Laszlo István Orbán and László Tihamér Orban
1.
STATIC GRAVITATION FIELD
Although gravitation was the first of the four basic interactions that science started to examine, still utmost attention is paid to experimental or theoretical research that aims to better understand the fundamentals of the gravitation. Here the authors would like to present a theoretical method that establishes relationship between the gravitation force and the cosmic neutrinos.
Let us first assume that the neutrinos that have finite rest masses, - this has been proved already through experiments - can be found everywhere and are evenly distributed in the Universe. According to this assumption, this medium consists of chaotic moving free neutrinos. The only possible interaction among them is the elastic collision.
Let assume that the gravitation effect is conveyed between two objects by the above mentioned neutrino medium. Thus, at the calculation of the gravitation force between these objects, one must determine the number of the neutrino that interacts with both objects, and the impulse variation transported by these neutrinos. The calculations are related to the unit of time.
Let discuss the case of two macroscopic objects with masses m_{1} and m_{2} that are separated by a distance r from each other.
These bodies at lowest level are composed by microscopic particles with the mass m_{n }and radius R_{n}.
The neutrinos will interact with the
microscopic particles around them, hence the sum of the micro-effects will result a macroscopic
effect.
Let us first determine the number of medium-elements making interaction during the
time-unit within a spherical microelement having the radius R_{n} (see Fig 1).
From the Fig.1 it can be concluded that only those
dN_{1 } neutrinos - within the
dN number of medium elements
being in a dV volume - can make interaction with
the sphere I
that have the direction of their velocity vector falling inside the solid angle
d Q_{1}.
The ratio between the solid angles will be equal to the ratio of the corresponding neutrino numbers since
the evenly distributed medium elements are in continuous chaotic motion. Thus one can write:
4 p | = | dN |
d W_{I} | dN _{I} |
dN_{I} = | d W_{I} |
dN | |
4 p |
where 4p is the value of the total solid angle in sterradian.
By definition,
dW_{1} = | pR_{n}^{2} |
r_{1}^{2} |
Introducing the notation
r _{N} | = | dN |
dV |
dN_{I} = | p R_{n}^{2} | r_{n} dV | = | R_{n}^{2} r_{n} | dV |
r_{1}^{2} | 4 p | 4r_{1}^{2} |
The total number of the elements in the medium ( N_{1} ) interacting in the time-unit with the sphere of radius R_{n}, can be obtained by summarizing the dN values contained in each sphere with the radius v _{h} this will result the integral below:
N_{I} = | dV = | 2 p | p | v_{h} | ||||
r _{N} R_{N}^{2} | r _{N} R_{N}^{2} | r_{I}^{2} sin Q_{1} dr_{I} d Q_{1} d j | ||||||
= | ||||||||
4r_{I}^{2} | 4 | r_{I}^{2} | ||||||
0 | 0 | R |
= p r _{N} R_{n}^{2} ( v_{h} - R_{n} )
where v_{h} is the propagation velocity of the wave in the medium. Presuming this environment behaves similar to all the elastic mediums, one can write: ; where v_{h} stands for the propagation velocity of the interactions in the given medium; while v_{k} stands for the mean value of the velocity for medium-elements.
The expression of the previous integral: N_{1}
= p
r_{N}
R_{n}^{2} ( v_{h} - R_{n} )
can be simplified if v_{n} >> R_{n }_{} thus:
N_{1} » p r _{N} R _{n}^{2} v_{h}
Lets annotate the microscopic particles mass with m_{n} and its
radius with R_{n}.
From the medium-particles ( N_{1} ) that collides with the
sphere I., only those ( N_{2} ) will collide
with the sphere II. that have their velocity vector pointing toward the
solid angle having the value:
d W _{2} = | p R_{n}^{2} |
r^{2} |
This deduction was based on the assumption that the neutrinos reflected from the sphere I. got in the neighborhood of sphere II. in a straight line, but in the reality due to numerous collisions, they make a zigzag path. When two medium-particles of identical masses have elastic impacts, it can be assumed that because of the changes in the velocities the particle a will continue to travel in straight direction. The ratio of the solid angles – similar to the previous case - will be equal to the ratio between the numbers of medium-elements:
4 p | = | N_{1} |
d W | N_{2} |
N_{2} = | d W _{2} | N_{1} |
4 p |
After performing the substitution the expression for N_{2} will be:
N_{2} = | p R_{n}^{2} | p r _{N} R_{n}^{2} v_{h} = | p | r _{N} R_{n}^{4} v_{h} | 1 |
4 p r^{2} | 4 | r^{2} |
Let us replace the I. and II. micro objects with the m1 and m2 macroscopic bodies.
The value of total collisions in the first approximation can be estimated as the number of
collisions multiplied with the number of elementary particles the macroscopic bodies is built up.
This is valid for the so-called weak gravitation fields, where the size of macroscopic bodies is
much less then the distance traveled by the wave in the unit of time. Furthermore the self-dashing
effect can be neglected that would appear for the particles building up the macroscopic object.
The number of particles for the macroscopic part object be calculated as
the ratio between m_{1} /m_{n} and m_{2} /m_{n}.
Therefore the N_{1} and N_{2} values should be
multiplied with these values. Thus the numbers of neutrinos (N_{1,2})
that make interaction between the two macroscopic bodies can be given by:
N_{1,2} = | m_{1} | m_{2} | p | r _{N} R_{n}^{4} v_{h} | 1 | = | p r _{N} R_{n}^{4} v_{h} | m_{1} m_{2} |
m_{n} | m_{n} | 4 | r^{2} | 4 m_{n}^{2} | r^{2} |
u_{1} = | m_{n} - m_{k} | v_{k} |
m_{n} + m_{k} |
where m_{k} is the mass and v_{k} the velocity of the neutrino, whereas the m_{n} is the mass of the microscopic particle.
The D p impulse variation of the medium element can be written as:
D p = m_{k} v_{k} - m_{k} u_{1} = m_{k} ( v_{k} - | m_{n} - m_{k} | v_{k} ) = | 2 m_{k}^{2} v_{k} |
m_{n} + m_{k} | m_{n} + m_{k} |
If m_{n} > > m_{k} and , then
D p = 2 | m_{k}^{2} v_{h} |
m_{n} |
(1). | F = N_{1,2} D p = | p r _{N} R_{n}^{4} v_{h} | m_{1} m_{2} | 2 m_{k}^{2} v_{h} | = | p | r _{N} m_{k}^{2} R_{n}^{4} v_{h}^{2} | m_{1} m_{2} | |
4 m_{n}^{2} | r^{2} | m_{n} | m_{n}^{3} | r^{2} |
that means:
(2)
if
(3)
The formula obtained is Newton's gravitation law for the case of weak gravitation fields. Based on the formula (3), the gravitation constant G can be determined through the properties of the neutrino medium and the particles that builds up the macroscopic objects.
It comes out from the presented deduction that a certain time is needed to convey the gravitation force between the objects. This time is determined by the speed of medium elements and the distance between the objects. Furthermore can be concluded that gravitation waves are generated whenever an object makes periodic or accelerating movement in the neutrino medium. Consequently, the gravitation effects and the gravitation waves are supposed to be transferred by the medium consisting from neutrinos.
A proof of this theory would be if substituting the experimental values into the formula the
constant value G = 6.67·10^{-11} Nm^{2}/kg^{2} be obtained
for the gravitational constant.
At this moment this issue cannot be determined so easily since there are no reliable data about
the cosmic neutrinos. Furthermore it is not determined yet whether nucleons, pions, quarks,
gluons or leptons can be used as parameters for m_{n} and R_{n}
in the previous formula calculating G.
It would be important to determine which neutrinos ('
) the medium consist of. It would be essential also to
find out if the medium's wave velocity (v_{h}) is equal with the speed of the light or
differs from it.
Based on various experiments, the neutrino's mass is less then 17 eV
»
3·10^{-35} kg. Using this value, and supposing that v_{h} = c ,
the density values
r
_{N} and
r
=
r
_{N} m_{k}
could be determined from the following formula:
The calculation was carried out with different m_{m} and R_{m} values ( see Table I. )
Table I : Estimation for the density of the neutrino medium
Elementary particle in bodies |
m_{n} kg |
R_{n} m |
GNm^{2}kg^{-2} |
m_{k} kg |
v_{h} ms^{-1} |
r_{N} part/m^{3} |
r kg/m^{3} |
Nucleons (protons, neutrons) |
1.67·10^{-27} |
10^{-15} |
6.67·10^{-11} |
3·10^{-35} |
3·10^{8} |
1.72·10^{21} |
5.18·10^{-14} |
Pions |
2.5·10^{-28} |
10^{-16} |
6.67·10^{-11} |
3·10^{-35} |
3·10^{8} |
5.79·10^{22} |
1.74·10^{-12} |
Quark |
5.88·10^{-28} |
10^{-16} |
6.67·10^{-11} |
3·10^{-35} |
3·10^{8} |
7.54·10^{23} |
2.26·10^{-11} |
Lepton |
9.1·10^{-31} |
2.8·10^{-15} |
6.67·10^{-11} |
3·10^{-35} |
3·10^{8} |
4.54·10^{9} |
1.36·10^{-25} |
From the data-table above the value
r
»
1.36·10^{-25} kg m^{-3} seems to be the most realistic one.
This value is the closest to the limit density 4.5 10^{-27} kg m^{-3}, which predicts the expansion or contraction of the Universe.
If the experiments will confirm the existence of the neutrino medium with the density of
r
»
10^{-25} kg m^{-3}. then the gravitation force may be interpreted as the
result of interactions between the neutrinos and electrons. Any other approach would give a
density value higher with 11-14 magnitudes.
2. THE "GRAVO-DYNAMIC" FORCE
Assuming the existence of neutrino medium, the shift of matter (flux of substance) may result forces similar to the magnetic forces generated by electric currents. This gravo-dynamic force should be extremely weak for speeds negligible in comparison with the speed of the light. Therefore the gravo-dynamic effects may be perceptible only on the cosmic scale having mass fluxes in the range of galaxies and cluster of galaxies.
Let suppose that two bodies m_{1} and m_{2} are moving in parallel at the speeds v_{1} and v_{2}. If v_{1} and v_{2} are negligible in comparison with v_{h} then only the transferred impulse-variation has to be recalculated since N_{1,2}, (in the first approximation) will be identical with those calculated in the static case.
The medium element will collide with m_{1} and m_{2} if:
v_{h}^{2} t^{2} = r^{2} + v_{2}^{2} t^{2},
then
cos a = | v_{2} t | = | v_{2} |
v_{h} t | v_{h} |
Those medium-elements will collide with the particle with mass m_{1}, that are moving with the
speed given by the expression v_{1}cos
a.
The velocity u_{1,1} of the medium-element (supposing elastic collision) after the
collision can be calculated from the energy and impulse conservation laws.
u_{1,1} = | 2 m_{n} v_{1} cos a + (m_{n} - m_{k}) v_{k} |
m_{n} + m_{k} |
The sum of the impulse variations transferred by the medium elements:
D p = m_{k} ( v_{k} - u_{1,1} ) =
= m_{k} ( v_{k} - | m_{n} - m_{k} | v_{k} - | 2 m_{n} v_{1} cos a | ) = | m_{k} | (2 m_{k} v_{k} - 2 m_{n} v_{1} cos a ) |
m_{n} + m_{k} | m_{n} + m_{k} | m_{n} + m_{k} |
Substituting the
cos a = | v_{2} |
v_{h} |
D p = | 2 m_{k}^{2} v_{k} | = | 2 m_{n} m_{k} | v_{1} v_{2} |
m_{n} + m_{k} | m_{n} + m_{k} | v_{h} |
If m_{n} > m_{k} and then
D p = | 2 m_{k}^{2} v_{h} | - | 2 m_{k} v_{1} v_{2} |
m_{n} | v_{h} |
The force between two moving bodies can be calculated also with the formula: F = N_{1,2} Dp that will result:
F = | p | r _{N} v_{h}^{2} m_{k}^{2} R_{n}^{4} | m_{1} m_{2} | - | p | r _{N} m_{k} R_{n}^{4} | m_{1} v_{1} m_{2} v_{2} |
m_{n}^{3} | r^{2} | 2 | m_{n}^{2} | r^{2} |
From the expression 3 and the above deducted formula one can have:
G_{d} = | p | r _{N} m_{k}^{2} R_{n}^{4} | = | 1 | m_{n} | G |
2 | m_{n}^{2} | m_{k} v_{h}^{2} |
Then one can get:
F = G | m_{1} m_{2} | - G_{d} | m_{1} v_{1} m_{2} v_{2} | = F_{s} - F_{d} |
r^{2} | r^{2} |
According to this formula, in addition to the static force ( F_{s} ) an additional, speed dependent dynamic force ( F_{d} ) will also arise.
Replacing the m_{1} v_{1} with I_{1} Dl_{1} and m_{2} v_{2} with I_{2} Dl_{2} , the gravo-dynamic force will have the form very similar to the electro-dynamics force.
F_{d} = G_{d} | I_{1} D l_{1} I_{2} D l_{2} |
r^{2} |
In the general case, when the vectors D l_{1} and D l_{2} have different angles with the r distance vector, the gravo-dynamic vector-force is calculated as a vector-product:
F_{d} = G_{d} I_{1} I_{2} | ( D l_{1} x r ) x D l_{2} |
r^{3} |
One can understand that in the case of moving objects, addition to the static gravitation field a dynamic field will also arise. The description of the dynamic gravitational (gravo-dynamic) field can be done similar to the electro-dynamic field. However while in the electro-dynamic case the ( I ) means electric current, in the case of gravo-dynamic force would mean mass-current-intensity whereas instead of permeability used in the electro-dynamic case one will use the gravitational constant G_{d}.
Since the relation between G_{d} and G can be given by the formula:
G_{d} = | 1 | m_{n} | G |
m_{k} v_{h}^{2} |
As long as the numeric values for v_{1} and v_{2} are very small in comparison with v_{h} ( v_{1} , v_{2} << v_{h} ), the “gravo-dynamic” force can be neglected in comparison with the static gravitational force. In addition, while the value of the m_{n} / m_{k} ratio is uncertain, further research is expected on this field.
Finally, let us emphasize that the assumption of neutrino field enables both to predict the gravitational effects and helps to better understand many other phenomenon of the quantum physics.