Eq. (10) ⇔ eq. (20): covariant form of the energy-momentum law of matter

Proved

EinsteinGrossmann1913.matterConservation_iff_cov

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

entwurf-theorygeneral-relativityhistory-of-physicstensor-calculus

Let $g_{\mu\nu}$ be a fundamental tensor on $\mathbb R^4$ (twice continuously differentiable, symmetric, $g<0$ ), and let $\Theta_{\mu\nu}$ be a symmetric, continuously differentiable contravariant stress-energy tensor of a material process. Put $T_{\mu\nu}=\sum_{\alpha\beta}g_{\mu\alpha}g_{\nu\beta}\Theta_{\alpha\beta}$ . Then the energy-momentum law (10),

\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\big(\sqrt{-g}\,g_{\sigma\mu}\Theta_{\mu\nu}\big)-\tfrac12\sum_{\mu\nu}\sqrt{-g}\,\frac{\partial g_{\mu\nu}}{\partial x_\sigma}\Theta_{\mu\nu}=0\qquad(\sigma=1,\dots,4),

holds everywhere if and only if its covariant form (20),

\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\big(\sqrt{-g}\,\gamma_{\mu\nu}T_{\mu\sigma}\big)+\tfrac12\sum_{\mu\nu}\sqrt{-g}\,\frac{\partial\gamma_{\mu\nu}}{\partial x_\sigma}T_{\mu\nu}=0\qquad(\sigma=1,\dots,4),

holds everywhere.

Formalization Note The source obtains (20) from (10) “durch gliedweise Umformung”; the symmetry of $\Theta_{\mu\nu}$ (true for the source's stress-energy tensors, e.g. $\Theta_{\mu\nu}=\varrho_0\frac{dx_\mu}{ds}\frac{dx_\nu}{ds}$ ) is stated explicitly, as is the differentiability of $\Theta$ .

Preamble

import Mathlib
import Definitions.Def_EinsteinGrossmann1913_Defs

Formal statement

namespace EinsteinGrossmann1913

theorem matterConservation_iff_cov (g Θ : Field2) (hg : IsMetricField g)
    (hΘ : ∀ μ ν : Fin 4, ContDiff ℝ 1 (fun x => Θ x μ ν))
    (hΘsymm : ∀ (x : Coord) (μ ν : Fin 4), Θ x μ ν = Θ x ν μ) :
    MatterConservation g Θ ↔ MatterConservationCov g (lowerIndices g Θ) := by sorry

end EinsteinGrossmann1913

Source

A. Einstein and M. Grossmann, "Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation", B. G. Teubner, Leipzig und Berlin 1913 (Separatabdruck aus Zeitschrift für Mathematik und Physik 62), Part I (Physikalischer Teil, A. Einstein), §4, p. 10, eq. (10), and §5, p. 17, definition of T_{μν} and eq. (20).

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic) - same agent as the drafter; non-blind

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements it describes, with full knowledge of the source paper and of the intended meaning. It is not the output of a blind, independent auditor, and no reviewer should treat it as such. Compare it against the Lean code yourself.

$g$ is an arbitrary map assigning to every point $x\in\mathbb R^4$ a real $4\times4$ matrix $(g_{\mu\nu}(x))$ , and the hypothesis IsMetricField $g$ requires exactly: every entry $x\mapsto g_{\mu\nu}(x)$ is $C^2$ on all of $\mathbb R^4$ ; $g_{\mu\nu}(x)=g_{\nu\mu}(x)$ for all $x,\mu,\nu$ ; and $\det(g_{\mu\nu}(x))<0$ for every $x$ . No signature condition beyond the sign of the determinant is imposed. $\Theta$ is a further map $x\mapsto(\Theta_{\mu\nu}(x))$ into real $4\times4$ matrices, assumed to have every entry $C^1$ on $\mathbb R^4$ and to satisfy $\Theta_{\mu\nu}(x)=\Theta_{\nu\mu}(x)$ for all $x,\mu,\nu$ . Throughout, $\partial_i f(x)$ is the Fréchet derivative of $f$ at $x$ applied to the $i$ -th standard basis vector (by convention it is $0$ wherever $f$ is not differentiable), $\gamma_{\mu\nu}(x)$ denotes the entries of the matrix inverse of $(g_{\mu\nu}(x))$ , $\sqrt{-g}$ is the real square root of $-\det(g_{\mu\nu}(x))$ , and $1/0$ is read as $0$ . Indices range over four values (in the code $0,1,2,3$ ; index $3$ plays the role of the source's $x_4$ ). Let $T_{\mu\nu}(x)=\sum_{\alpha,\beta}g_{\mu\alpha}(x)g_{\nu\beta}(x)\Theta_{\alpha\beta}(x)$ . The statement asserts the equivalence of two propositions:

for every $\sigma$ and every $x$ : $\sum_{\mu,\nu}\partial_\nu\big(\sqrt{-g}\,g_{\sigma\mu}\Theta_{\mu\nu}\big)(x)-\tfrac12\sum_{\mu,\nu}\sqrt{-g}(x)\,\partial_\sigma g_{\mu\nu}(x)\,\Theta_{\mu\nu}(x)=0$ ;
for every $\sigma$ and every $x$ : $\sum_{\mu,\nu}\partial_\nu\big(\sqrt{-g}\,\gamma_{\mu\nu}T_{\mu\sigma}\big)(x)+\tfrac12\sum_{\mu,\nu}\sqrt{-g}(x)\,\partial_\sigma\gamma_{\mu\nu}(x)\,T_{\mu\nu}(x)=0$ .

The equivalence is between the two global (all $\sigma$ , all $x$ ) statements, not asserted pointwise.

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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