Explicit causal-cone and radial-derivative checks in the Kruskal model at the horizon for s = 1 m

Fix the Kruskal datum of The Kruskal model datum: domain, implicit radial function, metric and time orientation with length scale $L_0:=1\mathrm{m}$ and Schwarzschild radius $s=L_0$; this sets the model parameter $m_{\ast }=c^2{L}_0\mathrm{/}(2G_N)$ and is a normalization choice, not a measurement. Write $\bar{r}:=r\mathrm{/}{L}_0$, $\bar{g}:=g_{\mathcal{K}}\mathrm{/}{L}_0^2$, and $\bar{A}:=A\mathrm{/}{L}_0^2=4e^{-\bar{r}}\mathrm{/}\bar{r}$, where $A=4s^3{e}^{-r\mathrm{/}s}\mathrm{/}r$ is the coefficient of The causal cone, the escape horizon and decreasing areal radius in the Kruskal model clause (a). The coordinates $U,V$ and angular coordinates are dimensionless. Fix ${\omega }_0$ in a spherical-polar chart with $0<{\vartheta }_0<\pi$, and let $N_{\mathcal{K}}={\mathrm{\partial }}_U+{\mathrm{\partial }}_V$ be the time orientation (Kruskal model properties: smooth radial function, Lorentzian metric, four-region isometries, vacuum and Kretschmann scalar). Take the point

$p_0:=(U,V,\omega )=(0,1,{\omega }_0)\in \mathcal{K}\mathrm{.}$

Then:

1. The radius and the cone coefficient at $p_0$. $U(p_0)V(p_0)=0$, so $\bar{r}(p_0)=1$ and $r(p_0)=L_0=s$ (the point lies on the horizon); thus $\bar{A}(p_0)=4e^{-1}$ and $A(p_0)=4L_0^2\mathrm{/}e$. Dividing the future causal criterion of The causal cone, the escape horizon and decreasing areal radius in the Kruskal model clause (a) by $L_0^2$, its normalized form at $p_0$ reads: $X$ is future-directed causal if and only if $X^U\ge 0$, $X^V\ge 0$, $X^U+X^V>0$ and $\frac{4}{e}{X}^U{X}^V\ge {\gamma }_{p_0}(X^{\omega },X^{\omega })$.
2. Four explicit tangent vectors at $p_0$.
   • $X_1={\mathrm{\partial }}_U+{\mathrm{\partial }}_V$: $X_1^U{X}_1^V=1\ge 0$ and $\frac{4}{e}\cdot 1\ge 0=\gamma (0,0)$, so $X_1$ is future causal; indeed $\bar{g}(X_1,X_1)=2{\bar{g}}_{UV}=-\bar{A}(p_0)=-\frac{4}{e}<0$, timelike.
   • $X_2={\mathrm{\partial }}_U$: $X_2^U=1$, $X_2^V=0$, so the inequality reads $0\ge 0$ and $X_2$ is future causal; moreover $\bar{g}(X_2,X_2)=0$, so ${\mathrm{\partial }}_U$ is future null at $p_0$; it is transverse to $U=0$, whose horizon generator direction is ${\mathrm{\partial }}_V$.
   • $X_3={\mathrm{\partial }}_V$: likewise $\bar{g}(X_3,X_3)=0$, $\bar{g}(X_3,N_{\mathcal{K}})=-\frac{\bar{A}(p_0)}{2}<0$; it is future null and tangent to the horizon generator.
   • $X_4={\mathrm{\partial }}_U-{\mathrm{\partial }}_V$: $X_4^U{X}_4^V=-1<0$ violates the criterion, and indeed $\bar{g}(X_4,X_4)=\bar{A}(p_0)=\frac{4}{e}>0$, spacelike.
3. Angular cut-off. For $X_5={\mathrm{\partial }}_U+{\mathrm{\partial }}_V+\omega$ with purely angular part $\omega$ one has $\bar{g}(X_5,X_5)=-\frac{4}{e}+{\gamma }_{p_0}(\omega ,\omega )$, so $X_5$ is future causal exactly when ${\gamma }_{p_0}(\omega ,\omega )\le \frac{4}{e}$. In the chart with round components ${\gamma }_{22}=1$, ${\gamma }_{33}={\sin }^2\vartheta$, the choices $\omega =(1,0)$ and $\omega =(2,0)$ give $\gamma =1<\frac{4}{e}$ (causal) and $\gamma =4>\frac{4}{e}$ (spacelike), because $1<e<4$; the cut-off is the exact inequality, not a numerical approximation.
4. Radial derivatives at $p_0$. By clause (c) of the causal lemma, ${\mathrm{\partial }}_U\bar{r}=-{\displaystyle \frac{V{e}^{-\bar{r}}}{\bar{r}}}$ and ${\mathrm{\partial }}_V\bar{r}=-{\displaystyle \frac{U{e}^{-\bar{r}}}{\bar{r}}}$ under the normalization $s=L_0$; at $p_0$ this gives ${\mathrm{\partial }}_U\bar{r}=-e^{-1}<0$ and ${\mathrm{\partial }}_V\bar{r}=0$, equivalently ${\mathrm{\partial }}_{U}r=-L_0{e}^{-1}$ and ${\mathrm{\partial }}_{V}r=0$. The normalized formula agrees with differentiating $UV=-(\bar{r}-1)e^{\bar{r}}$: at $(0,1)$, $VdU+UdV=-\bar{r}{e}^{\bar{r}}d\bar{r}$, so $dU=-ed\bar{r}$. Consequently the areal radius decreases along the future null direction ${\mathrm{\partial }}_U$ at the horizon, in accordance with the decreasing-radius clause of the causal lemma.

• What the example checks. The example instantiates the cone criterion and the radial derivatives at one explicit point with the model normalization $s=L_0=1\mathrm{m}$; accordingly its mass parameter is $m_{\ast }=c^2(1\mathrm{m})\mathrm{/}(2G_N)$. This is a choice of units and model parameter, not a measurement, and asserts no numerical value of $G_N$ or $c$ and no property of any observed system. The comparisons $1<4\mathrm{/}e<4$ are exact inequalities in the constant $e$.