Einstein field equations

The equations state that spacetime curvature is produced by stress-energy. The left-hand side, G_μν + Λg_μν, describes geometry through the metric, its derivatives, and the cosmological constant; the right-hand side, T_μν, describes energy density, momentum density, pressure, and stresses.¹ The metric g_μν determines intervals, clocks, rulers, and the paths of freely falling bodies. The Einstein tensor G_μν is constructed so that its covariant divergence vanishes, matching local conservation of stress-energy when matter obeys compatible equations.²

Although displayed as one tensor equation, the system represents ten symmetric component equations in four-dimensional spacetime, reduced by coordinate freedom and differential identities. It is nonlinear: gravity itself contributes to the gravitational field, unlike the linear field equations of Newtonian gravity in their usual form.

The equations predict that mass-energy changes the geometry through which matter and light move. In the weak-field, slow-motion limit, they reproduce Newton’s inverse-square law, while in stronger regimes they describe phenomena such as gravitational time dilation, light deflection, the perihelion advance of Mercury, gravitational waves, and black holes.²

Exact solutions are special cases selected by symmetry or idealized matter. The Schwarzschild solution describes the exterior field of a spherical, nonrotating body; the Kerr solution describes a rotating black hole; and the Friedmann equations arise from applying the field equations to a homogeneous and isotropic universe.³ Numerical relativity is required for many dynamical systems, including merging black holes, whose emitted gravitational waves have been directly observed.⁴

The field equations can be derived from the Einstein–Hilbert action, an action proportional to the spacetime integral of the Ricci scalar plus a cosmological-constant term. Varying the metric produces the geometric tensor on the left and, when matter is included, the stress-energy tensor on the right.¹ This variational formulation connects general relativity with the broader language of classical field theory.

The equations also encode a consistency condition: the contracted Bianchi identity makes the covariant divergence of the Einstein tensor vanish. Consequently, the matter equations must imply covariant conservation of T_μν. This does not mean that energy is always expressible as a single globally conserved number; in curved or expanding spacetimes, global gravitational energy can be subtle and may depend on boundary conditions and spacetime symmetries.⁵

The cosmological constant is not merely an optional correction: it is the simplest generally covariant term that can be added to the geometric side without introducing higher derivatives of the metric. Einstein introduced it in 1917 in an attempt to obtain a static universe, later abandoning that motivation; modern cosmology uses a positive effective cosmological constant to model accelerated expansion.³

In vacuum, the equations do not imply flat spacetime: setting T_μν = 0 still permits gravitational waves, black holes, and other curved solutions. Another subtlety is that the equations are underdetermined until coordinates, initial data, and matter laws are specified. Their initial-value formulation therefore involves constraints that the data must satisfy before evolution can be calculated.⁵ This combination of geometric elegance and analytical difficulty is why questions about singularities, cosmic censorship, and quantum gravity remain active research areas.

Metric tensor

A tensor field that determines spacetime distances, causal structure, and the readings of ideal clocks and rulers.

Stress-energy tensor

The tensor describing local energy density, momentum density, pressure, and shear stresses of matter and fields.

Einstein tensor

The divergence-free geometric tensor Gμν formed from the Ricci tensor and Ricci scalar.

Cosmological constant

The constant Λ multiplying the metric tensor; it represents a uniform vacuum-energy-like contribution in the field equations.

Bianchi identity

A differential identity for curvature that leads to the covariant divergence-free property of the Einstein tensor.