Existence from the Newton's law of universal gravitation and the Einstein theory of quanta Edit

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   acting on the probe mass   at the distance   is expressed by the formula


Because the gravity force is decreasing with the position distance as   it suggests absorption of the particles by the mass   from the total flux passing through the surface of the sphere  .

Writing the probe mass   as


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


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


where in agreement to the Einstein formula


and   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


where   is the number of the emitted gravitons in time.

We obtain


Comparing again with the universal gravity law we get



Additionally assuming


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


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


For example this frequency for the Earth with the mass   is   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   and   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   such that


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


if we only estimate   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  .

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   i.e.




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


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


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




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   and is the upper limit of the estimation of the rest mass. [1][2]

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




If from the symmetry of this formula we interpret   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


Inverting the Einstein formula for the relativistic mass


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

The recent experiments estimate the graviton mass for less then  .[3]

  1. Goldhaber, Albert S.; Nieto, Michael Martin (1974-02-15). "Mass of the graviton". Phys. Rev. D 9: 1119-1121. doi:10.1103/PhysRevD.9.1119. 
  2. Woodward, James F.; Crowley, Ronald J.; Yourgrau, Wolfgang (2006-03-15). "Mach's principle and the rest mass of the graviton". Phys. Rev. D 11: 1371-1374. doi:10.1103/PhysRevD.11.1371. 
  3. Abbott et al., B. P. (2016-02-12). "Observation of Gravitational Waves from Binary Black Hole Merger". Phys. Rev. Lett. 11: 061102-1-12. doi:10.1103/PhysRevLett.116.061102.