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From Newton's Universal Gravitation to Einstein's Geometric Theory of Gravity

Starting with Newton's law of universal gravitation, we generalize it step-by-step to obtain Einstein's geometric theory of gravity. Newton's gravitational potential satisfies the Poisson equation. We relate the potential to a component of the metric tensor by equating the nonrelativistic result of the principle of stationary proper time to the Lagrangian for a classical gravitational field. In the Poisson equation the Laplacian of the component of the metric tensor is generalized to a cyclic linear combination of the second derivatives of the metric tensor. In local coordinates it is a single component of the Ricci tensor. This component of the Ricci tensor is proportional to the mass density that is related to a single component of the energy-momentum stress tensor and its trace. We thus obtain a single component of Einstein's gravitational field equation in local coordinates. From the principle of general covariance applied to a single component, we obtain all tensor components of Einstein's gravitational field equation.