Einstein field equations imply local conservation of energy–momentum

Proved

EinsteinFieldEquationsD.local_energy_momentum_conservation

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

differential-geometryeinstein-field-equationsgeneral-relativity

Local conservation of energy and momentum. Let $g$ be a smooth, symmetric, nondegenerate metric on an open set $U\subseteq\mathbb{R}^D$ , let $G>0$ (Newton's constant) and $c>0$ (speed of light), put $\kappa=8\pi G/c^4$ , and suppose the Einstein field equations

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

hold on $U$ . Then the stress–energy tensor is covariantly conserved on $U$ :

\nabla_\beta T^{\alpha\beta}=0,

formalized in the index-lowered form $g^{\beta\lambda}\nabla_\lambda T_{\alpha\beta}=0$ .

Preamble

import Mathlib
import Definitions.Def_EinsteinFieldEquationsD_Defs

Formal statement

namespace EinsteinFieldEquationsD

theorem local_energy_momentum_conservation {D : ℕ} (g T : Tensor2 D) (U : Set (Coord D)) (Λ G c : ℝ)
    (hg : IsMetricOn g U) (hG : 0 < G) (hc : 0 < c)
    (hEFE : EinsteinFieldEquationsOn g T Λ (einsteinGravitationalConstant G c) U) :
    ∀ x ∈ U, ∀ a : Fin D, divergence g T x a = 0 := 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 'Features — Conservation of energy and momentum'

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: $T$ is an arbitrary matrix-valued function on $\mathbb{R}^D$ (no smoothness or symmetry assumed); $\Lambda$ is any real; $G$ and $c$ are reals with $G>0$ and $c>0$ ; $\kappa:=8\pi G/c^4$ . Hypothesis: for every $x\in U$ and all $a,b$ , $G_{ab}(x)+\Lambda g_{ab}(x)=\kappa T_{ab}(x)$ , where (careful: the same letter) $G_{ab}$ here is the Einstein tensor of $g$ , while the scalar $G$ is the gravitational constant.

Claim. For every $x\in U$ and every index $a$ ,

\sum_{b,l}g^{bl}(x)\Big(\partial_lT_{ab}(x)-\sum_s\big(\Gamma^s_{la}(x)T_{sb}(x)+\Gamma^s_{lb}(x)T_{as}(x)\big)\Big)=0,

with $\partial_l$ the Fréchet partial derivative (junk value $0$ where $T_{ab}$ is not differentiable) and $g^{bl}$ the matrix inverse of $g(x)$ .

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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