Finding the Schwarzschild metric from gravity’s Exterior and Interior Effacement algebra

In the author’s previous publications, a recursive linear algebraicmethod was introduced for obtaining (sans gravitational radiation) the full potential expansions for the gravitational metric field components and the Lagrangian for the general N-body system. Two apparent properties of gravity—Exterior Effacement and Interior Effacement—were defined and enforced to obtain the recursive algebra, especially for the motion-independent potential expansions of the general N-body situation. The linear algebraic equations of this method permit determination of the potential coefficients at any order n† of the expansions in terms of the lower order coefficients. To illustrate the capabilities of this algebraic method by enforcing exterior and interior effacement, and focusing on only a needed few potential series of the full motion-independent potential expansions, the complete exterior metric field for a single, spherically symmetric mass source is here obtained—the Schwarzschild metric field of general relativity (the Eddington PPN parameter γ = 1) as well as its generalization if the isotropic spatial metric potential’s linearized form is −gSS(γ, r) = 1 + 2γGm/c2r +.... with γ ≠ 1 are obtained.