Analytic Kerr coordinates across the chosen nondegenerate horizon

By the defining series and their infinite radii, the power-series analyticity theorem applies to sine and cosine at every real ${\theta }_0$. Their expansions about ${\theta }_0$ also give multivariable germs by assigning exponent zero to the other coordinates; their absolute convergence is unchanged. The coordinate projections and constants have finite power-series expansions.

The quantities $q^2$, $\mathrm{\Delta }$, ${\mathrm{\Sigma }}^2$ and all component numerators are finite sums and products of those analytic functions. Since $q^2\ge r^2>0$ on $D$, the quotient theorem applies everywhere on $D$ to the displayed components. Thus all components are real analytic, including at $\mathrm{\Delta }=0$, and no denominator contains $\mathrm{\Delta }$.

To verify that this analytic expression stays Lorentzian, write $s=\sin \theta$, $k=du-a{s}^{2}d\phi$, $\mathrm{\ell }=dr+a{s}^{2}d\phi$, and $f=(2mr-q^2)\mathrm{/}{q}^2$. Using ${\sin }^2\theta +{\cos }^2\theta =1$ gives ${\mathrm{\Sigma }}^2=(r^2+a^2)q^2+2mr{a}^2{s}^2$; substitution into the displayed expression yields $g=f{k}^2-(k\otimes \mathrm{\ell }+\mathrm{\ell }\otimes k)+q^{2}d{\theta }^2+q^2{s}^{2}d{\phi }^2$. The forms $(k,\mathrm{\ell },d\theta ,d\phi )$ are a coframe, since $du=k+a{s}^{2}d\phi$ and $dr=\mathrm{\ell }-a{s}^{2}d\phi$. Its first block is $\left(\begin{array}{cc}f& -1\\ -1& 0\end{array}\right)$, with eigenvalues $(f\pm \sqrt{f^2+4})\mathrm{/}2$, one positive and one negative. The other two diagonal coefficients are positive because $q^2>0$ and $s>0$. Hence the metric has signature $(-,+,+,+)$ on all of $D$. The smooth field $N={\mathrm{\partial }}_u+(1+f^2){\mathrm{\partial }}_r\mathrm{/}2$ has $g(N,N)=f-1-f^2=-(f-1\mathrm{/}2{)}^2-3\mathrm{/}4<0$, so it supplies a time orientation for the horizon conventions below.

On any radial interval where $\mathrm{\Delta }\mathrm{\ne }0$, choose local primitives $F^{\mathrm{\prime }}(r)=(r^2+a^2)\mathrm{/}\mathrm{\Delta }$ and $G^{\mathrm{\prime }}(r)=a\mathrm{/}\mathrm{\Delta }$ and set $t=u+F(r)$, $\varphi =\phi +G(r)$. The coordinate change has triangular Jacobian of determinant one. Substituting $du=dt-F^{\mathrm{\prime }}(r)dr$ and $d\phi =d\varphi -G^{\mathrm{\prime }}(r)dr$ in the outgoing expression gives $g=-\frac{q^2\mathrm{\Delta }}{{\mathrm{\Sigma }}^2}d{t}^2+\frac{{\mathrm{\Sigma }}^2{\sin }^2\theta }{q^2}(d\varphi -\frac{2amr}{{\mathrm{\Sigma }}^2}dt{)}^2+\frac{q^2}{\mathrm{\Delta }}d{r}^2+q^{2}d{\theta }^2$, exactly (3.2). Here ${\mathrm{\Sigma }}^2=(r^2+a^2)q^2+2mr{a}^2{s}^2>0$, so this comparison has no extra singular denominator. This is a local algebraic identification wherever $\mathrm{\Delta }\mathrm{\ne }0$, without a restriction on the sign of $2mr-q^2$. Steps 2.1 and 3.1 supply the regular outgoing-coordinate extension at the missing radial roots, rather than attributing that larger domain to the source's stated region $R$.

The components are independent of $u$ and $\phi$. Translation in either coordinate therefore preserves the metric locally, so the corresponding coordinate derivatives are Killing. Their commutator on every smooth function is zero because mixed coordinate derivatives commute. The proof makes no assertion about local Killing fields of a deformed metric.

At $r=r_{+}$, $\mathrm{\Delta }=0$ and $2m{r}_{+}=r_{+}^2+a^2$, so $2m{r}_{+}-q^2=a^2{\sin }^2\theta >0$ off the axis. Thus every such horizon point is in the source region $R$, and the analytic regularity just proved applies across the chosen horizon there.

The level $H$ is a nonempty smooth embedded hypersurface, since $r$ is a coordinate and $dr\mathrm{\ne }0$. The field $K=T+(a\mathrm{/}B)Z$ is Killing: its local flow is $(u,\phi )\mapsto (u+\tau ,\phi +(a\mathrm{/}B)\tau )$, which preserves the components from step 4.2. It is tangent to $H$ and nowhere zero, since its $u$ component is one. At $H$, use $2m{r}_{+}=B$ and $q_H^2=B-a^2{\sin }^2\theta$ in the displayed metric to obtain $g(K,{\mathrm{\partial }}_{\theta })=g(K,{\mathrm{\partial }}_u)=g(K,{\mathrm{\partial }}_{\phi })=0$ and $g(K,{\mathrm{\partial }}_r)=-1+(a^2\mathrm{/}B){\sin }^2\theta =-q_H^2\mathrm{/}B$. Thus $K^{\mathrm{♭}}=-(q_H^2\mathrm{/}B)dr$ as an ambient covector on $H$. In particular $K$ is a nonzero tangent vector orthogonal to every tangent vector of $H$, so the induced metric has nonzero radical and $H$ is null. Its nonzero-generator part is all of $H$, hence nonempty and dense. This verifies every Killing-horizon condition in [F7] on the time-oriented metric of step 3.1.

Keep $\mathrm{\Omega }=a\mathrm{/}B$ constant off $H$. Direct contraction of the displayed metric, using the Pythagorean identity, gives $g(K,K)=-\mathrm{\Delta }(1-a\mathrm{\Omega }{\sin }^2\theta {)}^2\mathrm{/}{q}^2+{\sin }^2\theta ((r^2+a^2)\mathrm{\Omega }-a{)}^2\mathrm{/}{q}^2$. On $H$, both $\mathrm{\Delta }$ and $(r^2+a^2)\mathrm{\Omega }-a$ vanish, while $1-a\mathrm{\Omega }{\sin }^2\theta =q_H^2\mathrm{/}B$. The differential of the squared second factor vanishes there; the differential of the first term is therefore $d(g(K,K))=-{\mathrm{\Delta }}^{\mathrm{\prime }}(r_{+})q_H^2{B}^{-2}dr$. This is an ambient differential: angular derivatives vanish because the norm restricts identically to zero on $H$, and the only remaining derivative is radial. Combining with $K^{\mathrm{♭}}=-(q_H^2\mathrm{/}B)dr$ gives ${\kappa }_H=-{\mathrm{\Delta }}^{\mathrm{\prime }}(r_{+})\mathrm{/}(2B)=-(r_{+}-m)\mathrm{/}B$. Since $r_{+}-m=\sqrt{m^2-a^2}>0$ and $B>0$, the scalar is nowhere zero. The chosen outgoing branch and generator normalization account for its negative sign; nondegeneracy uses its nonvanishing. The extension through $H$ and its inclusion in $R$ remain those of steps 2.1–5.1.