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particle is [320] 

<img src='content_image/316789.jpg'>

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 , 

<img src='content_image/316790.jpg'>

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), 

<img src='content_image/316791.jpg'>

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$_{µν}$ : 

<img src='content_image/316795.jpg'>

