The cosmological volume function

In a previous work, the regular cosmological volume function $τ_V$ 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, $τ_V$ 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.

💡 Research Summary

This paper establishes the continuous differentiability (C¹) of the cosmological volume function τ_V in several physically significant scenarios, leading to profound implications for spacetime geometry.

The cosmological volume function, defined as τ_V(p) = vol_g(I⁻(p)) (the volume of the chronological past of a point p), was introduced by the author as a temporal measure based on the concept of volume incompleteness, analogous to the more familiar cosmological time function τ based on length. A key motivation is to obtain a canonical time function derived solely from the Lorentzian metric g, without introducing arbitrary choices like a weighted measure. While previous work by Chruściel, Grant, and Minguzzi showed that one can always find some finite measure μ such that p → μ(I⁻(p)) is a C¹ temporal function, the choice of μ is non-unique. This paper proves that the natural choice—using the metric volume measure vol_g itself—already yields a C¹ temporal function τ_V under specific, well-motivated conditions.

The two main settings are:

1. Future Cauchy Development: If (M, g) is the future Cauchy development of an initial data set with future Cauchy surface S, then the cosmological volume function on I⁺(S) is regular and a C¹ temporal function (Theorem 1.1).
2. Cosmological Spacetimes with No Past Observer Horizons: If (M, g) is a causal spacetime with finite τ_V and satisfies the No Past Observer Horizons condition (meaning I⁺(γ)=M for every past-inextendible causal curve γ), then τ_V is regular and a C¹ temporal function (Corollary 1.2). This condition ensures the spacetime is globally hyperbolic with compact Cauchy surfaces, and every point has an entire Cauchy surface in its past.

The proof of Theorem 1.1 is technically intricate, adapting the arguments from