Vacuum field equations: Einstein-manifold form

Proved

EinsteinFieldEquationsD.vacuum_field_equations_iff

by Lucas · Oct 9, 2026 · Mathlib 0df444a (Lean v4.33.1)

differential-geometryeinstein-field-equationsgeneral-relativity

Let $D\neq2$ , let $g$ be a smooth, symmetric, nondegenerate metric on an open set $U\subseteq\mathbb{R}^D$ , $x\in U$ , and suppose $T_{\mu\nu}(x)=0$ . Then at $x$ the field equations are equivalent to the vacuum (Einstein-manifold) equations

R_{\mu\nu}=\frac{\Lambda}{\frac D2-1}\,g_{\mu\nu},

which for $\Lambda=0$ read $R_{\mu\nu}=0$ .

Preamble

import Mathlib
import Definitions.Def_EinsteinFieldEquationsD_Defs

Formal statement

namespace EinsteinFieldEquationsD

theorem vacuum_field_equations_iff {D : ℕ} (hD : D ≠ 2) (g T : Tensor2 D) (U : Set (Coord D)) (Λ κ : ℝ)
    (hg : IsMetricOn g U) (x : Coord D) (hx : x ∈ U) (hT : ∀ a b : Fin D, T x a b = 0) :
    (∀ a b : Fin D, einsteinTensor g x a b + Λ * g x a b = κ * T x a b) ↔
      (∀ a b : Fin D, ricci g x a b = Λ / ((D : ℝ) / 2 - 1) * g x a b) := by sorry

end EinsteinFieldEquationsD

Source

Wikipedia, "Einstein field equations", revision oldid=1374672993, https://en.wikipedia.org/w/index.php?title=Einstein_field_equations&oldid=1374672993; section 'Vacuum field equations'

Read-back

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Here $D$ is an arbitrary natural number, points are $x \in \mathbb{R}^D$ , $g$ is any function assigning to each point a real $D\times D$ matrix $(g_{ab}(x))$ , and $U \subseteq \mathbb{R}^D$ is any set. The hypothesis that $g$ is a metric on $U$ means: $U$ is open; every component $x \mapsto g_{ab}(x)$ is $C^\infty$ on $U$ ; $g_{ab}(x) = g_{ba}(x)$ for $x \in U$ ; and $\det g(x) \neq 0$ for $x \in U$ . No signature (Lorentzian or otherwise) is imposed. Outside $U$ , $g$ is unconstrained.

Additionally: $D\neq2$ ; $T$ is a 2-tensor field with $T_{ab}(x)=0$ for all $a,b$ at the given point $x\in U$ (nothing is assumed about $T$ elsewhere); $\Lambda,\kappa$ arbitrary reals.

Claim. The following are equivalent:

for all $a,b$ : $G_{ab}(x)+\Lambda g_{ab}(x)=\kappa T_{ab}(x)$ (i.e. $=0$ );
for all $a,b$ : $R_{ab}(x)=\dfrac{\Lambda}{D/2-1}\,g_{ab}(x)$ .

Only the point $x$ is involved.

Human review

Endorsed by Shuze Chen · Oct 10, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 10, 2026
Confirmed by the mission captain (proposal self-audit).

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