Existence and boost-coordinate description of Hawking-flow heights

Put $F(t,x)=t-u_q(x)$. Its level sets are stationary translates of the spacelike candidate graph. The positive $T$-lapse fixes $dF$ to be positive on future timelike vectors: $dF(T)=1$ and $T$ has positive normal component. Thus $dF(K)>0$ on each complete orbit under consideration. The mean value theorem on every finite orbit-parameter interval shows that $s\mapsto F({\varphi }_s^{K}x)$ is strictly increasing.

At $y\in S$, put $B_y=(dK{)}_y$. Because $K{\mathrm{\mid }}_S=0$, the connection term vanishes there, so this is ${\mathrm{\nabla }}^{(q)}K$ for every $q$, including the original member. It annihilates $T_{y}S$ and is invertible on each metric's normal plane by [F3]. Thus both normal planes equal the same rank-two image $\mathrm{im}{B}_y$. Their intersections with $T_y{\mathrm{\Sigma }}_1$ are the same line. Both ${\bar{\nu }}_q$ and ${\bar{\nu }}_1$ point into the same actual reference side $C$, so ${\bar{\nu }}_q=c_q{\bar{\nu }}_1$ for some $c_q>0$. The positive boost identities $B_y{\bar{\nu }}_q=\kappa {\bar{n}}_q$ and $B_y{\bar{\nu }}_1=\kappa {\bar{n}}_1$ then give ${\bar{n}}_q=c_q{\bar{n}}_1$. Choose any local inward candidate-boundary curve $\gamma (\lambda )$, with $\lambda \ge 0$, $\gamma (0)=y$ and ${\gamma }^{\mathrm{\prime }}(0)$ transverse to $T_{y}S$, using its supplied embedded boundary parametrization. Decompose ${\gamma }^{\mathrm{\prime }}(0)=\xi +U{\bar{n}}_q+V{\bar{\nu }}_q$, where $\xi \in T_{y}S$. Subtracting $\xi$ leaves a nonzero candidate tangent vector, so its spacelikeness gives $V^2-U^2>0$. The zero-section differential of the normal tubes identifies the derivatives of its original coordinates as $a^{\mathrm{\prime }}(0)=c_{q}U$ and $b^{\mathrm{\prime }}(0)=c_{q}V$. For $\lambda >0$ sufficiently small, $\gamma (\lambda )$ is in the stationary graph and hence in the original exterior $\mathcal{E}$. By [F9], $b(\lambda )>\mathrm{\mid }a(\lambda )\mathrm{\mid }$. Divide by $\lambda$ and take its one-sided limit to obtain $V\ge \mathrm{\mid }U\mathrm{\mid }$; together with $V^2-U^2>0$ this gives $V>0$. The candidate therefore enters the positive-$v$ side of the $g_q$ normal tube, where $v$ is the coefficient of ${\bar{\nu }}_q$. The reference enters that side by its definition. Only the differentials of the two normal tubes at $S$ have been compared; their coordinate maps are not identified away from $S$.

Commutation gives ${\varphi }_s^K={\varphi }_s^T\circ {\varphi }_{{\mathrm{\Omega }}_{H}s}^Z$ on the complete collar saturation. Since $T={\mathrm{\partial }}_t$ and $u_q$ is independent of $t$, $F({\varphi }_s^{K}x)=s+F({\varphi }_{{\mathrm{\Omega }}_{H}s}^{Z}x)$. For a positive common period $P$ of $Z$, the function $a\mapsto F({\varphi }_a^{Z}x)$ is continuous on the nonempty compact metric interval $[0,P]$ and is bounded there by [F7]. Periodicity extends that bound to every real $a$, hence to $a={\mathrm{\Omega }}_{H}s$. Thus $F({\varphi }_s^{K}x)$ tends to opposite infinities as $s\to \pm \mathrm{\infty }$.

In the normal tube from [F3], write $u$ for the coefficient of ${\bar{n}}_q$ and $v$ for that of ${\bar{\nu }}_q$. Projection of either spacelike section to $(v,\vartheta )$ has invertible derivative at $S$: its kernel would be a timelike tangent vector, whereas the section is spacelike. Apply [F4] to the projection of a local smooth extension of a boundary parametrization, obtaining $u=w(v,\vartheta )$ with $w(0,\vartheta )=0$. Step 1.2 identifies the actual section side of each such graph as $v\ge 0$. Local uniqueness glues these graphs, and compactness and the product-neighborhood assertion in [F7] give a common smaller collar. Spacelikeness gives $\mathrm{\mid }{\mathrm{\partial }}_{v}w(0,\vartheta )\mathrm{\mid }<1$, with no zero-slope requirement for the candidate. Define $h(v,\vartheta )={\int }_0^1{\mathrm{\partial }}_{v}w(av,\vartheta )da$. For $v>0$, [F5] applied to $a\mapsto w(av,\vartheta )$ gives $w(v,\vartheta )=vh(v,\vartheta )$; at zero this identity still holds and $h(0,\vartheta )={\mathrm{\partial }}_{v}w(0,\vartheta )$. On closed coordinate rectangles, successive applications of [F6] differentiate this integral in every parameter. These derivatives are jointly continuous: for any resulting continuous integrand $H(a,p)$ and fixed $p_0$, the open set $\mathrm{\mid }H(a,p)-H(a,p_0)\mathrm{\mid }<\eta$ contains $[0,1]\times \{p_0\}$. Compactness of $[0,1]$ and the tube lemma in [F7] make this bound uniform in $a$ near $p_0$, and the integral-error bound in [F6] gives continuity of its integral. Applying this to all derivative orders proves that $h$ is jointly smooth, including at $v=0$ using the local smooth extensions.

Choose finite parameters with opposite signs of $F({\varphi }_s^{K}x)$ using step 1.3. The intermediate value theorem on that closed parameter interval gives a zero, and strict increase from step 1.1 gives uniqueness. At the zero the derivative in $s$ is $dF(K)>0$, so [F4] applied in interior charts gives smooth dependence of that zero on $x$. Uniqueness makes these local solutions agree, proving interior smoothness of $u_q^K$.

For each section, [F7] on $S$ gives $m={\max }_{\vartheta }\mathrm{\mid }h(0,\vartheta )\mathrm{\mid }<1$. Fix $m<\theta <1$ and shrink by the tube lemma to obtain $\mathrm{\mid }h\mathrm{\mid }<\theta$ on a common collar. Put $R(v,\vartheta )=v\sqrt{1-h(v,\vartheta {)}^2}$; then ${\mathrm{\partial }}_{v}R(0,\vartheta )=\sqrt{1-h(0,\vartheta {)}^2}\ge \sqrt{1-m^2}$. A further use of the tube lemma gives ${\mathrm{\partial }}_{v}R>c=\frac{1}{2}\sqrt{1-m^2}>0$ there. Choose $\epsilon >0$ with the closed interval $0\le v\le \epsilon$ in that collar. The mean value theorem gives strict increase and $R(\epsilon ,\vartheta )>c\epsilon$; the intermediate value theorem gives the unique inverse $v=b(r,\vartheta )$ for $0\le r<c\epsilon$. By [F4], this inverse is smooth, using local extensions also at zero. Thus the rapidity is $\mathrm{arctanh}h(b(r,\vartheta ),\vartheta )$, and the normal coordinates are $u=r\sinh (\text{rapidity})$, $v=r\cosh (\text{rapidity})$. This constructs the smooth functions ${\alpha }_q$ and ${\beta }_q$ on a common smaller collar for the two sections. The reference tangent is ${\bar{\nu }}_q$, so its slope is zero and ${\alpha }_q(0,\vartheta )=0$.

The $K$-flow fixes each $\vartheta \in S$, is an isometry of $g_q$, and commutes with the normal exponential map wherever the displayed expressions are defined. Its differential adds $\kappa s$ to normal rapidity while preserving $r$ and $\vartheta$. Consequently the local intersection of the two rapidity graphs occurs at $s=({\beta }_q-{\alpha }_q)\mathrm{/}\kappa$. Their boundary values are bounded on $S$ by [F7], and continuity and the tube lemma give bounded rapidities on a smaller collar. Throughout this bounded flow segment the normal-vector coefficients are bounded by a constant times $r$, so a further shrinking keeps the entire segment in the tube. This local intersection is the unique global intersection from step 2.2. Its smooth formula extends $u_q^K$ to $r=0$ and gives $u_q^K{\mathrm{\mid }}_S={\beta }_q(0,\vartheta )\mathrm{/}\kappa$.