I am having a problem with Minkowski space. When I configure a space with my current code I need to multiply by the square root of the length² of the natural normal to a facet of a zone. The purpose of this is for the two zones that share the facet to agree on a coordinate system in which to express vectors being transferred across the facet. With a Minkowski space the length² may be negative. When there is a null vector in the facet the length² may be zero while the normal is certainly not zero.

This may solve the problem. But back to the old problem.

I now think that the neat trick of using rows of the contravariant metric tensor is suitable only for positive definite metrics. I suspect we need a full fledged theory of affine connections on a complex. In the continuous case the non definite metric determines a unique affine connection. I am not yet sure about how to represent an affine connection (but see this) or how to derive it from an indefinite metric.

Transferring a vector across a facet should be a job for an affine connection. The non-positive-definite quadratic forms that define a metric for Minkowski space seem also to provide a perfectly normal affine connection. I need to understand this better.

An affine connection is much heavier than a metric; it has n²(n+1)/2 reals. My intuition is that a linear transformation between pairs of neighboring zones suffices. A positive definite metric does this and so does a pseudo metric, but I am not yet sure of the math. Computationally if each zone x carried for each neighbor y a representation of the neighbor’s novel basis element, in the bcc of x, then this logic would move a vector across quickly.

In more detail each facet would have a distinguished vertex, known and agreed upon by both neighbors. (Probably the vertex with the greatest global index) Each neighbor would speak and listen as if that vertex were its own origin. The neighbor’s novel basis element is that unique edge from the origin of the shared facet to the neighbor’s unique unshared vertex.

Consider 2D space with metric signature (− +). This simple covering of 2D space with triangles has two classes:

{(<i, j>, <i+1, j>, <i+1, j+1>) | i & j are integers}
{(<i, j>, <i, j+1>, <i+1, j+1>) | i & j are integers}

Together they tile the plane. Suppose that

g_ij

1	0
0	−1

The lengths of the edges (<i, j>, <i+1, j+1>) are all 0 yet this is fine tiling which we must allow. Suppose that vertex numbers for vertex <i, j> is 1000 i + j.

To be continued

I think that perhaps a zone must be constrained to have just one time like edge in order to conform to some von Neumann-Richtmeyer like stability threshold. Perhaps this is contingent on a Cauchy like initial value computation. This is perhaps better developed here.

Affine Notes

An Intuitive Explanation of the Affine Connection