Smooth Regular Cosmological Time Function

• Smooth regular cosmological time functions are defined using the supremum of Lorentzian distances, ensuring finiteness and decay to zero along every past-inextendible curve.
• They act as temporal functions with unit timelike gradients, yielding spacelike level sets and inducing a null distance that metrizes the manifold topology.
• The collapse of diameter in level sets provides a precise metric singularity criterion, linking classical Big Bang models to modern metric convergence analysis.

A smooth regular cosmological time function is the smooth case of the cosmological time introduced by Andersson, Galloway, and Howard on a time-oriented, connected Lorentzian manifold $(M,g)$. It is defined by

$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$

where $d_g(p,q)$ is the Lorentzian length of a longest future-directed causal curve from $p$ to $q$, and it is called regular when it is finite everywhere and tends to $0$ along every past-inextendible future-directed causal curve toward its past end. When $\tau_g$ is smooth and satisfies $g(\nabla\tau_g,\nabla\tau_g)=-1$, it becomes a temporal function with spacelike level sets, admits a metric realization through null distance, and supports a precise singularity criterion: collapse of the diameters of the level sets as $t\to0$ forces the null-distance completion to have a one-point initial boundary (Nigri, 9 Jul 2025).

1. Lorentzian definition and regularity

For a time-oriented, connected Lorentzian manifold of dimension $n+1$, the cosmological time function is defined pointwise by the Lorentzian distance to the causal past: $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$0 In general, $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$1 may be infinite or discontinuous. The regularity condition consists of two requirements: $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$2 for all $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$3, and for every past-inextendible future-directed causal curve $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$4, one has $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$5 as $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$6 (Nigri, 9 Jul 2025).

Under regularity, Andersson, Galloway, and Howard prove that $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$7 is a continuous time function, strictly increasing along future-causal curves. They also show that $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$8 exists almost everywhere, and that through each $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$9 there passes a unique future-directed timelike unit-speed geodesic, called a generator,

$d_g(p,q)$0

When $d_g(p,q)$1 is smooth on $d_g(p,q)$2, Proposition 5.3 in the null-distance paper yields

$d_g(p,q)$3

so $d_g(p,q)$4 is past-directed timelike of unit magnitude. In that case, $d_g(p,q)$5 is called a smooth regular cosmological time function (Nigri, 9 Jul 2025).

This notion isolates the regime in which the Lorentzian distance-based cosmological time is not merely continuous or Lipschitz, but differentiable enough to control both foliation geometry and induced metric structures. The smoothness assumption is therefore not ornamental: it is the threshold used to relate causal, Riemannian, and completion-theoretic properties in a unified framework.

2. Temporal functions, anti-Lipschitz behavior, and null distance

The smooth regular cosmological time function sits inside the broader hierarchy of causal time functions. A continuous $d_g(p,q)$6 is a time function if it is strictly increasing along every future-directed causal curve. If $d_g(p,q)$7 is differentiable, then $d_g(p,q)$8 is either past-directed causal or zero at each point. A temporal function is a $d_g(p,q)$9 function whose gradient is everywhere past-directed timelike (Nigri, 9 Jul 2025).

A second characterization uses the local anti-Lipschitz property. For each $p$0, there must exist a neighborhood $p$1 and a constant $p$2 such that

$p$3

where $p$4 is any Riemannian distance on $p$5. For $p$6-functions, the null-distance paper establishes the equivalence

$p$7

Thus every smooth regular cosmological time function is, in particular, a temporal function and therefore defines a genuine metric through null distance (Nigri, 9 Jul 2025).

Given any time function $p$8, the associated null distance is defined by first assigning to a piecewise causal curve $p$9 the null length

$q$0

and then setting

$q$1

This is always a semi-metric, with two fundamental properties: $q$2 and

$q$3

Moreover, $q$4 is a metric inducing the manifold topology if and only if $q$5 is locally anti-Lipschitz. Sormani and Vega’s null distance therefore converts appropriate time functions into metric objects adapted to Lorentzian causal structure (Nigri, 9 Jul 2025).

3. Geometry of level sets and comparison with intrinsic Riemannian distance

If $q$6 is smooth and $q$7 never vanishes, each level set

$q$8

is a smooth spacelike hypersurface. The induced Riemannian metric on $q$9 is denoted

$0$0

with associated Riemannian distance $0$1. For smooth regular cosmological time, these level sets are the constant-$0$2 slices of the canonical foliation (Nigri, 9 Jul 2025).

The central geometric estimate in the null-distance analysis applies under a constant-norm condition. If on some level set $0$3 one has

$0$4

then for all $0$5,

$0$6

The proof proceeds locally. Near a point $0$7, one chooses normal coordinates sending $0$8 to $0$9 and a tangent direction to $\tau_g$0. For a short segment in $\tau_g$1, one then constructs a piecewise null path formed by two null segments meeting at an auxiliary point $\tau_g$2. In these approximately Minkowski coordinates, the null length is asymptotically $\tau_g$3 times the spatial length, uniformly as the segment length tends to $\tau_g$4. A Lebesgue-number covering argument promotes this local estimate to arbitrary pairs on the whole slice (Nigri, 9 Jul 2025).

For a smooth regular cosmological time function, the normalization $\tau_g$5 means $\tau_g$6. Hence on every slice $\tau_g$7,

$\tau_g$8

This gives a direct comparison between the causal-metric structure encoded by null distance and the intrinsic Riemannian geometry of the level sets. The same work states that smooth regular cosmological time functions provide a canonical foliation of globally hyperbolic, and hence in particular stably causal, spacetimes by constant-$\tau_g$9 hypersurfaces which are Cauchy slices; their gradients furnish a global time-flow by past-directed unit timelike normals, and the associated null distance metrizes the manifold topology while capturing causal separation (Nigri, 9 Jul 2025).

4. Level-set collapse and the Big Bang singularity criterion

In the program of Sakovich and Sormani, a spacetime exhibits a Big Bang singularity if, roughly speaking, its initial boundary in the null-distance completion is a single point. The null-distance paper proves a theorem confirming their Conjecture 3.7 in the setting of smooth regular cosmological time (Nigri, 9 Jul 2025).

Let $g(\nabla\tau_g,\nabla\tau_g)=-1$0 admit a smooth regular cosmological time $g(\nabla\tau_g,\nabla\tau_g)=-1$1, and write

$g(\nabla\tau_g,\nabla\tau_g)=-1$2

with induced metric $g(\nabla\tau_g,\nabla\tau_g)=-1$3. If the diameters of the level sets $g(\nabla\tau_g,\nabla\tau_g)=-1$4 with respect to $g(\nabla\tau_g,\nabla\tau_g)=-1$5 shrink to zero as $g(\nabla\tau_g,\nabla\tau_g)=-1$6, then the unique extension $g(\nabla\tau_g,\nabla\tau_g)=-1$7 to the metric completion $g(\nabla\tau_g,\nabla\tau_g)=-1$8 has initial level set

$g(\nabla\tau_g,\nabla\tau_g)=-1$9

consisting of exactly one point $t\to0$0. This is the precise form of the Big Bang singularity conclusion proved in Theorem 5.2 (Nigri, 9 Jul 2025).

The proof uses the slice estimate described above. Since $t\to0$1, one has $t\to0$2 on each $t\to0$3, and hence the null-distance diameters of the slices also converge to zero. Standard Hausdorff-diameter continuity in metric completions then implies that the family $t\to0$4 converges in Hausdorff distance to the initial level set $t\to0$5. A limit of sets whose diameters collapse to zero must be a singleton, so the initial boundary reduces to one point. Equivalently, all generators $t\to0$6 collapse to a single endpoint as $t\to0$7 (Nigri, 9 Jul 2025).

The significance of this theorem lies in the translation of singularity formation into metric convergence. The paper explicitly places the result in the spirit of Gromov–Hausdorff limits: a “crushing” of spatial slices to zero diameter becomes a completion-theoretic statement about the initial boundary. It also states that this framework recovers classical Big Bang singularity theorems, including Geroch’s, within a metric-convergence formulation (Nigri, 9 Jul 2025).

5. Limits of regularity and geometric control below the smooth regime

A common misconception is that regular cosmological time is automatically smooth. The more cautious statement is that regularity alone yields continuity and strong causal monotonicity, but not $t\to0$8-regularity. Under the regularity hypothesis, the cosmological time is continuous, globally Lipschitz, and has two one-sided derivatives almost everywhere; the strongest conclusion under mere regularity is continuity together with almost-everywhere differentiability properties, not a global $t\to0$9 theorem (Galloway et al., 2024).

The same source emphasizes that no theorem there upgrades $n+1$0 to $n+1$1 or better under the bare regularity assumptions. What it does provide instead is geometric information about the level sets in a weak sense. If $n+1$2 is future timelike geodesically complete and either admits a compact Cauchy slice or has spacelike future causal boundary, then every level set

$n+1$3

is a topological Cauchy hypersurface. Moreover, if $n+1$4 in timelike directions and $n+1$5 is regular, then each $n+1$6 has mean curvature bounded above in the support sense by

$n+1$7

These results show that substantial geometric control remains available even when smoothness fails (Galloway et al., 2024).

A related comparison appears in García-Heveling’s study of the cosmological volume function. There the classical cosmological time $n+1$8 is described as length-based, continuous, strictly increasing, with almost everywhere second-derivatives, but in general only locally Lipschitz and failing to be $n+1$9 precisely along its timelike cut-locus. The same summary states that smoothing $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$00 to $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$01 typically requires curvature bounds or mean-convexity. This clarifies why the adjective “smooth” in smooth regular cosmological time function marks a genuinely restrictive hypothesis rather than a routine consequence of regularity (García-Heveling, 12 Dec 2025).

6. Related canonical time functions and neighboring notions

The smooth regular cosmological time function belongs to a broader family of canonical temporal constructions in Lorentzian geometry. One important comparison object is García-Heveling’s regular cosmological volume function

$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$02

Under either of two hypotheses—restriction to $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$03 for a $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$04 future Cauchy surface $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$05, or a causal spacetime with no past observer-horizons and finite $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$06 everywhere—$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$07 is regular and $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$08 temporal. Because $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$09 is timelike, the metric splits canonically as

$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$10

and this induces a canonical Wick-rotated metric

$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$11

The same comparison notes that both $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$12 and $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$13 vanish only at the singularity under regularity, but differ structurally: $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$14 is length-based, whereas $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$15 is volume-based, and $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$16 need not even be Lipschitz in general without extra hypotheses (García-Heveling, 12 Dec 2025).

A distinct but related literature uses York time, defined from the mean extrinsic curvature of a spacelike slicing rather than Lorentzian distance. In one formulation,

$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$17

with $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$18; in a homogeneous flat cosmology this gives

$\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$19

For a flat forever-expanding universe, $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$20 at the big bang and approaches $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$21 or another finite non-positive value as $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$22, depending on the asymptotic behavior of the potential. The reduced Hamiltonian description shows that a smooth extension through $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$23 requires the condition $\tau_g(q):=\sup\{\,d_g(p,q)\mid p\le q\,\},$24 (Roser, 2014). This suggests that “cosmological time” in the literature refers not to a single canonical construction, but to several non-equivalent time parameters—distance-based, volume-based, and mean-curvature-based—each adapted to different problems.

Within that landscape, the smooth regular cosmological time function is distinguished by three linked features. First, it is defined directly from Lorentzian distance to the past. Second, when smooth, it is automatically temporal with unit timelike gradient. Third, its associated null distance produces a metric whose completion can encode singularity formation via collapse of the Riemannian diameters of the level sets. Those features explain its central role in recent attempts to bridge global causal theory, metric geometry, and singularity analysis (Nigri, 9 Jul 2025).