Towards a Geodesic Flow Bundle Formalism of General Relativity: Riemannian case

TL;DR

This paper introduces a formalism that describes Riemannian geometries through geodesic flow bundles and infinitesimal triangles, enabling direct calculation of geodesics from curvature fields without metric solving.

Contribution

It develops a novel approach using geodesic flow bundles and spherical triangles to analyze Riemannian geometries, generalizing classical trigonometric laws to variable curvature fields.

Findings

Validated against the sphere geometry.

Extended cosine and sine laws to variable curvature.

Derived a product integral analogue of the fundamental theorem of calculus.

Abstract

Gravity is a phenomenon which arises due to the space-time geometry. The main equations that describe gravity are the Einstein equations. To understand the consequences of these field equations we need to calculate the free particle worldlines to the geometries, which solve these field equations e.g. the Schwarzschild metric solves the Einstein equations and we would need to solve the geodesic equations to this metric. If we were to describe the space-time geometry in terms of geodesics instead of the metric, we could skip the step of solving for the metric and solve for the geodesics directly. In this work we have developed a formalism doing that: we use the bundle of the arclength parametrized geodesics (geodesic flow bundle GFB) from all points in the manifold to describe a Riemannian geometry. Our formalism uses infinitesimal spherical triangles as generating elements, to solve geometric problems. We relate the geodesic flow bundle to Gaussian curvature and develop a method to calculate the geodesics geometrically starting from a Gaussian curvature field. The result amounts to a generalization of the cosine- and sine-laws for constant curvature to varying curvature fields. In this work we restrict ourselves to the Riemannian case and positive curvature fields. We expand the triangulation problem in power series and calculate the main result up to second order. The method itself could be extended to treat more generic cases in particular pseudo-Riemannian geometries which relate directly to Einsteins equations. We test our results against the sphere and perform consistency checks for some examples of varying curvature fields. As a by-product we generalize the notion of integration to products and derive a relation analogue to the main theorem of calculus, for product integrals.