The Reciprocal of the Fundamental Theorem of Riemannian Geometry

TL;DR

This paper introduces Ricardo's formula, an inversion of the fundamental theorem of Riemannian geometry for analytic Christoffel symbols, providing a framework to derive metrics from geodesic data, relevant for gravity experiments.

Contribution

It presents the first explicit inversion formula for Christoffel symbols without extra assumptions, enabling metric reconstruction from unparameterized geodesics.

Findings

Ricardo's formula mathematically inverts the fundamental theorem of Riemannian geometry.

A procedure to derive Christoffel symbols from unparameterized geodesics is outlined.

A complete framework for metric extraction from measurements is proposed.

Abstract

The fundamental theorem of Riemannian geometry is inverted for analytic Christoffel symbols. The inversion formula, henceforth dubbed Ricardo's formula, is obtained without ancillary assumptions. Even though Ricardo's formula can mathematically give the full answer, it is argued that the solution should be taken only up to a constant conformal factor. A procedure to obtain the Christoffel symbols out of unparameterized geodesics is sketched. Thus, a complete framework to obtain the metric out of measurements is presented. The framework is suitable for analysis of experiments testing the geometrical nature of gravity.