An intuitive and simplified formulation of general relativistic gravity in freely falling frames

Abstract

General relativistic gravity in freely falling frames is formulated using undergraduate calculus and matrix algebra. The key to the formulation involves retaining Newton’s first law of motion in the local spacetime of freely falling frames. The formulation shows that the relative acceleration of two particles in the freely falling frame does not depend directly on their relative velocity. Also, the 4 × 4 matrix relating the relative acceleration of two particles to their relative spacetime displacement closely resembles the physics of Newtonian gravity, implying that the local gravitational field obeys Laplace’s equation. The results of this paper and a previous publication [L. R. Miller, Phys. Essays 35, 202 (2022)] provide an intuitive approach that enables the use of lower-level mathematics to arrive at Einstein’s theory of gravity in empty space.