= Analytic extension of a Majumdar–Papapetrou horizon
Near a distinct centre of mass $M$, write $H=M/r+h+rF(r,\theta,\phi)$, where $h$ is the regular part evaluated at that centre and $F$ is analytic in local angular charts. Set $A=M+hr$, $Q=rH=A+r^2F$ and $dv=dt+(h+M/r)^2dr$. The metric becomes
$$
ds^2=-\frac{r^2}{Q^2}dv^2+2\frac{A^2}{Q^2}dv\,dr+F\frac{(Q+A)(Q^2+A^2)}{Q^2}dr^2+Q^2d\Omega_2^2.
$$
All coefficients extend analytically across $r=0$, with $Q(0)=M>0$ and two-dimensional determinant $-1$. The surface is null, its generator is $\partial_v$, and the double zero of its squared norm gives zero surface gravity. The construction uses a constant $h$ so that the coordinate differential is integrable despite the nonspherical regular part of $H$.
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