53 Chapter 2: Motivation particle is [320] where ω$_{d}$ is the volume of a d -dimensional sphere of radius 1 ( ω$_{1}$ = 2 π , ω$_{2}$ = 4 π , etc.). We see that n = 4 gives the usual 1 /r potential. Eq. (2.21) yields the n -dimensional gravitational force on a test mass m , φ ( ⃗r ) = − κ $^{2}$Mc3 2( n − 2) ω$_{(}$$_{n}$$_{−}$$_{2)}$ 1 | ⃗r | n − $_{3}$, (2.21) ⃗ F ( ⃗r ) = − m ⃗ ∇ φ ( ⃗r ) = − 8( n − 3) πG ( n ) N mM ( n − 2) ω$_{(}$$_{n}$$_{−}$$_{2)}$ ⃗r | ⃗r | n − $_{1}$. (2.22) ds 2 = −$^{(}$ 1 − 2 GM r ) dt 2 +$^{(}$1 − 2 GM r ) − 1 dr 2 + r $^{2}$d Ω$_{2}$ . (2.23) Taking L$_{M}$ = 0 and a mass M at the origin leads to the usual Schwarzschild solution in 4 D (with G ≡ G (4) N , and ℏ = c = 1 for the rest of the thesis), There is a nonsingular event horizon at r = 2 GM corresponding to g$_{rr}$ switch- ing sign and a true singularity at r = 0 (which can be seen from the invariant scalar R $^{µνρσ}$R$_{µνρσ}$ = 48( GM ) $^{2}$/r$^{6}$; other scalars that can be examined for sin- gularities are the Ricci scalar R = R µ µ and the Ricci tensor squared, R $^{µν}$R$_{µν}$ ). Now consider an n = 4 gravity action including the E&M Lagrangian, which is simply a two form field strength squared with a proportionality con- stant, L$_{E&M}$ = − (1 / 4) F $^{µν}$F$_{µν}$ : S =$^{∫}$ d $^{4}$x √ − g$^{(}$ 1 16 πG (4) N R − 1 4 F 2 2 ) . (2.24)