Local charged Birkhoff rigidity away from critical areal gradients

By [F2], $d\varpi =0$. Any two points in a connected smooth manifold can be joined by a piecewise smooth path: the set reachable from a point by such paths and its complement are both unions of coordinate-ball neighborhoods, so connectedness makes the reachable set the whole patch. Along each smooth path segment $\sigma$, $(\varpi \circ \sigma {)}^{\mathrm{\prime }}=d\varpi ({\sigma }^{\mathrm{\prime }})=0$, so the one-variable mean-value theorem makes $\varpi$ constant along the segment. Thus $\varpi =m$ on the patch.

Let $X^A:={ϵ}_2^{AB}{\mathrm{\nabla }}_B^{q}r$, where indices on ${ϵ}_2$ are raised with $q^{-1}$. The volume form is nondegenerate, so $X\mathrm{\ne }0$ because $dr\mathrm{\ne }0$. Antisymmetry gives $X(r)=0$. In an oriented orthonormal frame with ${ϵ}_2=-e^0\wedge e^1$, direct contraction gives $q(X,X)=-(r_1{)}^2+(r_0{)}^2=-z$. By [F2] the Hessian is ${\mathrm{\nabla }}^{q}dr=\frac{1}{2}({\mathrm{\Delta }}_{q}r)q$, and [F3] gives ${\mathrm{\nabla }}^q{ϵ}_2=0$. Hence ${\mathrm{\nabla }}_A{X}_B=\frac{1}{2}({\mathrm{\Delta }}_{q}r){ϵ}_{BA}$, which is antisymmetric in $A,B$. In local coordinates, [F9] and the Lie-derivative definition give $({\mathcal{L}}_{X}q{)}_{AB}=X^C{\mathrm{\partial }}_C{q}_{AB}+q_{CB}{\mathrm{\partial }}_A{X}^C+q_{AC}{\mathrm{\partial }}_B{X}^C$. Expanding ${\mathrm{\nabla }}_A^q{X}_B+{\mathrm{\nabla }}_B^q{X}_A$ with $X_B=q_{BC}{X}^C$ and the Christoffel formula for ${\mathrm{\nabla }}^q$ in [F3] gives the same expression: the coefficient of $X^C$ reduces to ${\mathrm{\partial }}_C{q}_{AB}$ by $2{\mathrm{\Gamma }}^D{}_{AB}{q}_{DC}={\mathrm{\partial }}_A{q}_{BC}+{\mathrm{\partial }}_B{q}_{AC}-{\mathrm{\partial }}_C{q}_{AB}$. Thus ${\mathcal{L}}_{X}q=0$. Let ${\widehat{X}}_{(y,p)}:=(X_y,0)$ be the lift to $\mathcal{M}$; its local flow is $(y,p)\mapsto ({\mathrm{\Phi }}_t(y),p)$, so ${\mathcal{L}}_{\widehat{X}}{\pi }_S^{\ast }{\gamma }_{S^2}=0$. Applying the defining Lie-derivative formula to the warped metric yields ${\mathcal{L}}_{\widehat{X}}g={\pi }_U^{\ast }({\mathcal{L}}_{X}q)+2(r\circ {\pi }_U)(Xr){\pi }_S^{\ast }{\gamma }_{S^2}=0$. Thus the lifted vector field is Killing for the full four-dimensional metric, and in the constructed coordinates it is the lift $K$ of ${\mathrm{\partial }}_t$.

The smooth local flow theorem [F4] gives a local flow ${\mathrm{\Phi }}_t$ of the nonvanishing field $X$. To construct a transverse curve, choose a chart $(y^0,y^1)$ with ${\mathrm{\partial }}_{y^0}r\mathrm{\ne }0$ at the point; the map $(y^0,y^1)\mapsto (r(y^0,y^1),y^1)$ has invertible differential there. The inverse function theorem [F5], after restricting the chart, therefore gives a smooth curve $\sigma (s)$ through the point with $r(\sigma (s))=s$. Define $\mathrm{\Psi }(t,s):={\mathrm{\Phi }}_t(\sigma (s))$ on a sufficiently small product neighborhood. The identity $X(r)=0$ gives $r(\mathrm{\Psi }(t,s))=s$. At the initial point the two derivatives of $\mathrm{\Psi }$ are $X$ and ${\sigma }^{\mathrm{\prime }}(s)$; they are independent because $dr(X)=0$ and $dr({\sigma }^{\mathrm{\prime }})=1$. Thus $d\mathrm{\Psi }$ is invertible, and [F5] makes $(t,r)$ valid local coordinates with $X={\mathrm{\partial }}_t$.

Since $X={\mathrm{\partial }}_t$ and ${\mathcal{L}}_{X}q=0$, each coordinate coefficient is independent of $t$, so $q=A(r)d{t}^2+2B(r)dtdr+C(r)d{r}^2$. From step 1.2, $A=q(X,X)=-z\mathrm{\ne }0$. Both $A$ and $B$ are smooth functions of $r$, so $B\mathrm{/}A$ is smooth on the local $r$-interval. Define $G(r):={\int }_{r_0}^{r}B(s)\mathrm{/}A(s)ds$. Applying [F6] on each compact subinterval containing $r_0$ and $r$ gives $G^{\mathrm{\prime }}=B\mathrm{/}A$, first making $G$ continuously differentiable. Since its derivative $B\mathrm{/}A$ is smooth, repeated differentiation shows $G$ is smooth. The coordinate change $(t,r)\mapsto (\tilde{t},r):=(t+G(r),r)$ and its inverse $(\tilde{t},r)\mapsto (\tilde{t}-G(r),r)$ are therefore smooth. Substituting $dt=d\tilde{t}-G^{\mathrm{\prime }}(r)dr$ makes the cross-term coefficient $2(B-A{G}^{\mathrm{\prime }})=0$. The coordinate $r$ is unchanged, so $q^{rr}=q^{-1}(dr,dr)=z$; for a diagonal two-metric this implies $q_{\tilde{t}\tilde{t}}=-z$ and $q_{rr}=z^{-1}$.

Solving $\varpi =m$ from step 1.1 for $z$ gives $z=1-2m\mathrm{/}r+(q_e^2+q_m^2)\mathrm{/}{r}^2-\mathrm{\Lambda }{r}^2\mathrm{/}3=:f(r)$. Substituting this in step 3.1 gives the stated metric; $f\mathrm{\ne }0$ by [A2]. In these diagonal coordinates $\det q=-1$, so the metric volume form has the form ${ϵ}_2=sd\tilde{t}\wedge dr$ for a locally constant $s\in \{-1,1\}$. Raising indices gives ${ϵ}_2^{\tilde{t}r}=-s$, and hence $X^{\tilde{t}}={ϵ}_2^{\tilde{t}r}{\mathrm{\partial }}_{r}r=-s$. Since the construction gives $X={\mathrm{\partial }}_{\tilde{t}}$, it follows that $s=-1$. Thus ${ϵ}_2=-d\tilde{t}\wedge dr$ and the Maxwell form in [F1] is exactly the stated field. Since $q({\mathrm{\partial }}_{\tilde{t}},{\mathrm{\partial }}_{\tilde{t}})=-f$, its causal character has the claimed sign. For $\mathrm{\Lambda }=0$, this is precisely the signature and field normalization of [F7]. Renaming $\tilde{t}$ as $t$ gives the coordinates in the Statement. No extension across $f=0$ or global uniqueness follows from these local coordinates.