Causal Curves and Time Orientation

A causal curve is a possible track for matter or light once a time orientation has chosen which half of every light cone is future. That sentence contains two logically separate ingredients: the Lorentzian metric supplies the cones, while a global choice of cone component supplies the arrow “future-directed.” Keeping them separate prevents coordinate time, proper time, and causal orientation from being confused.

Required background. Lorentzian Metrics and Causal Cones supplies the sign convention and pointwise classification of tangent vectors; Smooth Manifolds and Tensors supplies tangent spaces, vector fields, and differentiable curves. Helpful background. Geodesics and Jacobi Fields clarifies the extra differential equation that distinguishes geodesics from general causal curves.

Cones and a time orientation

Let $(M,g)$ be a connected Lorentzian spacetime with signature $(-,+,+,+)$. A nonzero tangent vector $v\in T_pM$ is timelike, null, or spacelike according as

$$g(v,v)<0,\qquad g(v,v)=0,\qquad g(v,v)>0.$$

The timelike vectors at $p$ form two connected cones. The metric alone does not label either one future. A time orientation is a continuous choice of one component at every point. Equivalently, it is the choice of a continuous nowhere-vanishing timelike vector field $T$. With $T$ declared future-directed, a causal vector $v$ is future-directed precisely when

$$g(v,T)<0,$$

and past-directed when the sign is positive. The inequality follows from the adopted signature: in an orthonormal frame with both vectors future timelike, their Lorentzian inner product is negative. Replacing $T$ by another future timelike field does not change the chosen cone component.

Time orientability is a global condition. Every sufficiently small coordinate neighborhood admits a consistent local choice, but those choices need not glue around a nontrivial loop. Passing to the time-orientation double cover repairs the choice at the price of changing the spacetime. Thus “future” cannot in general be inferred from the sign of an arbitrarily named coordinate $x^0$.

Curves that respect the cones

For a piecewise regular $C^1$ map $\gamma:I\to M$, the elementary definition says that $\gamma$ is future-directed timelike, null, or causal when its tangent on every regular piece has the corresponding character and orientation. Regular means that the tangent is nonzero; constant curves are not classified as timelike by a vacuous tangent test. “Causal” includes both timelike and null tangents. A null curve is more restrictive: all its tangents on regular pieces must be null. A curve may change continuously between timelike and null segments and remain causal, but then it is not a null curve.

Modern causal theory normally admits locally Lipschitz curves. Such a curve is differentiable almost everywhere; in a regular parametrization it is future causal when that almost-everywhere tangent is nonzero and lies in the future causal cone. One can use arc length for an auxiliary positive-definite metric to remove artificial pauses. With appropriate common parametrizations, this curve class supports the limit-curve theorems needed for compactness arguments. For a smooth metric, it gives the same endpoint causal relation as piecewise-smooth curves. Reflexivity of that relation is added separately by declaring each event related to itself. Precise regularity choices and their consequences are developed in Minguzzi 2019, §§2.1–2.4.

An orientation-preserving reparametrization leaves causal character unchanged. If $s=s(\lambda)$ is increasing, then

$$\frac{d\gamma}{d\lambda}=\frac{ds}{d\lambda}\frac{d\gamma}{ds}$$

lies in the same cone. A decreasing reparametrization reverses future and past. No preferred parameter is part of the definition of a causal curve. Proper time is available along timelike curves, while an affine parameter is additional structure for a null geodesic. An arbitrary null curve need not be geodesic, and an arbitrary timelike causal curve can represent an accelerated observer.

The distinction matters operationally. A massive point particle has a future timelike worldline, an ideal light ray follows a future null geodesic in geometric optics, and the boundary of causal influence is often generated by null geodesic segments. The cone condition says what propagation is kinematically allowed; the geodesic equation says what freely falling propagation does.

Minkowski checks

In two-dimensional Minkowski spacetime,

$$ds^2=-dt^2+dx^2,$$

choose $T=\partial_t$. A tangent $\dot\gamma=(\dot t,\dot x)$ is future causal exactly when

$$\dot t>0, \qquad \lvert\dot x\rvert\leq \dot t.$$

The strict inequality gives a timelike tangent; equality gives a null tangent. For example, $\gamma(\lambda)=(\lambda,v\lambda)$ is future timelike for $\lvert v\rvert<1$ and future null for $\lvert v\rvert=1$. The curve $(\lambda,\sin\lambda/2)$ is timelike wherever it is regular, even though it is not a geodesic. Conversely, $(\lambda,2\lambda)$ has increasing coordinate time but is spacelike. Increasing $t$ is therefore not, by itself, the causal test.

For any future causal curve between endpoints $p$ and $q$ in this chart,

$$t(q)-t(p)\geq \lvert x(q)-x(p)\rvert.$$

This follows by integrating $\lvert\dot x\rvert\leq\dot t$. It is the simplest global check that a cone-respecting tangent produces an endpoint inside or on the future light cone. The converse also holds in flat spacetime by the straight segment, but it can fail in a restricted domain when that segment leaves the manifold. Endpoint relations therefore depend on the whole spacetime, not only on the two tangent cones at the endpoints.

Regularity and common pitfalls

The basic cone definition survives for a continuous Lorentzian metric, but familiar “push-up,” boundary, and maximizing results can require stronger regularity. Below local Lipschitz regularity, causal bubbling can separate notions that coincide in smooth spacetimes. A theorem about $I^+$, $J^+$, or limiting curves must therefore state its metric regularity rather than silently importing the smooth theory; Minguzzi and Sánchez 2008, §2, pp. 3–17 gives the smooth hierarchy used in the next pages.

Three errors recur. First, a causal curve is not synonymous with a geodesic: acceleration changes the tangent along a curve without moving it outside the cone. Second, a null tangent does not give positive proper time; the Lorentzian length of a null segment is zero, so its natural geodesic parameter is affine rather than proper time. Third, time orientation does not guarantee causality. It consistently labels future cones, but a future-directed causal curve may still close in a spacetime with causal pathology.

Exercise

Let $\gamma(\lambda)=(a\lambda,\lambda^2)$ in two-dimensional Minkowski spacetime, restricted to $-L\leq\lambda\leq L$, with $a>0$. Determine when the entire curve is future timelike, future causal, or neither.

Solution

The tangent is $(a,2\lambda)$. It is future-directed because $\dot t=a>0$. Its squared norm is

$$g(\dot\gamma,\dot\gamma)=-a^2+4\lambda^2.$$

It is timelike at a point when $2\lvert\lambda\rvert<a$ and causal when $2\lvert\lambda\rvert\leq a$. Therefore the entire restricted curve is future timelike if $2L<a$, future causal with null tangents at the endpoints if $2L=a$, and not causal on the whole interval if $2L>a$. No geodesic equation was used: this is purely a cone test.

With admissible curves defined, Chronological and Causal Futures turns their existence into the relations $I^\pm$ and $J^\pm$.

References

• Minguzzi, Ettore, “Lorentzian Causality Theory”, Living Reviews in Relativity 22, article 3, §§2.1–2.4, 2019.
• Minguzzi, Ettore, and Miguel Sánchez, “The Causal Hierarchy of Spacetimes”, in Recent Developments in Pseudo-Riemannian Geometry, §2, pp. 3–17 in the arXiv version, European Mathematical Society, 2008.