54 Chapter 2: Motivation 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, ds 2 = − Δ dt 2 + Δ − $^{1}$dr2 + r $^{2}$d Ω 2 $_{2}$, (2.25) Δ = 1 − 2 GM r + GQ 2 r 2 . (2.26) r$_{±}$ = GM ± $^{√}$G $^{2}$M2 − GQ $^{2}$; (2.27) There are now two special radial coordinates at which g$_{rr}$ = 0, 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. Following [321, 322] consider the bosonic part of a supergravity action in 10 dimensions, a slight generalization of Eq. (2.24): p branes S = 1 (2 π ) $^{7}$l8 s ∫ d $^{10}$x √ − g$^{(}$e − 2 $^{φ}$( R + 4( ∇ φ ) $^{2}$) + 2 (8 − p )! F 2 p +2 ) , (2.28)