The key ingredi-ent, introduced by Reiris in [23], is the monotonicity of the Hawking energy(equivalent to the Misner-Sharp energy in spherical symmetry) on untrappedregions.

We only need to compute the null expansions of thespheres in term of the mass and the charge

Wesay the data iselectrovacuumifμM= 0 andj= 0.

Theξibe one of the Killing vectors that generate the groupSO(3), then we say thethe initial data set isspherically symmetricif£ξhij=£ξKij=£ξμ=£ξji= 0,(100)for all the generatorsξofSO(3),