 Analytic extension of a Majumdar–Papapetrou horizon

Near a distinct centre of mass $M$, write $H=M/r+h+rF(r,\theta ,\varphi )$, where $h$ is the regular part evaluated at that centre and $F$ is analytic in local angular charts. Set $A=M+hr$, $Q=rH=A+r^{2}F$ and $dv=dt+(h+M/r{)}^{2}dr$. The metric becomesAll 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 $H$.