Geodesics in the Generalized Schwarzschild Solution

TL;DR

This paper derives the most general stationary, spherically symmetric vacuum metric in general relativity, explores its geodesics, and discusses various coordinate systems, emphasizing its pedagogical value over the standard Schwarzschild metric.

Contribution

It provides a unified derivation of the general form of the spherically symmetric vacuum metric and analyzes particle and photon trajectories within this framework.

Findings

Derived the most general form of the stationary, spherically symmetric vacuum metric.

Obtained explicit particle and photon geodesics for the general metric.

Showed the general metric's simplicity makes it suitable for teaching purposes.

Abstract

Since Schwarzshild discovered the point-mass solution to Einstein's equations that bears his name, many equivalent forms of the metric have been catalogued. Using an elementary coordinate transformation, we derive the most general form for the stationary, spherically-symmetric vacuum metric, which contains one free function. Different choices for the function correspond to common expressions for the line element. From the general metric, we obtain particle and photon trajectories, and use them to specify several time coordinates adapted to physical situations. The most general form of the metric is only slightly more complicated than the Schwarzschild form, which argues effectively for teaching the general line element in place of the diagonal metric.