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Spacetime Curvature

Mass and energy warp the fabric of spacetime, causing objects to move along curved paths – what we perceive as gravity.
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The statement of the theorem

Let gμνg_{\mu\nu} be the metric tensor defining the geometry of a four-dimensional spacetime manifold (M,g)(\mathcal{M}, g). The curvature of this spacetime is quantified by the Riemann curvature tensor RμνρσR_{\mu\nu\rho\sigma}. The Ricci tensor, RμνR_{\mu\nu}, is obtained by contracting the Riemann tensor: Rμν=RμρνρR_{\mu\nu} = R^{\rho}_{\mu\rho\nu}. The scalar curvature RR is the trace of the Ricci tensor: R=gμνRμνR = g^{\mu\nu} R_{\mu\nu}. The Einstein tensor, GμνG_{\mu\nu}, which represents the geometric side of the field equations, is defined as: Gμν=Rμν−12gμνRG_{\mu\nu} = R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R \n\nThis geometric curvature is directly related to the distribution of mass and energy, represented by the Stress-Energy-Momentum tensor TμνT_{\mu\nu}. The Einstein Field Equations (EFE) establish this fundamental relationship:\n\nGμν=8πGc4TμνG_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \n\nwhere GG is Newton's gravitational constant and cc is the speed of light. The equation dictates that the curvature of spacetime (GμνG_{\mu\nu}) is proportional to the density and flux of energy and momentum (TμνT_{\mu\nu}).