From Newton to Hawking and back: Newtonian analogies in spherical spacetimes, gravitoelectromagnetism and Hawking mass

Analogies occurring in the literature between spherically symmetric General Relativity and Newtonian physics are discussed, focusing on gravito-electromagne-tism (which contains Newtonian gravity as its 'electrostatics'). In spherical symmetry, the gravitoelectric potential contains the Misner-Sharp-Hernandez mass, to which the Hawking quasilocal mass reduces in this situation. It is shown that, according to observers defined rather naturally, the Hawking energy receives contributions from the electric part of the Weyl curvature but not from its magnetic part, making it 'Newtonian' in a well-defined sense. The gravitoelectric potential in spherical symmetry then becomes completely Newtonian.This article is part of the theme issue 'Newton, <i>Principia</i>, Newton Geneva Edition (17th-19th) and modern Newtonian mechanics: heritage, past & present'.