Length monotonicity of the squared displacement: once a future causal curve in a normal neighborhood has a future timelike position vector, the displacement radius grows at least as fast as the accumulated Lorentzian length

Assume Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) for the absolutely continuous causal-curve calculus and the supplied causal results used below.

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $p\in M$ and let $U_p={\exp }_p(W_p)$ be a normal neighborhood of $p$ as in Normal neighborhoods, convex normal sets, the position vector and the squared-displacement function clause (a). Let $\sigma :[a,b]\to U_p$ be a future causal curve in the class of Causal curves, chronological and causal reachability, and inextendibility (locally absolutely continuous, with future causal nonzero velocity almost everywhere), and write $v(t):={\exp }_p^{-1}(\sigma (t)),P(t):=d({\exp }_p{)}_{v(t)}(v(t)),Q(t):=g_p(v(t),v(t))$ for $t\in [a,b]$. Suppose there is $t^{\ast }\in [a,b]$ such that $v(t^{\ast })$ is future timelike. For any locally absolutely continuous future causal curve $\eta :[u,v]\to M$, define its Lorentzian length by $L_g(\eta ):={\int }_u^v\sqrt{-g_{\eta (s)}({\eta }^{\mathrm{\prime }}(s),{\eta }^{\mathrm{\prime }}(s))}ds,$ where the derivative is taken almost everywhere and the integral is the Lebesgue integral. This integral is finite on a compact parameter interval: the curve image is compact, a finite chart subdivision has bounded metric coefficients on each piece, and each coordinate derivative is integrable by local absolute continuity. The radicand is nonnegative almost everywhere by causality. On piecewise-$C^1$ future timelike curves this agrees with the length in Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. Then:

(a) Propagation. For every $t\in [t^{\ast },b]$ with $\sigma ([t^{\ast },t])\subseteq U_p$, the vector $v(t)$ is future timelike; consequently $Q(t)<0$, and $R(t):=\sqrt{-Q(t)}>0$.

(b) Length inequality. For all $t^{\ast }\le t_1<t_2\le b$ with $\sigma ([t_1,t_2])\subseteq U_p$, $R(t_2)-R(t_1)\ge L_g\left(\sigma {\mathrm{\mid }}_{[t_1,t_2]}\right)\mathrm{.}$ In particular $R$ is nondecreasing on $[t^{\ast },b]$.

(c) Derivative formula. The map $t\mapsto Q(t)$ is absolutely continuous and $\frac{d}{dt}Q(t)=2g_{\sigma (t)}(P(t),{\sigma }^{\mathrm{\prime }}(t))$ at almost every $t\in [a,b]$.

• Why the timelike hypothesis at a single time suffices. The reverse Cauchy–Schwarz inequality is applied with the future timelike vector $P(t)$, whose future-directedness is granted by Gauss' Lemma and Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed; the direction cannot change while $Q<0$, and $Q$ can only decrease, so one timelike value propagates forward as long as the curve remains in the normal neighborhood.

• Sharpness. Equality in (b) holds along radial reparametrizations of the geodesic from $p$; the inequality is the local maximality of geodesics in its integrated form and is used in Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic.