---
tag: 7C2WP33E10
kind: definition
name: Einstein field equations
msc: [83C05, 35Q76]
defines: 3MCZ97SW6V
uses: [BY4QYYS788, 4GTNM0WHDH, ZHX2VYQDZ3, WX302T93TW]
sources:
  - url: https://en.wikipedia.org/wiki/Einstein_field_equations
---

## Definition

Let $M$ be a four-dimensional manifold, let $\Lambda \in \R$ be a constant, the cosmological constant, and let $\kappa > 0$ be a constant.
Let $T$ be a given symmetric $(0,2)$-[[tensor field|WX302T93TW]] on $M$, the stress–energy tensor.
A [[Lorentzian metric|BY4QYYS788]] $g$ on $M$ satisfies the *Einstein field equations* if
$$\mathrm{Ric}_g - \tfrac{1}{2} R_g \, g + \Lambda g = \kappa T.$$
Here $\mathrm{Ric}_g$ is the [[Ricci curvature|4GTNM0WHDH]] and $R_g$ is the [[scalar curvature|ZHX2VYQDZ3]] of $g$.
In physical units $\kappa = 8 \pi G / c^4$, with $G$ the gravitational constant and $c$ the speed of light.
The *vacuum Einstein equations* are the case $T = 0$ and $\Lambda = 0$.

## Remarks

Taking the trace with respect to $g$ in dimension four shows that the vacuum equations are equivalent to $\mathrm{Ric}_g = 0$.
In local coordinates the equations read $R_{\mu\nu} - \tfrac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}$, ten equations for the ten components of $g$.

Example: Minkowski space, $\R^4$ with $g = -dt^2 + dx^2 + dy^2 + dz^2$, is flat, so $\mathrm{Ric}_g = 0$ and it solves the vacuum equations.

Non-example: de Sitter space has $\mathrm{Ric}_g = \Lambda_0 g$ with a constant $\Lambda_0 > 0$, so it does not solve the vacuum equations.
It solves the equations with $T = 0$ and cosmological constant $\Lambda = \Lambda_0$.

The left side has zero divergence by the contracted Bianchi identity, which forces the conservation law $\nabla^\mu T_{\mu\nu} = 0$.
Yvonne Choquet-Bruhat proved that the vacuum equations have a well-posed initial value problem for suitable initial data.
The Riemannian analogue of $\mathrm{Ric}_g = \lambda g$ defines an [[Einstein manifold|41S2EZ0KR3]].