Lorentzian length as a refined sum of local separations, and upper semicontinuity of the length under uniform convergence

(Fixed local branches over the compact image.) Choose finitely many adapted charts $(V_j,x_j)$ and open sets $A_j$ covering the image of $\alpha$, with compact closures $K_j\subset V_j$. Shrink these sets using the joint exponential inverse of [L2], so that for each $j$ there is ${\rho }_j>0$ with the following properties. The joint exponential is defined and injective on all $(p,v)$ with $p\in K_j$ and $\mathrm{\mid }v{\mathrm{\mid }}_{x_j}\le {\rho }_j$; its smooth inverse branch $v_j(p,q)$ is defined near $K_j\times K_j$ and $\mathrm{\mid }{v}_j(p,q){\mathrm{\mid }}_{x_j}<{\rho }_j\mathrm{/}2$ there. Its radial geodesics stay in $V_j$ for $0\le \lambda \le 1$. These choices follow by taking a closed product inside the open inverse-function domain near each diagonal point, then shrinking the coordinate ball: $v_j(p,p)=0$, and the geodesic with zero velocity is constant. Smoothness on these closed products gives uniform finite bounds, with one constant $C$ enlarged over the finite family, such that $\mathrm{\mid }{v}_j(p,q){\mathrm{\mid }}_{x_j}\le C\mathrm{\mid }{x}_j(q)-x_j(p)\mathrm{\mid },$ $v_j(p,q)=x_j(q)-x_j(p)+R_j(p,q),\mathrm{\mid }{R}_j(p,q)\mathrm{\mid }\le C\mathrm{\mid }{x}_j(q)-x_j(p){\mathrm{\mid }}^2,$ where the tangent vector is expressed in the coordinate basis at $p$. The second estimate follows by integrating the second-variable derivative along the coordinate segment twice: its value at $q=p$ is the identity and its derivative is bounded on the compact product. Likewise, bounded first derivatives of the geodesic evaluation give $\mathrm{\mid }{x}_j({\exp }_p(\lambda v_j(p,q)))-x_j(p)\mathrm{\mid }\le C\mathrm{\mid }{v}_j(p,q)\mathrm{\mid }(0\le \lambda \le 1)\mathrm{.}$ A Lebesgue number for the finite parameter cover ${\alpha }^{-1}(A_j)$ ensures that every sufficiently short parameter subarc lies in some $A_j$.

(The selected inverse cannot leave this branch.) Let a short subarc $\alpha ([u,v])$ lie in $A_j$, and suppose also that it lies in any strictly convex normal ball $U$, with $p=\alpha (u)$. The selected lift $z(t):={\exp }_{p,U}^{-1}(\alpha (t)),u\le t\le v,$ is continuous by [L2], with $z(u)=0$. Here the subscript $U$ specifies the inverse selected by that normal ball, not a different exponential map. If $\mathrm{\mid }z(t){\mathrm{\mid }}_{x_j}$ ever reached ${\rho }_j$, let $t_{\ast }$ be its first such parameter. At $t_{\ast }$ both $z(t_{\ast })$ and $v_j(p,\alpha (t_{\ast }))$ are in the fixed injectivity domain of step 1.1 and have the same exponential image. They must agree, contradicting the strict bound $\mathrm{\mid }{v}_j\mathrm{\mid }<{\rho }_j\mathrm{/}2$. Consequently $\mathrm{\mid }z(t)\mathrm{\mid }<{\rho }_j$ throughout, and the same injectivity gives ${\exp }_{p,U}^{-1}(\alpha (t))=v_j(p,\alpha (t))(u\le t\le v)\mathrm{.}$ Thus the inverse and joining geodesic selected by each $U_i^N$ agree with the fixed branch for all sufficiently fine partitions. No bound on the sizes, chart constants or injectivity radii of the varying $U_i^N$ is used.

(One integrable control.) Fix a finite subdivision $a=r_0<\cdots <r_m=b$ with each block image in one $A_j$ and define $w(t)$ on that block to be the Euclidean norm of $(x_j\circ \alpha {)}^{\mathrm{\prime }}(t)$, setting it to zero at the finitely many subdivision points. Local absolute continuity gives $w\ge 0$, $w\in L^1[a,b]$. There is a uniform constant $C$ such that, in any selected chart containing a point of a block, $\mathrm{\mid }(x_k\circ \alpha {)}^{\mathrm{\prime }}\mathrm{\mid }\le Cw$ almost everywhere on the overlap. Indeed the transition derivatives are bounded on the intersections of the finitely many compact sets $K_j,K_k$; apply the chain rule. Thus, even when a fine partition interval $I_i=[t_i,t_{i+1}]$ crosses a fixed block boundary, finite integral additivity gives $\mathrm{\mid }{x}_k(\alpha (t_{i+1}))-x_k(\alpha (t_i))\mathrm{\mid }\le C{\int }_{I_i}w$ for the chart assigned to that entire short subarc. The metric components are bounded on the same compact sets, so the original Lorentzian density $f(t)=\sqrt{-g({\alpha }^{\mathrm{\prime }}(t),{\alpha }^{\mathrm{\prime }}(t))}$ satisfies $f\le Cw$.

(Interpolants and uniform control.) For any of the promised partitions, write $h_i=t_{i+1}-t_i$, $p_i=\alpha (t_i)$ and let $v_i$ be the inverse selected by $U_i^N$. Concatenate its radial geodesics, affinely parametrized on $I_i$, to obtain ${\gamma }_N$. The endpoints are distinct because the adapted time of [L3] strictly increases along the original subarc. The cone theorem [L1] therefore makes each radial segment future causal; its length is $d_{U_i^N}(p_i,p_{i+1})$ and its constant Lorentzian speed is $f_N(t)=\frac{\sqrt{-g_{p_i}(v_i,v_i)}}{h_i}(t\in \mathrm{int}{I}_i)\mathrm{.}$ By steps 1.1-2.1 and 2.2, for all sufficiently large $N$, $0\le f_N(t)\le C{w}_N(t),w_N(t):=\frac{1}{h_i}{\int }_{I_i}w,{\int }_a^b{w}_N={\int }_a^{b}w\mathrm{.}$ The same estimates put the geodesic and the original subarc in the fixed chart, with their coordinate discrepancy bounded by $C{\int }_{I_i}w$. Absolute continuity of the integral of $w$ makes this bound tend to zero uniformly as the mesh tends to zero. Finite chart comparison therefore also proves ${\gamma }_N\to \alpha$ uniformly. Crucially all constants belong to the fixed finite charts, not the independently varying normal balls.

(Almost-everywhere speed convergence.) Outside a null set, $t$ is a Lebesgue point of $w$ and of every local coordinate derivative of $\alpha$ where its chart is defined (use countable compact exhaustions of each open parameter preimage of $V_j$), and is neither a fixed subdivision point nor any partition point. The latter points form a countable union of finite sets. For the interval $I_i$ containing such $t$, [L4] gives $w_N(t)\to w(t)$ and the coordinate secant velocity tends to the coordinate velocity of $\alpha$ at $t$. Step 1.1 has the precise error bound $\mid \frac{v_i}{h_i}-\frac{x_j(p_{i+1})-x_j(p_i)}{h_i}\mid \le C{h}_i{w}_N(t{)}^2⟶0.$ Here $w_N(t)$ is bounded at this Lebesgue point; there is no assertion of a uniform $o(h_i)$ error for an arbitrary absolutely continuous curve. Since only finitely many charts are possible, these pointwise conclusions hold even if their assigned index varies with $N$. Continuity of $g$, with $p_i\to \alpha (t)$, gives $f_N(t)\to f(t)$ almost everywhere.

(Fatou proves the full refined-sum identity.) Choose the common $C$ of step 3.1 also large enough that $f\le Cw$. The nonnegative functions $C{w}_N-f_N$ converge almost everywhere to $Cw-f$. Fatou gives $C{\int }_a^{b}w-L_g(\alpha )\le {\mathrm{lim\hspace{0.17em}inf}}_N(C{\int }_a^{b}w-{\int }_a^b{f}_N),$ hence ${\mathrm{lim\hspace{0.17em}sup}}_N\int f_N\le L_g(\alpha )$. Conversely, [L1] on every original subarc gives $L_g(\alpha )\le {\sum }_i{d}_{U_i^N}(p_i,p_{i+1})={\int }_a^b{f}_N\mathrm{.}$ The two inequalities prove (a), including its independently varying-ball quantifier and independence of both the partitions and the chosen balls.

(Upper semicontinuity.) Choose a fixed finite buffered normal-ball cover of $\alpha ([a,b])$ and successively finer partitions such that each compact subarc is contained in one cover ball. For any fixed such partition, uniform convergence puts the corresponding subarcs of ${\alpha }_k$ in those same open balls for every sufficiently large $k$. By [L1] and continuity of the finitely many selected inverse functions [L2], ${\mathrm{lim\hspace{0.17em}sup}}_k{L}_g({\alpha }_k)\le {\sum }_i{d}_{U_i}(\alpha (t_i),\alpha (t_{i+1}))\mathrm{.}$ Now refine the fixed-cover partitions and use (a), already proved without using (b), to make the right side tend to $L_g(\alpha )$. This proves (b).

The arbitrary-ball identity is step 5.1 and upper semicontinuity is step 6.1, proving both promised claims.