Introduction

We humans live in a physical universe and we strive to understand how it operates. In this task mathematical models have been used with great success. I shall here present a mathematical model which is intended to represent physical space.

A most fundamental property of space is that two different locations are at a nonzero distance from each other. We are accustomed to make this statement in only three dimensions, and the statement itself commingles levels of mathematical abstraction, but mathematical engines of vast power exist in spaces where this statement is true and are absent when it is false. Specifically, for a mathematical model of physical space I use a Hausdorff space, in which this statement can be made.

It is a major theorem that every Hausdorff space is a metric space, and every metric space is a Riemannian manifold (and conversely). The theorems about Fredholm integration and Green's function apply to Riemannian manifolds.

A major portion of Einstein's General Theory of Relativity is to identify the acceleration of gravity, and other terms such as gravitational magnetism, with particular Christoffel symbols, which are coefficients in the covariant differentiation of tensors. Since every Hausdorff space has Christoffel symbols, then (commingling the symbol and the referent) every Hausdorff space has gravity, which for mathematical reasons plays an essential role in the motion of the represented objects, and no reasonable mathematical model of physics could be constructed that lacked gravity, whether or not gravity were known to exist empirically.

In the rest of Einstein's theory, the density of matter and the stresses in it are set equal to a certain combination of derivatives of Christoffel symbols. The exact source term does not have ironclad mathematical backing. Also the theory clearly requires an indefinite metric called the Minkowski metric, whereby timelike separations are at a positive distance; spacelike (tachyonic) separations are negative; and points along a ray of light are at zero distance from each other. With the resulting topology the space is not a Hausdorff space, not a metric space and not a Riemannian manifold; it is junk, and physical results can only be recovered by ignoring the topology induced by the Minkowski metric. A major focus of this paper will be to discover what the Minkowski metric really means.

An analogous physical model can be formulated using Fredholm integration of the generalized Laplacian operator. The Christoffel symbols are the potential, and the corresponding source term is conserved because of a generalization of Ricci's theorem.