content_image/295436.jpg content_image/295437.jpg content_image/295438.jpg content_image/295443.jpg
For a point particle of mass M and charge Q located at the origin variation of this action leads to the Reissner-Nordstr¨ om metric, 

There are now two special radial coordinates at which g$_{rr}$ = 0, 

<img src='content_image/295436.jpg'>

<img src='content_image/295437.jpg'>

<img src='content_image/295438.jpg'>

there can be zero, one, and two solutions to this equation. For GM 2 < Q 2 there are no solutions, which leads to a naked singularity at the origin (under very general conditions, the cosmic censorship conjecture prevents these from forming from gravitational collapse). For GM 2 > Q 2 there are two event horizons, both of which are removable coordinate singularities. The r$_{+}$ = r$_{−}$ single solution case defines an extremal black hole. 

## p branes 

Following [321, 322] consider the bosonic part of a supergravity action in 10 dimensions, a slight generalization of Eq. (2.24): 

<img src='content_image/295443.jpg'>

