Einstein field equations

The Einstein field equations relate spacetime curvature to the distribution of matter and energy. They are nonlinear equations at the heart of general relativity.

Einstein tensor: A symmetric tensor built from spacetime curvature whose covariant divergence vanishes. It represents the geometric side of the equations while satisfying local energy-momentum conservation.
General relativity: Einstein’s theory of gravity, in which gravitation is described through spacetime geometry. The field equations are its central dynamical law.
Schwarzschild metric: The vacuum solution describing the exterior gravitational field of a spherical, nonrotating mass. It predicts the spacetime outside an idealized star or nonrotating black hole.
Albert Einstein: A German-born theoretical physicist who developed special and general relativity. He formulated the field equations and published their final form in 1915.
Newtonian gravity: A theory in which masses exert an instantaneous attractive force described by a gravitational potential. It is the weak-field limit of the equations, but not their relativistic account of gravity.
Stress–energy tensor: A tensor describing the density and flow of energy and momentum, as well as stresses. It supplies the matter and energy content that shapes spacetime curvature.
Riemann curvature tensor: A tensor that measures the intrinsic curvature of a space or spacetime. The equations use contractions of curvature to express how spacetime bends.
Kerr metric: The vacuum solution describing the spacetime around a rotating, uncharged black hole. It models strong gravity when angular momentum cannot be neglected.
David Hilbert: A German mathematician whose work shaped modern mathematics and mathematical physics. He developed a variational formulation of gravitational field equations around the time Einstein completed his.
Scalar–tensor theory: A class of gravity theories in which a scalar field supplements the spacetime metric as a gravitational degree of freedom. These theories add gravitational dynamics beyond those encoded by the Einstein equations.

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