_{1}

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.

Astrophysical standard model has confirmed to accept the cosmological constant in relating to dark energy―undetectable particles same as dark matter. The matter distribution inside the universe is roughly dark fluid 95 percent and normal matter 5 percent. Different notions such as dark energy, dark matter, aether, pure space and others are of the same entity. They are different manifestations of the same dark fluid aether, just like the extension and compression of a spring [

Or, after contraction of

Properties of these equations had been studied by various authors [

For

Operating

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

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 ^{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:

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

the earth’s rotation

the earth’s radius

the earth’s mass

being the latitude of Massachusetts where Pound and Rebka performed their experiments at Harvard. Upon substituting the acceleration

Thus, the spring constant of the earth

From Equation (10), there exists a point or a spherical shell at a distance of

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 [

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

If fifth force does exist, the total time T must take longer than the classical one without the spring term [

The Binet Equation (53) yields the solution [

where D is a constant. Setting the cosine part to zero, the spring of the sun is

Mercury | Venus | Earth | Mars | |
---|---|---|---|---|

r (10^{9} m) from sun | 58 | 108 | 149 | 224.9 |

Period T (days) | 89 | 224.7 | 365.25 | 686.98 |

2.7 | 3.7 | 4.4 | 5.3 | |

Jetzer [ | Cardona [ | Iorio [ | Adkins [ | Tsang | |
---|---|---|---|---|---|

Mercury | 10^{−}^{24} | - | 10^{−}^{24} | - | 10^{−}^{14} |

Venus | 10^{−}^{22} | - | 10^{−}^{22} | - | 10^{−}^{15} |

Earth | 10^{−}^{25} | - | 10^{−}^{25} | - | 10^{−}^{16} |

Mars | 10^{−}^{25} | - | 10^{−}^{25} | - | 10^{−}^{16} |

General | - | 10^{−}^{25} | - | 10^{−}^{25} | - |

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 [

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 [

Equation (28) can be reduced to, upon integration:

which is just the Hubble’s law having the Hubble constant

or ^{8.5} m/s. Superluminal recession of galaxies is acceptable by some cosmologists [

It is already known that the cosmological constant is the answer of dark matter [

・ aberrations in the observed velocity and distance are unavoidable [

・ 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:

We have studied the rotation curves of galaxies in

Nearly all these rotation curves yield the same (for detail see [

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

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

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

The potential can be written as

where A and B need to be determined in short range since coupling is involved. Obviously, for

and

Among the 4 pairs solution after solving the above, the most logical pair is

Instead, through the electron-positron scattering, the upper limit of

resulting to a value less than 1 eV: too small to affect the fine structure of hydrogen spectrum.

As both the Coulomb and Newton’s inverse square law are analogous to one another, the gravitational field from Equation (42) becomes

where

Including the spring term, the new Binet equation can be written as

To solve for the above Equation (53), we followed the same procedures as in [

a) Cornell potential

where

b) Natural log potential

where

c) Spring theory (Equation (45) + spring term)

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

There are many combinations of a and b in Equation (57). For instance, for charmonium,

Revisiting the equations from (40) to (45), we come to something interesting:

・ total field energy of a charge particle with radius R

・ total self energy of a charge with radius R

For R = 0, none of the above tends to infinity. In a book written by Sapogin [

Tsang, L.M. (2016) Spring Theory as an Approach to the Unification of Fields. Journal of Modern Physics, 7, 2219-2230. http://dx.doi.org/10.4236/jmp.2016.715192