56 Chapter 2: Motivation r$_{±}$ characterize the mass and charge of the p brane by where d$_{p}$ is a numerical factor given by M = 1 (7 − p )(2 π ) $^{7}$d$_{p}$ g 2 $_{s}$l 8 s $^{(}$(8 − p ) r 7 − p + − r 7 − p − ) (2.33) N = 1 d$_{p}$g$_{s}$l 7 − p s ( r$_{+}$r$_{−}$ ) 7 − p 2 , (2.34) d$_{p}$ = 2 5 − $^{p}$π 5 − p 2 Γ$^{(}$ 7 − p 2 ) . (2.35) ds 2 = $^{√}$f$_{+}$ ( ρ )$^{(}$ − dt 2 + p ∑ i =1 dx $^{i}$dxi ) + f$_{+}$ ( ρ ) 3 $_{2}$− 5 − p 7 − p dρ 2 + ρ $^{2}$f$_{+}$ ( ρ ) 1 $_{2}$− 5 − p 7 − p d Ω 2 8 − $_{p}$. (2.36) r 7 − p = ρ 7 − p − r 7 − p + (2.37) The interpretation for p branes is exactly the same as for charged black holes: r$_{−}$ > r$_{+}$ gives a naked singularity at r = r$_{−}$ ; r$_{+}$ > r$_{−}$ a horizon and removable coordinate singularity at r = r$_{+}$ . For an extremal p brane r$_{+}$ = r$_{−}$ and Eq. (2.30) becomes Changing coordinates to a distance beyond the horizon