Diagonal Metrics of Static, Spherically Symmetric Fields The Geodesic Equations and the Mass-Energy Relation from the Coordinate Perspective

TL;DR

This paper derives geodesic equations for static, spherically symmetric fields, analyzes gravitational acceleration from a coordinate perspective, and generalizes the mass-energy relation to diagonal metrics, providing insights into gravitational forces without repulsion.

Contribution

It introduces a coordinate-based approach to geodesic equations and extends the mass-energy relation to diagonal metrics, offering new perspectives on gravitational acceleration and force expressions.

Findings

No gravitational repulsion found for Schwarzschild and exponential metrics.

Derived motion equations for test particles in static, spherically symmetric fields.

Generalized mass-energy relation to account for location-dependent light speeds.

Abstract

The geodesic equations for the general case of diagonal metrics of static, spherically symmetric fields are calculated. The elimination of the proper time variable gives the motion equations for test particles with respect to coordinate time and an account of gravitational acceleration from the coordinate perspective. The results are applied to the Schwarzschild metric and to the so-called exponential metric. In an attempt to add an account of gravitational force from the coordinate perspective, the special relativistic mass-energy relation is generalized to diagonal metrics involving location dependent and possibly anisotropic light speeds. This move requires a distinction between two aspects of the mass of a test particle (parallel and perpendicular to the field). The obtained force expressions do not reveal gravitational repulsion for the Schwarzschild metric and for the exponential metric.