Gravitation of ultra-relativistic barionic matter. Implications for weak equivalence principle.

Previously it was found that the non-barionic matter inside the stars will be gravitating much stronger per  mc² compare to the barionic matter (2*n² stronger, where n is the effective refraction coefficient for the light in the fully ionized plasma) [1]. For the high-energy gamma-quanta the effective gravitation is only 2 times stronger (this is because of the general relativity prediction  for the deviation of the light beam near the sun being twice the Newton value - the experimental fact confirming general theory of relativity).

However, any barionic matter composed of ordinary particles (electrons, protons, neutrons, neutrinos) which is the main composition of the star (fully ionized plasma) when heated to tens of million degrees (Suns core) will be closer to relativistic compare to cold barionic matter. Ultra-relativistic matter will have the energy-pulse relation E=pc (here E-energy, p=pulse, c- speed of light) exactly like photons and thus it should behave exactly like photons and gravitate like photons (twice the Newton value). 

Indeed, accurate calculations based on general relativity were performed few decades ago and confirmed, that the ultra-relativistic particle  will be deflected by the gravity exactly like photons (very small difference is present, of course, because they do have rest mass) [2].

However, the star is almost entirely composed from very hot matter (fully ionized plasma at temperature of ten of millions degrees). Application of the formula output procedure described in [1] to the ultra-relativistic particle (it is exactly like photons but the refraction coefficient is 1, the particle is moving with ~ speed of light) will lead to the increase of the gravitational force experienced by the star as a whole. 

Exactly how big is this contribution? The rest energy of the proton is 1.5*10exp(-10) J and for electron it is 8.2*10exp(-14) J. Assuming the kinetic energy of the electron or proton is still governed by the Boltzman formula E=3/2*kT, the kinetic energy of the particles inside the star core would be 2.1*10exp(-16) J - still to small to claim the particles are ultra-relativistic.

In this case the gravitational properties of the star barionic matter will be between the Newton limit (cold barionic matter) and Einstein limit (ultra-relativistic barionic matter). Application of the formula from [2] for the intermediate region lead to the following approximation: for the slow particles the gravity increase would be proportional to 1+v²/c²=1+mv²/mc²=1+2E_k/E_o

For the protons this means the change in gravity of only 1.4*10exp(-6). That would be too little to explain the presence of dark matter (most probably another, non-barionic matter responsible, see [1, 3,4], but clearly means that weak equivalence principle is not valid for star considered as a whole. 

References.

1. http://tipikin.blogspot.com/2019/10/stars-are-full-of-trapped-light-may.html

2. D.Fargion "Deflection of Massive Neutrinos by Gravitational Fields" // Lettere al Nouvo Cimento, Vol.31, No 2, 1981

https://www.researchgate.net/publication/

227245454_Deflection_of_massive_neutrinos_by_gravitational_fields

3. http://tipikin.blogspot.com/2019/12/light-matter-attraction-as-driving.html

4. http://tipikin.blogspot.com/2019/09/accelerated-rotation-of-star-because-of.html

5.