Selected ingoing RN extension and time orientation — Examples

For each $r>0$, $r^{2}f(r)=r^2-2Mr+Q^2=(r-M{)}^2+(Q^2-M^2)>0.$ Therefore $f(r)>0$ and the quadratic has no real root. Since $g^{-1}(dr,dr)=f$ by [F1], every positive constant-$r$ level has spacelike conormal, not a null conormal. This proves there is no positive radial root or constant-$r$ null level in this example.

Fix any $r_0>0$, $v_0$ and sphere point ${\omega }_0$. For $r\ge r_0$ put $v(r)=v_0+{\int }_{r_0}^r\frac{2}{f(s)}ds\mathrm{.}$ The integrand is smooth and positive, and [F2] gives $v^{\mathrm{\prime }}(r)=2\mathrm{/}f(r)$. The curve $\gamma (r)=(v(r),r,{\omega }_0)$ has tangent $2f(r{)}^{-1}{\mathrm{\partial }}_v+{\mathrm{\partial }}_r=2f(r{)}^{-1}n$, a positive multiple of the future null field in [F1]. Reparameterizing by $s=r-r_0\in [0,\mathrm{\infty })$ gives the path domain in [L1]. It is defined for every finite $r\ge r_0$ and reaches arbitrarily large radii under [L1]. Thus every point belongs to the escape set.

Step 1.2 includes every point in the escape set; its complement is empty, so its relative boundary is empty. Step 1.1 separately excludes a constant-$r$ null level. These are algebraic facts about this model and do not imply censorship or collapse conclusions.

Substituting $q_e^2+q_m^2=M^2$ gives $f(r)=1-\frac{2M}{r}+\frac{M^2}{r^2}=(1-M\mathrm{/}r{)}^2\mathrm{.}$ Thus $f(M)=f^{\mathrm{\prime }}(M)=0$, so the positive root is double. By [F1], its surface gravity is $f^{\mathrm{\prime }}(M)\mathrm{/}2=0$.

The $v,r$ block in [F3] has determinant $-1$ for every $f$, and at $r=M$ the sphere metric $M^2\gamma$ is positive definite. Hence the metric is nondegenerate and Lorentzian at the double root. The extra chart in [F2] assumes a simple root and $\kappa >0$, both of which fail by step 1.1; that construction therefore supplies no extremal bifurcation chart.

Steps 1.1 and 2.1 establish the double-root, zero-surface-gravity case and the regular ingoing metric while delimiting the simple-root construction. No maximal extension or physical existence conclusion follows.

In RN, [F1] gives ${\int }_{S_r}\mathcal{F}=4\pi q_m\mathrm{\ne }0$. If $S_r=\mathrm{\partial }c$ for a finite smooth singular 3-chain in the field domain, then $d\mathcal{F}=0$ by the local potential identities in [F1] and $d^2=0$. Applying [F2] would give $4\pi q_m={\int }_{S_r}\mathcal{F}={\int }_{c}d\mathcal{F}=0,$ a contradiction. Thus $S_r$ is nonbounding.

The conditional flux interface [L1] gives ${\int }_{S_r}{\mathcal{F}}_{\beta }=4\pi q\sin \beta \mathrm{\ne }0$. By [L2], the unrotated field is a smooth source-free geometric electrovac datum on the full-sphere exterior, and its constant dual rotation is closed on the annulus. If $S_r=\mathrm{\partial }c$ for a finite smooth singular 3-chain there, [F2] gives ${\int }_{S_r}{\mathcal{F}}_{\beta }={\int }_{c}d{\mathcal{F}}_{\beta }=0$, a contradiction. A global smooth real one-form potential would likewise give zero flux by Stokes and $\mathrm{\partial }{S}_r=0$. The north and south stereographic sphere charts, each omitting one pole, give product coordinate domains $\mathbb{R}\times (R_0,R_1)\times {\mathbb{R}}^2$, which are convex and hence star-shaped about any of their points. Pull back ${\mathcal{F}}_{\beta }$ to either domain, apply [L3] with $k=2$, and transport the resulting primitive back. This supplies smooth real patch potentials covering the annulus. Thus the KN sphere is nonbounding and its field is locally but not globally exact, conditionally on [L1-L2].

If a global smooth potential $\mathcal{A}$ existed, then [F2] and $\mathrm{\partial }{S}_r=0$ would give $4\pi q_m={\int }_{S_r}d\mathcal{A}={\int }_{\mathrm{\partial }{S}_r}\mathcal{A}=0,$ contradicting step 1.1's nonzero flux. The smooth patch potentials in [F1] show that the obstruction is global exactness, not local existence.

The RN conclusion follows from the current RN local-gauge proof and finite-chain Stokes. The KN conclusion remains conditional on the exact batch-21 flux, full-sphere verification and duality interfaces and their draft proofs. Neither yields a regular magnetic source or a quantization condition.

At $\theta =\pi \mathrm{/}2$, the norm in [F1] vanishes when $r^2-2Mr+q^2=0$. Since $M^2-q^2=(M^2-a^2-q^2)+a^2>0$, its outer root is the real value $r_E=M+\sqrt{M^2-q^2}$.

The outer horizon radius in [F2] is $r_H=M+\sqrt{M^2-a^2-q^2}$. Put $u=\sqrt{M^2-q^2}$ and $w=\sqrt{M^2-a^2-q^2}$. Then $u,w\ge 0$ and $(u-w)(u+w)=a^2>0$. Since $u+w>0$, we have $u>w$, hence $r_E>r_H$. Therefore these two level sets are distinct at the equator, refuting the universal claim.

At the equatorial horizon, [F1] gives $g(T,T)=a^2\mathrm{/}{r}_H^2>0$, whereas [F2] gives a null horizon generator. Thus the stationary field itself is spacelike there and is not the horizon generator. On the rotation axis $\sin \theta =0$ the displayed outer radii coincide; this is the boundary case and does not restore the universal claim. The deductions remain conditional on [F1-F2]'s stated interfaces.