**Chapter
9**

**THE MASS OF THE
DARK MATTER WG AND TWO EXPERIMENTAL DATA.
**

**9.1 The principle for calculating the “WG” mass;
and the numerical relation of WG 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 WG 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_{W}
= 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 “WG 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 “WG” gives use to collapse of the “WG 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 “WG cloud”
assure that there exist three Stable states in the universe.

- The
fulfilled orbital state
- The
space like state (the “B body “as a Core oscillates violently, and the
surrounding “WG cloud” disappears).
- The
coupled state, i.e. Two “B body” shares a common “WG 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 “WG” 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 “WG cloud”
in a proton because they have same internal core of the “B body”. So I can
estimate the number of “WG” original particle by the following method.

The “WG 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 “WG” 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
“WG” 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 “WG” in the
“WG 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 “WG 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 “WG”:

m_{w}
=3.63 6 × 10^{-45} kg.

**9.2
The experimental test and the verification of the theoretical mass of
“WG” ― 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 “WG” 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 μ. Only providing an upper limit on
μ (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_{λ} (potential
function) and its first order derivative ∂A_{λ /} ∂x _{ν.}
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 μ^{2}A_{r}A_{ν
}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. Ε _{λνρσ
}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_{μ}

(9.3-6)

Where
( ’Alembertian). The equation
determines the electromagnetic potential A_{ν} 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 μ = 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 (μ≠ 0) in
a vacuum. The free plane wave solution of equation (9.2-6) in a vacuum (absent
charge and current) is

A_{ν }~ exp{i(k·r
– ωt)}
(9.2-2.1)

Where wave vector k ( |k|
≡ 2 π ⁄ λ,
λ is wave length), angular frequency ω and mass satisfy

k^{2} - ω^{2}
⁄ C^{2 } = - μ^{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

μ = ω ⁄ |k|
= c (1 - μ^{2} c^{2} ⁄ ω^{2 }) ^{–1/2
}(9.2-2.3)

The group speed is
defined as

v _{k }= d ω ⁄ d |k|
= c (1 -
μ^{2} c^{2} ⁄ ω^{2 }) ^{–1/2}
(9.2-2.4)

Because the mass of photon μ is a finite constant, when ω→∞
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 ω = μ c,
k = 0 the free electromagnetic wave does not propagating. Otherwise when ω
< μ 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 ω › μ 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 ω_{1}, ω_{2}, assuming ω_{1},ω_{2}
> μ 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 (μ ^{2 }c ^{2
}/ ω_{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 μ. 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, α: the angle between k and B.
In inter galaxy space the magnetic field is very weak ω_{β}
can be ignored. Therefore equation (X.2-3.2) gives the dispersion effect of
electromagnetic field in plasma.

V_{g} = d ω ⁄ d |k|
= c (1 – ω _{ρ}^{2} ⁄ ω^{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 μ.

**
**

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
λ_{2}^{2} – λ_{1}^{2} ≈ 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
≤ 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
≤ 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 ω _{p }/ c = 4π 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 “WG “ light matter. In this respect,
the mentioned experiments have been significant to verify our assertion
sufficiently.