Causal curves, chronological and causal reachability, and inextendibility

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), with the pointwise causal classification of Timelike, null, spacelike and causal vectors of a Lorentz vector space and the equivalence of "future-directed" with "pairing negatively with $N$" supplied by Classification of future and past causal vectors by a timelike field. All curves below take values in $M$; the physical reading of the model is not part of this definition.

(a) The curve class. Let $I\subseteq \mathbb{R}$ be an interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) and let $\gamma :I\to M$ be a continuous map, where $I$ has its relative topology. A chart representative on a parameter neighborhood $J\subseteq I$ is $x\circ \gamma {\mathrm{\mid }}_J$, where $(U,x)$ is a smooth chart and $\gamma (J)\subseteq U$ (Smooth manifolds and their smooth charts, $C^r$ and smooth maps between smooth manifolds). The map $\gamma$ is locally absolutely continuous when for each $t\in I$ there are $\delta >0$ and a chart $(U,x)$ such that $\gamma (I\cap (t-\delta ,t+\delta ))\subseteq U$ and every coordinate of $x\circ \gamma$ is absolutely continuous on every compact subinterval of $I\cap (t-\delta ,t+\delta )$ in the sense of Absolute continuity on a compact interval. This includes one-sided neighborhoods at included endpoints. Equivalently, for every compact subinterval $[a,b]\subseteq I$ and every chart $(U,x)$ with $\gamma ([a,b])\subseteq U$, the coordinates of $x\circ \gamma$ are absolutely continuous on $[a,b]$. The equivalence and independence of the chart and smooth atlas are proved in Chart independence, concatenation, reparametrization and transitivity for causal curves ↗ clause (a); continuity is part of both formulations.

(b) Velocity and causal curves. Let $\gamma$ be locally absolutely continuous and let $t$ be an interior point of $I$ at which some — equivalently every — chart representative is differentiable. The velocity ${\gamma }^{\mathrm{\prime }}(t)\in T_{\gamma (t)}M$ is the derivation whose components in a chart $(U,x)$ around $\gamma (t)$ are ${\gamma }^{\mathrm{\prime }}(t)={\sum }_{a=0}^3\frac{d}{dt}(x^a\circ \gamma )(t){\mathrm{\partial }}_a{\mathrm{\mid }}_{\gamma (t)}$ (The velocity derivation of a smooth curve, Derivations at a point and the tangent space); it is independent of the chart by the chain rule, as recorded in Chart independence, concatenation, reparametrization and transitivity for causal curves ↗ clause (b). A future causal curve is a locally absolutely continuous $\gamma$ such that the set of $t\in I$ for which ${\gamma }^{\mathrm{\prime }}(t)$ exists, is nonzero and is future causal is of full measure in $I$ (its complement meets every compact subinterval of $I$ in a set of Lebesgue measure zero). A future timelike curve (also called future chronological) is defined in the same way with "future causal" replaced by "future timelike". Past-directed versions are defined with $g_p(v,N_p)>0$. Since a future causal nonzero vector is nonzero by definition, the velocity is required to be nonzero at almost every parameter value; in particular no nondegenerate subinterval of $I$ can be mapped to a single point: a constant chart representative has derivative zero everywhere in its interior, contradicting the full-measure nonzero-velocity requirement.

(c) Reachability. For events $p,q\in M$ define:

• $p\le q$, read "$q$ is in the future of $p$", when $p=q$, or when there are $a<b$ and a future causal curve $\gamma :[a,b]\to M$ with $\gamma (a)=p$ and $\gamma (b)=q$;
• $p\ll q$, read "$p$ chronologically precedes $q$", when there are $a<b$ and a future timelike curve $\gamma :[a,b]\to M$ with $\gamma (a)=p$ and $\gamma (b)=q$.

The causal future and chronological future of a subset $S\subseteq M$ are $J^{+}(S):={\bigcup }_{p\in S}{J}^{+}(p),I^{+}(S):={\bigcup }_{p\in S}{I}^{+}(p),$ where $J^{+}(p):=\{q\in M:p\le q\}$ and $I^{+}(p):=\{q\in M:p\ll q\}$; past versions are $J^{-}(S):=\{p:\mathrm{\exists }q\in S,q\in J^{+}(p)\}$ and $I^{-}(S):=\{p:\mathrm{\exists }q\in S,p\ll q\}$. By the displayed definition $p\in J^{+}(p)$ for every $p$. Whether $p\in I^{+}(p)$ is a property of the model, not a convention: $p\ll p$ holds exactly when there are $a<b$ and a future timelike curve $\gamma :[a,b]\to M$ with $\gamma (a)=\gamma (b)=p$, that is, exactly when a closed future timelike curve through $p$ exists. No irreflexivity of $\ll$ is asserted, and closed timelike curves are not excluded by the definition.

(d) Inextendibility. A locally absolutely continuous causal curve $\gamma :I\to M$ is future-inextendible when $\gamma$ does not converge to any point of $M$ as its parameter tends to $\sup I$, that is, there is no $x\in M$ with $\gamma (t)\to x$ as $t\to \sup I$; in particular a curve whose parameter range is unbounded above is future-inextendible exactly when it does not converge as $t\to +\mathrm{\infty }$. Past-inextendible is defined dually with $\inf I$, and $\gamma$ is inextendible when it is both. This is the convention of Minguzzi's Definition 2.16; for causal curves parametrized by arclength of a complete auxiliary Riemannian metric it agrees with "the domain cannot be extended", and a causal curve that converges at an endpoint can be extended by a short geodesic segment, as proved in Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal ↗ clauses (a) and (b). A causal curve passes through $p$ when $p\in \gamma (I)$.

(e) Scope. The definitions are mathematical. They record no postulate about nature, no measurement and no dynamical law; the physical reading of $(M,g,N)$ and of the causal structure as the model's signalling structure is adopted in the framework postulates and is not asserted here.