The Einstein Field Equations

Everything so far has been kinematic: gravity is the curvature of a Lorentzian manifold, freely falling bodies follow geodesics, and tidal effects are the Riemann tensor. What remains is the dynamics — the law that says how matter curves spacetime. That law is the Einstein field equations:

$$\boxed{G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }.}$$

Geometry on the left ($G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$, plus a cosmological term), matter on the right ($T_{\mu \nu }$, the stress–energy tensor). This page builds both sides, fixes the constant $8\pi G$ from the Newtonian limit, argues uniqueness, and reads off the first physical consequences. We use $c=1$, mostly-plus signature.

The two sides

The previous page supplied the left-hand side: the Einstein tensor $G_{\mu \nu }$, the unique symmetric, divergence-free (${\nabla }_{\mu }{G}^{\mu \nu }=0$) tensor built from the metric and its first two derivatives, linear in the second derivatives.

The right-hand side is the stress–energy tensor $T_{\mu \nu }$, the source. It is the symmetric $(0,2)$-tensor collecting the density and flux of energy and momentum:

• $T_{00}$ — energy density $\rho$;
• $T_{0i}$ — energy flux / momentum density;
• $T_{ij}$ — momentum flux, i.e. stresses (pressure on the diagonal, shear off it).

Two canonical forms recur:

• Perfect fluid (matter with density $\rho$, isotropic pressure $p$, four-velocity $u^{\mu }$): $$T_{\mu \nu }=(\rho +p)u_{\mu }{u}_{\nu }+p{g}_{\mu \nu },$$ the workhorse for stars and cosmology. Dust is the $p=0$ case.
• Electromagnetic field (the covariant Maxwell field on a curved background): $$T_{\mu \nu }=F_{\mu \alpha }{F}_{\nu }{}^{\alpha }-\frac{1}{4}{g}_{\mu \nu }{F}_{\alpha \beta }{F}^{\alpha \beta },$$ which is traceless — radiation curves spacetime but sources $R$ differently from matter.

The physical content of the source side — a fuller catalogue of $T_{\mu \nu }$, its conservation, and the energy conditions that constrain "reasonable" matter — is developed in the planned matter/ pages.

Reading the equation

The field equations are a coupled system of ten nonlinear, second-order PDEs for the ten components of $g_{\mu \nu }$. Key features:

• Nonlinear. $G_{\mu \nu }$ depends nonlinearly on $g$ (through $\Gamma \Gamma$ terms in Riemann). Physically: gravity gravitates — the gravitational field carries energy, which itself sources further curvature. There is no superposition of solutions, unlike Maxwell theory.
• Constrained, not just evolutionary. The contracted Bianchi identity ${\nabla }_{\mu }{G}^{\mu \nu }=0$ means four of the ten equations are constraints on initial data rather than evolution equations; correspondingly four of the ten metric components are pure coordinate gauge. This is the entry point to the initial-value (ADM) formulation, planned in foundations/.
• Matter tells geometry, geometry tells matter. Wheeler's slogan: "spacetime tells matter how to move [geodesics]; matter tells spacetime how to curve [these equations]." The two are coupled — $T_{\mu \nu }$ depends on $g$, which is determined by $T_{\mu \nu }$.
• Built-in conservation. Because ${\nabla }_{\mu }{G}^{\mu \nu }\equiv 0$ identically, the equations force ${\nabla }_{\mu }{T}^{\mu \nu }=0$: local energy–momentum conservation is a consequence of the geometry, not an extra assumption.

Fixing the constant: the Newtonian limit

The coefficient $8\pi G$ is not free — it is fixed by demanding that weak, static, slowly-varying gravity reproduce Newton. Take:

1. Weak field: $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$, $|h|\ll 1$, keep first order.
2. Slow motion: particle four-velocity $u^{\mu }\approx (1,0)$, so the geodesic equation becomes ${\ddot{x}}^i\approx -{\Gamma }^i{}_{00}\approx -\frac{1}{2}{\partial }_i{h}_{00}$.
3. Static, non-relativistic source: $T_{\mu \nu }$ dominated by $T_{00}=\rho$, with $p\ll \rho$.

Matching the geodesic equation to Newton's ${\ddot{x}}^i=-{\partial }_i\Phi$ identifies $h_{00}=-2\Phi$, i.e. $g_{00}=-(1+2\Phi )$ — the metric-as-potential relation. Taking the trace-reversed field equations in this limit reduces the $00$-component to

$${\nabla }^2\Phi =4\pi G\rho ,$$

which is exactly the Newtonian Poisson equation — provided the constant on the right of the field equations is $8\pi G$. So $8\pi G$ is the unique choice that recovers Newtonian gravity, and $G$ is Newton's constant. (Restoring units, the coefficient is $8\pi G/c^4$.)

Uniqueness: Lovelock's theorem

Is $G_{\mu \nu }$ the only possible left-hand side? Essentially yes. Lovelock's theorem states: in four spacetime dimensions, the only symmetric, divergence-free $(0,2)$-tensor built from the metric and its first two derivatives, and linear in the second derivatives, is

$$a{G}_{\mu \nu }+b{g}_{\mu \nu }$$

for constants $a,b$. Identifying $a$ with $1$ (absorbed into $8\pi G$) and $b=\Lambda$ gives exactly the Einstein tensor plus a cosmological term. Under mild, physically motivated assumptions the field equations are therefore unique — there is no freedom to add curvature-squared or higher terms without either raising the derivative order, breaking general covariance, or leaving four dimensions. (Those extensions are the subject of modified-gravity theories.)

The same equations follow from a variational principle: extremizing the Einstein–Hilbert action $S=\frac{1}{16\pi G}\int R\sqrt{-g}{d}^{4}x+S_{\text{matter}}$ with respect to $g^{\mu \nu }$ yields $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$, with $T_{\mu \nu }=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{matter}}}{\delta g^{\mu \nu }}$. This derivation, and the boundary term it requires, is developed in the planned foundations/action.md.

The cosmological constant

The term $\Lambda g_{\mu \nu }$ is permitted by Lovelock's theorem and by ${\nabla }_{\mu }{g}^{\mu \nu }=0$ (the metric is covariantly constant). Einstein introduced $\Lambda$ (1917) to allow a static universe, then discarded it after Hubble's expansion. It returned with the 1998 discovery of accelerating cosmic expansion: a small positive $\Lambda \sim 10^{-52}{\mathrm{m}}^{-2}$ fits the data. Moved to the right-hand side, it acts as a perfect fluid with equation of state $p=-\rho$ — dark energy / vacuum energy:

$$G_{\mu \nu }=8\pi G(T_{\mu \nu }+T_{\mu \nu }^{\Lambda }),T_{\mu \nu }^{\Lambda }=-\frac{\Lambda }{8\pi G}{g}_{\mu \nu }.$$

The enormous mismatch between this observed value and naive quantum-field-theory vacuum-energy estimates is the cosmological constant problem, a headline clue in the search for quantum gravity.

First solutions and predictions

The nonlinear equations are hard, but symmetry yields exact solutions whose predictions define the classic tests. (Each is developed in its own planned solutions/ page.)

Schwarzschild — the field of a spherical mass

The unique static, spherically symmetric vacuum solution ($T_{\mu \nu }=0$, $\Lambda =0$) is the Schwarzschild metric:

$$d{s}^2=-(1-\frac{2GM}{r})d{t}^2+(1-\frac{2GM}{r}{)}^{-1}d{r}^2+r^{2}d{\Omega }^2.$$

By Birkhoff's theorem it is the only such solution — any spherical mass has this exterior field, and a spherically pulsating star emits no gravitational waves. Its geodesics give the four classical tests:

At $r=2GM$ (the Schwarzschild radius) the metric coefficients degenerate: this is the event horizon of a black hole — a coordinate singularity (finite Kretschmann scalar), removable by better coordinates, unlike the genuine curvature singularity at $r=0$. Rotating (Kerr) and charged generalizations, horizons, and thermodynamics are planned in solutions/ and blackholes/.

Cosmology — FLRW

Imposing spatial homogeneity and isotropy gives the Friedmann–Lemaître–Robertson–Walker metric

$$d{s}^2=-d{t}^2+a(t{)}^2[\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\Omega }^2],$$

with scale factor $a(t)$ and spatial curvature $k\in \{-1,0,+1\}$. The field equations reduce to the Friedmann equations for $a(t)$, sourced by a perfect fluid; they predict cosmic expansion, redshift, the Big Bang, and — with $\Lambda$ — late-time acceleration. This is the backbone of modern cosmology; details in solutions/flrw.md.

Gravitational waves

Linearizing about flat space, $g_{\mu \nu }={\eta }_{\mu \nu }+h_{\mu \nu }$, the vacuum equations become a wave equation $\square {\bar{h}}_{\mu \nu }=0$ (in harmonic gauge): ripples in the metric propagating at $c$, with two transverse-traceless polarizations. Predicted by Einstein (1916), inferred from the Hulse–Taylor binary pulsar's orbital decay, and directly detected by LIGO (2015) from merging black holes. The SR bridge to field theory foreshadows the massless spin-2 quantum (the graviton); the classical theory is planned in waves/.

Summary

• The Einstein field equations $G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G{T}_{\mu \nu }$ equate geometry (Einstein tensor) to matter (stress–energy).
• $T_{\mu \nu }$ collects energy density, momentum, and stress; perfect-fluid and electromagnetic forms are canonical.
• Ten coupled nonlinear second-order PDEs; nonlinear because gravity gravitates; four are constraints (Bianchi/gauge); conservation ${\nabla }_{\mu }{T}^{\mu \nu }=0$ is automatic.
• The constant $8\pi G$ is fixed by the Newtonian limit (${\nabla }^2\Phi =4\pi G\rho$); Lovelock's theorem makes the left-hand side essentially unique; the same equations follow from the Einstein–Hilbert action.
• The cosmological constant $\Lambda$ is the permitted extra term, now identified with dark energy.
• Exact solutions — Schwarzschild (classical tests, black holes), FLRW (cosmology), linearized (gravitational waves) — define the theory's confirmed predictions.

Next

This completes the core spine. The planned continuations develop the source side (stress–energy and energy conditions), the exact solutions in detail (Schwarzschild, Kerr, FLRW), black holes and their thermodynamics, gravitational waves, the variational and 3+1 formulations, and the frontier where general relativity meets quantum theory.