RN curvature invariants and the C2 extension obstruction

The scalar curvature and both polynomial invariants are the exact identities in [F1]. If $Q\mathrm{\ne }0$, then $4Q^4\mathrm{/}{r}^8\to +\mathrm{\infty }$ as $r\downarrow 0$; if $Q=0$ and $M\mathrm{\ne }0$, then $48M^2\mathrm{/}{r}^6\to +\mathrm{\infty }$.

If $M=Q=0$, then $q_e=q_m=0$, $f=1$ and $\mathcal{F}=0$ by [F3]. The map $\mathrm{\Phi }(x^0,r,p)=(x^0,rp)$ is a smooth diffeomorphism onto $\mathbb{R}\times \{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }\in J\}$, with inverse $(x^0,x)\mapsto (x^0,\mathrm{\mid }x\mathrm{\mid },x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })$ since $r>0$. As $p\cdot p=1$ implies $p\cdot dp=0$, the Euclidean metric pulls back to $d{r}^2+r^2{\gamma }_{S^2}$. Thus $g={\mathrm{\Phi }}^{\ast }(-(d{x}^0{)}^2+{\sum }_{i=1}^3(d{x}^i{)}^2)$. In these Cartesian coordinates the zero connection is torsion-free and metric-compatible, hence is the metric-selected connection; [F4] gives zero curvature, which is the asserted flatness. The invariants vanish by [F1]. This covers the remaining parameter case without inferring flatness from vanishing scalar contractions.

Suppose, for contradiction, that a $C^2$ nondegenerate extension $(\tilde{M},\tilde{g},j)$ and an added point $p\mathrm{\notin }j(U)$ realize the approach of a sequence from [A1], so $j(x_n)\to p$. This is exactly the setup in [F2]. By step 1.1 one of the two polynomial complete curvature contractions diverges along that sequence in the corresponding charge case, contradicting the obstruction criterion in [F2]. Hence no such extension realizes the approach. This does not create a point source or assert geodesic incompleteness.