## A The Volume Tensor

Here we list a number of general identities for the completely antisymmetric volume tensor in dimensions. The most useful are those involving the tensor product (including, as needed, contractions over indices), of the volume tensor with itself [114]:

where is the number of minus signs in the metric (e.g.  for spacetime). We have used the variation of the volume tensor with respect to the metric in the actions principle presented in Sections 8, 9, and 10. We will here derive this variation using the identities above as applied to four-dimensional spacetime ( and ).

Start by writing Equation (360) as

vary it with respect to the metric, and then contract the result with to find
where we have used
If we now contract with we find
and thus

The last thing we need is the variation of the determinant of the metric, since it enters directly in the integrals of the actions. Treating the metric as a matrix, and “normalizing” the by dividing by its one independent component, the determinant is given by

The right-hand-side is proportional to the left-hand-side of Equation (362) and thus
It is not difficult to show