Identity (12a): the gravitational energy-momentum identity

Proved

EinsteinGrossmann1913.gravitational_energy_momentum_identity

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, with $g=\det(g_{\mu\nu})<0$ ), let $\gamma_{\mu\nu}$ be its reciprocal, and let $\kappa\neq0$ . With $\vartheta_{\mu\nu}$ the stress-energy tensor of the gravitational field (eq. (13)) and $\Delta_{\mu\nu}(\gamma)$ the operator of eq. (15), for every $\sigma=1,2,3,4$ and every point,

\sum_{\mu\nu}\frac{\partial}{\partial x_\nu}\Big\{\sqrt{-g}\,g_{\sigma\mu}\,\kappa\vartheta_{\mu\nu}\Big\}=\tfrac12\sum_{\mu\nu}\sqrt{-g}\,\frac{\partial g_{\mu\nu}}{\partial x_\sigma}\Big\{-\Delta_{\mu\nu}(\gamma)+\kappa\vartheta_{\mu\nu}\Big\}.

This is the identity (12) of the paper rewritten with the abbreviations (13) and (15). It is a pure differential identity in $g_{\mu\nu}$ , valid for every metric field, and it is the step from which the field equations (18) and the conservation law (19) are read off.

Formalization Note $\kappa\vartheta_{\mu\nu}$ does not depend on $\kappa$ ; $\kappa\neq0$ is assumed because $\vartheta_{\mu\nu}$ is defined with the factor $1/(2\kappa)$ .

Preamble

import Mathlib
import Definitions.Def_EinsteinGrossmann1913_Defs

Formal statement

namespace EinsteinGrossmann1913

theorem gravitational_energy_momentum_identity (κ : ℝ) (hκ : κ ≠ 0) (g : Field2)
    (hg : IsMetricField g) (σ : Fin 4) (x : Coord) :
    ∑ μ : Fin 4, ∑ ν : Fin 4,
        pd ν (fun y => sqrtNegDet g y * g y σ μ * (κ * gravStressEnergy κ g y μ ν)) x
      = (1 / 2) * ∑ μ : Fin 4, ∑ ν : Fin 4,
          sqrtNegDet g x * pd σ (fun y => g y μ ν) x
            * (-entwurfDelta g x μ ν + κ * gravStressEnergy κ g x μ ν) := 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, pp. 15–16, eqs. (12), (13), (15) and (12a).

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 $\kappa$ be a real number with $\kappa\neq0$ . $g$ 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. Fix an index $\sigma$ and a point $x\in\mathbb R^4$ . 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 $x_4$ ). The statement asserts

\sum_{\mu,\nu}\partial_\nu\Big(y\mapsto\sqrt{-g}(y)\,g_{\sigma\mu}(y)\,\kappa\,\vartheta_{\mu\nu}(y)\Big)(x)=\frac12\sum_{\mu,\nu}\sqrt{-g}(x)\,\partial_\sigma g_{\mu\nu}(x)\,\Big(-\Delta_{\mu\nu}(x)+\kappa\,\vartheta_{\mu\nu}(x)\Big),

where

\vartheta_{\mu\nu}=-\frac{1}{2\kappa}\sum_{\alpha,\beta,\tau,\rho}\Big(\gamma_{\alpha\mu}\gamma_{\beta\nu}\,\partial_\alpha g_{\tau\rho}\,\partial_\beta\gamma_{\tau\rho}-\tfrac12\gamma_{\mu\nu}\gamma_{\alpha\beta}\,\partial_\alpha g_{\tau\rho}\,\partial_\beta\gamma_{\tau\rho}\Big)

and

\Delta_{\mu\nu}=\sum_{\alpha,\beta}\frac{1}{\sqrt{-g}}\,\partial_\alpha\big(\gamma_{\alpha\beta}\sqrt{-g}\,\partial_\beta\gamma_{\mu\nu}\big)-\sum_{\alpha,\beta,\tau,\rho}\gamma_{\alpha\beta}g_{\tau\rho}\,\partial_\alpha\gamma_{\mu\tau}\,\partial_\beta\gamma_{\nu\rho}.

The identity is asserted for every such $g$ , every $\sigma$ and every $x$ ; no matter tensor or field equation is involved.

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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