Modification of Einstein’s Gravitational Field Equation and Cracking of Foundational Difficulties in Physics

Abstract In the framework of gravitational theory of general relativity, this article has systematically and radically solved the problem of galaxy formation and some significant cosmological puzzles. A flaw with Einstein’s equation of gravitational field is firstly corrected and the foundations of general relativity are perfected and developed, and space-time is proved to be infinite, expansion and contraction of universe are proved to be in circles, the singular point of big bang is naturally eliminated, celestial bodies and galaxies are proved growing up with cosmic expansion, for example Earth’s mass and radius at present increase by 1.2 trillion tons and 0.45mm, respectively in a year, in response to which geostationary satellites rise by 2.7mm.


Introduction
Though general relativity obtains considerable success, some significant problems such as the problem of singular point, the problem of horizon, the problem of distribution and existence of dark matter and dark energy, the problem of the formation of celestial bodies and galaxies, the mystery of solar neutrino, as well as the problem of asymmetry of particle and antiparticle, always are not solved naturally and satisfactorily. These problems long remain implies strongly that the fundamentals of general relativity have flaw and needs further perfection. For the purpose, this paper begins with determining the vacuum solution of Einstein's field equation in the background coordinate system, then by correcting rationally Einstein's field equation from an all new perspective these problems get removed.

The static metric of spherical symmetry in background coordinate system
In this paper light's speed 1 c = . According to general relativity, for the static and spherically symmetric case, in the standard coordinate system (Weinberg, 1972;Peng and Xieng, 1998), the correct form of line element outside gravitational source is given by Hereτ is proper time, M is the total mass of gravitational source; l is called standard radial coordinate, doesn't have clear physical significance and only in the far field is approximately viewed as true radius. In order to describe clearly motion of particle and enable general relativity to link up with other theories of physics and to compare results one another, it is necessary to transform (1) into the form expressed in background coordinates. Hence we take ( ) l l r = . Here r is defined as background coordinate (ZHou, 1982;Zhou, 1983;Fock, 1964) and refers to true radius which are said and used usually. , , t θ ϕ are standard coordinates and can also be viewed as background coordinates, which represent true time and angle. In the following we try to determine ( ) l l r = by the introduction of an additional transformation equation, and such operation is allowed is because metric tensor satisfies Bianchi identity and if a metric is a solution of field equation in one coordinate system it is also a solution under arbitrary coordinate transformation.
According to general relativity the dynamical equation of particle outside source is geodesic where 0 x t = , and indexes , , , , , 0,1,2,3 v λ µ σ α β = . Equation (2) (2) When a particle of mass m is moving along radius in the static gravitational field of spherical symmetry, giving consideration to the speed, in the background coordinate system, in the far field (weak field) the radial component of Eq.
(2) should reduce to the following relativistic equation (3) rather than others 2 d dr mGM m dt dt r where m refers to relativistic dynamic mass, namely . Why the radial component should reduce to (3) is that (3) stands for the equality of gravitational mass and inertial mass and also stands for the speed of light is the limit one. In order to enable it to reduce to (3) we may introduce a transformation equation as follows The correctness of Eq. (4) will be seen later, it determines a coordinate transformation of l r → .
By means of separating variables, the solution of Eq. (4) is easily given by Here constant 1 C is determined by the continuity of function ( ) l l r = on boundary of source, and the back equation (23) Note that now ( ) l l r = is already a specific function of r , which is determined by Θ .
Consequently, we conclude that (5) is the appropriate line element which satisfies the requirements completely.
As a serious emphasis, we must point out that using directly l r = in (1) gives another exact solution, namely the following (7) However, in accordance with (7) the corresponding geodesic can't reduce to (3) in weak field, instead it reduces to Proof: (7) provides 00 2 1 GM g r = − + (1 2 ) GM GM r r Γ = − ( 1 ) (1 2 ) Eq. (8) not only goes against the elementary principle of equality of gravitational mass and inertial mass but also leads to incorrect conclusion that gravitational field becomes repulsive one for a particle whose speed exceeds 0.58c. Hence Eq. (8) must be wrong, and implies (7) can't describe high speed and has shortcoming as compared with (5).
Note that the angle of orbital precession of Mercury described by (5) is still the same as that described by line element (7) (Peng and Xieng, 1998), it doesn't change under the transformation of radial coordinates. In a word, (5) is the correct line element expressed in background coordinate system And again, though general relativity doesn't exclude to use other coordinates, we must use background coordinates when we take geodesic to compare with Newtonian gravitational law because the meaning of every term in the geodesic is not clear in other coordinates and the comparison cannot realize.

Modification of Einstein's gravitational field equation
It is seen from the above discussions that in the case of weak field approximation, Tg µν . Note that the coupling constant γ relates to the form of weak field approximation metrics g µν and is determined in the course of solving weak field approximation metrics, and the change of the metrics means that the coupling constant γ need change too. And now we set out to reconfirm the coefficient γ by solving weak field approximation metrics g µν .
Here T µν ( ) Omitting the terms of less than 2 ( ) o h we have (Weinberg, 1972) and adding up the above two equations In view of (3) After a certain calculation we arrive in 11 00 11 00 11 00 1 , , On the other hand, for the weak field case Bianchi identity can give the ordinary conservation law , 0 T µ ν µ = . Proof: For the static case, using , , , And again, as a special case of spherical symmetry, namely we infer from (10) a very useful result p ρ = − which can be regarded as the form of pressure in weak field in the case of that ρ is homogeneous. It is obviously too subjective to take pressure for zero in advance, in fact, by serious calculation we see that the pressure is negative where matter exists and the place where matter exists turns out to be so-called pseudo-vacuum (Gondolo and Fresse, 2003;Guth, 1981). This is a new important result which isn't in agreement with traditional opinion.
To sum up, we can conclude that in any coordinate system gravitational field equation is revised as where 4 replaces previous 8 − , obviously Eq. (11) preserves general covariance. Of course, (1) and (5) satisfy Eq. (11) because both p and ρ vanish outside source, Eq. (11) becomes 0 R µν = .

Applications and tests of Eq. (11) in cosmology
With l as standard radial coordinate, in the co-moving coordinates Friedmann-Robertson-Walker metric is given by (Weinberg, 1972;Sawangwit and Shanks, 2005).
Here n represents the density of particle (galaxy) number. Since ρ is assumed homogeneous, we may use the weak field condition p = −ρ , and substituting it into Eq. (13) yields 0 which is the most appropriate expression of energy conservation in infinite spacetime and indicates the singular point of big bang did not exist. In addition, (13) implies the mass of galaxy changing with cosmic expansion since n ρ stands for per particle mass. And further, the solution of Eq. (12), namely expanding factor, is given by Here A is a positive constant. So far cosmic expansion and contraction are proved to be in circles. Note that (15)shows that the expansion of the present universe is decelerating but not accelerating this fact agrees to the newest conclusions (Dominik J, 1993). Now we compute the relation between distance and red-shift. May as well put 0 ( ) 1 a t = , the light from a galaxy to us satisfies (Weinberg,1972) Note that the subscript "0" refers to the present-day values. . (Linder, 2003;Hamuy, 2003;Alcaniz, 2004 ) No , an , and further we have the following relation of reshift Z and universe time t Here t is the time at which photons was given out from the celestial body. The relation can be used to evaluate low limit of celestial body age.

Exact interior solution of Eq. (11) and mechanism of celestial body's expansion.
In the case of static spherical symmetry, inside a celestial body (gravitational source), with l as standard radial coordinate the exact interior solution of Eq. (11) is given by . In order to determine the interior form of (20) in background coordinates, Eq. (4) is naturally extended as inside source Obvious under the transformation of Eq.
Here ( ) l l r = is a specific function of r , which is determined by Eq. (21). Line element (22) On the other hand, the conserved law gives out On the boundary the gravity acceleration should be continual, according to (2), using (4), (5), (21),  As an emphasis, we must point out that when (1) or (5) (26) that when celestial bodies expand with cosmic expansion its density may be unchanging in the case of 0 p ρ + = .
So far, we deduce that bursts of celestial bodies and formation of earthquakes originate from unceasing accumulation of inside matter and change of distribution; and it is the negative pressure that gets matter in celestial body continuously product.

Cracking of the problem of dark matter
The negative pressure as important gravitational source is invisible , and it is the negative pressure that appears as the form of dark matter and leads to the phenomenon of missing mass, or say that so-called dark matter is just the negative pressure, this fact is showed as follows.
Speaking generally, within a galaxy the metric field is weak field, and when a galaxy is treated as a celestial body of spherical symmetry, according to the discussion in section 3, within the galaxy ( 0 e r r ≤ ≤ ) pressure 0 p const = ≠ . And from (10) From (27) it is seen that when ( ) m r looks even on the verge of zero near the centre of the galaxy the speed v can become high, too, and this explains so-called missing mass. Again, from (27) (27) and (29), and it is in conformity with recent observational results (Cayrel et al., 2001).
So far, we conclude that so-called dark matter is just the effect of the negative pressure, and the dark matter (Genzel et al., 2006;Li, 2008, Gaugh, 2008 puzzle has naturally been solved. Of course, so-called dark energy problem is also removed since cosmological constant is reconfirmed as zero and the concept of dark energy becomes unnecessary in the new amendment.

Fig. 2. The velocity distribution diagram 7 Motion in centre field and galaxy formation
Equation (14) indicates that not only space is expanding but also celestial bodies or galaxies themselves, that is, like a expanding balloon, the ink prints on it proportionally expand also. To illuminate galaxy formation clearer we investigate the motion in centre field. Let M denote mass of centre body. Generally speaking，its gravitational field is weak, geodesic reduces to Newton's law, for a object moving around the centre body we have where r is the radius of round orbit, T is revolution period. Noticing M to be variable now and to satisfy (18) and using (30) we infer where the final term is explained as perturbation and gravitational radiation. For instance, apply (31) to the motion round today's Earth, for geostationary satellite, neglecting perturbation and gravitational radiation, namely taking 0 0 dT = we find that its orbit radius will increase by 0 2.7 r ∆ = mm in a year. And for the motion of Moon, observations show that its orbit radius increases by 0.38cm in a year today, then using (31) we conclude that the orbit period 0 T of Moon will slow by 0.0001s in a year today. When (31) is used to the edge of a spiral galaxy, it is concluded that the terminuses of spiral arms gradually stretch outward. Of course, other points near the terminuses continuously follow and form involutes. See the following figure 3.
Expanding centre Gradually stretching out arm

Fig. 3. Sketch map of formation and evolution of spiral arms
Eq. (31) means that separating speed from centre lies on v rH = neglecting perturbation and radiation damp.
It is important to realize that the spin of a system is the composition of orbit motion of many particles, spin and orbit motion do not have essential difference. And for celestial body's expansion, lying on v rH = means its spin period not to change.
Note that the existence of equation (31) doesn't mean the destruction of conservation of angular momentum because mass M is connected with the factor ( ) a t , which embodies the interaction among galaxies, the non-conservation of angular momentum of individual galaxy is admitted.
Again, the fact that space, celestial bodies and galaxies simultaneously expand proportionally links the homogeneity of today's universe in a large range with that of early universe in a small range, because the large range is just the amplification of early the small range. Background radiation has proven early universe to be homogeneous in quite small range. Therefore our conclusion is in accordance with observations.
The following figure 4 is the global picture of galaxy evolution and distribution under const ρ = in different stages, the earlier, the smaller and the denser. Figure 5 is the picture of galaxies seen by today's telescope, and the farer, the earlier and the evener. Naturally, the microwave background radiation measured today is the compositive effect of various photons emitted by innumerable remote galaxies, whose distances to us are unidentifiable, which comprised infinitely deep thin gas and can absorb any frequency photon and therefore possess black body character.
Earlier Early Today

Fig. 4. The global picture of galaxy evolution and distribution in different stages
Background radiation source Galaxy Telescope

Fig. 5. The actual picture of galaxies seen by today's telescope
Note that the horizon at moment So-called horizon puzzle or homogeneity puzzle does not exist in the present theory framework at all. Naturally, the microwave background radiation measured today is the compositive effect of various photons emitted by unnmberable galaxies remote, whose distances to us are unidentifiable，which comprised infinitely deep thin gas and could absorb any frequency photon and therefore possess black body feature. Note that the state that horizon vanishes is unobservable though ( ) 0 h d t = for 0 t = , because any observation needs a lag of time t ∆ 8 Quantum process of continuous creation of matter in celestial bodies P ρ = − tells us that the negative pressure in celestial bodies is actually a negative energy field, and p and ρ excite with each other and generate simultaneously. Connecting with particle physics it is naturally deduced that in celestial bodies many particle-antiparticle pairs (including neutron and antineutron, proton and antiproton, electron and positron and so on) can ceaselessly produce and annihilate, the antiparticles lie in negative energy level(can't be observed ), the particles lie in positive energy level, and the absolute value of energy of particle and antiparticle is equal. Let t ∆ denote the lifetime of a kind of particle-antiparticle pairs, namely the average time from production to annihilation, according to uncertain principle the range E ∆ of energy satisfies which shows that instantaneous energy of new particle may be very high. Note that not all of the particles annihilate as soon as they come into being, only those which don't not have opportunity in the time t ∆ to react with the surrounding particles or to collide and change their energy can annihilate, once the reaction with other particles or the collision occur the annihilation no longer carry out, and in this case the negative energy field detains a negative energy antiparticle while the particle becomes constituents of matter. Therefore, the negative energy field is too a quantum field to consist of various negative energy antiparticles. Of course, an antiparticle of negative energy ε − can be excited to positive energy ε by a meson of energy 2ε and becomes the antiparticle that can be observed. For no other reason than that many antiparticles lie in negative energy level and can't be observed, we perceive that particles and antiparticles aren't symmetrical. As a result of general relativity, equation (14) in section 4 exposes already that matter and antimatter are symmetrical.
Obviously the negative pressure field provides energy source of star radiation, not only thermal nuclear reactions, therefore the mystery of solar neutrino doesn't exist in the new theory framework.
And considering of tunneling effect in quantum theory, many nuclear reactions are able to complete slowly in celestial bodies even if the temperature ( average kinetic energy of particles ) is low, which implies that in the case of low temperature elements can also compose. As for what kind of nuclear reaction is in evidence, this depends on temperature of celestial bodies. And as a result, the abundance of elements in a celestial is the effect of various nuclear reaction for long time.
For a celestial body of temperature T, we may as well treat all atoms in it as an open thermodynamic system, whose giant distribution function according to quantum statistics is (33)decides the abundance of elements in a celestial body. Observations of astronomy show that element abundance is different for different celestial, which is consistent with (33).
Observations of astronomy show that the abundance of elements is in accordance seen from large scope, which implies both temperature and chemical potential are uniform seen from large scope.
Observations of astronomy show that all elements in other body can be found out on the earth, which implies that the origin of various elements is the same, namely originate production and annihilatation of particle-antiparticle pairs.