Have we been able to prove gravitation between a free electron and a free electron in space through direct experiments?
The gravitational attraction between two electrons is given by Gm2/r2, where G is the gravitational constant, m is the electron mass and r is the distance between the two electrons. The numerical value of the numerator is approximately 5.5 × 10−71.
The Coulomb repulsion between two electrons is given by ke2/r2, where k is the Coulomb constant and e is the electron's charge. The numerical value of the numerator is approximately 2.3 × 10−28.
The ratio between these two forces is approximately 4.2 × 1042. That is, the gravitational attraction introduces a correction to the electrostatic repulsive force after the 42nd decimal digit. To detect this small correction, we would need to measure the force between two electrons with a precision of at least 43 decimal digits. This is much, much smaller than anything we can measure presently or in the foreseeable future. So the answer to your question is a firm no.
The gravitational attraction between two electrons is given by Gm2/r2, where G is the gravitational constant, m is the electron mass and r is the distance between the two electrons. The numerical value of the numerator is approximately 5.5 × 10−71.
The Coulomb repulsion between two electrons is given by ke2/r2, where k is the Coulomb constant and e is the electron's charge. The numerical value of the numerator is approximately 2.3 × 10−28.
The ratio between these two forces is approximately 4.2 × 1042. That is, the gravitational attraction introduces a correction to the electrostatic repulsive force after the 42nd decimal digit. To detect this small correction, we would need to measure the force between two electrons with a precision of at least 43 decimal digits. This is much, much smaller than anything we can measure presently or in the foreseeable future. So the answer to your question is a firm no.
The answer is no. But if we prove gravitational attraction between a free electron and the earth, we can simultaneously prove length contraction and gravitational attraction between two electrons. The later will be an indirect proof. Practical feasibility of this experiment is not clear.
If we experiment with a free electron and prove only 1% change in wavelength, that would prove the length contraction. If you fire an electron through a vacuum tube parallel to the surface of the earth and measure its momentum at departure and after travelling distance d, then de Broglie wavelength = Planck constant / momentum. Radial gravitational acceleration will be uniform through out distance d. Change in the wavelength can be found. Next we rotate this apparatus by 90 deg. and fire the electron vertically upwards and again find the change in the wavelength. This time electron will experience the gravitational potential difference. If you can detect even 1% difference between the two computed change in wavelengths, you would have proved length contraction. One can clearly see that the momentum is bound to be affected by the variation in gravitational potential and if it doesn't, then that would mean violation of Newton's inverse square law of gravitation. So this will be indirect proof of gravitational attraction between two electrons.
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