Two static axisymmetric solutions in Entangled Relativity

Abstract

Entangled Relativity is a non-linear reformulation of General Relativity that prevents the existence of genuine vacuum spacetimes and which possesses one less defining constant than General Relativity. From two existing static and axisymmetric solutions of General Relativity, one in vacuum and one with a scalar field, we derive two new static and axisymmetric solutions of Entangled Relativity in the Einstein frame. However, once written in the entangled conformal frame—where the non-linearity of the action is manifest—, such solutions would correspond to vacuum solutions. Since Entangled Relativity is ill-defined in vacuum, as the theory enforces the Machian property that spacetime is entirely determined by matter and cannot exist without it, we first embed these solutions in a magnetic (or electric) field à la Melvin, before taking the \(T_{\mu \nu } \rightarrow 0\) limit. As for all other solutions of Entangled Relativity, we show that this limit is perfectly well-defined and finite, despite the fact that a vacuum solution, from the outset, is ill-defined.