Differentiability and other properties of the cosmological volume function

In a previous work, the regular cosmological volume function ${\tau }_V$ was introduced as an alternative to the regular cosmological time function of Andersson, Galloway, and Howard. Building on work by Chruściel, Grant and Minguzzi, in this paper we show that in many cases of interest, ${\tau }_V$ is a continuously differentiable temporal function. This leads to a canonical splitting of the metric tensor, and induces a canonical "Wick-rotated" Riemannian metric. We also provide some further results and examples related to the cosmological time and volume functions.