Temporal normalization makes the null geodesic flow compact over compact bases

For smooth local fields put $K(X,Y,Z)=Xg(Y,Z)+Yg(X,Z)-Zg(X,Y)+g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X)$. Expanding the product rules gives $K(fX,Y,Z)=fK(X,Y,Z)$, $K(X,Y,fZ)=fK(X,Y,Z)$ and $K(X,fY,Z)=fK(X,Y,Z)+2X(f)g(Y,Z)$: the derivatives of $f$ from the bracket terms cancel those from the three metric derivatives, except the two copies of $X(f)g(Y,Z)$ in the last identity. Thus $K(X,Y,\cdot )$ is a covector. Nondegeneracy defines a unique ${\mathrm{\nabla }}_{X}Y$ by $2g({\mathrm{\nabla }}_{X}Y,Z)=K(X,Y,Z)$. In a chart it is $\frac{1}{2}{\sum }_{i,j}{g}^{ij}K(X,Y,{\mathrm{\partial }}_j){\mathrm{\partial }}_i$, which is smooth: the inverse matrix entries are the adjugate entries divided by the nonzero determinant. The defining intrinsic identity makes these chart fields agree on overlaps. The three scalar identities and real bilinearity give precisely the affine-connection rules.

Set $a=\sqrt{-g^{-1}(dv,dv)}>0$ and $n=-{\mathrm{grad}}_{g}v\mathrm{/}a$. Then $g(n,n)=-1$, $dv(n)=a$, and the positive future sign makes $n$ future. The smooth projection $X\mapsto X+g(X,n)n$ has image $H=n^{\perp }=\ker dv$ and kernel $\mathbb{R}n$. Projecting coordinate fields and selecting a locally independent $(m-1)$-tuple gives local smooth frames of $H$. The orthogonal splitting $TE=\mathbb{R}n\oplus H$ and inertia imply $h=g{\mathrm{\mid }}_H$ is positive definite. Applying Gram--Schmidt in each such frame is smooth since every normalization denominator is positive, and gives smooth orthonormal frames. Every nonzero future null vector is uniquely $w=b(n+e)$ with $b=-g(w,n)>0$, $e\in H$, and $h(e,e)=1$. Since $dv(w)=ab$, the map $(p,e)\mapsto (p,a^{-1}(n+e))$ is a smooth bundle isomorphism from $S_h(H)$ onto ${\mathcal{N}}_v$, with inverse $e=aw-n$. For any compact $B\subset E$, take finitely many closed compact subsets $B_i$ covering $B$, each contained in an orthonormal trivialization (shrink coordinate neighborhoods inside the trivializations first). The sphere bundle over each $B_i$ is $B_i\times S^{m-2}$, hence compact, so its finite union over $B$ is compact. The same holds for ${\mathcal{N}}_v$; no vector there is zero, since $dv(w)=1$. If $m=1$, $H$ has rank zero and ${\mathcal{N}}_v$ is empty, with all flow assertions vacuous.

Expansion of the same six terms gives $K(X,Y,Z)-K(Y,X,Z)=2g([X,Y],Z)$ and $K(X,Y,Z)+K(X,Z,Y)=2Xg(Y,Z)$. Consequently the connection is torsion-free and metric-compatible. Conversely those two properties, substituted into the six terms, give $K(X,Y,Z)=2g({\mathrm{\nabla }}_{X}Y,Z)$, proving uniqueness. With indices ranging over all $1,\dots ,m$, evaluation on coordinate fields yields ${\mathrm{\Gamma }}_{jk}^i=\frac{1}{2}{\sum }_{\mathrm{\ell }}{g}^{i\mathrm{\ell }}({\mathrm{\partial }}_j{g}_{k\mathrm{\ell }}+{\mathrm{\partial }}_k{g}_{j\mathrm{\ell }}-{\mathrm{\partial }}_{\mathrm{\ell }}{g}_{jk})$ and $({\mathrm{Hess}}_{g}v{)}_{jk}={\mathrm{\partial }}_j{\mathrm{\partial }}_{k}v-{\sum }_i{\mathrm{\Gamma }}_{jk}^i{\mathrm{\partial }}_{i}v$. In particular the coefficients and Hessian are smooth in every finite dimension, and $D_{s}w$ has components $(w^i{)}^{\mathrm{\prime }}+{\sum }_{j,k}{\mathrm{\Gamma }}_{jk}^i(p)(p^j{)}^{\mathrm{\prime }}{w}^k$. This supplies the dimension-independent metric connection and geodesic equation needed here.

For $m\ge 2$, define the lifted equation in each tangent-bundle chart by the following formula, where repeated indices are summed: [step 2.1, step 1.2] $(p^i)\mathrm{\prime }=w^i,(w^i)\mathrm{\prime }=-{\mathrm{\Gamma }}_{jk}^i{w}^j{w}^k-({\mathrm{Hess}}_{g}v)(w,w)w^i\mathrm{.}$ Its intrinsic form is $D_{s}w=-({\mathrm{Hess}}_{g}v)(w,w)w$, so the chart fields agree and are smooth. Write $H_v=({\mathrm{Hess}}_{g}v)(w,w)$. The derivatives of the two constraints are $(dv(w))\mathrm{\prime }=H_v(1-dv(w))$ and $(g(w,w))\mathrm{\prime }=-2H_{v}g(w,w)$, using metric compatibility. They vanish on ${\mathcal{N}}_v$, so the field is tangent to its smooth constraint manifold; future direction is retained there by continuity and the nonzero normalization. Along its base curves $d(v(p))\mathrm{/}ds=1$.

To check the geodesic-path assertion, along a lifted orbit choose $\lambda (s)=\lambda (s_0)\exp ({\int }_{s_0}^s{H}_v(r)dr)>0$ and introduce $t$ by $dt\mathrm{/}ds=1\mathrm{/}\lambda (s)$. Then $dp\mathrm{/}dt=\lambda w$ and $D_t(dp\mathrm{/}dt)=\lambda (\lambda \mathrm{\prime }-\lambda H_v)w=0$. This gives an increasing affine parameter locally. Conversely, for an affine future null geodesic $c(t)$ put $q=dv(c\mathrm{\prime })>0$ and use $s=v(c(t))$ as parameter. Then $w=c\mathrm{\prime }\mathrm{/}q$ and $dq\mathrm{/}dt=({\mathrm{Hess}}_{g}v)(c\mathrm{\prime },c\mathrm{\prime })$ give $D_{s}w=-({\mathrm{Hess}}_{g}v)(w,w)w$ and $dv(w)=1$. Thus exactly the stated geodesic paths lift to this field, with the temporal parameter differing from the flow parameter by an additive constant.

Apply [F1] to the smooth field on ${\mathcal{N}}_v$. Its maximal flow domain is open and its flow is smooth. Suppose an orbit $z(s)$ has a finite right endpoint $b$ and its terminal portion has base in compact $B$. By step 1.2 that portion lies in compact ${\mathcal{N}}_v{\mathrm{\mid }}_B$. At every point of this compact set, local existence gives a neighborhood and a flow defined for $\mathrm{\mid }t\mathrm{\mid }<{\epsilon }_i$ uniformly on that neighborhood. Take a finite subcover and let $\epsilon ={\min }_i{\epsilon }_i>0$. Choose $s_0<b$ in the terminal portion with $b-s_0<\epsilon \mathrm{/}2$. A neighborhood containing $z(s_0)$ supplies a solution until $s_0+\epsilon >b$; uniqueness glues it to the original orbit. This contradicts maximality. Reversing the parameter handles a finite left endpoint. This is continuation in temporal time and asserts no global affine completeness.