Compact timelike-length bounds restrict to open subdomains

Use [F11] and [F4] to equip $B$ and $W$ with their restricted smooth structures. A chart of the ambient manifold restricts first to $B$ and then to $W$; the resulting coordinate functions and transition maps are unchanged wherever defined. For the inclusion $W\hookrightarrow B$ the coordinate map is the identity on an open set, so its tangent map is the identity in these coordinates by [F12]. Consequently the restrictions of $g$ and $N$ have the same coefficient functions, signature, regularity and time orientation. This supplies the smooth-manifold and tangent identifications used below.

Fix $p$ and apply [F13] to the four-dimensional space $T_{p}B$ of [F11]. In an orthonormal basis write $N_p=(n^0,n_s)$, reversing the time basis vector if necessary so that $n^0>\mathrm{\mid }{n}_s\mathrm{\mid }$. The component containing $N_p$ is then $T_{+}$ of [F13]. If $v\in \mathrm{cl}(T_{+})\setminus \{0\}$, then $v^0\ge \mathrm{\mid }{v}_s\mathrm{\mid }$ and $v^0>0$, so $g_p(v,N_p)=-v^0{n}^0+{\sum }_{i=1}^3{v}^i{n}^i\le -v^0(n^0-\mathrm{\mid }{n}_s\mathrm{\mid })<0$ by the finite Cauchy-Schwarz inequality in [F13]. On $\mathrm{cl}(T_{-})\setminus \{0\}$ the pairing is positive by applying this calculation to $-v$. Every nonzero causal vector is in exactly one of those two closed cones by [F1] and [F13], proving the equivalence asserted in the Statement. All pairings are intrinsic evaluations of bilinear forms. Their coordinate expressions vary continuously with $p$ and the vector components because $g$ and $N$ have continuous coefficients. Thus the strict future inequality defines an open condition, restricted to the causal vectors, at precisely the stated regularities. The pointwise argument chooses a basis only at the fixed point; no field of bases, smooth orientation field, or stronger choice principle is used.

Let $t\in (a,b)$, put $p=c(t)$, and suppose $y\circ c$ is differentiable at $t$ in one smooth chart $y$ about $p$. For any other smooth chart $x$ about $p$, continuity of $c$ permits restriction to a parameter neighborhood whose image lies in the overlap. The transition $x\circ y^{-1}$ is smooth, so the Euclidean chain rule [F10] gives differentiability of $x\circ c$ at $t$ and $(x^i\circ c{)}^{\mathrm{\prime }}(t)={\sum }_{\alpha =1}^4{A}_{\alpha }^i(y^{\alpha }\circ c{)}^{\mathrm{\prime }}(t)$, where $A_{\alpha }^i=\mathrm{\partial }(x^i\circ y^{-1})\mathrm{/}\mathrm{\partial }{y}^{\alpha }$ is evaluated at $y(p)$. Apply [F12] to the smooth identity map on the overlap, represented in these two charts, to obtain ${\mathrm{\partial }}_{y^{\alpha }}{\mathrm{\mid }}_p={\sum }_i{A}_{\alpha }^i{\mathrm{\partial }}_{x^i}{\mathrm{\mid }}_p$. The coordinate basis in [F11] makes each proposed velocity a derivation in $T_{p}B$, and substitution gives ${\sum }_i(x^i\circ c{)}^{\mathrm{\prime }}(t){\mathrm{\partial }}_{x^i}{\mathrm{\mid }}_p={\sum }_{\alpha }(y^{\alpha }\circ c{)}^{\mathrm{\prime }}(t){\mathrm{\partial }}_{y^{\alpha }}{\mathrm{\mid }}_p$. Interchanging the two charts proves equivalence of derivative existence as well as equality of the vectors. Thus the Statement defines an intrinsic velocity at every interior differentiability time. Here [F12] concerns the smooth identity map and [F10] concerns Euclidean maps differentiable at the one time under consideration.

Fix an eligible curve $c$. By local coordinate Lipschitz continuity, each parameter value has a relative open interval on which the image is in one chart and its coordinates satisfy one Lipschitz bound. Shrink such intervals to closed relative intervals $J_k$ whose relative interiors still cover $[a,b]$ and on each of which that chart and bound remain valid. By [F8] finitely many suffice. For each component $x^i$ of its coordinate curve on $J_k$, extend $x^i$ to $\mathbb{R}$ by keeping it constant beyond the two endpoints of $J_k$. The extension ${\tilde{x}}^i$ is continuous and Lipschitz with the same bound. For integers $n\ge 1$, the difference quotient $q_n^i(t)=n({\tilde{x}}^i(t+1\mathrm{/}n)-{\tilde{x}}^i(t))$ is continuous, hence Borel measurable by [F6], and Lebesgue measurable by [F5]. Its limsup $v_k^i$ is measurable by [F6] and finite everywhere because $\mathrm{\mid }{q}_n^i\mathrm{\mid }$ is bounded by that Lipschitz constant. At every interior parameter where $x^i$ is differentiable, $v_k^i=(x^i{)}^{\mathrm{\prime }}$. Choose a measurable null set $E$ containing the endpoints of these finitely many intervals and all exceptional times for the velocity and timelikeness conditions; such a set exists by [F7]. No differentiability theorem is being inferred from the Lipschitz definition: derivative existence almost everywhere is part of the stated curve class. The case $a=b$ has the zero integrand and zero integral.

On $J_k$, every coefficient $g_{ij}(c(t))$ in its chart is continuous, hence measurable by [F5] and [F6]. Combine these coefficients with the measurable finite coordinates $v_k^i$ into one measurable finite-dimensional map, using [F6]. The real function of these coefficients and coordinates given by $h_k(t)=\sqrt{\max \{0,-{\sum }_{i,j=1}^4{g}_{ij}(c(t))v_k^i(t)v_k^j(t)\}}$ is measurable: the map performing the displayed arithmetic, maximum and square root is continuous, so [F6] applies to its composition. Make a disjoint Borel partition by assigning each point to the first interval $J_k$ that contains it. Define $h=h_k$ on its assigned piece outside $E$ and $h=0$ on $E$. Each piece is measurable and there are finitely many, so preimages under this piecewise definition are finite unions of measurable sets. Thus $h$ is nonnegative measurable. Outside $E$, step 1.3 identifies the coordinate derivatives with the velocity defined in the Statement, and $g(\dot{c},\dot{c})<0$, so $h=\sqrt{-g(\dot{c},\dot{c})}$. For overlapping coordinates $x,y$, put $A_{\alpha }^i=\mathrm{\partial }{x}^i\mathrm{/}\mathrm{\partial }{y}^{\alpha }$. By [F12] the coordinate tangent bases satisfy ${\mathrm{\partial }}_{y^{\alpha }}={\sum }_i{A}_{\alpha }^i{\mathrm{\partial }}_{x^i}$. Bilinearity in [F11] gives $g_{\alpha \beta }^{(y)}={\sum }_{i,j}{A}_{\alpha }^i{A}_{\beta }^j{g}_{ij}^{(x)}$. At each differentiability time outside $E$, the chain rule [F10] gives ${\dot{x}}^i={\sum }_{\alpha }{A}_{\alpha }^i{\dot{y}}^{\alpha }$, hence ${\sum }_{\alpha ,\beta }{g}_{\alpha \beta }^{(y)}{\dot{y}}^{\alpha }{\dot{y}}^{\beta }={\sum }_{i,j}{g}_{ij}^{(x)}{\dot{x}}^i{\dot{x}}^j$. This derives the tensor transformation rule and the scalar's chart independence without differentiating $g$ or $N$. This establishes the claimed measurable length integrand with the stated null-set convention.

Changing the chart intervals, their assignment, or the measurable exceptional null set gives another nonnegative measurable representative $h^{\mathrm{\prime }}$ equal to $h$ outside a measurable null set $Z$: take the union of the two exceptional sets and the finite interval endpoints. If $s$ is a nonnegative simple minorant of $h$, the function equal to $s$ off $Z$ and zero on $Z$ is a simple minorant of $h^{\mathrm{\prime }}$ with the same simple integral; its removed part is supported on $Z$ and has integral zero by [F9]. Taking the supremum over $s$ in [F3] gives $\int h\le \int h^{\mathrm{\prime }}$. Interchanging the representatives gives the reverse inequality. Hence the length is independent of all these auxiliary choices, including the measurable null set used for the zero values.

If $C\subset W$ is compact in $W$, the topology induced on $C$ from $W$ is the topology induced from $B$: both consist of the same iterated intersections of open sets with $C$. Thus $C$ is compact in $B$ by the intrinsic convention [F2]. The restricted atlas in step 1.1 gives exactly the same coordinate representatives of a curve into $C$ in either domain. Local coordinate Lipschitz continuity may be tested in this atlas: changing to any compatible chart composes with a smooth transition, which is locally Lipschitz by [F10]. The product of its local Lipschitz bound and that of the coordinate curve is a bound for the new coordinate representative. By step 1.2 the future condition is the pointwise pairing inequality at the given regularities. The velocity defined in the Statement is unchanged by this identification, since its coordinate derivatives and coordinate tangent bases agree by steps 1.1 and 1.3. The inequalities $g(\dot{c},N)<0$ and $g(\dot{c},\dot{c})\le -{ϵ}_c$ are unchanged under the identity tangent map. Choose the same chart intervals and exceptional set in step 2.1 for both domains; they give identical $h$ and therefore identical length. Step 3.1 removes dependence on that choice.

Apply the assumed bound $L_C$ in $B$ to each such curve into $C$. Step 4.1 shows that it is exactly an eligible curve in $B$ with the same length and the same individual constant ${ϵ}_c$. The bound $L_C$ is therefore common to every eligible curve in $C$ in $W$, with no extra uniformity imposed on ${ϵ}_c$. This proves the restriction property.