Revised Newtonian Formula of Gravity and Equation of Cosmology in Flat Space-Time Transformed from Schwarzschild Solution

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.


Introduction
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, singularity 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 [3]. The common understanding about the spatial curvature of the R-W metric is wrong. This idea would impose great influence on cosmology. Due to this result, many conclusions in the cosmology such as the densities of dark material and dark energy should be re-estimated.
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 [4]. It means that the Friedmann equation is equivalent to the Newtonian theory actually. It is only suitable for describing the process of low speed expansion of the universe, but not for the process of high speed expansion.
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 revised 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 Newtonian 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 including 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.

Revised Newtonian Formula of Gravity
Based on the Schwarzschild

Revised Newtonian Formula of Gravity
According to general relativity, the Schwarzschild metric (external solution) is Here  and are constants. By cancelling L ds from the formulas, we can obtain In which  is eigen time, t is coordinate time. Then, Here is the angular momentum of unit mass. (6) is just the conservation formula of angel momentum.

L
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 d , we get 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 0 , 0 r  and 0 represent the space-time coordinates of flat spacetime, due to the invariability of t 2 ds , we have 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 0 , r r  0    and get the relation between times and t 0 t by considering (4), we get and from (8) Comparing with (4), we have Comparing with (14), we get This is just the formula of time delay in special relativity. The result verifies the rationality of (18). Let at last, we write (18) as It is the revised Newtonian formula of gravity based on general relativity. In the formula, 0 is the static mass of moving particle and the center static mass 0 m M has spherical symmetry. Angle momentum makes gravity larger but speed makes it smaller. The result is equivalent to replace particle's static mass with following effective mass in the Newtonian theory 2 2 We can call m as the motion mass of gravity which is related to object's speed and angle momentum.

The Motion of Particle in Gravitational Field with Spherical Symmetry
For simplicity, we only discuss the motion of a particle moves along the radius vector direction with 0 L  . In this case, by considering (23), (25) becomes by multiplying on both sides of (27), the potential energy of the particle in gravitational field is dr   The dynamic energy of particle is . So the law of energy conservation of a particle in the gravitational field can be written as , we get the classic law of energy conservation in the Newtonian theory of gravity In the situation of , we calculate the problem in the weak field with 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 , we have and . Suppose that the particle is at point It is obvious that every thing is normal within the region . The particle is monotonously accelerated by gravitation. There is no any singularity in the whole spacetime. When particle is at the original point It indicates that the speed of particle tends to have light's speed in vacuum at point . Acceleration is also finite. So within the region , the motion of particle with static mass is continuous. Only at point the force acted on particles becomes infinite. But this kind of singularity appears in any theories in which particles are considered with infinite small size, and have nothing to do with space-time singularity. The singularity of the Schwarzschild solution is eliminated.

The Fiedmann Equation is Equivalent to the Newtonian Theory of Gravity
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 is scalar factor, is curvature constant factor, ( ) R t   is the universe material density and is the intensity of pressure. By eliminating form (39), we obtain the Fiedmann equation 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 . However, the Newtonian theory of gravity is only suitable for the motions with low speeds. For the high speed expansion of the universe, it is unsuitable. The Fiedmann equation needs relativity revision due to this fact. 0

 
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 , in the direction of sphere radius, the Newtonian equation of gravity is For the expansive sphere, by considering co-moving coordinate   r R t r  in which r has nothing to do with time, (41) becomes (42) is the same as the first formula of (39) when . Because mass is invariable in the expansive process, we have Substituting (43) in (42) and taking the integral, we obtain (40). In this case, integral constant is equivalent with curvature constant in the R-W metric.
 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.

The R-W Metric Violates the Principle of Invariance of Light's Velocity
According to the principle of cosmology, the universe is uniform and isotropy.
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 r increases with time, enough long time later, light's speed may greatly exceed its speed in vacuum. 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 , let 0 On the other hand, as we known that coordinate r has no meaning of measurement in curved space. What is meaningful is proper distance. Suppose that an observer stays at the original point of coordinate system, the definition of proper distance for the R-W metric between observer and light's source is [5].
        It is completely different from the R-W metric (44) when . The metric (53) seems to be curved but is flat essentially. According to the principle of the Riemannian geometry, if we can find a method to turn a curved space into flat, the original space is flat essentially. If we can not find such method, the original space is a curved space in essence. It is obvious that we can not find a transformation to turn (52) into (45) when , the spatial part of (45) can not be flat! On the other hand, the four dimensional metric in which three dimensional space has a constant curvature  is 2 Let 0   , we reach (53) rather then (45). Therefore, if we use co-moving coordinate to describer the expansive universe in which the space is flat, we should use (53), rather than (45). If we describe the expansive universe with constant curvature, we should use (55), rather than (44). Another result of using the R-W metric in cosmology is that it leads to the united universe time. In the R-W metric, 00 1 g  indicates that we have the same time for any spatial point in the expansive universe. This obviously violates special relativity. Because there is a relative motion speed between two objects in the expansive universe, there exists time delay between them according to special relativity. It is actually the result of the Newtonian mechanics to use the united universe time in cosmology. This is another reason why we say that the Friedmann equation is equivalent to the Newtonian mechanics.
However, it is easy to prove that if we use flat spacetime metric (53) in the Einstein's equation of gravity, the Einstein's tensor would become zero with 0 R   . In this way, we can not describe the gravity field of the expansive universe. Therefore, both the R-W metric and the flat space-time metric are unsuitable for cosmology. We should look for other proper methods to describe the expansive universe.

Dynamic Energy Momentum Tensor Should Be Used in Cosmology
The energy momentum tensor of ideal liquid is used in cosmology with the form is the four dimensional velocity. In the standard cosmology, we take and It means that we take static energy momentum tensor energy in the Einstein's equation of gravity without considering material's velocity. 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 between observer and luminous material. If co-moving co- 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 . The forth dimensional velocities 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 on the right sides but not on the left sides, we obtain the Fiedmann equation. But we can not do it in this way. Because  is a constant, we have three ways to make (65) tenable. The first is to let which describes the static universe. By considering the observation fact of the Hubble redshift, this is improper. The second is to take simultaneously 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 should be related to p r . In this 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.

Velocity, Acceleration and Initial
Conditions of the Universe Expansion

Velocity and Acceleration of the Universe Expansion
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. Therefore, 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 static mass of sphere with radius R is According to the Newtonian theory, gravity acted on a small object located at point r with mass is [6] 0 m 0 2 The formulas indicate that when mass 0 is located outsider the sphere with , the gravity acted on it is equal to that when the spherical mass is centralized at the center of sphere. When mass 0 is located inside the sphere with , the gravity acted on it is only related to r m r R  m r R  M , having nothing to do with the mass distributed outside the radius r.
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 with static mass 0 and speed r V along radius direction. Suppose that r satisfies (22) approximately, we use (22) to describe object's effective mass. According to (23) Because (81) can not be integrated directly, we need approximate method. When x is very small ( 1 ), by developing (76) into the Taylor's series in the region 0  , we obta Substituting the formulas in (78) and (85), we obtain the formula of acceleration and speed of the universe expansion

Initial Condition 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. However, 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 The integral of (93) is

Red Shifts of Cosmology and Hubble
Diagram of Supernova

The Red Shift of Ia Supernova
In Figure 1, the curved line with 0 and  0. represents practical relation between red shift and distance of Ia supernova at the early period of time t. According to photometry measurement, the density of luminous material in the universe is about 0 0.7     kg/m 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 27 2 10 0     kg/m 3 . In Figure 1, we take B 5.5 m   5log d L in which d L is luminosity distance with unit length . But the concept of luminosity distance is unnecessary in this paper for our discussion is based on flat space-time. So we need to transform 6 3.09 m 2 2 10 pc 10   d L to real distance r. The curved line in Figure 2 shows the relations between red-shifts, distances and parameters of initial condition of Ia supernova. The vertical coordinate is the values of      g K r for different objects. It is also unnecessary for us to introduce the concept of dark energy by introducing the effect of initial conditions. For