Graviton

Existence from the Newton's law of universal gravitation and the Einstein theory of quanta
The graviton existence can be naively predicted already without any advanced quantum field theory by rewriting the Newton's law of universal gravitation as the effect of collisions of hypothetical quantum particles with the positive energy but transferring as the result of the collision strange negative momentum e.g. particles with the negative inertial mass causing in nonelastic collisions the reaction opposite than normally i.e. casing the negative pressure (attraction but not the repulsion). While some elastic collisions may not alter the energy of the gravitons for example they may really have the positive momentum but the neutrons or protons may act as they were active inside for example if there where Maxwell's demons inside them bouncing the gravitons momentum rigidly back with infinitely heavy tennis rockets from the forth dimension before the fully non-elastic absorption.

The gravitational force from the large-source mass $$M$$ acting on the probe mass $$m$$ at the distance $$r$$ is expressed by the formula
 * $$F =G \frac{M m}{r^2}$$

Because the gravity force is decreasing with the position distance as $$1/r^2$$ it suggests absorption of the particles by the mass $$m$$ from the total flux passing through the surface of the sphere $$4 \pi r^2$$.

Writing the probe mass $$m$$ as

$$m = \rho S \lambda_0 $$

where $$S$$ is the section of the probe mass, $$\rho$$ is its density and $$\lambda_0$$ is its length or the length of the free path of the absorbed graviton and assuming that the gravitational mass $$M$$ is radiatively "evaporating" very slowly and exponentially according to the Einstein formula emitting gravitons i.e.

$$M= M_0 e^{-\gamma t}$$

we can write then the gravity law as the absorption of the momentum flux

$$F = 4 \pi \kappa \frac{d P}{dt} \frac{1}{4 \pi r^2} \rho S \lambda_0 $$

where in agreement to the Einstein formula

$$\frac{d P}{dt} = -\frac{d(M c^2)}{c dt}  \approx \frac{M_0 c^2 \gamma}{c} $$

and $$\kappa$$ is for now an unknown coefficient of the momentum transfer. Because the gravitons are going the be Einstein energy quanta like photons from the other side we have

$$\frac{d P}{dt} = \frac{\hbar \omega}{c} \frac{d N}{dt} $$

where $$N$$ is the number of the emitted gravitons in time.

We obtain

$$\frac{d N}{dt} = \frac{M_0 c^2 \gamma}{\hbar \omega}$$

Comparing again with the universal gravity law we get

$$\kappa=\frac{G}{c \gamma}$$

$$F = \frac{4 \pi G \hbar}{c^2} \frac{\omega}{\gamma} \frac{d N}{dt} \frac{1}{4 \pi r^2} \rho S \lambda_0$$

Additionally assuming

$$\omega= \gamma$$

the gravitons emission occurs with the giant half of the so called Zitterbewegung frequency

$$\frac{dN}{dt} = \frac{M_0 c^2}{\hbar}$$

known from the Dirac equation and which action is balanced by the minimal value of the coupling constant entering the cross section which contains the small gravitational constant

$$\tilde G=\frac{4 \pi G \hbar}{c^2}=9.83547 \times 10^{-61}\frac{m^3}{s}$$.

For example this frequency for the Earth with the mass $$M_0=5.972 \times 10^{24} kg$$ is $$d N/dt= 5.09238 \times 10^{75}$$ gravitons per second. It means that with the weakness of the gravity gravitons have almost zero mass and energy. Of course because the gravitational masses in the Universe seem to remain constant both the $$\omega$$ and $$\gamma$$ are here almost immeasurably small. As it is seen according to this theory the gravitation between bodies gradually weakens but it happens however almost immeasurably slowly. Particularly defining the total cross section for the graviton recapturing as $$\sigma$$ such that

$$F = \frac{\hbar \omega}{c} \frac{d N}{dt} \frac{\sigma}{4 \pi r^2}$$

we get for the heaviest elementary particle of the normal matter neutron

$$\sigma = \frac{4 \pi G }{c} \frac{m_n}{\gamma}>2.03791 \times 10^{-27}m^2$$,

if we only estimate $$\gamma$$ as the inverse of the current age of the Universe counting from the Big Bang i.e., the cross section of the order of its spherical geometric section $$\pi a_0^2 = 2.01062 \times 10^{-30}m^2$$.

If we assume the opposite, that the cross section for the graviton capturing by neutron (or proton) is equal to its "seen" side surface calculated with the known neutron radius $$a_0$$ i.e.

$$\sigma = \pi a_0^2$$

and

$$F = p_a \frac{\hbar \omega}{c} \frac{d N}{dt} \frac{\pi a_0^2}{4 \pi r^2}$$

where $$p_a$$ is the probability of the non-elastic graviton absorption by the neutron (proton)

$$p_a = \frac{4 \pi G }{c} \frac{m_n}{\gamma \pi a_0^2}$$

and is equal to 1 i.e the graviton is always fully absorbed by the neutron if only it hits it we obtain the estimate for the speed of the graviton mass decay

$$\gamma = \frac{4  G }{c} \frac{m_n}{a_0^2}=2.32917 \times 10^{-15}s^{-1}$$

i.e. about 1 per one thousandth of the age of the universe (about 1 per 10 million years) and for the energy (mass) of the graviton

$$m_g = \frac{\hbar \omega}{c^2}\approx \frac{\hbar \gamma}{c^2}=\frac{4  G \hbar }{c^3}\frac{1}{a_0^2}=\frac{4}{\Lambda a_0^2}m_n=2.73149 \times 10^{-66}kg =1.53242  \times 10^{-30} eV/c^2$$.

where

$$\Lambda=\frac{c^3}{ G \hbar }=3.829 \times 10^{69}m^{-2}$$

is so-called cosmological constant (giant theoretical value).

This formula expresses the relativistic mass in motion with the velocity equal (the rest mass 0) or smaller than the speed of light $$c$$ and is the upper limit of the estimation of the rest mass.

As it is seen this relation is in some sense symmetric if one rewrites it as

$$m_g/b_0^2=m_n/a_0^2$$

where

$$b_0=\left (\frac{4}{\Lambda}\right)^{1/2} $$

If from the symmetry of this formula we interpret $$b_0$$ also as the geometric radius of the graviton and assume that the graviton at rest is build from the uniform matter with a density similar to that of the proton or the neutron we obtain the estimate for the rest mass

$$m_{0g}=\left (\frac{b_0}{a_0} \right)^3 m_n=1.10337 \times 10^{-85} kg= 6.19015 \times 10^{-50}eV/c^2$$

Inverting the Einstein formula for the relativistic mass

$$m_{g}=\frac{m_{0g}}{\sqrt{1-\frac{v^2}{c^2}}}$$

implies that the gravitons with such a small rest mass are moving with the velocity indistinguishably close to $$c$$.

The recent experiments estimate the graviton mass for less then $$1.2 \times 10^{-22} eV/c^2$$.