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The exterior Schwarzschild charts do not cover the horizon-crossing extension: the regular horizon point (v, r) = (0, s)

Statement refuted

Statement refuted: "Let $(M, g, N)$ be the Schwarzschild exterior datum with $s > 0$ and let $(M_{e x t}, g_{e x t})$ be the horizon-crossing extension of Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart. Then every point of $M_{e x t}$ lies in the image of the exterior datum under the embedding $Φ (x^{0}, r, ω) = (x^{0} + r_{*} (r), r, ω)$ , hence in the domain of some Schwarzschild (product) chart of the datum; equivalently, every point added by the extension is reached at a bounded value of the exterior coordinate $x^{0}$ ."

The counterexample is the point

$q_{0} : = (v, r, ϑ, φ) = (0, s, ϑ_{0}, φ_{0}) \in M_{e x t}, \sin ϑ_{0} \neq 0,$

on the level set ${r = s}$ , and the following three facts refute both forms of the statement:

$q_{0} \notin Φ (M)$ : the map $Φ$ preserves the function $r$ , so $r (Φ (p)) = r (p) > s$ for every $p \in M$ , whereas $r (q_{0}) = s$ .
$q_{0}$ is a regular point of a smooth Lorentzian manifold, not a boundary artifact: at $q_{0}$ one has $\det (g_{e x t}) = - s^{4} \sin^{2} ϑ_{0} \neq 0$ , the inertia is $(3, 1, 0)$ , and the Kretschmann scalar is $12 s^{2} / r^{6} = 12 / s^{4}$ , finite and nonzero.
Every sequence $p_{n} \in M$ with $Φ (p_{n}) \to q_{0}$ must have $r (p_{n}) \to s$ and exterior coordinate $x^{0} (p_{n}) = v_{n} - r_{*} (r (p_{n})) \to + \infty$ , where $v_{n} : = v (Φ (p_{n})) \to v_{0} = 0$ ; so the exterior coordinate $x^{0}$ is unbounded along the approach to $q_{0}$ , and the second form of the statement ("bounded $x^{0}$ ") fails as well.

Facts & Assumptions

Given: The exterior datum with $s > 0$ , its tortoise function $r_{*} (r) = r + s \log ∣ r / s - 1 ∣$ on $(s, \infty)$ , the extension $(M_{e x t}, g_{e x t})$ with chart $(v, r, ϑ, φ)$ , the embedding $Φ (x^{0}, r, ω) = (x^{0} + r_{*} (r), r, ω)$ , and the point $q_{0} = (0, s, ϑ_{0}, φ_{0})$ with $\sin ϑ_{0} \neq 0$ .

[L1]

$Φ$ restricts to a diffeomorphism of $M$ onto the open subset ${r > s}$ of $M_{e x t}$ and $Φ^{*} g_{e x t} = g$ ; on this subset the chart function $r$ of $M_{e x t}$ pulls back to the radial coordinate $r$ of the datum, and $v = x^{0} + r_{*} (r)$ (Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart clauses (a)-(c), The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clauses (b)-(c)).

[L2]

$g_{e x t}$ is smooth and nondegenerate of inertia $(3, 1, 0)$ on all of $M_{e x t}$ , $\det (g_{e x t}) = - r^{4} \sin^{2} ϑ$ in every spherical-polar product chart, and the Kretschmann scalar of $g_{e x t}$ equals $12 s^{2} / r^{6}$ on $M_{e x t}$ (Smooth horizon-crossing extension of the Schwarzschild exterior metric in the advanced chart clauses (a), (d), (e), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Smooth covariant 2-tensor fields on a smooth manifold).

[L3]

On $(s, \infty)$ the tortoise function satisfies $r_{*} (r) = r + s \log (r / s - 1)$ ; $\log$ is the inverse of the strictly increasing bijection $\exp : R \to (0, \infty)$ , so for every $M \in R$ the inequality $\log t < - M$ is equivalent to $t < e^{- M}$ for $t > 0$ ; hence $\log t \to - \infty$ as $t ↓ 0$ (The natural logarithm as the inverse of the exponential function, The exponential is a continuous bijection from $R$ onto $(0, \infty)$ , The exponential function is strictly increasing, The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Rational powers $a^{r}$ of a positive base, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Counterexample

technique · direct; use that $\Phi$ preserves $r$ to separate $q_0$ from the image, exhibit the regular data at $q_0$, and solve $v_0=x^{0}+r_{*}(r)$ for $x^{0}$ along $r\downarrow s$

1.1L1algebra

( $q_{0}$ is not in the image of the exterior datum.) Let $p \in M$ with $p = (x^{0}, r, ω)$ . By [L1], $Φ (p) = (x^{0} + r_{*} (r), r, ω)$ , whose third-to-second chart entry equals the radial coordinate $r (p)$ of the datum, and $r (p) > s$ because the domain of the datum is ${r > s}$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation clause (b)). At $q_{0}$ the extension chart has $r (q_{0}) = s$ ; if $q_{0} = Φ (p)$ then $s = r (q_{0}) = r (p) > s$ , a contradiction. Hence $q_{0} \notin Φ (M)$ , and since every product chart of the datum is a chart of $M$ , no Schwarzschild chart of the datum contains $q_{0}$ . This is clause 1.

1.2L2algebra

( $q_{0}$ is a regular point.) By [L2], $\det (g_{e x t}) (q_{0}) = - s^{4} \sin^{2} ϑ_{0}$ , which is nonzero because $s > 0$ and $\sin ϑ_{0} \neq 0$ ; the inertia at $q_{0}$ is $(3, 1, 0)$ , and the Kretschmann scalar there is $12 s^{2} / s^{6} = 12 / s^{4}$ , finite and positive. So $q_{0}$ is an interior point of a smooth nondegenerate Lorentzian four-manifold on which all stated curvature data are finite; in particular the failure of the product charts to cover $q_{0}$ is not due to $q_{0}$ being a singularity or a boundary point of the extension. This is clause 2.

2.1L1L3step 1.1algebra

(The exterior coordinate is unbounded.) Let $p_{n} \in M$ be a sequence with $Φ (p_{n}) \to q_{0}$ . By [L1] the first chart coordinate of $Φ (p_{n})$ is $v_{n} = x^{0} (p_{n}) + r_{*} (r (p_{n}))$ and the second is $r_{n} = r (p_{n})$ ; convergence to $q_{0} = (0, s, ϑ_{0}, φ_{0})$ forces $r_{n} \to s$ and $v_{n} \to 0$ . Then $x^{0} (p_{n}) = v_{n} - r_{*} (r_{n})$ . Since $r_{n} > s$ for all $n$ and $r_{n} \to s$ , the argument $r_{n} / s - 1$ tends to $0^{+}$ , so by [L3] $\log (r_{n} / s - 1) \to - \infty$ and hence $r_{*} (r_{n}) = r_{n} + s \log (r_{n} / s - 1) \to - \infty$ ; therefore $x^{0} (p_{n}) = v_{n} - r_{*} (r_{n}) \to + \infty$ . More explicitly, for every bound $B$ there is $n_{0}$ with $r_{*} (r_{n}) < - B - ∣ v_{n} ∣$ for $n \geq n_{0}$ , whence $x^{0} (p_{n}) > B$ : the exterior coordinate $x^{0}$ is unbounded above along the approach. This is clause 3.

3.1L1step 1.1step 1.2step 2.1construct∎

(An approach exists.) For each integer $n \geq 1$ set $r_{n} = s + s / n$ and $p_{n} = (- r_{*} (r_{n}), r_{n}, ω_{0})$ , with $ω_{0}$ the fixed angular point of $q_{0}$ . Then $p_{n} \in M$ and $Φ (p_{n}) = (0, r_{n}, ω_{0}) \to q_{0}$ , so the limiting assertion of step 2.1 is nonvacuous. Together with steps 1.1 and 1.2, this refutes both claimed properties at the specified regular horizon point.

Remarks

What the counterexample says. The horizon point $q_{0}$ lies in the closure of $Φ (M)$ in $M_{e x t}$ but not in $Φ (M)$ ; it is a regular point of the extension and requires a genuinely different chart, whose existence is the content of the cited extension lemma. The refuted statement is not that the exterior chart fails somewhere — the cited lemma proves its regularity on $r > s$ — but that the charts of the exterior datum suffice for the extension. No value of $s$ , $c$ , $G_{N}$ or $m_{*}$ , no numerical integration and no measurement occurs.

The exterior Schwarzschild charts do not cover the horizon-crossing extension: the regular horizon point (v, r) = (0, s)

Statement refuted

Facts & Assumptions

Counterexample

Remarks

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