It is well accepted that gravitomagnetic field is extremely weak and just recently was measured using satellites. However, one of the problems of the physics - the weakness of the gravitational force - may have the answer connected with the presence of very heavy gravitational-antigravitational pairs of virtual particles in the quantum vacuum [1]. Very much like the electrostatic force is limited by the presence of electron-positron virtual pairs in the quantum vacuum, the gravitational field is so weak because the very massive virtual pairs included antigravitational particle are polarized by the gravity of the tested particle and thus attenuates the force perceptibly [1].
But Higgs boson is not fermion from ordinary spin definition and seemingly should not have any antigravitational counterpart. The hypothesis is the presence of one more quantum number, close to spin but revealed by the mechanical moment decoupled from charge. Indeed, Einstein-De Haas experiment revealed that the reverse of the orbital magnetic moment forces the macroscopic object to rotate, thus connecting directly the mechanical moment and magnetic moment. For the orbital moment this experiment demonstrates full mechanical moment. But situation is not so obvious for spin of elementary particles. There is undoubtedly the mechanical moment coupled with electric charge (or electric current) and the flip of the spin will mean the flip of mechanical moment. However, the uncoupled mechanical moment may be still present. Imagine the fast rotating dielectric ball with superconducting strip on equator. The current in such a superconducting strip will be responsible for the magnetic moment of the object (magnetic spin) and cooper pairs would be responsible for the mechanical moment associated with such a magnetic spin. However, the main mechanical moment may be as large as possible and may have different direction. Flip of the current will reveal in this situation only the coupled part of the mechanical moment while the main moment will stay unnoticed. Only gravitomagnetic field will reveal the total mechanical moment of the elementary particle (for macroscopic object, of course, much simpler experiment will work).
The hope of this idea is that such experiment may reveal another quantum number, second spin, which would mean that the Higgs boson is not real boson, but only partial boson (with respect to magnetic spin) and still may have the gravitational antiparticle, thus explaining the weakness of the gravitational force.
How to measure such a spin? At first it would be necessary to evaluate, whether it is possible to measure the usual spin of say electron using any modern day equipment. The only object generating strong enough gravitomagnetic field would be rotating Earth [2].
The easiest experiment to be done is gravitomagnetic Stern-Gerlach experiment: the spin will exert the force in the gradient of gravitomagnetic field. The gradient of the gravitomagnetic force would be (for Earth):
Bg=[G/(5*c*c)]*[M/r]*[2π/T]
dBg/dr=-[G/(5*c*c)]*[M/(r*r)]*[2π/T]
Here Bg is gravitomagnetic field of the rotation ball (Earth), G is gravitational constant, c is speed of light, r is the distance from the center of the Earth, M is the mass of the Earth (5.97*10exp(24) kg), T is the period of the rotation (1 day or 86400 seconds).
The known mechanical moment of the elementary particle would be spin of electron, which is S=h/(4*pi). Here h is Planks constant and pi is 3.14159.
For the electron traveling in the gravitomagnetic field gradient the force between the spin up and spin down particles would be F=2S*dBg/dr (electron will travel in space away from the Earth). Since the gradient is varying with the distance as 1/r^2 law, instead of integration of the force along the path for crude evaluation the distance r is taken to be 8 thousands kilometers (the experiment starts at 6.3 thousands kilometers and ends at 16.3 thousands kilometers). The force for the electron would be:
F=2S*dBg/dr=[h/(2π)]*[G/(5*c*c)]*[M/r*r]*[2π/T]
F=1*10exp(-55) Newton
For the electron traveling with velocity of 0.1 m/s, the distance is 10000 km (1*10exp(7) meters), the time of travel is 1*10exp(8) seconds (~3 years). Using mass of electron 9.1*10exp(-31) kg and simple formula L=a*t*t/2 the expected separation of the electrons due to ordinary spin at the end of travel would be 5.5*10exp(-10) m (5.5 Angstrom - measurable at modern technology).
More accurate double integration will give the similar result:
L=[1/v*v]*[h/(2*π)]*[G/(5*c*c)]*[M/m]*[2π/T]*ln(R1/Ro)
Here v is the velocity of the electron, m is the mass of electron, Ro and R1 are distance from the center of the Earth at the start and at the finish, correspondingly. L is 6.9 Angstrom.
The largest problem is here: the electron is subject to the magnetic field of the Earth and will travel in a circle around the Earth magnetic field line. The only way to compensate such rotation is to put the compensating electric field in the opposite direction during the whole travel of the electron from start to finish. Despite the keeping the whole satellite all the time around the traveling pulse of electrons is an expensive task, it may be done provided the electrons are moving very slow. The electrons and the satellites may, of course rotate around the Earth in the equatorial plane (otherwise such satellite would not work), slowly moving away from the Earth as the bunch of electrons travels away at a speed of 10 cm/s and the separation due to the gravitomagnetic field gradient accumulates.
Another problem would be the presence of the magnetic field gradient (so the classical Stern-Gerlach experiment would be performed in Earth magnetic field gradient). It is possible to carry the compensating magnetic field gradient on the same satellite as well. Actually while the full compensation of the classical Stern-Gerlach splitting would not be possible, the idea here to see the additional splittings in the final picture. The presence of such splittings would mean the presence of one more quantum number - second spin for elementary particles.
Second spin will allow to hypothesize the presence of heavy particle-gravitational antiparticle virtual pairs in the quantum vacuum (like Higgs - antigravitational Higgs pairs proposed in [1]) and explain the gravitational constant in a way similar to the electrostatic one.
References.
1.http://tipikin.blogspot.com/2019/12/quantum-vacuum-application-to-gravity.html
2.https://en.wikipedia.org/wiki/Gravitoelectromagnetism
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