The Geometrization of Maxwell’s Equations and the Emergence of Gravity

Assuming the geometry of nature is Riemannian with four dimensions, the classical Maxwell equations are shown to be a derivable consequence of a single equation that couples the Maxwell tensor to the Riemann-Christoffel curvature tensor. This geometrization of the Maxwell tensor extends the interpretation of the classical Maxwell equations, for example, giving physical quantities such as charge density a geometric definition. Including a conserved energy-momentum tensor, the entirety of classical electromagnetism is shown to be a derivable consequence of the theory. The coupling of the Riemann-Christoffel curvature tensor to the Maxwell tensor also leads naturally to the emergence of gravity which is consistent with Einstein’s equation of General Relativity augmented by a term that can mimic the properties of dark matter and/or dark energy in the context of General Relativity. In summary, the proposed geometrization of the Maxwell tensor puts both electromagnetic and gravitational phenomena on an equal footing, with both being tied to the curvature of space-time. Using specific solutions to the proposed theory, the unification brought to electromagnetic and gravitational phenomena as well as the relationship of those solutions with the corresponding solutions of the classical Maxwell and Einstein field equations are examined.