55 Chapter 2: Motivation where φ is the dilaton field, F$_{p}$$_{+2}$ is a p +2 form field strength, and l$_{s}$ is the string length; the string length l$_{s}$ is related to the string tension (2 πα $^{′}$)− 1 by l 2 s = α $^{′}$. This form is suggestive of the result G (10) N = 8 π $^{6}$g2 $_{s}$l 8 s [320], where g$_{s}$ is given by the vacuum expectation value of the dilaton field, g$_{s}$ = exp( φ ). Just as a 0 dimensional point charge sources a 1 form potential A which creates a 2 form field strength F = dA in E&M, this p + 2 form is sourced by a p dimensional object, a p brane. Similarly to E&M, setting a spherically symmetric charge at the origin the integral of the flux of the p + 2 form yields the charge: Extremizing the above action for a p brane with the p parallel dimensions denoted by x $^{i}$, the radial distance away from the p brane denoted by ρ , and the angular dimensions encoded in d Ω, yields the metric ∫ S 8 − p ∗ F$_{p}$$_{+2}$ = N. (2.29) ds 2 = − f$_{+}$ ( ρ ) $^{√}$f$_{−}$ ( ρ ) dt 2 + $^{√}$f$_{−}$ ( ρ ) p ∑ i =1 dx $^{i}$dxi + f$_{−}$ ( ρ ) − 1 $_{2}$− 5 − p 7 − p f$_{+}$ ( ρ ) dρ 2 + ρ $^{2}$f$_{−}$ ( ρ ) 1 $_{2}$− 5 − p 7 − p d Ω 2 8 − $_{p}$; (2.30) e − 2 φ = 1 g 2 s f$_{−}$ ( ρ ) − p − 3 2 , (2.31) f$_{±}$ ( ρ ) = 1 − $^{(}$r$_{±}$ ρ ) 7 − p . (2.32)