Einstein Field Equations

The Einstein field equations are the equations governing gravitation in general relativity. They relate the curvature of spacetime to the distribution of matter, radiation, and other sources of energy and momentum. Albert Einstein presented their definitive form on November 25, 1915. Unlike Newtonian gravity, which treats gravity as a force acting within space, these equations make spacetime geometry itself a dynamical physical field. (einstein.caltech.edu)

Mathematical form

A standard formulation, including the cosmological constant, is

Here $g_{\mu\nu}$ is the metric tensor, which determines spacetime intervals; $R_{\mu\nu}$ is the Ricci tensor; $R=g^{\mu\nu}R_{\mu\nu}$ is the scalar curvature; and $T_{\mu\nu}$ is the stress–energy tensor. The constants $G$ and $c$ are Newton’s gravitational constant and the speed of light. Greek indices range over the four spacetime coordinates. Ricci curvature is obtained by contracting the Riemann curvature tensor. (damtp.cam.ac.uk)

The combination

is called the Einstein tensor, allowing the compact expression $G_{\mu\nu}+\Lambda g_{\mu\nu}=(8\pi G/c^4)T_{\mu\nu}$. Although this looks like one equation, it represents ten component equations because the tensors are symmetric. They form coupled, nonlinear, second-order partial differential equations for the metric. Sign conventions for curvature differ among textbooks, so formulas must be interpreted together with their definitions. (damtp.cam.ac.uk)

Geometry and physical sources

The metric describes a four-dimensional manifold with Lorentzian geometry. It determines clock readings, spatial distances, and light cones, thereby specifying which events can causally influence one another. The mathematical framework is differential geometry, but the metric is not merely a fixed background: its behavior is determined by the field equations. (damtp.cam.ac.uk)

The source tensor includes energy density, momentum density, energy flux, pressure, and stresses. Thus gravitational sources are not restricted to rest mass: radiation and electromagnetic fields also contribute. Moreover, the source tensor generally depends on the metric, so geometry and matter must be solved together rather than treated as entirely separate inputs. (damtp.cam.ac.uk)

Once a metric is known, an ideal freely falling test particle follows a geodesic. This distinguishes the field equations, which determine geometry, from the equations governing motion within that geometry. The test-particle approximation neglects the particle’s own influence on the gravitational field. (ned.ipac.caltech.edu)

Conservation and dynamical structure

A geometric identity, the contracted Bianchi identity, implies

Because the metric’s covariant derivative vanishes and $\Lambda$ is constant, the field equations require

This expresses local covariant conservation of energy and momentum. It does not generally provide a uniquely defined total energy for an arbitrary curved spacetime; global conserved quantities require additional structure, such as suitable symmetries or asymptotic conditions. (damtp.cam.ac.uk)

The ten component equations are constrained by these identities and by coordinate freedom. In a decomposition into space and time, four equations constrain admissible initial data, while the remaining equations describe evolution after coordinate conditions are specified. Consequently, ten metric components do not correspond to ten independently propagating physical quantities. (ned.ipac.caltech.edu)

Action formulation

The equations can also be derived from the Einstein–Hilbert action. In units where $c=1$, its gravitational part is

where $g=\det(g_{\mu\nu})$. Adding a matter action and varying the total action with respect to the inverse metric produces the field equations, with appropriate treatment of boundary terms. Variation of the matter action defines the stress–energy tensor. This formulation connects gravity with the variational methods used throughout theoretical physics. (damtp.cam.ac.uk)

Newtonian limit and gravitational waves

For weak, approximately static fields and slowly moving matter, with negligible cosmological effects, the equations recover Newtonian gravity. The time-time metric component encodes the Newtonian potential $\Phi$, and the field equations reduce to

where $\rho$ is mass density. Matching this limit fixes the coupling coefficient $8\pi G/c^4$. (ned.ipac.caltech.edu)

Writing $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, with a small perturbation about flat spacetime, yields linearized equations. In suitable coordinate conditions, their vacuum solutions include gravitational waves traveling at light speed. Linearization allows approximate superposition, whereas exact solutions generally cannot be added because the full equations are nonlinear. (damtp.cam.ac.uk)

Vacuum and important solutions

When $T_{\mu\nu}=0$ and $\Lambda=0$, the equations reduce to $R_{\mu\nu}=0$. This does not require the full Riemann tensor to vanish: empty regions can retain tidal curvature or carry gravitational radiation. Flat Minkowski spacetime is one vacuum solution, but not the only one. (damtp.cam.ac.uk)

The Schwarzschild solution describes the vacuum exterior of a spherically symmetric body and also a nonrotating black hole. The Kerr solution describes a rotating, uncharged black hole. Their curvature demonstrates why absence of local matter does not imply absence of gravity. (damtp.cam.ac.uk)

In cosmology, imposing large-scale homogeneity and isotropy reduces the field equations to the Friedmann equations for cosmic expansion. The cosmological constant can remain on the geometric side or be represented as a vacuum-energy contribution to the source tensor. With $\Lambda\neq0$, vacuum instead satisfies $R_{\mu\nu}=\Lambda g_{\mu\nu}$. (damtp.cam.ac.uk)