Introduction

The goal of this paper is to create one mathematical structure for gravity and electromagnetism that can be quantized. The difference between gravity and electromagnetism is one of the oldest problems facing physics, going back to studies of electromagnetism in the eighteenth century. Gravity was the first inverse square law, discovered by Isaac Newton. After twenty years of effort, he was able to show that inside a hollow, massive shell, the gravitational field would be zero. Ben Franklin, in his studies of electricity, demonstrated a similar property for an electrically charged ball inside an insulated, conducting can. Joseph Priestly realized this meant that the electrostatic force was governed by an inverse square law like gravity. Due to his experimental work, Coulomb got the credit for the electrostatic force law modeled on Newton's law of gravity.[5]

Over a hundred years later, Einstein started from the tensor formalism of electromagnetism on the road to general relativity. Instead of the antisymmetric electromagnetic field strength tensor, Einstein realized a change in the symmetric metric tensor would be governed by symmetric tensors. There is a precedence for transforming mathematical structures between gravity and electromagnetism.

Previous efforts to use 4-potentials for gravity report failures due in part to considering analogies that were too close to electromagnetism. Although free to choose the sign of a mass current, researchers chose the same sign used for electromagnetism where like charges repel.[4,13] Other efforts created an attractive gravitational force, but forgot Einstein's insight that a symmetric tensor was required.[11, Exercise 7.2] Perhaps the greatest barrier however was the inverse squared potential that solved the field equations suggest a non-physical, inverse-cubed force law.

In this paper, analogies to electromagnetism are used for gravity, respecting differences. The mass current always has a sign opposite to electric current. The second-rank gravitational field strength tensor is symmetric. An inverse distance potential is physically relevant, so a normalized, perturbation of a 4-potential near that classical result will be the focus. By staying close to electromagnetism, particularly by working with a linear field theory, the difficulty of quantizing gravity may dissolve.