Rough Lorentzian Metrics and Causality

A continuous Lorentzian metric still has light cones and assigns lengths to locally Lipschitz causal curves, but several smooth equivalences can fail. In particular, causal curves need not be approximable by timelike curves, null generators may branch, and the Christoffel symbols may not exist as continuous functions. Low-regularity causality must therefore begin with curve-based definitions and state the regularity needed for every later geodesic claim.

Required background. Analysis for Mathematical Relativity supplies almost-everywhere derivatives and compactness; The Einstein Equation gives the geometric setting; PDEs and Green Functions for Gravitation supplies limit-curve intuition.

Helpful background. Causality and Global Structure provides the smooth causal hierarchy that is being weakened here.

Causal curves without Christoffel symbols

Let $g$ be a continuous Lorentzian metric with a chosen time orientation. A locally Lipschitz curve $\gamma:I\to M$ is future causal if its almost-everywhere tangent satisfies

$$g(\dot\gamma,\dot\gamma)\le0$$

and is nonzero and future directed almost everywhere. Rademacher’s theorem provides that almost-everywhere tangent. These conditions use the continuous cone field and do not require a connection.

For chronological futures on this page, use piecewise $C^1$ future timelike curves, with nonzero timelike one-sided tangents at corners. On compact segments their tangents stay uniformly inside the light cones. Define $p\ll q$ and $I^+(p)$ using these curves, and $p\le q$ and $J^+(p)$ using the causal Lipschitz curves above. Merely requiring a Lipschitz tangent to be timelike almost everywhere is another possible convention at low regularity; one must not assume all such chronological futures coincide. The Lorentzian length

$$L_g(\gamma)=\int_I\sqrt{-g(\dot\gamma,\dot\gamma)}\,ds$$

is meaningful for causal Lipschitz curves. The time separation is the supremum of these lengths over curves from $p$ to $q$, with value zero when no such curve exists.

This curve-first approach is more robust than defining causality through geodesics. For a merely continuous metric, the connection formula contains derivatives of $g$ and has no classical meaning.

Regularity thresholds for geodesics

The geodesic equation in coordinates is

$$\ddot x^\mu+\Gamma^\mu{}_{\alpha\beta}(x) \dot x^\alpha\dot x^\beta=0.$$

If $g\in C^{1,1}$, then the Christoffel symbols are locally Lipschitz, so standard ODE theory gives locally unique geodesics depending continuously on their initial data. If $g\in C^1$, the Christoffels are continuous: Peano existence applies, but uniqueness can fail. If $g$ is locally Lipschitz, the Christoffels exist only almost everywhere and are essentially bounded; generalized Carathéodory or Filippov solutions can be considered, but the chosen solution concept becomes part of the theorem.

Thus a continuous metric can support a strong curve-based causal theory even though “the null geodesic through this vector” is undefined or nonunique. Conflating those levels imports smooth assumptions invisibly.

Push-up and causal bubbles

For smooth metrics, a causal curve with a timelike segment can be deformed into a timelike curve. This push-up property underlies familiar identities between boundaries of chronological and causal futures. It need not hold for continuous metrics.

A causal bubble is an open region reachable by causal curves from $p$ but not by timelike curves from $p$. Such regions arise for cone fields with sub-Lipschitz variation: null solutions of the associated differential inclusion can branch and enclose a region. Chruściel and Grant construct and analyze these examples in Chruściel–Grant 2012, §§1–2 and 4–6. Their examples show that continuity alone does not recover all smooth causal intuition.

Locally Lipschitz metrics are sufficiently regular for important push-up results, though borderline statements depend on the precise definition of causal curve. A useful proof technique approximates $g$ by smooth metrics $g^-_\varepsilon$ and $g^+_\varepsilon$ with respectively narrower and wider cones. Cone control, not just uniform component convergence, is essential.

A bubble one can calculate

Use dimensionless local coordinates $(u,x)$ and choose $0\lt\lambda\lt1$. The metric

$$g=-\bigl[du+(1-|u|^\lambda)dx\bigr]^2+dx^2$$

is continuous, Lorentzian with determinant $-1$, and has future time orientation $\partial_u$. In a sufficiently small neighborhood of the origin, the future right-going null equation is

$$\frac{du}{dx}=|u|^\lambda.$$

It admits a curve that waits on $u=0$ until any $x=c\ge0$, then leaves:

$$u_c(x)= \begin{cases} 0,&0\le x\le c,\\ \bigl[(1-\lambda)(x-c)\bigr]^{1/(1-\lambda)},&x\ge c. \end{cases}$$

Consequently every point with $x\gt0$ and $0\lt u\lt[(1-\lambda)x]^{1/(1-\lambda)}$ is causally reachable from the origin. A future timelike curve, however, satisfies $du\gt u^\lambda dx$ in $u\gt0$. Integration gives $u^{1-\lambda}\gt(1-\lambda)x$, excluding that open wedge. Allowing a curve to move left does not defeat the inequality: the other side of the future cone also forces $du\gt0$ there. This is the explicit bubbling mechanism in Chruściel–Grant 2012, Example 1.11.

To see the failure of push-up, first follow $u=0$ to $x=c\gt0$, then a short vertical timelike segment. Choose its endpoint inside the wedge. The concatenated curve is causal and has a timelike segment, but no entirely timelike curve from the origin reaches its endpoint. The smooth push-up inference fails because the null ODE at $u=0$ is nonunique.

Global hyperbolicity at continuous regularity

A robust definition combines causal control with compactness: the spacetime is non-totally imprisoning, and every diamond

$$J^+(p)\cap J^-(q)$$

is compact. Under the continuous-metric theory, this gives a limit-curve theorem strong enough to obtain length-maximizing causal curves between causally related points. Sämann establishes the relevant equivalences and maximizer results in Sämann 2016, §§3–4.

The maximizer is initially a causal Lipschitz curve. Calling it a geodesic requires an additional regularity theorem. At smooth regularity a maximizing timelike curve is a timelike geodesic and a maximizing null curve is a null pregeodesic; below $C^{1,1}$, those conclusions require care.

Global hyperbolicity is stable under suitable cone-widening perturbations, but it is not a consequence of pointwise Lorentzian signature alone. Compact diamonds can fail because curves escape to infinity, while non-imprisonment can fail in compact regions.

What survives and what fails

Object	Continuous metric	Additional regularity commonly needed
Timelike and causal Lipschitz curves	Defined from the cone field	None beyond continuity
Lorentzian length and time separation	Defined	Compactness for existence of maximizers
Smooth push-up identities	Can fail through bubbling	Locally Lipschitz control for standard recovery results
Classical Christoffel symbols	Not defined	$C^1$ metric for continuity
Unique geodesic flow	Not guaranteed	Typically $C^{1,1}$

The next page imposes weak differentiability sufficient to define curvature as a distribution. That requirement is logically independent of many curve-based causal definitions.

Solved diagnostic: cones without a connection

For the continuous metric below, derive the causal-speed inequality and explain why it survives even when no classical geodesic equation exists.

Solution

Consider the continuous metric

$$g=-dt^2+a(t,x)^2dx^2,$$

where $a$ is continuous and strictly positive. Parametrize a future curve by $t$. It is causal exactly when

$$\left|\frac{dx}{dt}\right|\le\frac{1}{a(t,x)}$$

almost everywhere. This inequality defines causal curves even if $a$ is nowhere differentiable. By contrast, the geodesic equation contains derivatives of $a$ and is not classically defined. The example cleanly separates cone causality from connection-based dynamics.

References

• Chruściel, Piotr T., and James D. E. Grant. “On Lorentzian Causality with Continuous Metrics.” Classical and Quantum Gravity 29 (2012): 145001. arXiv:1111.0400.
• Sämann, Clemens. “Global Hyperbolicity for Spacetimes with Continuous Metrics.” Annales Henri Poincaré 17 (2016): 1429–1455. DOI.