Causal cone domination in adapted charts and Lipschitz-graph bounds for causal curves

(Projection and full adapted shear.) Choose [L1]'s adapted chart centered at $p$ and write $e_a={\mathrm{\partial }}_{y^a}{\mathrm{\mid }}_p$, with $e_0=n_p$. The subspace $E=\mathrm{span}(e_1,e_2,e_3)$ complements $\mathbb{R}{n}_p$, so [L2]'s projection restricts to an isomorphism from $E$ onto $n_p^{\perp }$. In particular $T(n_p)=0$, not $n_p$. Put $w_k=e_k+g_p(e_k,n_p)n_p$. Finite Gram-Schmidt yields an orthonormal basis $v_i={\sum }_k{C}^k{}_i{w}_k$ of the rest space with $C$ invertible. Define new coordinates by the inverse linear transformation $y^k={\sum }_i{C}^k{}_i{x}^i$ and $y^0=x^0+{\sum }_i{a}_i{x}^i$, where $a_i={\sum }_k{C}^k{}_i{g}_p(e_k,n_p)$. Its block-triangular matrix is invertible. The chain rule gives ${\mathrm{\partial }}_{x^0}={\mathrm{\partial }}_{y^0}=n$ throughout the chart and ${\mathrm{\partial }}_{x^i}{\mathrm{\mid }}_p=a_i{n}_p+{\sum }_k{C}^k{}_i{e}_k=v_i$. Hence $g_{00}\equiv -1$, $g_{0i}(p)=0$ and $g_{ij}(p)={\delta }_{ij}$. A coordinate ball centered at zero inside this chart gives the adapted properties of (a).

(Positive time component on a compact temporal locus.) Let $K_0\subset P$ be compact and nonempty. In the $z$ coordinate trivialization define $\mathcal{Z}=\{(q,v):q\in K_0,\mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}}=1,g_q(v,v)\le 0,g_q(v,n_q)\le 0\}$. It is closed in the compact product $K_0\times S^3$ by continuity. At each such vector the last inequality is strict by [L3], so $\mathcal{Z}$ consists exactly of normalized nonzero future causal vectors and is nonempty. For $w=\mathrm{\nabla }{z}^0$, set $\lambda =\sqrt{-g(w,w)}>0$ and $m=-w\mathrm{/}\lambda$, a future unit timelike vector on $K_0$. The exact sign is $v^0=d{z}^0(v)=g(w,v)=-\lambda g(m,v)>0$ by [L3]. Thus the continuous time-component function has a positive compact minimum $\beta ={\min }_{\mathcal{Z}}{v}^0>0$. No minimum over points where the gradient may be null is taken.

(Uniform compact cone bounds; (e).) For any nonzero future causal $v$ based in $K_0$, normalization $u=v\mathrm{/}\mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}}$ puts $(q,u)$ in step 1.2's compact cone. Consequently $v^0\ge \beta \mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}}$ and $\mathrm{\mid }{v}_s\mathrm{\mid }\le \mathrm{\mid }v{\mathrm{\mid }}_{\mathrm{e}}\le {\beta }^{-1}{v}^0$. Choose $L=\max \{1,{\beta }^{-1}\}$, giving all bounds of (e). If $K_0$ is empty, take $L=1$. If the gradient is past timelike on the whole closure of $U_0⋐U^{\mathrm{\prime }}$, apply this result to $K_0=\overline{U_0}$. For merely pointwise temporal $U_0$, openness from [L4] provides local precompact neighbourhoods with closure in $P$, while each compact subset of $U_0$ is already an admissible $K_0$. This proves exactly the stated local and compact uniform scope.

(One adapted chart with uniform outer and inner cones.) At $p$ in step 1.1's chart, $\mathrm{\nabla }{x}^0=-n_p$. By [L4], choose a closed coordinate ball inside the chart on which this gradient remains past timelike. Also shrink the ball so the coordinate Euclidean operator norm of $g(q)-\eta$ is less than $1\mathrm{/}4$ on its closure. The compact bound of step 2.1 supplies $K\ge 1$ for both the full spatial norm and full coordinate norm on this ball, proving (a)-(b). If $\mathrm{\mid }e{\mathrm{\mid }}_{\mathrm{e}}=1$ and ${\eta }_{16}(e,e)\le 0$, then $\mathrm{\mid }{e}_s{\mathrm{\mid }}^2\le (e^0{)}^2\mathrm{/}16$, $(e^0{)}^2\ge 16\mathrm{/}17$, and $\eta (e,e)\le -15\mathrm{/}17$. Therefore $g_q(e,e)\le -15\mathrm{/}17+1\mathrm{/}4<0$ throughout the ball. Positive scaling proves every nonzero ${\eta }_{16}$-causal vector is $g$-timelike there. All properties hold on the same open coordinate ball $U=x^{-1}(B_r)$.

(Absolutely continuous graph bound; (c).) Assume the declared Countable and Dependent Choice premises and let $\gamma :I\to U$ be future causal. On any compact $[s,t]\subset I$, smooth composition and [L5] make its coordinate functions absolutely continuous; by step 3.1, their almost-everywhere derivatives satisfy $v^0>0$ and $\mathrm{\mid }{v}_s\mathrm{\mid }\le K{v}^0$. The fundamental theorem then gives $x^0(\gamma (t))-x^0(\gamma (s))={\int }_s^t{v}^0>0$. For the spatial vector, the triangle inequality for its component integrals gives $\mathrm{\mid }{x}_s(\gamma (t))-x_s(\gamma (s))\mathrm{\mid }\le {\int }_s^t\mathrm{\mid }{v}_s\mathrm{\mid }\le K{\int }_s^t{v}^0$. These are the claimed bounds. Strict increase and continuity make $x^0(\gamma (I))$ an interval and give a unique spatial value for each of its time values; the increment estimate makes that function $K$-Lipschitz in the Euclidean norm. No smooth inverse time parametrization is assumed.

(Chronology from the flat minorant; (d).) For $q,q^{\mathrm{\prime }}\in U$ satisfying the stated strict ${\eta }_{16}$ inequality, put $d=x(q^{\mathrm{\prime }})-x(q)$. If $d^0>0$, the straight coordinate segment $x(q)+td$, $0\le t\le 1$, stays in the convex ball and its nonzero tangent is $g$-timelike by step 3.1. Its positive $d{x}^0$ component makes it future-directed, since a past timelike vector pairs negatively with the past timelike gradient $\mathrm{\nabla }{x}^0$ and so has negative $d{x}^0$ component. Its coordinate representative is affine, hence absolutely continuous directly; in any other chart it is smooth and has bounded derivative on each sufficiently small compact segment, giving a Lipschitz and hence absolutely continuous representative, with finite subdivision handling a larger compact segment. Thus it belongs to the stated causal-curve class and is a smooth future timelike curve from $q$ to $q^{\mathrm{\prime }}$. Reversing the segment when $d^0<0$ proves the other chronological direction. This establishes the rest of (d), without an absolutely continuous integration argument.

The chart and cone bounds are steps 1.1 and 3.1, the causal graph estimate is step 4.1, the minorant chronology is steps 3.1 and 4.2, and the general compact temporal bounds are step 2.1. They prove (a)-(e), with exactly the declared choice uses and temporal compactness hypotheses.