On the world function of the Schwarzschild field

Abstract

The representation of the world function Ω of the Schwarzschild field as a power series is investigated. Initially we concern ourselves with a neighborhood of the event horizon. The symmetries of the metric are invoked effectively to reduce the number of independent variables upon which Ω depends from eight to four, and to show that when these are sufficiently small in magnitude Ω is an analytic function of them. A fairly large number of the early terms of the power series for Ω is found explicitly. The condition that one is to remain sufficiently close to the event horizon is then relaxed, it being merely stipulated that the endpoints shall be sufficiently close to each other. Finally, using other variables, the early terms of a series for Ω are obtained for the case in which the endpoints are restricted to lie outside the event horizon and sufficiently close to each other.