Einstein-Maxwell theory

The Einstein-Maxwell system couples the metric to an electromagnetic field. In geometrized units its source-free equations are $G_{ab}=8\pi T_{ab}^{\mathrm{EM}}$, ${\nabla }_a{F}^{ab}=0$ and $dF=0$, with $T_{ab}^{\mathrm{EM}}=(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})/(4\pi )$. Electrovacuum need not have vanishing Ricci tensor, so a Komar integral need not be unchanged between different enclosing surfaces.

The four-dimensional Majumdar–Papapetrou solution is a static multi-black-hole Einstein-Maxwell solution with $H=1+{\sum }_i{M}_i/|\mathbf{x}-{\mathbf{x}}_i|$ and positive geometrized masses $M_i$. The electromagnetic potential can be written $A=\pm (H^{-1}-1)dt$, with a common sign of charge. Its mass at infinity is ${\sum }_i{M}_i$. Each distinct centre is an extremal horizon of zero surface gravity rather than a pointlike naked metric singularity.

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 $H$.