Cauchy Time Functions and Metric Splitting

Global hyperbolicity does more than guarantee that some rough slice meets every causal history. It supplies a smooth scalar clock whose level sets are smooth spacelike Cauchy hypersurfaces, and it turns the spacetime into a global product $\mathbb R\times\Sigma$ with an orthogonal, generally time-dependent metric. The result is a geometric splitting theorem, not a claim that the spacetime is stationary or that the clock is unique.

Required background. Geodesics and Jacobi Fields supplies inextendible causal curves and their parametrizations. Helpful background. Lorentzian Metrics and Causal Cones fixes the sign of a timelike gradient; Causality and Global Structure provides the global setting; Partial Differential Equations and Green Functions explains why a smooth foliation is analytically useful.

Time, temporal, and Cauchy functions

A time function is a continuous function $t:M\to\mathbb R$ that is strictly increasing along every future-directed causal curve. Its existence characterizes stable causality, but not global hyperbolicity. A temporal function is smooth and has an everywhere timelike gradient. With signature $(-,+,+,+)$ and a function increasing toward the future, $\nabla t$ is past-directed, because for every future causal tangent $v$,

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

A time or temporal function is Cauchy when its level sets are Cauchy hypersurfaces; we use the standard normalization in which its range is all of $\mathbb R$. Equivalently, along every inextendible future-directed causal curve $\gamma$, $t\circ\gamma$ tends to $-\infty$ toward the past end and $+\infty$ toward the future end. Strict increase guarantees at most one crossing of a level, while the two limiting behaviors guarantee at least one.

These adjectives are not interchangeable. On Minkowski spacetime with one point deleted, the coordinate $t$ still has timelike gradient and is temporal. Yet a causal curve can be inextendible by approaching the missing point, so it need not cross levels on the other side of the deletion. The function is not Cauchy. A good local clock does not fill a global causal hole.

The smooth splitting theorem

Assume that $(M,g)$ is a smooth, globally hyperbolic, time-oriented spacetime without boundary. Then there exist a smooth manifold $\Sigma$, a smooth Cauchy temporal function $\tau:M\to\mathbb R$, a positive smooth function $\beta$, and a smoothly varying family of Riemannian metrics $h_\tau$ on $\Sigma$ such that

$$M\simeq\mathbb R\times\Sigma, \qquad g=-\beta(\tau,x)\,d\tau^2+h_\tau(x).$$

Every slice $\Sigma_c=\{\tau=c\}$ is a smooth spacelike Cauchy hypersurface. Conversely, the existence of a Cauchy hypersurface—or of a Cauchy temporal function—implies global hyperbolicity. This is the smooth form of the Geroch splitting theorem, with the exact statement and construction in Bernal and Sánchez 2005, Theorem 1.1, pp. 44–49 and the equivalent causal formulations in Minguzzi and Sánchez 2008, Theorem 3.78, pp. 48–49.

The displayed metric has no shift term because the product lines are chosen orthogonal to the slices. Writing $X=\partial_\tau$ in these adapted coordinates,

$$g(X,X)=-\beta, \qquad g(X,Y)=0$$

for every $Y$ tangent to $\Sigma_c$. The future-directed unit normal is $n=\beta^{-1/2}X$. General coordinates on the same spacetime can reintroduce a shift; zero shift here is a feature of the constructed orthogonal identification, not an invariant absence of frame dragging or motion.

How the construction works

The proof architecture has three distinct stages. First, compact causal diamonds yield a continuous Geroch time function. Volume functions of chronological pasts and futures vary monotonically, and compactness prevents an inextendible causal curve from remaining at a bounded time value. Their levels are topological Cauchy hypersurfaces.

Second, one smooths without losing causality or the Cauchy property. Naive coordinate convolution is unsafe: it can make the gradient null or spacelike and can destroy the global barrier behavior of levels. Bernal and Sánchez instead assemble smooth temporal step functions in neighborhoods of successive Cauchy slices. Convexity of each time cone ensures that a locally finite positive combination retains a timelike, consistently oriented gradient. Upper and lower barriers force the resulting function toward opposite infinities along every inextendible causal curve.

Third, take one level $\Sigma=\tau^{-1}(0)$ and follow the vector field

$$X=\frac{\nabla\tau}{g(\nabla\tau,\nabla\tau)}.$$

Because $g(\nabla\tau,\nabla\tau)<0$, $X$ is future-directed and satisfies $d\tau(X)=1$. Its integral curves are orthogonal to every level set. The Cauchy property ensures that each such curve meets each level exactly once, so its flow gives the global diffeomorphism $\mathbb R\times\Sigma\to M$. Moreover,

$$\beta=-\frac{1}{g(\nabla\tau,\nabla\tau)}>0,$$

and restricting $g$ to a level gives the positive-definite metric $h_\tau$. This derives the orthogonal form rather than assuming it.

What the splitting does not imply

The factors $\beta$ and $h_\tau$ may depend on $\tau$. Therefore the vector $\partial_\tau$ need not be Killing, the slices need not be isometric to one another, and the spacetime need not be static or stationary. The theorem provides global kinematics suitable for evolution; it does not solve the Einstein equation or impose any dynamics on $h_\tau$.

The splitting is also nonunique. If $F:\mathbb R\to\mathbb R$ is a smooth increasing diffeomorphism with $F'\gt0$, then $F\circ\tau$ remains Cauchy temporal; different Cauchy temporal functions can also generate different foliations. No preferred observer congruence, Hamiltonian, positive-frequency decomposition, or quantum vacuum follows from global hyperbolicity alone.

Nor must an arbitrarily prescribed achronal set be a smooth level. A compact achronal set can be extended to a Cauchy hypersurface under the theorem’s hypotheses; stronger smooth and spacelike conclusions require corresponding assumptions on the prescribed set. A general topological Cauchy hypersurface can be included as a level of a smooth Cauchy function, but if it has null portions it cannot be a level of a temporal function there, because temporal levels are spacelike.

Finally, a timelike boundary changes the problem. One may still choose a useful global time and product coordinates, yet causal curves can enter from the boundary. The ordinary Cauchy-level statement no longer supplies prediction without boundary data, so this theorem is not an initial–boundary-value theorem in disguise.

Use in field evolution

On each slice, initial data can be organized as a field and its derivative along the future unit normal. Compact causal diamonds then control finite propagation between slices. This is the geometric setting for the existence of advanced and retarded Green operators for normally hyperbolic equations, developed on Why Global Hyperbolicity Matters for Field Propagation.

The splitting is especially valuable because it separates what is global from what is dynamical. The product manifold and Cauchy foliation come from causal geometry. The lapse $\sqrt\beta$, slice metrics $h_\tau$, constraints, and evolution equations depend on the chosen theory and gauge. Keeping those levels separate prevents the splitting theorem from being mistaken for a solution formula.

Exercise

Suppose a spacetime is already written globally as $g=-\beta d\tau^2+h_\tau$ with $\beta>0$ and each $h_\tau$ positive definite. Show that $\tau$ is temporal and its levels are spacelike. Why do these algebraic facts alone not prove that $\tau$ is Cauchy?

Solution

The inverse metric gives

$$\nabla\tau=-\beta^{-1}\partial_\tau, \qquad g(\nabla\tau,\nabla\tau)=-\beta^{-1}<0.$$

Thus the gradient is timelike and past-directed, so $\tau$ increases along every future causal curve. On a level set, $d\tau=0$ and the induced metric is $h_\tau$, which is positive definite; the level is spacelike.

These are pointwise statements. To be Cauchy, every inextendible causal curve must meet every level. The Minkowski half-plane

$$M=\mathbb R_t\times(0,\infty)_x, \qquad g=-dt^2+dx^2$$

even has the stated global product form, with $\beta=1$ and $h_t=dx^2$. Nevertheless, the null curve $(t,x)=(s,s)$, $0\lt s\lt\infty$, is inextendible in this manifold and misses every level $t\leq0$: its past end approaches the excluded point $(0,0)$. The spatial factor is incomplete, and causal curves can reach its missing end at finite $t$. A global orthogonal product therefore needs the additional Cauchy intersection property; the product notation alone does not supply it.

References

• Bernal, Antonio N., and Miguel Sánchez, “Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes”, Communications in Mathematical Physics 257, Theorem 1.1 and proof, pp. 44–49, 2005.
• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§3–4, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §3.11.3, pp. 46–49 in the arXiv version, European Mathematical Society, 2008.