---
title: The Einstein Field Equations
module: Curved Spacetime
moduleNumber: 5
lessonNumber: 6
order: 506
summary: >
  The field equation is assembled from a short list of requirements: a symmetric,
  divergence-free, second-order geometric tensor set proportional to the
  stress–energy tensor, with the coefficient fixed by the Newtonian limit. The
  cosmological constant is the one extra term the requirements allow. The
  Einstein–Hilbert action gives the same equation from a variational principle,
  and the coupled system closes the logic of the module: matter curves spacetime,
  and spacetime tells matter how to move.
topics: [Curved Spacetime]
draft: false
sources:
  - book: Hartle
    ref: "Gravity, Ch. 21 — Curvature and the Einstein Equation; Ch. 22 — The Source of Curvature"
  - book: Carroll
    ref: "Lecture Notes on General Relativity, §4 — Gravitation"
---

Every piece is now in place. The
[stress–energy tensor](/relativity/covariant-electrodynamics/covariant-maxwell-and-the-stress-energy-tensor)
supplies a symmetric, conserved description of matter and fields; the
[Einstein tensor](/relativity/curved-spacetime/curvature-riemann-and-geodesic-deviation)
supplies a symmetric, divergence-free description of geometry; the
[geodesic equation](/relativity/curved-spacetime/geodesics-and-the-geodesic-equation)
fixed the Newtonian limit and identified $g_{00}$ with the potential. This lesson
assembles the **Einstein field equations** by matching these objects, determines
the coupling constant from Newtonian gravity, adds the cosmological constant, and
rederives the whole equation from the Einstein–Hilbert action.

## The source of curvature

Newtonian gravity is sourced by mass density $\rho$ through Poisson's equation
$\nabla^2\Phi = 4\pi G\rho$. Relativity forbids so narrow a source: mass is only
one component of energy, energy and momentum mix under boosts, and pressure and
stress carry energy too. The correct source is the full **stress–energy tensor**
$T^{\mu\nu}$, a symmetric $(2,0)$ tensor whose components are the flux of the
$\mu$-th momentum component across a surface of constant $x^\nu$:

- $T^{00}$ is the energy density.
- $T^{0i} = T^{i0}$ is the energy flux / momentum density.
- $T^{ij}$ is the momentum flux — pressure on the diagonal, shear off it.

For a **perfect fluid** with rest-frame energy density $\rho c^2$ and isotropic
pressure $p$, moving with four-velocity $u^\mu$,

$$
T^{\mu\nu} = \Big(\rho + \frac{p}{c^2}\Big)u^\mu u^\nu + p\,g^{\mu\nu}.
$$

Local conservation of energy and momentum is the covariant statement

$$
\nabla_\mu T^{\mu\nu} = 0,
$$

which in flat spacetime reduces to the continuity and Euler equations of fluid
mechanics. Any geometric object set equal to $T^{\mu\nu}$ must therefore also be
symmetric and divergence-free, or conservation would fail.

$$
% caption: The stress-energy tensor block structure: energy density in the corner,
% momentum density and energy flux along the top row and left column, and the
% momentum-flux (stress) block filling the spatial part with pressure on the
% diagonal.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % 4x4 grid
  \foreach \i in {0,1,2,3,4} {
    \draw[black] (0,\i) -- (4,\i);
    \draw[black] (\i,0) -- (\i,4);
  }
  % energy density corner (top-left in matrix = high row)
  \fill[acc!22] (0,3) rectangle (1,4);
  \node[acc, anchor=center] at (0.5,3.5) {energy};
  % top row + left col (momentum density / flux)
  \node[black, anchor=center] at (2.5,3.5) {energy current};
  \node[black, anchor=center, rotate=90] at (0.5,1.5) {momentum};
  % stress block 3x3
  \draw[black, thick] (1,0) rectangle (4,3);
  \node[black, anchor=center] at (2.5,1.5) {stress};
  \node[black, anchor=north] at (2.0,-0.15) {pressure on diagonal};
\end{tikzpicture}
$$

## Building the equation from requirements

The field equation should read, schematically, "curvature $=$ constant $\times$
matter," with the left side a tensor built from the metric. A short list of
demands fixes it almost uniquely.

- **Tensorial.** Both sides are $(0,2)$ tensors, so the equation holds in every
  frame — the general-covariance requirement.
- **Symmetric.** The right side $T_{\mu\nu}$ is symmetric, so the left must be.
- **Second order.** To recover Poisson's equation (second-order in $\Phi$) and to
  match the Newtonian limit $g_{00} \sim \Phi$, the left side should involve at
  most second derivatives of the metric, and be linear in those.
- **Divergence-free.** Since $\nabla_\mu T^{\mu\nu} = 0$ identically, the left
  side must be divergence-free identically, for consistency at all times.

The most general symmetric $(0,2)$ tensor built from the metric and its first two
derivatives, linear in the second derivatives, and with vanishing divergence is a
linear combination of the Einstein tensor and the metric,

$$
G_{\mu\nu} + \Lambda g_{\mu\nu},
$$

where $\Lambda$ is a constant (the metric is divergence-free because $\nabla_\mu
g^{\mu\nu} = 0$ by metric compatibility). This is a theorem — Lovelock's theorem
identifies $G_{\mu\nu} + \Lambda g_{\mu\nu}$ as the unique such tensor in four
dimensions. Setting it proportional to $T_{\mu\nu}$ gives the **Einstein field
equations**,

$$
G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa\,T_{\mu\nu},
$$

with a coupling constant $\kappa$ to be fixed and the cosmological constant
$\Lambda$ deferred to the next section.

## Fixing the coupling from the Newtonian limit

The constant $\kappa$ is determined by requiring that the equation reduce to
Poisson's equation in the weak, slow, static limit, exactly the regime of the
[Newtonian-limit derivation](/relativity/curved-spacetime/geodesics-and-the-geodesic-equation).
Take $\Lambda = 0$ for this match. It is convenient to trace-reverse the equation:
contracting $G_{\mu\nu} = \kappa T_{\mu\nu}$ with $g^{\mu\nu}$ gives $R -
2R = \kappa T$, so $R = -\kappa T$, and substituting back,

$$
R_{\mu\nu} = \kappa\Big(T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}T\Big).
$$

In the weak static field with slow, low-pressure matter, the dominant
stress–energy component is $T_{00} = \rho c^2$ and the trace is $T \approx
-\rho c^2$, so the $00$ component reads $R_{00} \approx \tfrac{1}{2}\kappa\rho c^2$.
The relevant curvature component in the weak-field limit is $R_{00} \approx
\nabla^2\Phi/c^2$ — the time-time curvature is the Laplacian of the potential,
inherited from $g_{00} = -(1 + 2\Phi/c^2)$ and the Riemann–tidal correspondence.
Equating,

$$
\frac{\nabla^2\Phi}{c^2} = \tfrac{1}{2}\kappa\rho c^2.
$$

Poisson's equation $\nabla^2\Phi = 4\pi G\rho$ then forces

$$
\kappa = \frac{8\pi G}{c^4}.
$$

The field equations in their standard form are

$$
G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu},
\qquad
R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}.
$$

The factor $c^4$ in the denominator is enormous, so a large stress–energy produces
a tiny curvature: spacetime is stiff, which is why gravity is the weakest
interaction yet dominates on astronomical scales, where masses accumulate without
cancellation. In geometrized units $c = G = 1$ the coupling is simply $8\pi$.

$$
% caption: The field equation ties the geometric Einstein tensor to the matter
% stress-energy tensor through the coupling 8 pi G over c to the fourth, a
% constant fixed by demanding the weak-field limit reproduce Newton's Poisson
% equation.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[acc, thick, fill=acc!10] (0,0.9) rectangle (2.4,2.1);
  \node[acc, anchor=center] at (1.2,1.5) {Einstein tensor};
  \node[black!70, anchor=center] at (3.1,1.5) {=};
  \node[black, anchor=center] at (4.0,1.5) {coupling};
  \draw[black, thick] (3.5,1.1) rectangle (4.5,1.9);
  \node[black!70, anchor=center] at (5.0,1.5) {x};
  \draw[black, thick] (5.4,0.9) rectangle (7.8,2.1);
  \node[black, anchor=center] at (6.6,1.5) {stress-energy};
  \node[black, anchor=north] at (4.0,0.65) {coupling from Newtonian limit};
\end{tikzpicture}
$$

## The cosmological constant

The term $\Lambda g_{\mu\nu}$ is permitted by every requirement — it is symmetric,
constructed from the metric, and divergence-free — so consistency alone does not
forbid it. Einstein introduced it to allow a static universe and later regretted
it; modern cosmology reinstated it as the leading model for the observed
accelerating expansion. Moved to the right-hand side it looks like a source,

$$
G_{\mu\nu} = \frac{8\pi G}{c^4}\Big(T_{\mu\nu} - \frac{\Lambda c^4}{8\pi G}g_{\mu\nu}\Big),
$$

a perfect fluid with energy density $\rho_\Lambda c^2 = \Lambda c^4/(8\pi G)$ and
pressure $p_\Lambda = -\rho_\Lambda c^2$. This equation of state, pressure equal to
minus the energy density, is the vacuum energy that drives accelerated expansion.
Its magnitude is tiny, $\Lambda \sim 10^{-52}\,\text{m}^{-2}$, negligible on
solar-system and galactic scales and dominant only over cosmological distances,
which is why the Schwarzschild and orbit problems of the next module set $\Lambda =
0$ without measurable error. The interpretation and consequences belong to the
cosmology bridge; here it is the one term the derivation cannot rule out.

## The Einstein–Hilbert action

The requirement-matching argument gives the equation but hides its economy. The
modern derivation is variational: postulate an action and demand it be stationary.
The **Einstein–Hilbert action** is the simplest coordinate-invariant scalar built
from the metric,

$$
S_{\text{EH}} = \frac{c^4}{16\pi G}\int R\,\sqrt{-g}\;\d^4 x,
$$

with $R$ the Ricci scalar and $\sqrt{-g}\,\d^4 x$ the invariant volume element.
Adding a matter action $S_{\text{m}}$ whose variation defines the stress–energy
tensor,

$$
T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\frac{\delta S_{\text{m}}}{\delta g^{\mu\nu}},
$$

and requiring $\delta(S_{\text{EH}} + S_{\text{m}})/\delta g^{\mu\nu} = 0$
reproduces the field equations with $\Lambda = 0$. The variation of the
$\sqrt{-g}\,R$ term yields exactly $R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}R$; the
boundary terms from varying the Ricci scalar cancel against the Gibbons–Hawking
surface term. Including a constant term $-2\Lambda$ under the integral,
$\int (R - 2\Lambda)\sqrt{-g}\,\d^4 x$, restores the cosmological constant. The
action route shows that the field equation is not a guess but the Euler–Lagrange
equation of the unique lowest-order invariant, and it is the starting point for
every extension and quantization programme.

$$
% caption: Two routes to the same field equation. Matching a symmetric,
% divergence-free, second-order geometric tensor to the stress-energy tensor, or
% extremizing the Einstein-Hilbert action, both yield curvature proportional to
% matter.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,2.0) rectangle (2.6,3.2);
  \node[black, anchor=center] at (1.3,2.6) {requirements};
  \draw[black, thick] (0,-0.4) rectangle (2.6,0.8);
  \node[black, anchor=center] at (1.3,0.2) {action principle};
  \draw[acc, thick, fill=acc!10] (4.4,0.8) rectangle (7.4,2.0);
  \node[acc, anchor=center] at (5.9,1.4) {Einstein equation};
  \draw[->, black] (2.65,2.5) -- (4.4,1.7);
  \draw[->, black] (2.65,0.3) -- (4.4,1.1);
\end{tikzpicture}
$$

## Matter and geometry, coupled

The field equations close the logic the module opened with. Geometry is not a
fixed backdrop; it is a dynamical field $g_{\mu\nu}$ determined by its matter
content, and matter moves along the geodesics of the geometry it produces. The two
statements are coupled and nonlinear — the metric enters $T_{\mu\nu}$ (through
$u^\mu$, $\sqrt{-g}$, index-raising), and $T_{\mu\nu}$ sources the metric — which
is why exact solutions are rare and precious. Wheeler's summary compresses the
content: matter tells spacetime how to curve, and curved spacetime tells matter
how to move.

$$
% caption: The coupled loop of general relativity. The stress-energy of matter
% sources the metric through the field equation; the metric determines the
% geodesics along which the same matter moves, which changes the stress-energy.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,0) rectangle (2.6,1.2);
  \node[black, anchor=center] at (1.3,0.6) {matter};
  \draw[acc, thick, fill=acc!10] (5.0,0) rectangle (7.6,1.2);
  \node[acc, anchor=center] at (6.3,0.6) {geometry};
  \draw[->, black] (2.65,0.85) .. controls (3.8,1.6) .. (4.95,0.85);
  \node[black, anchor=south] at (3.8,1.4) {curves};
  \draw[->, black] (4.95,0.35) .. controls (3.8,-0.4) .. (2.65,0.35);
  \node[black, anchor=north] at (3.8,-0.2) {guides motion};
\end{tikzpicture}
$$

The first exact solution of these equations — the geometry outside a static
spherical mass, found by Schwarzschild within months of the field equations —
opens the next module, along with the orbits, light bending, and horizons that
turn the field equation into predictions the solar system and telescopes test.

[^hartle-efe]: **Hartle**, _Gravity_, Ch. 21–22 — the stress–energy tensor as the source of curvature, the Einstein equation assembled from its required properties, the Newtonian-limit determination of the coupling $8\pi G/c^4$, and the cosmological constant.

[^carroll-efe]: **Carroll**, _Lecture Notes on General Relativity_, §4 — the field equations from the requirements and from the Einstein–Hilbert action, the trace-reversed form, the definition of the stress–energy tensor by metric variation, and the role of the cosmological constant.