The cosmological volume function

TL;DR

This paper investigates the properties of the cosmological volume function, demonstrating its differentiability and implications for metric splitting and Wick rotation, with additional results and examples.

Contribution

It introduces the differentiability of the cosmological volume function and explores its consequences for metric splitting and Wick rotation in cosmology.

Findings

$ au_V$ is often continuously differentiable

Leads to a canonical metric splitting

Induces a Wick-rotated Riemannian metric

Abstract

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. 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.