Classroom reconstruction of the Schwarzschild metric

TL;DR

This paper presents a simplified, physically motivated approach to derive the Schwarzschild metric's weak-field limit, enabling classroom teaching of general relativity without complex differential geometry.

Contribution

It introduces a method to establish the Schwarzschild metric's weak-field limit using minimal assumptions and experimental data, bypassing the full field equations.

Findings

Determined metric coefficients with high accuracy from experimental data.

Predicted classical tests of general relativity using simplified derivations.

Connected Mercury's perihelion precession to metric parameters.

Abstract

A promising way to introduce general relativity in the classroom is to study the physical implications of certain given metrics, such as the Schwarzschild one. This involves lower mathematical expenditure than an approach focusing on differential geometry in its full glory and permits to emphasize physical aspects before attacking the field equations. Even so, in terms of motivation, lacking justification of the metric employed may pose an obstacle. The paper discusses how to establish the weak-field limit of the Schwarzschild metric with a minimum of relatively simple physical assumptions, avoiding the field equations but admitting the determination of a single parameter from experiment. An attractive experimental candidate is the measurement of the perihelion precession of Mercury, because the result was already known before the completion of general relativity. It is shown how to determine the temporal and radial coefficients of the Schwarzschild metric to sufficiently high accuracy to obtain quantitative predictions for all the remaining classical tests of general relativity.