A Generally Covariant Field Equation for Gravitation and Electromagnetism

Abstract

A generally covariant field equation is developed for gravitation and electromagnetism by considering the metric vector qμ in curvilinear, non-Euclidean spacetime. The field equation is $$R^\mu - {\text{ }}\frac{1}{2}Rq^\mu = kT^\mu ,$$ , where Tμ is the canonical energy-momentum four-vector, k the Einstein constant, Rμ the curvature four-vector, and R the Riemann scalar curvature. It is shown that this equation can be written as $$T^\mu = \alpha q^\mu ,$$ where α is a coefficient defined in terms of R, k, and the scale factors of the curvilinear coordinate system. Gravitation is described through the Einstein field equation, which is recovered by multiplying both sides by qμ. Generally covariant electromagnetism is described by multiplying the foregoing on both sides by the wedge qν. Therefore, gravitation is described by symmetric metricqμqν and electromagnetism by the anti-symmetric defined by the wedge product qμqν.