You asked the question whether the electromagnetic and gravitomagnetic interactions (forces) become one in the very strong field. First of all, they become not one!
The interaction between particles is always the addition of electromagnetism and gravitation. For instance for proton/elton the static gravity is (g∙mP)^2/e^2 ≈ 10^-42 time weaker than the static electric force and it is independently of r. For electron/positron we have the relation (g∙me)^2/e^2 ≈ 3.4∙10^-36. The r-dependence of both static forces is 1/r^2. Since the condensed matter is either proton-based, or elton-based, the relation of electromagnetism and gravitomagnetism depends of the mass density of matter and the density of the motions of the particles, that is mainly the density of protons/eltons and the density the static magnetic momentums of the bound two-particle systems. In the following we want consider the proton-based condensed matter. In very strong gravitational potential, at very high mass density of condensed matter and in case of uncoordinated magnetic momentums of the neutrons, we have to calculate the relation of the static gravity of the neighbor protons and the electric force of the elementary electric charge, e.
It should be noticed that during the static gravitational forces of neighbor protons are added together, the static electric forces neutralize itself. The addition of gravitational forces of the protons with a constant proton density, ρ, since the force is 1/r^2 dependent, is an integration of 1/r^2 about the whole space from the distance, d, of the neighbor protons to a big radius, R. The situation is similar to the inner of a hollow sphere. The static gravitational force is zero, however, the gravitational potential has a big constant value within the sphere.
Since the particles cannot approach to each other too close, two kind of “maximal mass density” can be derived.
The one kind of density follows from the sizes of stable neutron N0 = (P,e) and of the electron-neutrino νe = (e,p), The sizes are 0.702∙10^-13 cm, respectively 0.703∙10^-13 cm. The gravitational mass, mg, of these two-particle systems are mg(N0) = mP – me ≈ mP and mg(νe) = 0. If we calculate that in a cube with the side length 10^-13 cm there is one N0 and the cubes are dense packed, the mass density is
ρ(mg(N0)) ≈ mP/(10^-13 cm)^3 = 1.67∙10^-27/10^-36 kg/cm^3 = 1.67∙10^+9 kg/cm^3.
This is the mass density of neutron-stars. The gravitational potential within the neutron star is constant, however has a big value. The static gravitational force is zero. The relation of the static gravity force to the static electric force of e would be the same as given above.
Another situation arises for the mass density if we calculate the mass density for the size of the proton-neutrino, νP = (P,E) and for the size of the (P,e)-system, whereby the bound (P,e)-system would have the bound energy E(bound) = (mp + me)∙c^2. The sizes are 0.383∙10^-16 cm, and 0.382∙10^-16 cm, respectively. The gravitational mass of the (P,e)-system is mg(P,e) = mP – me ≈ mP and of the neutrino mg(νP) = 0. If we calculate that in a cube with a side length of 10^-16 one bound (P,e)-system resides, the mass density would be
ρ(mg(P,e)) ≈ mP/(10^-16 cm)^3 = 1.67∙10^-27/10^-48 kg/cm^3 = 1.67∙10^+21 kg/cm^3.
The relation of the static gravity to the static electric force would be the same as above; however, a considerable greater constant potential term must be added.
At both extreme situations for mass density dominate the static electric force about the static gravitational force.
We can also calculate the mass density of normal matter if we assume one proton is in a cube with side length of 10^-9 cm (approximately the distance between atoms with the sizes somewhat greater than 10^-8 cm)
ρ (matter) ≈ mP/(10^-9 cm)^3 = 1.67∙10^-27 kg /10^-27 cm^3 = 1.67 kg/cm^3.
In matter the static electric force dominate over the static gravitation force.
The presented calculations were simple for an orientation about the relation of the static electric and static gravitation force in any cases, using the knowledge of the sizes of two-particle system, I have given, and with the knowledge of the five natural constants, c, e, mP, me and g =(4∙π∙G)^1/2. The most experimental uncertainty is with g which can be calculated with the CODAT value of the universal gravitational constant G. However, the true value of G is 1.5% smaller than the CODATA value of G.
However, we did not have calculated until now the possibility of coordinated static magnetic momentums. Such a situation exists in nature; they are the pulsars, the quick rotated neuron stars. The calculation is not easy. For instance we must make some assumption about the thermodynamically situation of the star; about the temperature distribution within the neutron star. In any case, the static magnetic momentum of the instable neutron N = (P,e,p,e) is experimentally known, and the static magnetic momentum of the stable neutron N0 = (P,e) can be calculated.
I am expecting from my students that they learn, beside my conceptual simple theory I am teaching, to think self-employed. I am somewhat impatient if I recognize that they didn’t have learned, or are unable, thinking self-employed. I don’t expect that they should not make errors. Self-employed thinking means self calculations with simple inputs of my theory.
Simultaneously, I know that the overwhelming number of researcher in physics, in any case all reviewer of physical journals I have met, obviously cannot think self-employed. They are ready to accept theories of authoritarian researches, no matter if the theories are stupid or not. But, they cannot think independent and self-employed and they don’t criticize the accepted theories.
I am Hungarian. In our country the most people think broadly independent. Sometimes two people have tree different meaning. Authority is less asked in the own meaning. Probably, the Hungarian language is the dept. In this language you can formulate complicated things in a very easy exact fashion. You are not forced to use detrimental and complicated grammatical rules. I have another meaning about physics as the Hungarians Loránd Eötvös, J. v. Neumann, Wigner and Teller thought. In any case, I have another meaning as Planck, Einstein, Heisenberg, Feynman and the other authoritarian physical researches thought. And I have enhanced also the simple gravitational theory which Kepler, Galileo and Newton have left us over.
I am thinking, my theory is conceptual very simple. Furthermore, the theory is mathematically and physically consistent and correct. And I use only appropriate mathematical tools. Occam’s razor which I use is very sharp. Therefore, I believe that my theory describe Nature comprehensively. “Nature does nothing in vain.” said Sir Isaac Newton.