Einstein field equations

The Einstein field equations are the heart of general relativity, the rule that links matter to geometry. In compact form they read G = (8 pi G / c^4) T. The left side describes how spacetime is curved; the right side describes the energy, momentum and pressure of whatever fills it. Set them equal and you have a complete law of gravity.

The physicist John Wheeler summed them up beautifully: matter tells spacetime how to curve, and curved spacetime tells matter how to move. The constant out front, with its factor of c^4 in the denominator, is enormous, which is why it takes something as massive as a planet or star to bend spacetime by an amount we can easily notice. Gravity is geometry, but it is a very stiff geometry that resists being bent.

Because both sides are full geometric objects with many components, the single line above is really ten coupled equations, and they are notoriously hard to solve. Exact solutions exist only in special cases, such as the spacetime around a lone spherical star or a smooth expanding universe; for everything else physicists turn to approximations and supercomputers. From these equations come black holes, the expansion of the cosmos, and gravitational waves.