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Spacetime Curvature

This defines the fundamental concept of how gravity works in this theory.
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The statement of the theorem

The intrinsic curvature of a spacetime manifold (M,g)(M, g) is quantified by the Riemann curvature tensor RμνρσR_{\mu\nu\rho\sigma}. This tensor is defined in terms of the Christoffel symbols Γμνλ\Gamma^\lambda_{\mu\nu} (which are derived from the metric gμνg_{\mu\nu}) by:\n\nRμνρσ=12(∂ρ∂νgμσ−∂σ∂νgμρ−∂ρ∂μgνσ+∂σ∂μgνρ)R_{\mu\nu\rho\sigma} = \frac{1}{2} (\partial_{\rho} \partial_{\nu} g_{\mu\sigma} - \partial_{\sigma} \partial_{\nu} g_{\mu\rho} - \partial_{\rho} \partial_{\mu} g_{\nu\sigma} + \partial_{\sigma} \partial_{\mu} g_{\nu\rho})\n\nAlternatively, using the Christoffel symbols: Rμνρσ=gσλ(∂ρΓνλ−∂νΓρλ+ΓρλΓνσ−ΓνλΓρσ)R_{\mu\nu\rho\sigma} = g_{\sigma\lambda} (\partial_{\rho} \Gamma^\lambda_{\nu} - \partial_{\nu} \Gamma^\lambda_{\rho} + \Gamma^\lambda_{\rho} \Gamma^\sigma_{\nu} - \Gamma^\lambda_{\nu} \Gamma^\sigma_{\rho}).