The geometric origin of electric force

Every textbook on quantum field theory points out the formal, mathematical parallels between gauge theory and general relativity. In this paper, we make these parallels visual. The differential geometry behind general relativity can be visualized using an embedding space. We will use similar embedding techniques to show the spatial geometry associated with an electric field. As one might expect from the Lagrangian, electric fields have time-changing geometry. This paper focuses on exposing the geometrical origin of electric force by using both a near-exact solution and a more simple, physically insightful, approximate solution