If we set the charge of Reissner–Nordström soultion purely imaginary, the metric would be well defined:
ds2=(1−rsr−r2Qr2)dt2−(1−rsr−r2Qr2)−1dr2−r2dΩ2,d�2=(1−�s�−�Q2�2)d�2−(1−�s�−�Q2�2)−1d�2−�2dΩ2,
where rs=2M�s=2�, rQ=|Q|2�Q=|�|2 (in units G=c=k=1�=�=�=1). However, making the EM potential purely imaginary (A=i|Q|/rdt�=�|�|/�d�) would make the energy of EM field negative and therefore the solution is unphysical. Therefore the geometry is rarely considered in literature.
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