Strong Causality, Stable Causality, and Time Functions

Strong causality prevents causal curves from making arbitrarily small almost-closed returns, while stable causality requires the absence of causal loops to survive a controlled widening of every light cone. The bridge between geometry and order is a theorem: a spacetime is stably causal exactly when it admits a continuous time function, equivalently a smooth temporal function. This is a global existence result, not the choice of a convenient coordinate label.

Required background. Lorentzian Metrics and Causal Cones supplies conformal cones and cone widening; Smooth Manifolds and Tensors supplies gradients, neighborhoods, and smooth functions. Helpful background. Geodesics and Jacobi Fields supports the limit-curve interpretation of almost-closed causal behavior.

Strong causality and small neighborhoods

Let $(M,g)$ be a connected, time-oriented spacetime with a $C^2$ metric. Strong causality holds at $p$ if every neighborhood $U$ of $p$ contains a smaller neighborhood $V$ of $p$ such that any causal curve with endpoints in $V$ remains inside $U$. Equivalently, $V$ may be chosen causally convex in $M$: a causal curve whose endpoints lie in $V$ cannot leave and re-enter it.

This forbids an almost-closed causal curve through $p$. Causality alone rules out an exactly closed curve, but it permits curves to leave every small neighborhood of $p$ and return arbitrarily close to their starting event. Strong causality supplies neighborhood-scale control at every point. It implies distinction, while the converse fails.

The chronological order itself detects this rung. Alexandrov diamonds

$$I^+(a)\cap I^-(b)$$

always form a basis for a topology no finer than the manifold topology. Strong causality is equivalent to equality of these two topologies; it is also equivalent to the Alexandrov topology being Hausdorff. The neighborhood and topology characterizations, with their proof, appear in Minguzzi and Sánchez 2008, §3.6 and §3.6.1, pp. 26–29.

Strong causality has useful consequences for limiting curves. An inextendible causal curve cannot be partially imprisoned in a compact set: after leaving a suitable compact neighborhood, it cannot return infinitely often. The condition still does not guarantee compact causal diamonds or a Cauchy surface; those belong to global hyperbolicity.

Stable causality and wider cones

Write $g<\widetilde g$ when every nonzero $g$-causal vector is $\widetilde g$-timelike, with the same time orientation. The metric $\widetilde g$ has strictly wider light cones. The spacetime $(M,g)$ is stably causal if there exists such a $\widetilde g$ for which $(M,\widetilde g)$ remains causal.

This definition is equivalent to causality holding throughout a neighborhood of the conformal class of $g$ in the $C^0$ fine topology. “Fine” matters on a noncompact manifold: the permitted perturbation can vary from place to place rather than being controlled by one uniform coordinate bound. The cone-order formulation avoids unnecessary dependence on a particular component norm.

Stable causality implies strong causality. Intuitively, the space between the $g$ cones and the wider causal $\widetilde g$ cones provides room to prevent repeated near returns. The converse is not valid: standard cut-and-identify examples are strongly causal but develop closed causal curves under every global cone widening. The definitions, topology, and a separating example are given in Minguzzi and Sánchez 2008, §§3.8.2–3.8.3, pp. 36–39.

This stability is conformal but not merely local. Every point admits a neighborhood whose cones can be widened causally, even when no compatible widening exists over all of $M$.

Time and temporal functions

A time function is a continuous map $t:M\to\mathbb R$ that is strictly increasing along every future-directed nonconstant causal curve. A temporal function is a smooth map $T:M\to\mathbb R$ whose gradient is everywhere past-directed timelike. With the site signature,

$$g(\nabla T,\nabla T)<0,$$

and for every future-directed causal tangent $v$,

$$\frac{d}{d\lambda}T(\gamma(\lambda)) =dT(v)=g(\nabla T,v)>0.$$

Thus every temporal function is a time function. The reverse statement for a particular function is false: a smooth time function can have a null gradient somewhere. The theorem concerns existence—if some continuous time function exists, then some, possibly different, smooth temporal function exists.

Level sets of a time function are acausal because strict increase forbids a causal curve from meeting one level twice. They need not be Cauchy surfaces: strict order does not ensure that an inextendible causal curve crosses every level. A Cauchy temporal function imposes that stronger coverage condition and is developed in Cauchy Time Functions and Metric Splitting.

The stable-causality theorem

Under the hypotheses above, the following statements are equivalent:

1. $(M,g)$ is stably causal.
2. $(M,g)$ admits a time function.
3. $(M,g)$ admits a temporal function.

The proof architecture explains why the equivalence is nontrivial. Suppose first that $T$ is temporal. After a conformal normalization, one can split the metric along $\nabla T$ as

$$g=-dT^2+h,$$

where $h$ is positive definite on the subbundle tangent to the levels of $T$. For $\lambda>1$,

$$g_\lambda=-\lambda\,dT^2+h$$

has wider cones, while $T$ still increases along every $g_\lambda$-causal curve. Hence $g_\lambda$ is causal and temporal-function existence implies stable causality.

Conversely, assume a causal wider metric is available. One chooses a monotone family of intermediate cone structures and constructs past-volume functions from a smooth positive density with finite total integral. In particular, chronological boundaries have zero measure, so point masses cannot create spurious volume jumps. Individual volume functions can still lack continuity because the chronological sets jump, but averaging over the widening parameter yields a continuous function strictly increasing on every $g$-causal curve. This produces a time function. Finally, a smoothing theorem replaces the continuous time function by a smooth temporal function without losing strict increase. The complete modern equivalence, including the necessary smoothing step, is presented in Minguzzi 2019, §4.4, pp. 137–142 and Minguzzi and Sánchez 2008, Theorem 3.56, pp. 38–39.

The result does not imply global hyperbolicity, compactness of diamonds, geodesic completeness, or field-equation predictability. It identifies exactly the robustness of causal order under cone widening.

Exercise

On two-dimensional Minkowski spacetime define

$$T(t,x)=t-\sqrt{1+x^2}.$$

Show that $T$ is temporal, but its zero level is not a Cauchy surface.

Solution

The covector $dT$ has components $(1,-x/\sqrt{1+x^2})$. Its metric-dual gradient is

$$\nabla T=-\partial_t-\frac{x}{\sqrt{1+x^2}}\partial_x,$$

so

$$g(\nabla T,\nabla T) =-1+\frac{x^2}{1+x^2} =-\frac{1}{1+x^2}<0.$$

Its negative $t$ component makes it past-directed, so $T$ is temporal. The zero level is the spacelike graph $t=\sqrt{1+x^2}$. The inextendible future-directed null line $t=x$ never meets it, because $x=\sqrt{1+x^2}$ has no real solution. Therefore a temporal level can be acausal and edgeless without being Cauchy.

Continue to Causal Continuity and Causal Simplicity for continuity and closedness properties of the endpoint relations.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§4.3.6–4.5, pp. 130–149, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §§3.6–3.10, pp. 26–42 in the arXiv version, European Mathematical Society, 2008.