Horizon versus singularity in the Schwarzschild model: a chart breakdown, a smooth null surface and a curvature obstruction

The exterior datum of The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation is defined on the open domain $r>s$ only. At every point of this domain a product-chart frame remains a basis. The component $g_{11}=f(r{)}^{-1}=(1-s\mathrm{/}r{)}^{-1}$ diverges as $r\downarrow s$, and the chart contains no point with $r=s$, so there is no frame limit there in this chart. This is a statement about one component, not an invariant: the metric tensor is smooth and nondegenerate on $r>s$ and has the separate regular extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart.

Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart constructs on $M_{\mathrm{e}\mathrm{x}\mathrm{t}}=\mathbb{R}\times (0,\mathrm{\infty })\times S^2$ a smooth nondegenerate Lorentzian metric $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ whose restriction to $r>s$ is the exterior datum; the level set $r=s$ is a smooth null hypersurface with generator direction ${\mathrm{\partial }}_v$, and the Ricci tensor and Kretschmann scalar of $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ extend smoothly across it. In the Kruskal model (The Kruskal model datum: domain, implicit radial function, metric and time orientation, Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar) the full level set $\{r=s\}$ is $\{UV=0\}=\{U=0\}\cup \{V=0\}$. Each axis times $S^2$ is a smooth hypersurface: $dU$ and $dV$ never vanish, and the inverse base matrix has $g^{UU}=g^{VV}=0$, so each has null normal covector. They intersect transversely at the bifurcation sphere $\{U=V=0\}\times S^2$. Their tangent spaces there together span the full four-dimensional tangent space, so their union cannot be a smooth three-dimensional hypersurface at that sphere. The function $r$ and metric remain smooth there; the curvature invariant is $12\mathrm{/}{s}^4$ on both axes. For future escape to the right exterior, the boundary is specifically $\{U=0\}$, while $\{V=0\}$ is the past escape boundary. The one-way character in the model is the exact statement of The causal cone, the escape horizon and decreasing areal radius in the Kruskal model clause (d): no future-directed causal curve issued at a point with $U\ge 0$ reaches the quadrant $\{U<0,V>0\}$ with $r\to +\mathrm{\infty }$, while every point with $U<0$ admits such a curve along the future radial null geodesic at constant $U$. This is a causal property of the smooth model; it is not a divergence of the metric at the surface, and the model's time orientation is part of the datum.

The Kretschmann scalar of the exterior datum is $K=R_{abcd}{R}^{abcd}=12s^2\mathrm{/}{r}^6=\frac{48G_N^2{m}_{\ast }^2}{c^4{r}^6}$ (Orthonormal-frame curvature components and the Kretschmann scalar 12s^2/r^6 of the Schwarzschild exterior metric clause (c)): on $r>s$ it takes values in $(0,12\mathrm{/}{s}^4)$ and has the finite boundary value $12\mathrm{/}{s}^4$ as $r\downarrow s$; on the smooth extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart, whose domain has $r>0$ and on which the same expression $12s^2\mathrm{/}{r}^6$ is the Kretschmann scalar, it grows without bound as $r\downarrow 0$. In the Kruskal model it is smooth on $\mathcal{K}$ and grows without bound as $UV\to 1$, that is, as $r\downarrow 0$ (Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar clause (d)). The value $r=0$ is not attained on $\mathcal{K}$; the set $\{UV=1\}$ is not part of the model. The invariant is a scalar built from the metric, so it is chart-independent, and the following argument shows that $r=0$ cannot be appended as a regular $C^2$ metric boundary point: if a $C^2$ Lorentzian metric on a manifold containing $\mathcal{K}$ together with a boundary point $q$ approached with $r\to 0$ restricted to $g_{\mathcal{K}}$ on $\mathcal{K}$, then its curvature tensor would be continuous near $q$ and its Kretschmann scalar bounded there, contradicting the unboundedness of the same invariant along sequences in $\mathcal{K}$. The distinction between the two values of $r$ is therefore a statement about the smooth curvature of the model, not about the chosen coordinates.

• No singularity theorem and no collapse. Nothing above proves that any solution of the field law develops the region $0<r<s$, that it is realized by collapsing matter, or that any observed object is described by the model; the interior region and the second exterior region of the Kruskal model are geometric regions of that model only.
• No maximality or completeness. The extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart is not claimed maximal, and no geodesic completeness or incompleteness statement for $\mathcal{K}$ is made here; the maximal-development and boundary theorems are deferred to their own modules of the track.
• No asymptotic or global mass. The parameter $m_{\ast }$ is the model's parameter (The Schwarzschild exterior vacuum model: field law, static clock readings and the mass parameter clause (c)); no ADM-type mass at infinity and no boundary-asymptotic comparison is asserted on this page.
• No observation. The words horizon and singularity here refer to properties of the model; whether the model applies to any observed system is an empirical question belonging to experiment items and to the experimental-interface page, with apparatus, calibration, sampling and uncertainty qualifications.