Heuristic approach to the Schwarzschild geometry

Abstract

In this article I present a simple Newtonian heuristic for deriving a weak-field approximation for the spacetime geometry of a point particle. The heuristic is based on Newtonian gravity, the notion of local inertial frames [the Einstein equivalence principle], plus the use of Galilean coordinate transformations to connect the freely falling local inertial frames back to the ``fixed stars''. Because of the heuristic and quasi-Newtonian manner in which the spacetime geometry is obtained, we are at best justified in expecting it to be a weak-field approximation to the true spacetime geometry. However, in the case of a spherically symmetric point mass the result is coincidentally an exact solution of the full vacuum Einstein field equations -- it is the Schwarzschild geometry in Painleve--Gullstrand coordinates. This result is much stronger than the well-known result of Michell and Laplace whereby a Newtonian argument correctly estimates the value of the Schwarzschild radius -- using the heuristic presented in this article one obtains the entire Schwarzschild geometry. The heuristic also gives sensible results -- a Riemann flat geometry -- when applied to a constant gravitational field. Furthermore, a subtle extension of the heuristic correctly reproduces the Reissner--Nordstrom geometry and even the de Sitter geometry. Unfortunately the heuristic construction is not truly generic. For instance, it is incapable of generating the Kerr geometry or anti-de Sitter space. Despite this limitation, the heuristic does have useful pedagogical value in that it provides a simple and direct plausibility argument for the Schwarzschild geometry.