WHAT shape is an electron? The standard model of particle physics predicts that electrons are egg-shaped, but that the amount of distortion from a perfect sphere is so tiny that no existing experiment could possibly detect it. However, a rival theory called supersymmetry predicts that this egg-shaped distortion should be large enough to be detectable.Here is the Nature abstract:
The electron is predicted to be slightly aspheric1, with a distortion characterized by the electric dipole moment (EDM), de. No experiment has ever detected this deviation. The standard model of particle physics predicts that de is far too small to detect2, being some eleven orders of magnitude smaller than the current experimental sensitivity. However, many extensions to the standard model naturally predict much larger values of de that should be detectable3. This makes the search for the electron EDM a powerful way to search for new physics and constrain the possible extensions.The shape of the electron played a crucial role in the discovery of relativity. Lorentz's relativity theory was called "electron theory". He got the Nobel prize for it in 1902, but by 1905 many younger physicists were following Max Abraham's theory for various theoretical and experimental reasons. Poincare's 1905 paper explained the superiority of relativity theory in terms of the stresses that maintain the shape of the electron. Those stresses are now called Poincare stresses or pressure. The electric charges within the electron would normal repel each other, and you can imagine some sort of pressure from the aether holding the charges together.
Einstein also wrote a 1905 paper on Lorentz's relativity, but it is not clear that he understood how it compared to the alternatives. He seemed to mainly just trust that Lorentz and Poincare had gotten it right, and regurgitated what they said.
The shape of the electron is still a bit of a mystery. If this experiment can be done more accurately, maybe it can help eliminate supersymmetry and other silly theories.