Suppose we have a sphere of radius $r$ and mass m and a negatively charged test particle at distance d from its center, $d\gg r$. If the sphere is electrically neutral, the particle will fall toward the sphere because of gravity. As we deposit electrons on the surface of the sphere, the Coulomb force will overcome gravity and the test particle will start to accelerate away. Now suppose we keep adding even more electrons to the sphere. If we have n electrons, the distribution of their pairwise distances has a mean proportional to $r$, and there are $n(n-1)/2$ such pairs, so the binding energy is about $n^2/r$. If this term is included in the total mass-energy of the sphere, the gravitational force on the test particle would seem to be increasing quadratically with $n$, and therefore eventually overcomes the linearly-increasing Coulomb force. The particle slows down, turns around, and starts falling again. This seems absurd; what is wrong with this analysis?
Answer
The passive gravitational mass of the electron is, by experiment, less than 9% of the theoretical expected value, as you can see in my answer on this question.
The result was so unexpected that no one believed in the correcteness of the result. I believe that the active gravitational mass of the electron will also be of the same order (or zero).
One nice theory could collapse because of one single experiment.
I am very much impressed: Almost no question do not include the Black Hole in the response. May be the case that they atract votes up. On this site there are already 2000 references to this term.
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