Einstein-Maxwell system in $3+1$ form and initial data for multiple charged black holes

We consider the Einstein-Maxwell system as a Cauchy initial value problem taking the electric and magnetic fields as independent variables. Maxwell’s equations in curved spacetimes are derived in detail using a $3+1$ formalism and their hyperbolic properties are analyzed, showing that the resulting system is symmetric hyperbolic. We also focus on the problem of finding initial data for multiple charged black holes assuming time-symmetric initial data and using a puncturelike method to solve the Hamiltonian and the Gauss constraints. We study the behavior of the resulting initial data families, and show that previous results in this direction can be obtained as particular cases of our approach.

Figure 1

Numerical solution for the function $u$ along the $z$ axis for the case of equal masses and equal but opposite charges.

Figure 2

Height map in the $(\rho ,z)$ plane of the numerical solution for the function $u$.

Figure 3

Convergence of the Hamiltonian constraint. We plot the logarithm of the absolute value of the Hamiltonian constraint evaluated at 5 different resolutions, with each resolution rescaled by the corresponding factor in order to show second order convergence.

Figure 4

Similar to Fig. 1, but for the case of equal masses and one charge equal to zero.

Figure 5

Similar to Fig. 1 for a more generic case in which the back hole at $z=2$ has $M_1=1$, $Q_1=1/2$, while for the black hole at $z=-2$ we take $M_2=1/2$ and a series of values for the charge: $Q_2=-0.25,-0.1,0,0.1,0.25$.

Figure 6

Solution for $u$ in the case when $M_1=1$, $M_2=0.5$, $Q_1=0.5$, $Q_2=0.01475$. Notice that the kink at $z=-2$ is essentially gone.