2.1 General Relativity 6 where we are using Einstein summation convention. Going from a flat (Minkowski) manifold to the generically curved one of spacetime, which is an example of a pseudo-Riemannian manifold, we have the general symmetric metric tensor g$_{µν}$ (as opposed to the flat η$_{µν}$ ). The line element now becomes The essence of General Relativity, hereafter referred to as GR, is reflected in the Einstein Field Equation , which is where R$_{µν}$ is the Ricci Tensor , given by where R µ νβα is the Riemann Tensor , given by where Γ µ νβ is the Christoffel Symbol , given by R$_{µν}$ − 1 2 g$_{µν}$R = 8 πGT$_{µν}$ (2.4) ds 2 = g$_{µν}$dx $^{µ}$dxν (2.3) R$_{µν}$ ≡ R λ µλν (2.5) R µ νβα ≡ ∂$_{α}$ Γ µ νβ − ∂$_{β}$ Γ µ να + Γ µ $_{σα}$Γ σ νβ − Γ µ $_{σβ}$Γ σ να (2.6) Γ µ νβ = 1 2 g $^{αµ}$( ∂$_{β}$g$_{αν}$ + ∂$_{ν}$g$_{αβ}$ − ∂$_{α}$g$_{νβ}$ ) (2.7)