On the Geometrization of Quantum Mechanics, Frozen Stars, the Bohm-Poisson and Nonlinear Klein-Gordon Equations

We revisit the nonlinear Klein–Gordon-like equation that was proposed by us which captures how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be “trivial” but with important physical implications related to the physics of frozen stars and with Mach’s principle. The nonlinear Klein–Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We finalize by studying the higher-dimensional generalizations of the nonlinear Klein–Gordon-like equation and examine the relativistic Bohm–Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.