Explicit metric, orientation and curvature data of the advanced-chart extension at the horizon r = s

Let $(M,g,N)$ be the Schwarzschild exterior datum with $s>0$ and $f(r)=1-s\mathrm{/}r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), and let $(M_{\mathrm{e}\mathrm{x}\mathrm{t}},g_{\mathrm{e}\mathrm{x}\mathrm{t}})$ with chart $(v,r,\vartheta ,\phi )$ and time-orientation field $N_{\mathrm{e}\mathrm{x}\mathrm{t}}=A(r){\mathrm{\partial }}_v-{\mathrm{\partial }}_r$, $A(r)=\frac{f(r)+\sqrt{f(r{)}^2+4}}{2}$, be the advanced-chart extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart. Write $q_0:=(v_0,s,{\vartheta }_0,{\phi }_0)$ for a point on the level set $\{r=s\}$ with $\sin {\vartheta }_0\mathrm{\ne }0$. Then:

1. Metric components and nondegeneracy at the horizon. In the product chart, $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ has components $g_{vv}=-f(s)=0,g_{vr}=g_{rv}=1,g_{\vartheta \vartheta }=s^2,g_{\phi \phi }=s^2{\sin }^2{\vartheta }_0,\det (g_{\mathrm{e}\mathrm{x}\mathrm{t}})=-s^4{\sin }^2{\vartheta }_0<0.$ The two-dimensional block $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ in the $(v,r)$ directions is nonsingular with eigenvalues $+1$ and $-1$, and the angular directions are positive on $(0,\pi )$; hence $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ is nondegenerate of inertia $(3,1,0)$ at $q_0$ with the same signature $(-,+,+,+)$ as in the exterior.
2. Horizon generator and time orientation. At $q_0$ the coordinate field ${\mathrm{\partial }}_v$ satisfies $g_{\mathrm{e}\mathrm{x}\mathrm{t}}({\mathrm{\partial }}_v,{\mathrm{\partial }}_v)=-f(s)=0$ and $g_{\mathrm{e}\mathrm{x}\mathrm{t}}({\mathrm{\partial }}_v,{\mathrm{\partial }}_r)=1\mathrm{\ne }0$, so ${\mathrm{\partial }}_v$ is a null direction tangent to the level set $\{r=s\}$ (the generator direction of clause (e) of the lemma). At $r=s$ the orientation field is $A(s)=\frac{0+\sqrt{4}}{2}=1$, so $N_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathrm{\partial }}_v-{\mathrm{\partial }}_r$, with $g_{\mathrm{e}\mathrm{x}\mathrm{t}}(N_{\mathrm{e}\mathrm{x}\mathrm{t}},N_{\mathrm{e}\mathrm{x}\mathrm{t}})=-2<0,g_{\mathrm{e}\mathrm{x}\mathrm{t}}(N_{\mathrm{e}\mathrm{x}\mathrm{t}},{\mathrm{\partial }}_v)=-1<0,$ a timelike vector of squared norm $-2$ (not unit-normalized) whose pairing with the generator is negative: ${\mathrm{\partial }}_v$ is future-directed at the horizon.
3. Finite curvature at the horizon. By clause (d) of the lemma the Kretschmann scalar of $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ equals $12s^2\mathrm{/}{r}^6$ on all of $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$, so at $q_0$ its value is $12s^2\mathrm{/}{s}^6=12\mathrm{/}{s}^4$, finite and nonzero for every $s>0$.
4. The chart component still diverges. In the exterior product chart of the datum the component $g_{11}=f(r{)}^{-1}=(1-s\mathrm{/}r{)}^{-1}$ diverges as $r\downarrow s$, while clauses 1 to 3 show that the metric, its orientation and the Kretschmann invariant at the same value $r=s$ are finite and nondegenerate in the extension. The divergence is therefore a property of the exterior chart's coefficient, not of the geometry at $q_0$.

• What the example computes. The example evaluates the supplied extension formulas at $r=s$; it uses the lemma's identity $K=12s^2\mathrm{/}{r}^6$ on $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ rather than extending the exterior-domain computation, and no value of the metric outside the extension's domain $r>0$ is used. No numerical value of $s$, $c$, $G_N$ or $m_{\ast }$ and no measurement occurs.