Summarizing the discussion so far, matter is described by the energy-momentum tensor, which we now write as \begin{equation} \Tvec = T^i{}_j \sigma^j \ee_i = {*}\tau^i \ee_i \end{equation} and conservation of matter requires that $\Tvec$ be divergence-free, that is, that \begin{equation} d{*}\Tvec = \zero \end{equation} Meanwhile, it turns out that there is a unique divergence-free vector-valued 1-form which can be constructed from the curvature, namely the Einstein tensor \begin{equation} \GG = G^i{}_j \sigma^j \ee_i = {*}\gamma^i \ee_i \end{equation} and which satisfies \begin{equation} d{*}\GG = \zero \end{equation} Einstein's fundamental insight was to equate these two expressions; Einstein's equation is \begin{equation} \GG = 8\pi\Tvec = \frac{8\pi G}{c^2}\Tvec \end{equation} where the factor of $8\pi$ is chosen to ensure agreement with Newtonian theory in the appropriate limit. In more traditional tensor language, Einstein's equation is written as \begin{equation} G_{ij} = R_{ij} - \frac12 g_{ij} R = \frac{8\pi G}{c^2} T_{ij} \end{equation}

Curvature equals matter! The conservation of matter has become a consequence of the Bianchi identity!