Differentiability and Other Properties of the Cosmological Volume Function

Technical Summary: Differentiability and Other Properties of the Cosmological Volume Function

Problem Statement
In General Relativity, the lack of an observer-independent time coordinate is a fundamental challenge. While singularities can sometimes provide a natural time coordinate, standard constructions face limitations. The regular cosmological time function $\tau$, introduced by Andersson, Galloway, and Howard, measures the maximum proper time an observer can experience since a singularity (the Big Bang). However, $\tau$ is generally only locally Lipschitz and differentiable almost everywhere, failing to be $C^1$ in many cases due to the timelike cut locus.

An alternative, the cosmological volume function $\tau_V$, was recently introduced to measure time via the canonical volume of the chronological past, $\tau_V(p) := \text{vol}_g(I^-(p))$. While $\tau_V$ shares many structural properties with $\tau$ (such as defining a global hyperbolic splitting), its differentiability properties were previously unclear. A key issue is that the canonical volume measure $\text{vol}_g$ can be infinite, and while weighted measures can be chosen to ensure differentiability (as shown by Chruściel, Grant, and Minguzzi), such choices are arbitrary. The central problem addressed in this paper is whether the canonical (unweighted) cosmological volume function $\tau_V$ is a continuously differentiable ($C^1$) temporal function in physically relevant settings, thereby providing a canonical splitting of the metric tensor without arbitrary choices.

Methodology
The paper employs techniques from Lorentzian geometry and measure theory, adapting arguments from Chruściel, Grant, and Minguzzi regarding weighted measures to the specific case of the canonical volume measure.

1. Differentiation of Integrals: The core technical approach involves differentiating the integral defining $\tau_V(p) = \int_{I^-(p) \cap I^+(S)} d\text{vol}_g$. The author utilizes a Leibniz integral rule for differential forms, where the variation of the domain of integration (the chronological past) is analyzed.
2. Geometric Analysis of Boundaries: The boundary of the chronological past, $\partial I^-(p)$, is ruled by past-directed null geodesics. The author analyzes the "velocity" of this boundary using Jacobi fields along these null geodesics.
3. Handling Singularities: Special care is taken regarding the null cut locus. The paper establishes that the null cut locus has measure zero, ensuring it does not contribute to the volume integral's derivative, unlike the timelike cut locus which affects the differentiability of $\tau$.
4. Continuity Proofs: The proofs involve constructing candidate derivatives as integrals over the set $\mathring{E}^-(p)$ (the boundary of the past excluding the cut locus) and demonstrating the continuity of these maps using properties of the exponential map and the compactness of Cauchy surfaces.

Key Contributions and Results

• Main Theorem (Theorem 1.1): The paper proves that if $(M, g)$ is the future Cauchy development of a Cauchy surface $S$ (i.e., time is measured from initial data rather than a singularity), the cosmological volume function $\tau_V$ restricted to $I^+(S)$ is a regular $C^1$ temporal function.
• Corollary 1.2 (Cosmological Setting): Under the condition that the spacetime is past volume incomplete and has no past observer horizons (meaning every past-inextendible causal curve sees the entire spacetime, $I^+(\gamma) = M$), and assuming $\tau_V$ is finite, $\tau_V$ is a regular $C^1$ temporal function.
• Canonical Metric Splitting: The $C^1$ regularity of $\tau_V$ allows for a canonical decomposition of the metric tensor:
  $$g = -\beta(d\tau_V)^2 + h$$
  where $\beta$ is a continuous function and $h$ induces a continuous Riemannian metric on the level sets of $\tau_V$. This enables the construction of a "Wick-rotated" Riemannian metric $g_R := \beta(d\tau_V)^2 + h$ that is derived solely from the Lorentzian metric $g$ without arbitrary weighting choices.
• Counterexamples and Limitations:
  • Example 3.8: Demonstrates that if the Cauchy surface $S$ has lightlike portions, $\tau_V$ may fail to be $C^1$.
  • Example 3.9: Shows a spacetime where $\tau_V$ is not even locally Lipschitz, occurring when the spacetime cannot be cast as the future domain of dependence of a Cauchy surface (due to a spacelike boundary with specific conformal factors).
  • Example 3.10: Illustrates a case where $\tau_V$ is $C^1$ but not $C^2$, highlighting that higher-order differentiability is not guaranteed.
• Comparison with Cosmological Time ($\tau$):
  • The paper establishes that regularity of $\tau$ (finite and tending to zero along past-inextendible curves) does not automatically imply regularity of $\tau_V$, though conditions involving compact Cauchy surfaces and Ricci curvature bounds (Proposition 4.1) can ensure it.
  • Unlike $\tau$, the level sets of $\tau_V$ are not necessarily Cauchy surfaces for all values, even if $\tau_V$ is regular (Example 4.4).
  • The cosmological cut locus for $\tau$ is not necessarily closed (Example 4.7), contrasting with standard Riemannian cut loci.

Significance and Claims
The paper claims that in two specific but important settings—spacetimes arising from initial data and certain cosmological models without observer horizons—the cosmological volume function $\tau_V$ provides a canonical $C^1$ temporal function. This is significant because:

1. Canonicity: Unlike previous approaches that required arbitrary choices of weight functions to ensure differentiability, $\tau_V$ relies only on the metric $g$ and the canonical volume measure.
2. Metric Splitting: The $C^1$ property allows for a continuous splitting of the metric tensor, facilitating the definition of a Wick-rotated Riemannian metric $g_R$.
3. Convergence Studies: The constructed $g_R$ is well-suited for studying the convergence of sequences of spacetimes $(M_i, g_i)$ using established metric space convergence notions (e.g., Gromov–Hausdorff), as it avoids the non-differentiability issues inherent in $\tau$ or arbitrary measures.

The author explicitly notes that while $\tau_V$ is $C^1$ in these cases, it is generally not $C^2$, and its level sets do not always share the same Cauchy properties as those of $\tau$. The results are presented as establishing the differentiability of $\tau_V$ in "many cases of interest," rather than universally for all spacetimes.