Spring Theory as an Approach to the Unification of Fields

The cosmological constant is necessary to be retained in Einstein’s field equations with value depending on the mass of the source. An overview of the spring theory in astrophysics and cosmology is included in this paper. In short range force, the two interacting particles are point-like vertices connected by a bosonic spring. We also suspect that electron may contain negative sterile neutrino. The self energy of a point charge is not infinite so that renormalisation is not necessary.

along a vertical path.
Properties of these equations had been studied by various authors [6] [7] [8]. Pavelle [6] pointed out that Yang's pure space is non-physical unless the cosmological constant remains in Einstein's field equations which was later verified by Mielke [8]. We begin from the second Bianchi Identity Comparing with Equation (2), R is a constant. Now the Einstein's field equations with the cosmological constant can be written as We obtain in the 4-dimensional case indicating that Einstein's case is a special solution of Yang's pure space where the covariant derivative of the Ricci tensor in Einstein's case is zero but not in Yang's case.
Hence, the cosmological constant needs to be retained but to be re-named as spring constant since it behaves like a harmonic oscillator as we can see later. In a 3-dimensional space, a spring term is added into Newton's law of gravity: where k is the spring constant of the source while Λ is assigned as the spring constant of the universe which also known as the cosmological constant. Throughout this paper, only 3-dimensional springs are to be considered.

The Pound-Rebka Experiments
These famous experiments can be found in many textbooks (see Gravitation by Misner/Thorne/Wheeler). The main purpose was to measure the frequency changes of pho-tons under the earth's gravity. The Jefferson Physical Laboratory at Harvard used a 57 Fe source placed at a height of 22.6 m above the detector.
Data were obtained when the gamma ray dropped onto the detector: 11 3.5 10 eV h ν − ∆ = × (11) height dropped 22.6 m ∆ = (12)   The last equation shows that the spring breaks at the distance of 32,000 km away from us. Equation (10) gives a clear picture of the fifth force different from the Yukawa type [9]- [15]. However, we have pointed out that the Yukawa type of fifth force is non-logical at 0 a = and cannot predict Equations ((22), (23) and (24)).

The Spring of the Moon
The almost vacuum lunar surface provides a frictionless condition for a free falling test to verify the existence of the fifth force as well as to obtain the spring of the moon. The total time travelled by a free falling object through a height H can simply be found as is the moon's gravity.
If fifth force does exist, the total time T must take longer than the classical one without the spring term [16].

The Spring of the Sun
The Binet Equation (53) where D is a constant. Setting the cosine part to zero, the spring of the sun is  Table 1 can be found in many astronomy textbooks. From Table 2, we can see that the average value of k is higher than those from various authors.
There are two main reasons of difficulty in determining the value of the sun's k: • the value of the two terms inside the bracket of Equation (27) is so close to each other.
• planetary interaction has not been taking into account.
However, the spring term of Equation (26) contributes insignificantly in the perihelion shift of planetary motion as well as the bending of light while grazing the sun.

The Cosmological Constant of the Universe
There are 3 main parameters in any cosmological model, namely the cosmological constant, the Hubble constant and the matter density [22]. In such a large scale structure, 3 dimensional space is sufficient to depict the universe instead of general relativity [23], Milne [24] and McCrea [25] used Newtonian mechanics to derive the cosmological equations while Harrison used the first law of thermodynamics and equations of hydrodynamics [26].
In the beginning, all matters were compressed into a high density lump of universe followed by a release in such a way that all matters were sprung out by the spring(s) as governed by the equation which is the same approach as Konuschko [27] except the cosmological term was not considered in his paper. Now, Equation (28) can be reduced to, upon integration:  [28] where they related Λ to ACDM model. From the above data, total mass of the universe is 51 10 kg M  . Hence, the universe stops to accelerate or 26 10 m r  which is approximately the present radius of our universe. At least it is a comfortable signal to show the tendency of ceasing to accelerate. Interestingly, matter at the outer rim of the universe exceeds the speed of light, i.e. ~10 8.5 m/s. Superluminal recession of galaxies is acceptable by some cosmologists [29] [30].

The Missing Mass in the Rotation Curve of Galaxies
It is already known that the cosmological constant is the answer of dark matter [31], or more precisely, variable cosmological constant [32]. This is explicitly referring to the spring constant of the galaxy, but awaiting to be spelt out. Again, in such a large scale of structure, only approximate estimation can be achieved with the following assumptions: • aberrations in the observed velocity and distance are unavoidable [33] [34].
• the radius R of the cluster and the velocity can be estimated from the rotation curve.

= ×
In the quantum version of the virial theorem, the average value of the operator T in energy eigenstates in one dimension is given by where T is kinetic energy and V is potential energy. Since the angular velocity of the galaxies is very small: 15 10 s ω −  , the viral theorem holds.
We have studied the rotation curves of galaxies in Figure 1 and Figure 2 with the help of the virial theorem.
( ) Nearly all these rotation curves yield the same (for detail see [17]) It is clear that each mass has only one unique spring constant assigned to it. Strictly speaking. a flat curve means that the mass is still decreasing depending on 2 2 v kr − in the virial theorem but good approximation of k can be obtained even though v remains constant over several kpc's. We take two other papers as a comparison. Firstly, we consider Gessner's paper [37] who used general relativity to investigate 6 NGC's, found the mass of galaxies

The Electric Field
The electric field energy density W surrounding a charge q is proportional to the square of the field intensity E   Since a charge is always accompanied by its electromagnetic mass m δ , the total mass of a charge particle is M = mechanical mass + electromagnetic mass m δ . The above two masses are non-separable from each other. The relationship of the charge and field density is assumed as which seems to be reasonable to say the energy density of the source is proportional to the energy density of its surrounding field. Upon integration A and B are constants. Integrating over the whole space, and set A = charge q, the total energy which is just the Gauss Law except the right hand side of of Equation (42) is not e m in the case of electron. Thus The potential can be written as ( ) long range, 10 m 4 where A and B need to be determined in short range since coupling is involved. Ob- There were queries about the internal structure of electron in the last century [39]. Bonnor even raised the question "Does electron contain negative mass?" [40]. As already known [41] that the electron mass resulting to a value less than 1 eV: too small to affect the fine structure of hydrogen spectrum.

The Gravitational Field
As both the Coulomb and Newton's inverse square law are analogous to one another, the gravitational field from Equation (42) To solve for the above Equation (53), we followed the same procedures as in [42] and [43] to get Equation (26). Comparing the tests with general relativity, the spring term contributes insignificantly in the bending of light while grazing the sun whereas the perihelion shift of a planet gives  c) Spring theory (Equation (45) + spring term)

Spring in the Short Range Interaction
where a and b can be estimated roughly from the graph. However, the constant C is in fact the energy of the spring or rather say, the energy of the confined quarkonia. It follows that  [45]. In fact, tracing back to 1981 [46], or even earlier, the Cornell potential was recommended as the unified potential for quarkonia, mesons and baryons. We hereby encourage particle physicists to use spring theory.

Discussions
Revisiting the equations from (40) to (45), we come to something interesting: • total field energy of a charge particle with radius R ( ) For R = 0, none of the above tends to infinity. In a book written by Sapogin [47], it was mentioned that the classical theory of electromagnetism was fundamentally wrong.
The electric field at the centre is zero because E is a vector. Feyman pointed out that Coulomb's inverse square law fails at very short distance (see Feyman Lectures on Physics volume 2 chapter 5.8). Hence, renormalisation is not necessary. Perhaps short range Maxwell's equations can be furtherly elaborated towards a new branch of electromagnetism [48].