry. Its form is

In which

For light’s motion, we have

For the light’s source fixed at point

By considering (46) and (47), the velocity of light relative to observer located at the original point of reference frame is

The formula indicates that light’s velocity is related to the expansion speed of space and violates the principle of invariance of light’s speed.

In fact, at the moment when light is just emitted out, (48) is 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 observer, plus sign is taken. In this case, light’s speed exceeds its speed in vacuum. Especially, because

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 principle, 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 great, huge error will be caused.

As for the curve space with

or

On the other hand, as we known that coordinate

Here

nant material relative to observer is

considering (49), the velocity of light emitted by illuminant material moves in the curved space is

So (51) still violates the principle of invariance of light’s speed. In fact, the four dimensional metric of flat space-time is

by using co-moving coordinate

It is completely different from the R-W metric (44) when

On the other hand, the four dimensional metric in which three dimensional space has a constant curvature

by using co-moving coordinate in (54), we obtain

Let

Another result of using the R-W metric in cosmology is that it leads to the united universe time. In the R-W metric,

However, it is easy to prove that if we use flat space- time metric (53) in the Einstein’s equation of gravity, the Einstein’s tensor would become zero with

The energy momentum tensor of ideal liquid is used in cosmology with the form

Here

This is an excessively simplified approximation. In fact, there exist relative velocities between materials and observers in the expansive universe. The most basic fact for cosmology is the Hubble’s red shift, which is explained as the kinematical effect caused by relative velocities be- tween observer and luminous material. If co-moving co- ordinate

According general relativity, we can use arbitrary reference frame to describe the gravity field. By using common spherical coordinate system, the partial velocities of an object which moves along the radius direction are

To simplify discussion below, we use the R-W metric and take

Substituting them in the Einstein’s equation of gravity

We get the motion equations of cosmology

Substitute

Take

Here A and B are constants. From (67), we obtain

The result violates the Hubble law too. In addition, these velocities are inconsistent, so (68) is impossible. The third is to get the solution from (65)

Substitute (69) in (64) and (66), we have

by cancelling

The equation becomes so complicated that it is impossible to solve actually. On the other hand, because the right hand sides of (64)-(66) contain

way, the principle of cosmology can not hold again. The result means that we will be in dilemma if dynamic energy momentum tensor is used in cosmology.

Pioneers of cosmology must have considered this problem, so they had to use static energy momentum to establish the equation of cosmology. In the early stage of cosmology, the observed expansion speed of the universe was low, so the simplified motion equation could be suitable. When cosmology develops to now day’s level, we observe cosmic phenomena which take place in the high speed expansive processes such as the high red shift of supernova. The simplified Friedmann equation becomes unsuitable so that many difficulties appear in the standard cosmology at present. This is the main reason why we have to introduce the hypothesis of the accelerating expansion of the universe, dark energy and non-baryon dark material.

Because (72) can not be solved practically when dynamic energy momentum tensor is considered, we have to look for other method to describe the expansive universe. We prove below that based on the revised formula (25), the high red shift of supernova can be explained well. There- fore, we do not need the hypothesis of dark energy and the universe accelerating expansion again.

In principle, we can take the CMB as static reference to describe the universe expansion. Practically, we take the earth as static reference frame for convenience. Suppose that the universe material is distributed with spherical symmetry and uniform density

The formulas indicate that when mass

Suppose that the universe expands along the direction of radius. In the process, angle momentum L is equal to zero. We calculate gravity between a spherical shell with radius R and an object located at point

Here

On the other hand, according to special relativity, we have

Based on (76) and (77), we get the acceleration of an object located on spherical surface

The acceleration is just related to the mass inside the sphere, and unrelated to the mass outsider the sphere. We also consider (78) as the expansion speed of spherical surface with radius r. Let

Let

In the expansion process of the universe,

unchanged with

is

we get

Let

we have

Because (81) can not be integrated directly, we need approximate method. When x is very small (

By considering (80), (87) becomes

Substituting the formulas in (78) and (85), we obtain the formula of acceleration and speed of the universe expansion

In the discussion above, we assume that material is only acted by gravity. However, practical situation is that strong, weak and electromagnetic interactions could not be neglected in the early phase of the universe during which material density was great. Even more, some unknown interaction may exist.

According to the theory of Einstein’s theory, material may be compressed into infinite density by gravity. How- ever, infinite density is unimaginable. In fact, the author had proved that due to use the improper boundary condition of flat space-time in the gravity theory of curved space-time, the current theory of singularity black hole is wrong. By strict calculation based on the Einstein’s equation of gravity and curved boundary condition, singular black hole with infinity density do not exist [7,8]. By the same reasons, the fashionable idea that the universe originated from infinite small point is also impossible.

In order to avoid infinite density, we assume that there exist a certain mechanism so that material sphere with mass M can only be compressed to a finite radius r_{0}. In this way, the motion equation of the universe expansion should be revised as

Here

Here _{0} on which the spherical surface can not be contracted further. Meanwhile, by the action of

The integral of (93) is

Let

Under the condition

Here

According to the Doppler’s formula, when celestial body moves along radius direction, we have relation between speed and red shift

Suppose that observer is located at the origin point of flat reference frame, the distance between observer and celestial body is

According to (96), we have

The real distance between observer and celestial body is

Using (102) in (101) and taking the integral, we can obtain the relation in principle

In the formulas above,

connecting (100) and (103), we can determinate ^{?}^{11}, r_{0} = y_{0} × 10^{26} m, r_{1} = y_{1} × 10^{26} m and ^{?26} kg/m^{3}, we have

We use x as basic variable to calculate y_{0} and _{1} are input parameters. According to this paper, we actually deduce the initial situations of the universe expansion reversely based on the present observations of red shift and distances. In other words, as long as the initial conditions of the universe expansion are known, we can know its current situations.

In ^{3} at present day. Because there exist a great mount of non-luminous material, we suppose that practical material is 10 times more than luminous material and let ^{3}. In

The curved line in _{0}. For Z = 1 and_{1} = 0.67 × 10^{26} m, we obtain _{1} = 0.15 × 10^{26} m, we obtain r = 0.16 × 10^{26} and

In this way, we can explain the high red shift of Ia supernova well. The hypotheses of dark energy and the accelerating expansion of the universe become unnecessary. The universe began its expansion from a finite volume, rather than from a singularity.

In order to compare with the equations of cosmology, we now transform (97) and (98) to the form of the Friedmann equation. Suppose that the universe is a uniform

sphere with density

Here

Similarly, let

On the other hand, the Friedmann equation containing cosmic constant

Here

It is obvious that after (110) and (111) are used, revised equations in this paper are with the same form with the Freidmann equation. The differences are that

According to (97), we have

Let

We get

At present

We see that

In fact, only taking the first and last items in (98), we obtain the result of the Newtonian theory

Taking ^{?26} kg/m^{3} and_{0}. For the situation with Z = 1 and

For

According to the theory of nuclei synthesis in cosmology, relative density of baryon is

Take ^{?27} kg/m^{3} and^{?27} kg/m^{3} and get

We consider the universe as a material sphere with radius ^{3} and find its red shift is ^{3}, equal to the density of neutron star. According to the calculation before, the celestial body has moved to the position

The result is

Using (118) to calculates the time during which the universe radius expanses from 1.23 × 10^{26} m to 1.95 × 10^{26} m, the result is 13 billion years, so the time during which the radius of the universe expanses from ^{26} m is 17.8 billion years. This is just the universe age we consider at present. In the present cosmology, the universe age is estimated to be about 10 ~ 15 billion years, too short to the formation of galaxies [

By transforming the geodesic equation of the Schwarzs- child solution of the Einstein’s equation into flat space- time to describe, the revised Newtonian formula of gravity and the revised equation of cosmology are obtained. The singularity problem in the Einstein’s theory of gravity described in curved space-time is eliminated thoroughly.

Because using two improper and approximate conditions, the Freidmann equation becomes the result of the Newtonian theory of gravity actually. It is only suitable to describe the low speed expansive processes of the universe, unsuitable to describe the high speed expansion. The equation of cosmology needs relativity revision.

By using the revised Newtonian formula of gravity, the revised equation of cosmology is obtained. The high red-shift of supernova can be well explained. It is unnecessary for us to introduce the hypotheses of the universe accelerating expansion and dark energy. It is also unnecessary for us to assume that non-baryon dark material is 5 - 6 times more than normal baryon dark material if it exists actually. Many problems existing in cosmology including the problem of the universe age can be resolved well.

In this way, the theory of gravity returns to the traditional form of dynamic description and becomes normal one. The revised equation can be used as the foundation of more rational cosmology.