Differential Bianchi identity

Proved

EinsteinFieldEquationsD.differential_bianchi_identity

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

differential-geometryeinstein-field-equationsgeneral-relativity

Let $g$ be a smooth, symmetric, nondegenerate metric on an open set $U\subseteq\mathbb{R}^D$ . Then the Riemann tensor satisfies the differential (second) Bianchi identity on $U$ :

\nabla_\varepsilon R^\mu{}_{\alpha\beta\gamma}+\nabla_\beta R^\mu{}_{\alpha\gamma\varepsilon}+\nabla_\gamma R^\mu{}_{\alpha\varepsilon\beta}=0,

i.e. $R^\mu{}_{\alpha[\beta\gamma;\varepsilon]}=0$ .

Preamble

import Mathlib
import Definitions.Def_EinsteinFieldEquationsD_Defs

Formal statement

namespace EinsteinFieldEquationsD

theorem differential_bianchi_identity {D : ℕ} (g : Tensor2 D) (U : Set (Coord D)) (hg : IsMetricOn g U) :
    ∀ x ∈ U, ∀ e m a b c : Fin D,
      covDerivRiemann g x e m a b c + covDerivRiemann g x b m a c e
        + covDerivRiemann g x c m a e b = 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', derivation box, first display

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.

Claim. For every $x\in U$ and all indices $e,m,a,b,c$ ,

\nabla_e R^m{}_{abc}(x)+\nabla_b R^m{}_{ace}(x)+\nabla_c R^m{}_{aeb}(x)=0,

where $R^m{}_{abc}=\partial_b\Gamma^m_{ac}-\partial_c\Gamma^m_{ab}+\sum_s(\Gamma^m_{sb}\Gamma^s_{ca}-\Gamma^m_{sc}\Gamma^s_{ba})$ and

\nabla_e R^m{}_{abc}=\partial_e R^m{}_{abc}+\sum_s\big(\Gamma^m_{es}R^s{}_{abc}-\Gamma^s_{ea}R^m{}_{sbc}-\Gamma^s_{eb}R^m{}_{asc}-\Gamma^s_{ec}R^m{}_{abs}\big).

The sum is cyclic in the derivative index and the last two Riemann indices.

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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