A physics-first approach to the Schwarzschild metric

Abstract

As is well-known, the Schwarzschild metric cannot be derived based on pre-general-relativistic physics alone, which means using only special relativity, the Einstein equivalence principle and the Newtonian limit. The standard way to derive it is to employ Einstein's field equations. Yet, analogy with Newtonian gravity and electrodynamics suggests that a more constructive way towards the gravitational field of a point mass might exist. As it turns out, the additional physics needed is captured in two plausible postulates. These permit to deduce the exact Schwarzschild metric without invoking the field equations. Since they express requirements essentially designed for use with the spherically symmetric case, they are less general and powerful than the postulates from which Einstein constructed the field equations. It is shown that these imply the postulates given here but that the converse is not quite true. The approach provides a fairly fast method to calculate the Schwarzschild metric in arbitrary coordinates exhibiting stationarity and sheds new light on the behavior of waves in gravitational fields.