Heuristic approach to the Schwarzschild geometry

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.