Or, after contraction of and

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

For, we have

Operating on above, we have

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 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.

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 photons under the earth’s gravity. The Jefferson Physical Laboratory at Harvard used a ⁵⁷Fe source placed at a height of 22.6 m above the detector.

Data were obtained when the gamma ray dropped onto the detector:

which is only true at Harvard, or likewise, the state of Massachusetts. In 1965 Pound and Snider refined the apparatus so that the energy shifts on the upward and downward path gave the measured difference of

Since the first term of Equation (15) is known, the second term will immediately yield the deceleration of. Taking

the earth’s rotation (16)

the earth’s radius (17)

the earth’s mass (18)

being the latitude of Massachusetts where Pound and Rebka performed their experiments at Harvard. Upon substituting the acceleration and deceleration into Equation (10), we obtain the following two equations

Thus, the spring constant of the earth which, as expected, is different from the cosmological constant. The spring term is so-called because it behaves like an harmonic oscillator.

From Equation (10), there exists a point or a spherical shell at a distance of away from earth where the spring and the earth gravity cancel out each other to give:

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 and cannot predict Equations ((22), (23) and (24)).

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

where 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 Binet Equation (53) yields the solution [17]

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.

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:

which is just the Hubble’s law having the Hubble constant in agreement with many literatures [28] where they related to ACDM model. From the above data, total mass of the universe is. Hence, the universe stops to accelerate when

or 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] .

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:, 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 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 to and the spring constant to. The results matched our work. The second paper belonged to an Indian team led by B. Aryal [38] . They also used general relativity to investigate 15 NGC’s, found to, and the spring constant to which is negative. Based on the information in their paper including the value of M, we used the virial theorem to obtain a positive value of. Once again, in large scale structure, 3 dimension is sufficient to depict astrophysical phenomena.