Constant-radius Bertotti-Robinson product data

Write $S^2=\{p\in {\mathbb{R}}^3:p\cdot p=1\}$ with its usual smooth structure and induced round metric as in Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$ and The round metric on the sphere as an induced metric. Explicitly, $T_p{S}^2=\{u\in {\mathbb{R}}^3:p\cdot u=0\}$ and ${\gamma }_{S^2,p}(u,v):=u\cdot v$. Use the outward orientation, whose area form is ${\omega }_{S^2,p}(u,v):=\det (p,u,v)$, with determinant in the standard orientation of ${\mathbb{R}}^3$.

Let $S^2=\{n\in {\mathbb{R}}^3:\mathrm{\mid }n\mathrm{\mid }=1\}$ carry its induced unit round metric ${\gamma }_{S^2}$ and outward area form ${\omega }_{S^2}$. Let $L>0$ have length units. Choose either (i) $f(z)=z^2$ on an open $J\subset (0,\mathrm{\infty })$, or (ii) $f(z)=1+z^2$ on an open $J\subset \mathbb{R}$. The coordinate $z$ and $\tau$ are dimensionless. On ${\mathcal{M}}_{L,f}={\mathbb{R}}_{\tau }\times J_z\times S^2$, orient by $d\tau \wedge dz\wedge {\omega }_{S^2}$ and define

$g_{L,f}=L^2(-f(z)d{\tau }^2+f(z{)}^{-1}d{z}^2+{\gamma }_{S^2}),{\mathcal{F}}_L=-Ld\tau \wedge dz\mathrm{.}$

The smooth field $N=(L\sqrt{f}{)}^{-1}{\mathrm{\partial }}_{\tau }$ is unit timelike; it selects the future direction. This explicit metric-field tuple is the Bertotti-Robinson product data for this library. “Electrovac” refers to the equations in Geometric electrovac equations and flux normalization and is not asserted by this definition. The product has sphere radius $L$ at every base point.