Eq. (22): conservation for matter and gravitational field, covariant form

Proved

EinsteinGrossmann1913.total_conservation_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, $g<0$ ), $\kappa\neq0$ , and $T_{\mu\nu}$ a continuously differentiable covariant stress-energy tensor of matter satisfying the covariant energy-momentum law (20) and the covariant field equations (21). Then for every $\sigma$ and every point

\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\Big\{\sqrt{-g}\,\gamma_{\mu\nu}\,(T_{\mu\sigma}+t_{\mu\sigma})\Big\}=0.

This is the covariant analogue of (19): matter and gravitational field together satisfy conservation laws.

Formalization Note The source prints (22) as $\sum_\nu\frac{\partial}{\partial x_\nu}\{\sqrt{-g}\cdot\gamma_{\sigma\mu}(T_{\mu\nu}+t_{\mu\nu})\}=0$ , in which the index $\mu$ occurs twice but is not listed under the summation sign and the free index $\sigma$ sits on $\gamma$ . The statement here uses the index placement of (20) and (12b), from which the source says (22) follows “analog (19)”; reviewers should check this reading.

Preamble

import Mathlib
import Definitions.Def_EinsteinGrossmann1913_Defs

Formal statement

namespace EinsteinGrossmann1913

theorem total_conservation_cov (κ : ℝ) (hκ : κ ≠ 0) (g T : Field2) (hg : IsMetricField g)
    (hT : ∀ μ ν : Fin 4, ContDiff ℝ 1 (fun x => T x μ ν))
    (h20 : MatterConservationCov g T) (h21 : EntwurfFieldEquationsCov κ g T)
    (σ : Fin 4) (x : Coord) :
    ∑ μ : Fin 4, ∑ ν : Fin 4,
      pd ν (fun y => sqrtNegDet g y * gamma g y μ ν
        * (T y μ σ + gravStressEnergyCov κ g y μ σ)) x = 0 := 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), §5, p. 17, eq. (22) (index placement corrected, see Formalization Note).

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.

Let $\kappa\neq0$ . $g$ 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. $T$ is a map $x\mapsto(T_{\mu\nu}(x))$ into real $4\times4$ matrices with every entry $C^1$ on $\mathbb R^4$ (no symmetry assumed). 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 $x_4$ ). Hypotheses: (i) for all $\sigma,x$ : $\sum_{\mu,\nu}\partial_\nu(\sqrt{-g}\,\gamma_{\mu\nu}T_{\mu\sigma})(x)+\frac12\sum_{\mu,\nu}\sqrt{-g}(x)\,\partial_\sigma\gamma_{\mu\nu}(x)T_{\mu\nu}(x)=0$ ; (ii) for all $x,\mu,\nu$ : $-D_{\mu\nu}(x)=\kappa\,(t_{\mu\nu}(x)+T_{\mu\nu}(x))$ , with $D_{\mu\nu}$ , $t_{\mu\nu}$ as in the definition file. Conclusion: for the given $\sigma$ and $x$ ,

\sum_{\mu,\nu}\partial_\nu\Big(y\mapsto\sqrt{-g}(y)\,\gamma_{\mu\nu}(y)\,\big(T_{\mu\sigma}(y)+t_{\mu\sigma}(y)\big)\Big)(x)=0.

The tensor $T$ is not required to come from a contravariant $\Theta$ .

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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