**
The
fundamental gravitino and its physical dark world**

**
Chapter 9**

**
THE MASS OF THE DARK MATTER FG AND TWO EXPERIMENTAL SUPORTING DATA.**

**
9.1 The principle for calculating the ¡°FG¡± mass; and the numerical relation of
FG mass and the universe mass**

We believe, when two B bodies get close each other to a critical scale, a
repulsive elastic force will be produced. If the mass in the elastic radius is
defined as the mass of the B body and let n_{p}(n_{e})
be quantum number of full filled (space like) orbit B body, we have

r _{p} = r_{Bp} = r_{1} n_{p}^{2} ; r_{Be} =
r_{1} n_{e}^{2} (9.1-1)

Where r_{Bp} and
r_{Be} is
the radius of core of proton and electron respectively; r_{p} is
acting radius of charge of proton. From equation (8.2), when the main quantum
number large enough, the number of orbital FG of B body can be calculated by the
following formula:

( 9.1-2)

Because of sharp decrease of the force field, a reasonable approximation
is 2n^{3} /3.
Therefore we get

(9.1-3)

Where M _{p} and
M_{e} is
the mass of proton and electron respectively. Considering

(9.1-4)

And

(9.1-5)

We get

(9.1-6)

M_{u} =10^{53}kg (9.1-7)

Finally

m_{F} =
3.636 x 10^{-45} kg (9.1-8)

From the argument about the ¡°Basic body of particles¡± above mentioned, this ¡°B body¡± can be summarize as following.

The scale of mass of the stable ¡°B body¡± in the universe is unique.

There is ¡°FG cloud¡± surrounding the ¡°B body¡±, which is governed by the quantum mechanical relation. But, the degeneration of orbit induced by the gravitational force of orbital ¡°FG¡± so that comes into the collapse of the ¡°FG cloud¡±. I.e. the interval of level decrease and the number of orbit increase. A lot of orbit are constrained a very thin layer outside of¡± B body¡± with a large main quantum number.

In equation (9.1-2) the ¡°B body¡± with surrounding ¡°FG cloud¡± assure that there exist three Stable states in the universe.

1. The fulfilled orbital state

2. The space like state (the ¡°B body ¡°as a Core oscillates violently, and the surrounding ¡°FG cloud¡± disappears).

3. The coupled state, i.e. Two ¡°B body¡± shares a common ¡°FG orbital cloud¡±.

The three stable states correspond to the three stable elementary particles, namely proton, electron and neutron; they are most stables one.

The four properties of the ¡°B body¡± provide a method to estimate the mass of the ¡°FG¡± original particle. It is an approximate method for the order evaluation. In the following we are going to introduce the principle of the calculation.

As is well known, the mass of proton, electron has been measured correctly in physics. They correspond to the fulfilled orbit and space- like Orbit State of the ¡°B body¡±. It follows that the difference of mass between a proton and an electron is determined by the mass of the orbital ¡°FG cloud¡± in a proton because they have same internal core of the ¡°B body¡±. So I can estimate the number of ¡°FG¡± original particle by the following method.

The ¡°FG cloud¡± outside the ¡°B body¡± gives raise to the degeneration and even the collapse of the energy level by gravitational interaction. Therefore the relation of the ¡°FG¡± orbital radius r and the main quantum number are equivalent to the relation in the atom model. The Pauli exclusive principle is obviously suitable, because all of the characters of the ¡°FG¡± is obtained under the assumption that the Pauli exclusive principle is suitable. Therefore the quantum mechanics provides the method for calculating the relation between the number and main quantum number of the ¡°FG¡± in the ¡°FG cloud¡± directly. (In calculation we keep only the first order and ignored the term that is comparable with the lost part induced by the slight oscillation of ¡°B body¡±)

In addition the number density of protons is comparable with that of electron. Although electron is a state of space-like orbit, there is still a very thin layer of ¡°FG cloud¡±. In the calculation of the integrating value includes the region between the shell and the first orbit. Therefore we can use the mass ratio of proton and electron to replace the cubic ratio of their radius without any error.

According to the above-mentioned principle after appropriate mathematical treatment we can obtain the mass of ¡°FG¡±:

m_{F} =3.63
6 ¡Á 10^{-45} kg.

**
9.2 The experimental test and the verification of the theoretical mass of ¡°FG¡± ¨D
observation on the double star and pulsar**

When we review the experimental data recorded in file, there are at least
two of them that have verified the theoretical value of the ¡°FG¡± mass. The first
was an observation in 1960 by De Brogue on the Binary star another was in 1969
by Feinberg on the pulsar. They have found that the rest mass of a photon is
about 0.8 ´ 10^{-39} kg
and 10^{-44} kg.
The average value of them is about 10^{-42} kg.
It is completely consist with my theoretical value.

I attach the details about verifying and calculation of two observations as follows:

For a long time physicists are attempt to use various electromagnetic
phenomena to examine the validity of the Maxwell theory of the electromagnetism
and check whether the rest mass of photon is zero. These experiments check also
the principle of the constancy of the speed of light. So far the examinations
are based on the massive electromagnetic theory (Proca equation). Assuming the
Lorentz transformation valid, and giving up the phase angle gauge (U _{(1)} gauge)
invariant, with a extra term added to represent the rest mass of photon, this is
a modified Maxwell equation namely Proca equation. In this case the constant in
Lorentz transformation does not represent the speed of light in common sense,
but is a universal constant with dimension of speed. In the following we will
see that it is a limited speed of photon. In other words, when the frequency (or
energy) of photon tends to infinity, its speed tends to constant c. In such a
theory the invariant principle has been violated. We would like to introduce the
Proca equation for electromagnetic field. Then we will predict some effects of
rest mass of photon based on the Proca equation and exam it with experiments.
Unfortunately, so far non-of the processed experiments show positive effect of
rest mass of photon ¦Ì. Only providing an upper limit on ¦Ì (Goldhaber and Nieto
have reviewed it in more detail)

In the following, we¡¯d like to briefly introduce the Proca equation. As is
well known, that in the Lagrangian formula of the Maxwell theory of
electromagnetic field, the Lagrangian density of the field is a bilinear type
and invariant (scalar) under Lorentz transformation and phase (U_{(1) }gauge)
transformation, consisting of field variable A_{¦Ë} (potential
function) and its first order derivative ∂A_{¦Ë
/} ∂x _{¦Í.} From
the Lagrangian through a variation procedure we get the Maxwell equation. For
giving up the U_{(1)} gauge
invariant, we add a mass term ¦Ì^{2}A_{r}A_{¦Í }to
the Lagrangian. From the modified Lagrangian we get Proca equation, which is a
massive electromagnetic field equation. In Gauussian unit is

(9.3-1)

Where

(9.3-2)

Which satisfies

(9.3-3)

The Greek alphabets run from 1 to 4. ¦¥ _{¦Ë¦Í¦Ñ¦Ò }are
a unit complete anti-symmetry tensor.

Is vector potential, f is scalar potential, J is current density, r is charge density.

The four dimensional current density J_{v} is
conserved, satisfy conserved equation

(9.3 -4)

Differentials equation (9.3-1), and using equation (9.3 -2) and (9.3-4) we get

(9.3-5)

It implies the charge conserved condition (9.3-2) and Lorentz condition (9.3-5) are equivalent to each other.

Put (9.3-2) into (9.3-1) and using (9.3-5) we obtain the wave equation of
electromagnetic potential A_{¦Ì}

(9.3-6)

Where (
¡¯Alembertian). The equation determines the electromagnetic potential A_{¦Í} uniquely.

The three dimensional form of (9.3-1) ~ (9.3-6) are

(9.3-7a)

(9.3-7b)

(9.3-8a)

(9.3-8b)

(9.2-9a)

(9.2-9b)

(9.2-10a)

(9.2-10b)

(9.2-11a)

(9.2-11b)

Obviously, when ¦Ì = 0, the Proca equation is reduced to the Maxwell equation.

Proca first suggested the equation in 30¡¯s this century. It is a uniquely generalized Maxwell equation (remaining invariant under Lorentz transformation). Equations (9.2-7) ~ (9.2-11) are the foundation for examining the rest mass of photon by experiments. We would like to introduce further in the following.

The most direct consequence of the massive electromagnetic theory is the dispersion effect of light speed, due to the massive photon (¦Ì¡Ù 0) in a vacuum. The free plane wave solution of equation (9.2-6) in a vacuum (absent charge and current) is

A_{¦Í }~
exp{i(k¡¤r ¨C ¦Øt)} (9.2-2.1)

Where wave vector k ( |k| ¡Ô 2 ¦Ð ⁄ ¦Ë, ¦Ë is wave length), angular frequency ¦Ø and mass satisfy

k^{2} - ¦Ø^{2} ⁄
C^{2 } = -
¦Ì^{2} (9.2-2.2)

This is the dispersion relation of an electromagnetic wave in vacuum. The phase speed of a free electromagnetic wave is

¦Ì = ¦Ø ⁄ |k| = c (1 - ¦Ì^{2} c^{2} ⁄
¦Ø^{2 }) ^{¨C1/2 }(9.2-2.3)

The group speed is defined as

v _{k }=
d ¦Ø ⁄ d |k| = c (1 - ¦Ì^{2} c^{2} ⁄
¦Ø^{2 }) ^{¨C1/2} (9.2-2.4)

Because the mass of photon ¦Ì is a finite constant, when ¦Ø¡ú¡Þ the phase speed and group speed tend to a constant c. i.e. the constant c in Proca equation is a speed of free electromagnetic wave with its frequency tending to infinity.

From equation (9.2-2.1) and (9.2-2.2) we can see, when ¦Ø = ¦Ì c, k = 0 the free
electromagnetic wave does not propagating. Otherwise when ¦Ø < ¦Ì c, k^{2} <
0, k is an imaginary number, there appears an exponential damping phase factor
exp{- |k| r}. In this case the amplitude of the wave will exponential dumping.
Only when ¦Ø › ¦Ì c, the wave may propagate free of damping. The phase speed and
group speed is given by equations of (9.2-2.3) and (9.2-2.4) respectively.

The equation (9.2-2.4) implies that the propagating speed of electromagnetic wave with different frequency is different, which is called the dispersion. This phenomenon provides a possibility to determine the rest mass of photon (measure the speed of light signal with different frequency, or measure the time difference between light signals with different frequency, traveling through the same distance)

Consider two series of electromagnetic waves with different angular frequency ¦Ø_{1},
¦Ø_{2},
assuming ¦Ø_{1},¦Ø_{2} >
¦Ì c. In this case the speed difference of the waves is given by (9.2-2.4)

(9.2-2.5)

In last equation the (¦Ì ^{2 }c ^{2 }/
¦Ø_{2})_{ }^{2} and
smaller term have been ignored. In the same approximation, from (9.2-2.2) we get

(9.2-2.6)

Using (9.2-2.6) the ∆v can be expressed in term of wavelength

(9.2-2.7)

If the wave series travel through same distance L the time difference of them will be

(9.2-2.8)

Equations (9.2-2.5)~(9.2-2.8) is the start point for using the dispersion effect to determine the rest mass of photon.

(3). The time difference of the light of star reaching the earth

Equation (9.2-2.8) shown ∆t is proportion to L, as the distance longer the effect will be larger. We can use the time difference ∆t resulting from the light with different frequency travel through the same distance to determine the rest mass of photon ¦Ì. We can measure the time difference of electromagnetic waves emitted from a distant star with different frequency when they reach the earth. For example by using the binary or pulsar we can process such observation.

It is worth to emphasizing that the dispersion effect of light of a star could be explained as an effect of rest mass as well as the non-linear effect of the electromagnetic field or plasma effect. In the universe space between the star and the earth there exists very dilute universe medium (plasma). The dispersion due to the plasma is quite similar to that of the rest mass of photon. This is a main obstacle to using the dispersion effect to determine the rest mass of photon. Now we would like to introduce briefly the dispersion effect of electromagnetic wave in plasma.

In general the dispersion equation of electromagnetic wave in plasma is

(9.2-3.la)

(9.2-3.lb)

Where n is the number density of electron in the plasma, m the rest mass
of electron, B the magnetic induction field, ¦Á: the angle between k and B. In
inter galaxy space the magnetic field is very weak ¦Ø_{¦Â} can
be ignored. Therefore equation (X.2-3.2) gives the dispersion effect of
electromagnetic field in plasma.

V_{g} =
d ¦Ø ⁄ d |k| = c (1 ¨C ¦Ø _{¦Ñ}^{2} ⁄
¦Ø^{2 })^{ ½} (9.2-3.2)

Comparing (9.2-3.2) with (9.2-2.4) we can see that the form of dispersion effect due to the characteristic frequency of plasma is the same as that due to the rest mass of photon. We could not distinguish them if no other method can be used to acquire knowledge of the density of intergalactic plasma. This prevents us from using freely the dispersion effect of starlight to determine the rest mass of photon ¦Ì.

**
**

In 1940 de Brogue suggested a method to determine the rest mass of photon
by using the binary system. The binary consists of two stars (for instance named
S_{1},
S_{2}),
which rotates in an elliptical orbit. At some time the star S_{1} blocks
the star S_{2} that
we cannot see at that time. In a moment S_{2} reappears
from behind S_{1}.
We can measure the time difference for the optical wave, which emitted from star
S_{2} with
different frequency, to reach the earth. The data observed by de Brogue is ¦Ë_{2}^{2} ¨C
¦Ë_{1}^{2} ¡Ö
0.5x10^{-8} cm^{2}.
The distance between the binary and the earth is L ~ 10^{3} ly.
The time difference of two light waves with different color is ∆t ¡Ü 10^{-3}.
If the contribution from the rest mass of photon cannot be ignored, from
equation (6-2-2.8) we obtain

(9.2-3.3)

**
(b) The observation of pulsar**

The discovery of the pulsar provides a new method to examine the rest mass
of photon. Although the dispersion effect of the two series of light waves in a
pulse emitted from a pulsar is very small, but the distance between the pulsar
and the earth is so long that the time difference is big enough to be observed.
The dispersion effect of the radio wave emitted from a pulsar is generally
expressed in terms of effective average number density of electron. For the
pulsar NP0532 Staelin and et. Al.(1968) give ¡Ü
2.8 10^{-2} cm^{-3}

Feinbertg(1 969) assumed the observed dispersion effect of the pulsar
NPO532 was mainly caused by the rest mass of photon. In this situation comparing
(9.2-2.4) and (9.2-3.2) we get ¦Ø _{p }/
c = 4¦Ð e^{2} /mc.
The dispersion effect caused by the electron in the plasma is the same as that
cause by the rest mass of photon. Therefore we obtain:

(9.2-3.4)

Feinberg suggests this is a complement to the Schrodinger¡¯s static field method. It is worth to note that the quoted experimental data were accepted by the physical society. The reason was that at that time, the physics was constrained by the limited level of development. The observed time difference between the light wave with different frequencies, which arrives the earth at different time, and the dispersion effect of pulsar NPO 532 had been attributed ¡°mainly from the rest mass of photon¡± hypothetically. At present, the astrophysics has verified that the mass of the cosmic dust plus the mass of all of the baryon hold only 5% of ¡°dark matter¡± in the cosmic space. Actually, we have verified, that the 95% of the ¡°dark matter ¡°indeed is the ¡°FG ¡° light matter. In this respect, the mentioned experiments have been significant to verify our assertion sufficiently.