Just as Maxwell's equations govern how the electric and magnetic fields respond to charges and currents, Einstein Field Equation governs how the metric respond to energy and momentum.
There are traditionally two ways of deriving this equation:
- - first by some informal reasoning by analogy, close to what Einstein himself was thinking (refer to Einstein Tensor and Einstein Field Equations) and which begins with the realization that we would like to find an equation that supersedes the Poisson equation for the Newtonian potential
- - or by starting with an action and deriving the corresponding equations of motion (refer to Einstein-Hilbert action)
