Spherical orbit geometry and rotational invariance

Let $U$ be a nonempty, connected, simply connected, second-countable smooth two-manifold without boundary. Let $q$ be a smooth symmetric covariant $2$-tensor field on $U$ such that $q_y$ is nondegenerate of signature $(-,+)$ for every $y\in U$. Choose a smooth vector field $N$ on $U$ with $q(N,N)<0$ everywhere; the time orientation is the choice of the connected component containing $N_y$ in each timelike cone $\{v\in T_{y}U:q_y(v,v)<0\}$. Choose an orientation of $U$, and let ${ϵ}_2$ be its metric volume form, characterized by ${ϵ}_2(e_0,e_1)=1$ on every positively oriented $q$-orthonormal frame with $q(e_0,e_0)=-1$ and $q(e_1,e_1)=1$. Let $r:U\to (0,\mathrm{\infty })$ be smooth. Take $S^2=\{p\in {\mathbb{R}}^3:p\cdot p=1\}$ with the smooth structure and induced round metric fixed by the earlier dyonic Reissner–Nordström data definition cited below. Its tangent plane is $T_p{S}^2=\{u\in {\mathbb{R}}^3:p\cdot u=0\}$, its metric is ${\gamma }_{S^2,p}(u,v)=u\cdot v$, and its outward area form is $({\omega }_{S^2}{)}_p(u,v)=\det (p,u,v)$. Write $\wedge$ for the exterior product of alternating forms. In polar coordinates ${\omega }_{S^2}=\sin \theta d\theta \wedge d\varphi$. Put $\mathcal{M}:=U\times S^2$ and define its projections by ${\pi }_U(y,p):=y$ and ${\pi }_S(y,p):=p$. For a smooth map $F:X\to Y$ and a smooth covariant $2$-tensor field $h$ on $Y$, define its pullback by $(F^{\ast }h{)}_x(v,w):=h_{F(x)}(d{F}_{x}v,d{F}_{x}w),$ where $x\in X$, $v,w\in T_{x}X$, and $d{F}_x:T_{x}X\to T_{F(x)}Y$ is the derivative. This formula also defines pullback of a covariant $2$-form. Define $g:={\pi }_U^{\ast }q+(r\circ {\pi }_U{)}^2{\pi }_S^{\ast }{\gamma }_{S^2}\mathrm{.}$ It has Lorentzian signature $(-,+,+,+)$. The fibre over $y\in U$ is $S_y^2:={\pi }_U^{-1}(\{y\})=\{y\}\times S^2$. Its induced Riemannian metric is $r(y{)}^2{\gamma }_{S^2}$ and its area form is $r(y{)}^2{\omega }_{S^2}$; in polar angles $0<\theta <\pi$ and $0\le \varphi <2\pi$, its area is ${\int }_0^{2\pi }{\int }_0^{\pi }r(y{)}^2\sin \theta d\theta d\varphi =4\pi r(y{)}^2$. Thus $r(y)>0$ is the areal radius of this orbit sphere. Choose the four-dimensional orientation so that its metric volume form is ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=-(r\circ {\pi }_U{)}^2{\pi }_U^{\ast }{ϵ}_2\wedge {\pi }_S^{\ast }{\omega }_{S^2}\mathrm{.}$ In any chart $(x^0,r)$ for which ${ϵ}_2=-d{x}^0\wedge dr$, this gives ${\mathrm{v}\mathrm{o}\mathrm{l}}_g=d{x}^0\wedge dr\wedge r^2{\omega }_{S^2}$.

Define $SO(3):=\{R\in {\mathbb{R}}^{3\times 3}:R^{\mathsf{T}}R=I_3,\det R=1\}\mathrm{.}$ Each $R\in SO(3)$ acts on $S^2$ by $p\mapsto Rp$ and defines the smooth diffeomorphism ${\rho }_R:\mathcal{M}\to \mathcal{M}$, ${\rho }_R(y,p)=(y,Rp)$, whose inverse is ${\rho }_{R^{\mathsf{T}}}$. For nonnegative integers $a,b$, a smooth tensor field of type $(a,b)$ on $\mathcal{M}$ is a smooth section of $T_b^a\mathcal{M}:=(T\mathcal{M}{)}^{\otimes a}\otimes (T^{\ast }\mathcal{M}{)}^{\otimes b};$ at each $x\in \mathcal{M}$ its value is multilinear on $a$ covectors in $T_x^{\ast }\mathcal{M}$ and $b$ vectors in $T_x\mathcal{M}$, where $T_x^{\ast }\mathcal{M}:={\mathrm{Hom}}_{\mathbb{R}}(T_x\mathcal{M},\mathbb{R})$. Smoothness means that the components of the section in every smooth product chart are $C^{\mathrm{\infty }}$. For such a field $T$, define its pullback by the diffeomorphism ${\rho }_R$ by $({\rho }_R^{\ast }T{)}_x({\alpha }_1,\dots ,{\alpha }_a;v_1,\dots ,v_b)=T_{{\rho }_R(x)}({\alpha }_1\circ (d{\rho }_R{)}_x^{-1},\dots ,{\alpha }_a\circ (d{\rho }_R{)}_x^{-1};(d{\rho }_R{)}_x{v}_1,\dots ,(d{\rho }_R{)}_x{v}_b)$ where ${\alpha }_i\in T_x^{\ast }\mathcal{M}$, $v_j\in T_x\mathcal{M}$, and $(d{\rho }_R{)}_x:T_x\mathcal{M}\to T_{{\rho }_R(x)}\mathcal{M}$ is the derivative of ${\rho }_R$ at $x$. A real two-form is a smooth alternating tensor field of type $(0,2)$. The field $T$ is rotationally invariant if ${\rho }_R^{\ast }T=T$ for every $R\in SO(3)$; a two-form $\mathcal{F}$ is rotationally invariant in exactly this sense. This definition specifies a local warped-product model and a group action. No embedding into a larger spacetime, global potential, maximal extension, or field equation is assumed.

Dyonic Reissner-Nordström metric and field domains

The displayed projection is only a schematic: its circle stands for the entire round two-sphere fibre. The proof of no result relies on the drawing.

\begin{tikzpicture}[scale=1]
  \draw[rounded corners] (0,0) rectangle (2.6,1.7);
  \node at (1.3,1.32) {$U$};
  \fill (1.3,0.72) circle (1.5pt);
  \node[below] at (1.3,0.65) {$y$};
  \draw[rounded corners] (5.0,0) rectangle (9.2,1.7);
  \node at (7.1,1.35) {$\mathcal M=U\times S^2$};
  \draw (7.1,0.68) circle (0.43);
  \node at (7.1,0.68) {$S^2_y$};
  \draw[->] (5.0,0.55) -- (2.6,0.55)
    node[midway,above] {$\pi_U(y,p)=y$};
  \node[below] at (7.1,0.08) {$S^2_y=\pi_U^{-1}(\{y\})=\{y\}\times S^2$};
\end{tikzpicture}