Trace of the Einstein field equations

Proved

EinsteinFieldEquationsD.trace_of_field_equations

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 $U\subseteq\mathbb{R}^D$ and suppose the field equations $G_{\mu\nu}+\Lambda g_{\mu\nu}=\kappa T_{\mu\nu}$ hold on $U$ . Then, taking the trace with respect to the metric,

R-\frac D2R+D\Lambda=\kappa T\qquad\text{on }U,

where $R$ is the scalar curvature and $T=g^{\mu\nu}T_{\mu\nu}$ .

Preamble

import Mathlib
import Definitions.Def_EinsteinFieldEquationsD_Defs

Formal statement

namespace EinsteinFieldEquationsD

theorem trace_of_field_equations {D : ℕ} (g T : Tensor2 D) (U : Set (Coord D)) (Λ κ : ℝ) (hg : IsMetricOn g U)
    (hEFE : EinsteinFieldEquationsOn g T Λ κ U) :
    ∀ x ∈ U, scalarCurvature g x - ((D : ℝ) / 2) * scalarCurvature g x + (D : ℝ) * Λ
      = κ * metricTrace g T x := 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 'Equivalent formulations', 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.

Additionally $T$ is an arbitrary 2-tensor field (any matrix-valued function, no regularity or symmetry assumed), and $\Lambda,\kappa$ are arbitrary reals (including $\kappa=0$ ). Hypothesis: for every $x\in U$ and all $a,b$ , $G_{ab}(x)+\Lambda g_{ab}(x)=\kappa T_{ab}(x)$ .

Claim. For every $x\in U$ ,

R(x)-\frac D2R(x)+D\Lambda=\kappa\sum_{a,b}g^{ab}(x)T_{ab}(x),

with $R=\sum_{a,b}g^{ab}R_{ab}$ and $D$ viewed as a real number.

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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