The gravitational self-energy of a spherical shell
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According to Einstein’s mass-energy equivalence, a body with a given mass extending in a large region of space, will get a smaller mass when confined into a smaller region, because of its own gravitational energy. The classical self-energy problem has been studied in the past in connection with the renormalization of a charged point particle. Still exact consistent solutions have not been thoroughly discussed in the simpler framework of Newtonian gravity. Here we exploit a spherically symmetrical shell model and find two possible solutions, depending on some additional assumption. The first solution goes back to Arnowitt, Deser and Misner (1960). The second is new. When applied to a spherical shell of a given “bare” mass M0, both solutions lead to a vanishing “renormalized” mass for a vanishing radius R of the shell. As a consequence the condition for the existence of a Newtonian black hole will never be met for finite R . When applied to the e.m. mass of a pure static electric charge the second solution yields a new vanishingly small value (10−55cm) for the classical electron radius.