Einstein field equations
G_{μν} = (8πG/c⁴) T_{μν}. Ten coupled nonlinear partial differential equations relating spacetime geometry (left) to matter-energy distribution (right). The defining equations of general relativity. Published November 1915 by Einstein; near-simultaneously derived by Hilbert via the Einstein-Hilbert action.
Definition
The Einstein field equations G_{μν} = (8πG/c⁴) T_{μν} are the ten coupled nonlinear partial differential equations that constitute the dynamical content of general relativity. The left-hand side, the Einstein tensor G_{μν} = R_{μν} − (1/2) R g_{μν}, is a functional of the metric tensor g_{μν} and its first two derivatives; the right-hand side, the stress-energy tensor T_{μν}, encodes the distribution of energy and momentum in matter and non-gravitational fields. The equations relate the geometry of spacetime to its matter content — in John Wheeler's slogan, "matter tells spacetime how to curve, spacetime tells matter how to move." Their nonlinearity is essential: the gravitational field carries energy and momentum, and that energy itself sources further gravity, so the equations are not amenable to superposition the way Maxwell's equations or Newton's law of gravity are.
Einstein presented the equations in their final form on 25 November 1915 to the Prussian Academy of Sciences, in his paper Die Feldgleichungen der Gravitation; David Hilbert independently derived equivalent equations five days earlier, on 20 November, by varying the Einstein-Hilbert action S = (c⁴/16πG) ∫ R √(−g) d⁴x. The publication-priority footnote is delicate but the consensus is that Einstein established the physical theory while Hilbert independently gave it a variational derivation. The equations admit a one-parameter family of generalisations through the addition of a cosmological constant term, G_{μν} + Λ g_{μν} = (8πG/c⁴) T_{μν}, which Einstein included in 1917 (and later regretted) and which observational cosmology has since 1998 found to be non-zero — Λ ≈ 1.1 × 10⁻⁵² m⁻², the dark-energy contribution. The equations are the single most quoted equation of twentieth-century physics and remain experimentally verified to high precision in every regime that has been probed, from solar-system orbits to binary-pulsar timing to gravitational-wave inspirals. See the topic page einsteins-field-equations for the full derivation and discussion.