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Einstein And Matter

1 · Prerequisites

2 · Summary

This page states the classical field law of the framework and the matter models it couples to. Sixteen items are mathematical definitions and conditional deductions inside a time-oriented smooth Lorentzian four-manifold. Five are physics items: two adopted presentations of the field law, two model-conditional theorems and one scope note. No item reports a measurement, and none asserts that the model or its postulates describe nature.

Volume, divergence and flux. The metric volume form ${v o l}_{g} = \sqrt{∣ \det g ∣} d x^{0} \land \dots \land d x^{3}$ , the current flux $Φ_{J}$ through oriented $3$ -faces and finite-chain charges are defined in The metric volume form, current flux through 3-faces and finite-chain charges; the chart-independence of the volume form, its value $1$ on positively oriented orthonormal frames, and the unit normal and density of a non-null hyperplane are proved in Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane. The density identity $\nabla_{a} J^{a} = w^{- 1} \partial_{a} (w J^{a})$ is The derivative of the metric density and the chart formula for the divergence of a vector field, and the divergence theorem on finite smooth singular chains with the flux balance for divergence-free currents $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B)$ and the non-null face formula are Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.

Variation and the Hilbert tensor. Stability of Lorentzian signature under small compactly supported perturbations (Lorentzian metrics are stable under small compactly supported symmetric perturbations) and the coefficient-vanishing lemma (A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes) are the analytic inputs for the first variation of the Einstein-Hilbert density and the timelike length functional (First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional). For an action domain containing every compactly supported metric perturbation, the Hilbert stress-energy tensor is defined by $δ S_{m} = - \frac{1}{2 c} \int T_{a b} δ g^{a b} {v o l}_{g}$ (The Hilbert stress-energy tensor of a metric-dependent action) and shown to be unique (Uniqueness of the Hilbert stress-energy tensor). Killing fields, their geodesic contraction, the conserved current built from a divergence-free symmetric tensor and the on-shell conservation of the Hilbert tensor are treated in Killing fields of a Lorentzian metric and Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor.

Matter branches. The scalar-field model with its action difference, equation $□_{g} φ - V^{'} (φ) = 0$ and candidate stress-energy tensor is The scalar-field matter model of the classical gravitational framework; the metric variation of its action is represented by that tensor and its conservation identity $\nabla^{a} T_{a b} = K (□_{g} φ - V^{'} (φ)) \partial_{b} φ$ is proved in First variation, Hilbert tensor and conservation of the scalar-field model. The dust model with its continuity and geodesic system is The dust matter model: rest density, four-velocity, stress-energy and dust system and the perfect-fluid model with its conservation equations is The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy; the exact equivalence of $\nabla^{a} T_{a b} = 0$ with those systems is Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations.

The field law. The action formulation, its Einstein-Hilbert density with coupling $κ = 8 π G_{N} / c^{4}$ , its supplied scalar-field matter action and its variational principle are adopted in Adopted classical gravitational action and variational field law; prescribed dust and perfect-fluid sectors use the direct-equation presentation is adopted in Adopted Einstein field equation in the direct-equation presentation. The two are identified by Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling for the supplied action and its Hilbert tensor: stationarity of the total action difference under compactly supported metric variations holds exactly when $G_{a b} + Λ g_{a b} = κ T_{a b}$ . The consistency conditions of that law — necessary conservation, the trace identity, the parameter-variation identity, the flat vacuum solution and the rest-frame readings of the energy conditions — are collected in Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid, and the scope note Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations fixes the status of the adopted and derived claims and the obligations deferred to later pages. The companion page gr-einstein-and-matter-examples computes the volume form and flux, conservation, Killing currents and energy-condition verdicts on flat-space examples and two counterexamples.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The metric volume form, current flux through 3-faces and finite-chain charges

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $M$ carry an orientation in the sense of Oriented smooth manifolds and oriented charts (so that its charts that are oriented in that orientation have orientation-preserving transition maps), and let $\nabla$ be its metric-selected connection. This item fixes the metric volume form, the flux of a current through oriented $3$ -faces and finite-chain charges. The orientation of $M$ is an independent datum: it is not determined by the metric and not by the time orientation.

(a) Metric volume density and volume form. Let $(U, x)$ be an oriented chart, with metric components $g_{a b} = g (\partial_{a}, \partial_{b})$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a)). Define the metric volume density

$w : = \sqrt{∣ \det (g_{a b}) ∣} > 0 on U,$

a strictly positive smooth function because $g_{a b}$ is smooth and $g_{p}$ is nondegenerate. In the chart define the local $4$ -form

${v o l}_{g, U} : = w d x^{0} \land d x^{1} \land d x^{2} \land d x^{3},$

with the exterior product of The wedge product of differential forms and the coordinate covector fields of Local coordinate expression for a differential form. That these local forms agree on chart overlaps and therefore define one smooth nowhere-vanishing $4$ -form ${v o l}_{g}$ on $M$ — the metric volume form — is proved in Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane ↗ clause (a), together with the reading statement of clause (b) there: in every oriented chart ${v o l}_{g} = w d x^{0} \land \dots \land d x^{3}$ , and on every positively oriented $g_{p}$ -orthonormal basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ one has ${v o l}_{g, p} (e_{0}, e_{1}, e_{2}, e_{3}) = 1$ .

(b) Current flux through oriented $3$ -faces and chains. Let $J$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle). The current $3$ -form of $J$ is the contraction

$ι_{J} {v o l}_{g},$

the interior product of the $4$ -form ${v o l}_{g}$ with $J$ (Interior product of a form by a vector field).

By a smooth singular $k$ -simplex of $M$ — the convention of the library definition def-smooth-singular-simplex (differential-geometry track), restated here so this page is self-contained — we mean a map $σ : Δ^{k} \to M$ whose domain is the standard $k$ -simplex $Δ^{k} = {t \in R^{k + 1} : t_{i} \geq 0, \sum_{i} t_{i} = 1},$ for which there exist an open set $O \subseteq A^{k} = {t \in R^{k + 1} : \sum_{i} t_{i} = 1}$ containing $Δ^{k}$ and a smooth map $\overset{ˉ}{σ} : O \to M$ with $\overset{ˉ}{σ} ∣_{Δ^{k}} = σ$ ; the extension is not part of the simplex data, and degenerate parametrizations are allowed. With the integration of forms over such simplices (Integral of a form over a smooth singular simplex), a $3$ -face is a smooth singular $3$ -simplex $σ$ , and the flux of $J$ through $σ$ is the real number

$Φ_{J} (σ) : = \int_{σ} ι_{J} {v o l}_{g} .$

A finite smooth singular $k$ -chain — the convention of the library definition def-smooth-singular-chain-and-cochain-complexes, restated here — is a finite formal real linear combination $Σ = \sum_{i} c_{i} σ_{i}$ of smooth singular $k$ -simplices. For such a $3$ -chain the flux of $J$ through $Σ$ is defined additively,

$Φ_{J} (Σ) : = \sum_{i} c_{i} Φ_{J} (σ_{i}),$

a definition independent of the written expression for $Σ$ , because a finite formal sum is evaluated by adding the finitely many coefficients and inserting a zero coefficient changes nothing. $Φ_{J} (Σ)$ is the charge of the current $J$ assigned to $Σ$ when the model reads $J$ as a conserved current; that reading belongs to the physical postulates and is not part of this definition.

(c) Non-null faces and their unit normals. A $3$ -dimensional linear subspace $H \subseteq T_{p} M$ is nondegenerate for $g_{p}$ when the restriction of $g_{p}$ to $H$ is nondegenerate, equivalently when $H \cap H^{⊥} = {0}$ in the orthogonality of eta-orthogonality and orthogonal complements in a Lorentz vector space. A smooth singular $3$ -simplex $σ$ is non-null at a parameter point $t$ when the span of the three velocity vectors $σ_{*} \partial_{1} ∣_{t}, σ_{*} \partial_{2} ∣_{t}, σ_{*} \partial_{3} ∣_{t}$ has dimension $3$ and is nondegenerate at $σ (t)$ , and non-null when this holds at every parameter point. For a non-null face the unit normal $ν$ at that point is the unique vector of $T_{σ (t)} M$ with $ν \in H^{⊥}$ , $g (ν, ν) \in {- 1, + 1}$ and $(ν, σ_{*} \partial_{1}, σ_{*} \partial_{2}, σ_{*} \partial_{3})$ a positively oriented basis of $T_{σ (t)} M$ ; its existence, uniqueness and its sign $ε : = g (ν, ν)$ are proved in Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane ↗ clause (c), together with the density identity of clause (c) used to read $ι_{J} {v o l}_{g}$ on such a face. Rank-deficient faces and rank-three faces with degenerate induced metric are not assigned a unit normal; their flux is defined by the $3$ -form $ι_{J} {v o l}_{g}$ itself, as in clause (b).

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) whose underlying manifold carries an orientation (Oriented smooth manifolds and oriented charts), and let $w$ and the local forms ${v o l}_{g, U}$ be as in The metric volume form, current flux through 3-faces and finite-chain charges clause (a).

(a) Chart independence. Let $(U, x)$ and $(V, y)$ be oriented charts with $x = φ \circ y$ on the overlap of their domains, where $φ$ is the smooth transition map; write $g_{a b}$ and $g_{a b}^{'}$ for the metric components in the two charts and $w = \sqrt{∣ \det (g_{a b}) ∣}$ , $w^{'} = \sqrt{∣ \det (g_{a b}^{'}) ∣}$ . Then

$\det (g_{a b}^{'}) = (\det D φ)^{2} \det (g_{a b} \circ φ), w^{'} = (\det D φ) (w \circ φ),$

and the local forms agree on the overlap,

$φ^{*} (w d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}) = w^{'} d y^{0} \land d y^{1} \land d y^{2} \land d y^{3} .$

Consequently there is a unique smooth nowhere-vanishing $4$ -form ${v o l}_{g}$ on $M$ — the metric volume form — whose restriction to the domain of every oriented chart $(U, x)$ is $w d x^{0} \land d x^{1} \land d x^{2} \land d x^{3}$ ; it is a smooth section with nowhere-zero values and positive on the given orientation.

(b) Value on orthonormal frames. For every $p \in M$ and every positively oriented ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ with $g_{p} (e_{A}, e_{B}) = diag (- 1, 1, 1, 1)_{A B}$ ,

${v o l}_{g, p} (e_{0}, e_{1}, e_{2}, e_{3}) = 1.$

(c) Non-null hyperplanes: normal and density. Let $p \in M$ , let $H \subseteq T_{p} M$ be a $3$ -dimensional subspace that is nondegenerate for $g_{p}$ , and let $v_{1}, v_{2}, v_{3}$ be an ordered basis of $H$ . Then $H^{⊥}$ is one-dimensional and nondegenerate and $H = (H^{⊥})^{⊥}$ . Let $u \in H^{⊥} ∖ {0}$ and write $ν : = u / \sqrt{∣ g_{p} (u, u) ∣}$ , so that $ε : = g_{p} (ν, ν) \in {- 1, 1}$ . If $δ \in {- 1, 1}$ is the sign for which $(δ ν, v_{1}, v_{2}, v_{3})$ is a positively oriented basis of $T_{p} M$ , then $ν^{*} : = δ ν$ is the unit normal to $H$ adapted to the ordered basis $(v_{1}, v_{2}, v_{3})$ ; it satisfies $g_{p} (ν^{*}, ν^{*}) = ε$ , $g_{p} (ν^{*}, h) = 0$ for every $h \in H$ , and it is the unique vector with these two properties for which $(ν^{*}, v_{1}, v_{2}, v_{3})$ is positively oriented. Moreover

${v o l}_{g, p} (ν^{*}, v_{1}, v_{2}, v_{3}) = \sqrt{∣ \det (g_{p} (v_{i}, v_{j})) ∣} .$

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with an orientation, oriented charts $(U, x)$ , $(V, y)$ with transition $φ$ , a point $p \in M$ , a three-dimensional nondegenerate subspace $H \subseteq T_{p} M$ with ordered basis $v_{1}, v_{2}, v_{3}$ , and the objects $w$ , ${v o l}_{g, U}$ of The metric volume form, current flux through 3-faces and finite-chain charges clause (a).

[L1]

The metric components transform as $g_{a b}^{'} (y) = \partial_{a} φ^{c} (y) \partial_{b} φ^{d} (y) g_{c d} (φ (y))$ (Chart independence and the transformation law of a smooth covariant 2-tensor field, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L2]

For square matrices of one size, $\det (A B) = \det A \det B$ and $\det (A^{T}) = \det A$ ; the determinant of $diag (- 1, 1, 1, 1)$ is $- 1$ (For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ , For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

[L3]

Pullback of forms is functorial and commutes with the wedge product; in coordinates, $φ^{*} (d x^{0} \land \dots \land d x^{3}) = \det (D φ) d y^{0} \land \dots \land d y^{3}$ for a smooth map of open subsets of $R^{4}$ , by the coordinate expression of pullback and of the wedge product (The pullback of a differential form, The wedge product of differential forms, Local coordinate expression for a differential form). Oriented charts of an oriented manifold have orientation-preserving transitions, so $\det D φ > 0$ (Oriented smooth manifolds and oriented charts).

[L4]

At every point there is a $g_{p}$ -orthonormal ordered basis $E_{0}, E_{1}, E_{2}, E_{3}$ with $g_{p} (E_{A}, E_{B}) = diag (- 1, 1, 1, 1)_{A B}$ ; exchanging two of its spacelike vectors preserves orthonormality and the position of the negative entry and reverses its orientation, so such a basis may be chosen positively oriented (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Alternating $k$ -linear maps).

[L5]

$⊥$ is the $g_{p}$ -orthogonal complement of eta-orthogonality and orthogonal complements in a Lorentz vector space: $S^{⊥} = {v : g_{p} (v, s) = 0 \forall s \in S}$ is a linear subspace, $S \subseteq (S^{⊥})^{⊥}$ , and $S^{⊥} = (span S)^{⊥}$ .

[L6]

Choose a basis $h_{1}, h_{2}, h_{3}$ of $H$ and extend it to a basis $h_{1}, h_{2}, h_{3}, e_{4}$ of the four-dimensional space $T_{p} M$ . Any $λ \in H^{*}$ extends to a functional $\tilde{λ} \in T_{p} M^{*}$ by assigning, for example, $\tilde{λ} (e_{4}) = 0$ . In this basis the matrix of $g_{p}$ is invertible because $g_{p}$ is nondegenerate; hence the linear map $T_{p} M \to T_{p} M^{*}$ , $v \mapsto g_{p} (v, \cdot)$ , is represented by an invertible matrix and is bijective. Thus $\tilde{λ} = g_{p} (v, \cdot)$ for a unique $v$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model clause (a)); this proves the extension-and-representation fact used below by finite-dimensional basis extension and an invertible matrix.

Proof

technique · direct; transform the metric determinant and the coordinate volume form between oriented charts, evaluate the result on orthonormal frames, and reduce the hyperplane statement to a finite coordinate computation

1.1L1L2L3algebra

(Determinant transformation.) On the overlap, [L1] reads $g^{'} = D φ^{T} (g \circ φ) D φ$ in matrix form. By [L2], $\det g^{'} = \det (D φ)^{2} \det (g \circ φ)$ . Since $\det D φ > 0$ by [L3] and $w, w^{'} > 0$ by definition, taking square roots gives $w^{'} = (\det D φ) (w \circ φ)$ .

1.2L5L6algebra

(The orthogonal complement is one-dimensional.) Let $L : T_{p} M \to H^{*}$ be $L (v) = g_{p} (v, \cdot) ∣_{H}$ . Then $\ker L = H^{⊥}$ . For each $λ \in H^{*}$ , [L6] constructs an extension $\tilde{λ} \in T_{p} M^{*}$ and the invertible matrix of $g_{p}$ gives a unique $u \in T_{p} M$ with $\tilde{λ} = g_{p} (u, \cdot)$ ; hence $L (u) = λ$ and $L$ is surjective. Rank-nullity gives $\dim H^{⊥} = 4 - \dim H^{*} = 1$ . Applying the same basis-extension and nondegenerate-matrix argument to the one-dimensional space $H^{⊥}$ shows $\dim (H^{⊥})^{⊥} = 3$ .

1.3L5algebra

(Coordinate computation.) Choose a positively oriented $g_{p}$ -orthonormal basis $E_{0}, \dots, E_{3}$ with $g_{p} (E_{A}, E_{B}) = diag (- 1, 1, 1, 1)_{A B}$ as in [L4] and write vectors as their coordinate columns. Let $c$ be the coordinate column of $ν^{*}$ and let $w_{i}$ be the coordinate columns of $v_{i}$ . Since $g_{p} (ν^{*}, v_{i}) = 0$ , the matrix $η = diag (- 1, 1, 1, 1)$ satisfies $w_{i}^{T} η c = 0$ for $i = 1, 2, 3$ , and $c^{T} η c = ε$ . Let $W$ be the $4 \times 3$ matrix with columns $w_{1}, w_{2}, w_{3}$ and $M$ the $4 \times 4$ matrix with columns $c, w_{1}, w_{2}, w_{3}$ . Then $M^{T} η M = (\begin{array}{cc} ε & 0 \\ 0 & Q \end{array}), Q = (g_{p} (v_{i}, v_{j}))_{i, j = 1}^{3} .$

2.1step 1.1L3algebra

(Agreement of local forms.) By [L3], $φ^{*} (d x^{0} \land \dots \land d x^{3}) = \det (D φ) d y^{0} \land \dots \land d y^{3}$ . Multiplying by $w \circ φ$ and substituting step 1.1 gives $φ^{*} (w d x^{0} \land \dots \land d x^{3}) = w^{'} d y^{0} \land \dots \land d y^{3}$ , the displayed identity. In particular the two local forms have the same value at every point of the overlap under the identification by $φ$ .

2.2step 1.2L5algebra

(Nondegeneracy and double complement.) If $u \in H \cap H^{⊥}$ , then $g_{p} (u, h) = 0$ for every $h \in H$ , and nondegeneracy of the restriction of $g_{p}$ to $H$ forces $u = 0$ . Hence $H \cap H^{⊥} = {0}$ , the sum $H + H^{⊥} = T_{p} M$ is direct by dimension, and the restriction of $g_{p}$ to $H^{⊥}$ is nondegenerate: if $u \in H^{⊥}$ satisfies $g_{p} (u, u^{'}) = 0$ for all $u^{'} \in H^{⊥}$ , then $g_{p} (u, \cdot)$ vanishes on $H$ and on $H^{⊥}$ and hence on $T_{p} M$ , so $u = 0$ . Finally $H \subseteq (H^{⊥})^{⊥}$ by [L5] and both have dimension $3$ by steps 1.2 and the definition of $H$ , so $H = (H^{⊥})^{⊥}$ .

2.3step 1.3L2algebra

( $M$ is invertible and a block identity.) The columns of $M$ form a basis of $T_{p} M$ : the $w_{i}$ are linearly independent because $H$ has dimension $3$ , and $c \notin span (w_{1}, w_{2}, w_{3})$ since $c$ is $η$ -orthogonal to that span while $c^{T} η c = ε \neq 0$ . Hence $\det M \neq 0$ , and $Q$ is invertible because $H$ is nondegenerate. Taking determinants in step 1.3 with [L2] and expanding the block determinant in the first coordinate gives $\det (M^{T} η M) = \det M^{2} \det η = - \det M^{2}$ and $\det (M^{T} η M) = ε \det Q$ , so $\det M^{2} = - ε \det Q = ∣ \det Q ∣$ . Therefore $∣ \det M ∣ = \sqrt{∣ \det Q ∣}$ .

3.1step 2.1L3given

(Gluing.) Each local form is smooth and nowhere vanishing on its chart domain because $w, w^{'} > 0$ are smooth and $d x^{0} \land \dots \land d x^{3}$ is a nowhere-vanishing smooth $4$ -form. By step 2.1 the local forms agree on every overlap, so they define a unique smooth section ${v o l}_{g}$ of $⋀^{4} T^{*} M$ with the stated local expressions; the transition identity also shows that ${v o l}_{g}$ evaluates positively on the positively oriented frames of the orientation at every point.

3.2step 2.2L4algebra

(Existence and uniqueness of the adapted normal.) Let $u$ span $H^{⊥}$ , put $ν = u / \sqrt{∣ g_{p} (u, u) ∣}$ ; then $g_{p} (ν, ν) \in {- 1, 1}$ , a value depending only on $H$ because replacing $u$ by a nonzero multiple does not change it. Choose $δ \in {- 1, 1}$ so that $(δ ν, v_{1}, v_{2}, v_{3})$ is positively oriented; such a sign exists and is unique because the two choices give opposite orientations and the orientation is a choice of one of the two rays of $⋀^{4} T_{p} M$ . Uniqueness of $ν^{*} = δ ν$ as a vector with $g_{p} (ν^{*}, ν^{*}) = \pm 1$ , $ν^{*} ⊥ H$ and $(ν^{*}, v_{1}, v_{2}, v_{3})$ positively oriented is immediate: any such vector is a nonzero multiple of $u$ , the last condition fixes its sign, and the unit condition fixes its modulus.

4.1step 3.1L2L4algebra

(Value on orthonormal frames.) Fix $p$ and a positively oriented $g_{p}$ -orthonormal basis $e_{0}, \dots, e_{3}$ as in [L4], take an oriented chart at $p$ , and let $E = [e_{A}^{i}]$ be the matrix whose columns are the coordinate components of the $e_{A}$ in the chart basis. Orthonormality is $E^{T} G E = diag (- 1, 1, 1, 1)$ with $G = (g_{i j} (p))$ , so [L2] gives $\det (E)^{2} \det G = - 1$ , that is, $∣ \det E ∣ = 1 / w (p)$ . Positivity of the orientation of $(e_{A})$ and of the chart gives $\det E > 0$ , hence $\det E = 1 / w (p)$ . By the local expression of step 3.1 and the defining property of wedge products, ${v o l}_{g, p} (e_{0}, \dots, e_{3}) = w (p) \det E = 1$ .

5.1step 4.1step 3.2step 2.3algebra∎

(Value of the volume form.) By step 4.1 applied to the basis $E_{A}$ , the coordinate expression of ${v o l}_{g}$ from step 3.1 gives ${v o l}_{g, p} (ν^{*}, v_{1}, v_{2}, v_{3}) = \det M$ , the determinant of the coordinate matrix of the four vectors. By construction of $ν^{*}$ in step 3.2 the tuple is positively oriented, so $\det M > 0$ ; by step 2.3, $\det M = \sqrt{∣ \det Q ∣} = \sqrt{∣ \det (g_{p} (v_{i}, v_{j})) ∣}$ . This is the displayed formula and completes the proof.

Remarks

Orientation versus time orientation. The construction uses the orientation of $M$ only; the time orientation enters neither ${v o l}_{g}$ nor the sign of the unit normal. Replacing the orientation by its opposite replaces $ν^{*}$ by $- ν^{*}$ and ${v o l}_{g}$ by $- {v o l}_{g}$ , leaving $ι_{J} {v o l}_{g}$ unchanged up to the corresponding sign convention on chains.
Null hyperplanes. A three-dimensional subspace $H$ on which $g_{p}$ is degenerate has $H^{⊥} \subseteq H$ ; it has no unit normal and the density identity above fails, which is why null faces require the three-form itself (The metric volume form, current flux through 3-faces and finite-chain charges clause (c)).

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

The derivative of the metric density and the chart formula for the divergence of a vector field

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Christoffel symbols of an affine connection, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), let $(U, x)$ be a smooth chart, and define the local metric density by $w : = \sqrt{∣ \det (g_{a b}) ∣}$ on $U$ , using the matrix of metric components in this chart. This definition requires no orientation or volume form. Write $g^{a b}$ for the inverse-metric components (The inverse metric, musical isomorphisms and index raising) and $Γ^{a}_{b c}$ for the Christoffel symbols of $\nabla$ in the chart.

(a) Derivative of the density. At every point of $U$ ,

$\partial_{a} w = \frac{1}{2} w g^{b c} \partial_{a} g_{b c} .$

(b) Trace of the Christoffel symbols. At every point of $U$ ,

$Γ^{b}_{b a} = w^{- 1} \partial_{a} w .$

(c) Divergence formula. For every smooth vector field $J$ on $M$ , at every point of $U$ ,

$\nabla_{a} J^{a} = \partial_{a} J^{a} + Γ^{a}_{a b} J^{b} = w^{- 1} \partial_{a} (w J^{a}),$

where $\nabla_{a} J^{a} = g^{a b} \nabla_{b} J_{a}$ is the divergence of $J$ contracted with the metric and $J^{a}$ are the components of $J$ (A smooth vector field is a smooth section of the tangent bundle).

The identity of (c) is a chart formula for a chart-independent scalar: the right-hand side is independent of the chart, as the left-hand side is.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , a smooth chart $(U, x)$ , the locally defined density $w = \sqrt{∣ \det (g_{a b}) ∣}$ , and a smooth vector field $J$ .

[L1]

In a chart the Christoffel symbols are $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ and satisfy $Γ^{a}_{b c} = Γ^{a}_{c b}$ (Christoffel symbols of an affine connection, Torsion freedom and metric compatibility in chart components, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

The extension of $\nabla$ to tensor fields has coordinate components $J^{a}_{; c} = \partial_{c} J^{a} + Γ^{a}_{d c} J^{d}$ for a vector field $J$ , and $\nabla g = 0$ , $\nabla g^{- 1} = 0$ ; the divergence is $\nabla_{a} J^{a} = J^{a}_{; a} = g^{a b} J_{a; b}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), The inverse metric, musical isomorphisms and index raising).

[L3]

The components $g^{a b}$ of the inverse metric are smooth and satisfy $g^{a c} g_{c b} = δ^{a}_{b}$ (The inverse metric, musical isomorphisms and index raising).

[L4]

For a differentiable matrix curve $A$ with invertible values, $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ (The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ ).

[L5]

There is a $g_{p}$ -orthonormal basis of $T_{p} M$ in which the matrix of $g_{p}$ is $diag (- 1, 1, 1, 1)$ ; consequently $\det (g_{a b} (p)) \neq 0$ and in fact $\det (g_{a b} (p)) < 0$ at every $p$ (Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

Proof

technique · direct; differentiate the determinant along a coordinate curve, then substitute the Christoffel formula

1.1L3L4L5algebra

(Determinant derivative.) Fix $p \in U$ with coordinates $x (p)$ and fix an index $a$ . Define the matrix curve $A (s) : = (g_{b c} (x (p) + s e_{a}))$ for small $∣ s ∣$ , where $e_{a}$ is the $a$ -th standard basis vector; its entries are differentiable at $s = 0$ and, since $\det A (0) = \det (g_{b c} (p)) \neq 0$ by [L5] and the determinant is continuous, $A (s)$ is invertible for small $∣ s ∣$ . By [L4], $\partial_{a} \det (g_{b c}) (p) = \det (g_{b c} (p)) tr (A (0)^{- 1} A^{'} (0)) = \det (g_{b c} (p)) g^{b c} (p) \partial_{a} g_{b c} (p)$ , using [L3].

2.1step 1.1L5algebra

(Derivative of the square root.) By [L5], $\det (g_{b c}) < 0$ on $U$ , so $∣ \det (g_{b c}) ∣ = - \det (g_{b c})$ and $w = (- \det (g_{b c}))^{1 / 2}$ . Differentiating the square root and substituting step 1.1, $\partial_{a} w = \frac{1}{2} (- \det (g_{b c}))^{- 1 / 2} (- \partial_{a} \det (g_{b c})) = \frac{1}{2} w g^{b c} \partial_{a} g_{b c}$ , which is claim (a).

3.1step 2.1L1L3algebra

(Christoffel trace.) By [L1], $Γ^{b}_{b a} = \frac{1}{2} g^{b c} (\partial_{a} g_{b c} + \partial_{b} g_{c a} - \partial_{c} g_{b a})$ . Relabelling the dummy indices $b \leftrightarrow c$ in the third term and using the symmetry of $g_{b a}$ and of $g^{b c}$ turns it into $- \frac{1}{2} g^{b c} \partial_{b} g_{c a}$ , which cancels the second term; hence $Γ^{b}_{b a} = \frac{1}{2} g^{b c} \partial_{a} g_{b c} = w^{- 1} \partial_{a} w$ by claim (a).

4.1step 3.1L2algebra∎

(Divergence.) By [L2], $\nabla_{a} J^{a} = \partial_{a} J^{a} + Γ^{a}_{a b} J^{b}$ . Substituting claim (b), $\nabla_{a} J^{a} = \partial_{a} J^{a} + w^{- 1} (\partial_{b} w) J^{b} = w^{- 1} (w \partial_{a} J^{a} + J^{a} \partial_{a} w) = w^{- 1} \partial_{a} (w J^{a})$ , which is claim (c) and completes the proof.

Remarks

Density reading. The function $w$ is not a scalar: for a change from chart $x$ to chart $y$ , put $A = D (x \circ y^{- 1})$ on the overlap. The metric transformation law gives $g_{y} = A^{T} g_{x} A$ and hence $w_{y} = ∣ \det A ∣ w_{x}$ (Chart independence and the transformation law of a smooth covariant 2-tensor field, For same-sized finite square matrices over a commutative ring, $\det (A B) = \det (A) \det (B)$ ). Writing $J_{x}^{a}$ and $J_{y}^{a}$ for the components of $J$ in the two charts at the same point, the quantities $w J^{a}$ form a vector density, meaning that $w_{y} J_{y}^{a} = ∣ \det A ∣ (A^{- 1})^{a}_{b} w_{x} J_{x}^{b}$ . By claim (c), $\partial_{y^{a}} (w_{y} J_{y}^{a}) = w_{y} \nabla_{a} J^{a} = ∣ \det A ∣ \partial_{x^{b}} (w_{x} J_{x}^{b})$ . Thus this coordinate divergence is a scalar density, meaning a coefficient transforming by $∣ \det A ∣$ , rather than a chart-independent scalar; $w^{- 1} \partial_{a} (w J^{a})$ is the chart-independent scalar. In oriented charts, the invariant top form represented by $\partial_{a} (w J^{a}) d x^{0} \land \dots \land d x^{3}$ is the expression used in Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.
Sign conventions. No orientation is used in this lemma; claims (a)-(c) hold on every chart of every time-oriented smooth Lorentzian four-manifold.

TheoremStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with an orientation, metric-selected connection $\nabla$ , metric volume form ${v o l}_{g}$ and current flux $Φ_{J}$ as in The metric volume form, current flux through 3-faces and finite-chain charges, and let $J$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle). Write $\nabla_{a} J^{a}$ for the divergence of $J$ .

(a) Exterior derivative of the current form. As smooth $4$ -forms on $M$ ,

$d (ι_{J} {v o l}_{g}) = (\nabla_{a} J^{a}) {v o l}_{g} .$

(b) Flux balance (Stokes). For every finite smooth singular $4$ -chain $C$ of $M$ — a finite formal real sum of smooth singular $4$ -simplices, the library convention def-smooth-singular-chain-and-cochain-complexes restated in The metric volume form, current flux through 3-faces and finite-chain charges clause (b) — and the flux $Φ_{J}$ of the induced boundary $3$ -chain (Integral of a form over a smooth singular simplex, Stokes theorem for smooth singular chains),

$Φ_{J} (\partial C) = \int_{C} (\nabla_{a} J^{a}) {v o l}_{g} .$

Consequently, if $\partial C = Σ_{2} - Σ_{1} + B$ for smooth singular $3$ -chains $Σ_{1}, Σ_{2}, B$ , then

$Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B) + \int_{C} (\nabla_{a} J^{a}) {v o l}_{g} .$

In particular, if $\nabla_{a} J^{a} = 0$ on $C$ then $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B)$ , and if moreover $Φ_{J} (B) = 0$ then $Φ_{J} (Σ_{2}) = Φ_{J} (Σ_{1})$ : equal charges need a vanishing side flux, not merely a divergence-free current.

(c) Non-null face formula. Let $σ$ be a smooth singular $3$ -simplex — a map $σ : Δ^{3} \to M$ admitting a smooth extension to a neighbourhood of the standard $3$ -simplex inside its affine plane, the convention restated in The metric volume form, current flux through 3-faces and finite-chain charges clause (b) — that is non-null at a parameter point $t$ (The metric volume form, current flux through 3-faces and finite-chain charges clause (c)), with $v_{i} : = σ_{*} \partial_{i} ∣_{t}$ spanning the hyperplane $H$ and with the unit normal $ν^{*}$ adapted to the ordered basis $(v_{1}, v_{2}, v_{3})$ as in Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane clause (c); write $ε : = g_{σ (t)} (ν^{*}, ν^{*}) \in {- 1, 1}$ . Then

$σ^{*} (ι_{J} {v o l}_{g}) (\partial_{1}, \partial_{2}, \partial_{3}) ∣_{t} = ε g_{σ (t)} (J, ν^{*}) \sqrt{∣ \det (g_{σ (t)} (v_{i}, v_{j})) ∣} .$

Equivalently, the tangent vector $J$ is decomposed as $J = ε g (J, ν^{*}) ν^{*} + w^{'}$ with $w^{'} \in H$ , and only the normal component contributes to the flux through the face.

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with an orientation and metric-selected connection $\nabla$ , the metric volume form ${v o l}_{g}$ and flux $Φ_{J}$ of The metric volume form, current flux through 3-faces and finite-chain charges, a smooth vector field $J$ , and a smooth singular $3$ -simplex $σ$ with parameter point $t$ .

[L1]

$(M, g, N)$ has a metric-selected connection $\nabla$ with chart components $Γ^{a}_{b c}$ and divergence formula $\nabla_{a} J^{a} = w^{- 1} \partial_{a} (w J^{a})$ in every chart, where $w = \sqrt{∣ \det g_{a b} ∣}$ (The derivative of the metric density and the chart formula for the divergence of a vector field).

[L2]

In an oriented chart, ${v o l}_{g} = w d x^{0} \land \dots \land d x^{3}$ , and $ι_{J} {v o l}_{g} = \sum_{a} (- 1)^{a} w J^{a} d x^{0} \land \dots \hat{d x^{a}} \dots \land d x^{3}$ , because $(ι_{J} ω) (X_{1}, \dots) = ω (J, X_{1}, \dots)$ (The metric volume form, current flux through 3-faces and finite-chain charges clause (a), Interior product of a form by a vector field, Interior product on alternating covectors, Local coordinate expression for a differential form).

[L3]

In an oriented chart the current form of $J$ is $ι_{J} {v o l}_{g} = \sum_{a = 0}^{3} (- 1)^{a} w J^{a} d x^{0} \land \dots \hat{d x^{a}} \dots \land d x^{3}$ by [L2]. Applying the coordinate formula for the exterior derivative of a $3$ -form, only the derivative in the omitted $a$ -coordinate contributes to the $a$ th summand; moving $d x^{a}$ into its ordered position supplies a second factor $(- 1)^{a}$ . Thus $d (ι_{J} {v o l}_{g}) = \sum_{a = 0}^{3} \partial_{a} (w J^{a}) d x^{0} \land d x^{1} \land d x^{2} \land d x^{3} .$ This is a derivative of the current $3$ -form, not of the top-degree volume $4$ -form (The exterior derivative by the invariant vector-field formula, The wedge product of differential forms).

[L4]

Stokes' theorem for smooth singular chains: for a finite smooth singular $k$ -chain $c$ — a finite formal real sum of smooth singular $k$ -simplices with the signed face differential, the convention of the library definition def-smooth-singular-chain-and-cochain-complexes — and a smooth $(k - 1)$ -form $ω$ , $\int_{c} d ω = \int_{\partial c} ω$ ; integration over chains is additive in the chain (Stokes theorem for smooth singular chains, Integral of a form over a smooth singular simplex, The metric volume form, current flux through 3-faces and finite-chain charges clause (b)).

[L5]

For a smooth singular simplex $σ$ — a smooth map $Δ^{k} \to M$ admitting a smooth extension to a neighbourhood of $Δ^{k}$ in its affine plane, the convention restated in The metric volume form, current flux through 3-faces and finite-chain charges clause (b) and the library definition def-smooth-singular-simplex — and a form $ω$ , the pullback $σ^{*} ω$ is smooth and $(σ^{*} ω) (\partial_{1}, \dots, \partial_{k}) ∣_{t} = ω_{σ (t)} (σ_{*} \partial_{1} ∣_{t}, \dots, σ_{*} \partial_{k} ∣_{t})$ (The pullback of a differential form).

[L6]

For a three-dimensional nondegenerate subspace $H \subseteq T_{p} M$ with ordered basis $v_{1}, v_{2}, v_{3}$ : the adapted unit normal $ν^{*}$ satisfies $H = (ν^{*})^{⊥}$ , $g (ν^{*}, ν^{*}) = ε \in {- 1, 1}$ and ${v o l}_{g, p} (ν^{*}, v_{1}, v_{2}, v_{3}) = \sqrt{∣ \det (g (v_{i}, v_{j})) ∣}$ (Chart independence of the metric volume form, its value on orthonormal frames, and the unit normal and density of a non-null hyperplane clause (c), eta-orthogonality and orthogonal complements in a Lorentz vector space).

Proof

technique · direct; compute the derivative of the current form in an oriented chart, apply Stokes' theorem to chains, and evaluate the pullback on a non-null face by decomposing the current into normal and tangential parts

1.1L1L2L3algebra

(Chart computation.) In an oriented chart use [L2] and [L3]: $d (ι_{J} {v o l}_{g}) = \sum_{a} \partial_{a} (w J^{a}) d x^{0} \land \dots \land d x^{3} = (\sum_{a} \partial_{a} (w J^{a})) {v o l}_{g} / w$ , and [L1] rewrites $\sum_{a} \partial_{a} (w J^{a}) = w \nabla_{a} J^{a}$ . Hence $d (ι_{J} {v o l}_{g}) = (\nabla_{a} J^{a}) {v o l}_{g}$ in the chart. Since both sides are globally defined smooth $4$ -forms and every point lies in an oriented chart, the identity holds on all of $M$ ; this proves (a).

1.2L6algebra

(Face decomposition.) At the parameter point $t$ put $p : = σ (t)$ and let $ν^{*}$ be the adapted unit normal of [L6], so that $H = (ν^{*})^{⊥}$ and $ε = g_{p} (ν^{*}, ν^{*})$ . Define $α : = ε g_{p} (J, ν^{*})$ and $w^{'} : = J - α ν^{*}$ . Then $g_{p} (w^{'}, ν^{*}) = g_{p} (J, ν^{*}) - α ε = 0$ , so $w^{'} \in (ν^{*})^{⊥} = H$ .

2.1step 1.1L4given

(Stokes flux balance.) Let $C$ be a smooth singular $4$ -chain and apply [L4] with $k = 4$ and $ω = ι_{J} {v o l}_{g}$ : $\int_{\partial C} ι_{J} {v o l}_{g} = \int_{C} d (ι_{J} {v o l}_{g}) = \int_{C} (\nabla_{a} J^{a}) {v o l}_{g}$ by step 1.1. By the definition of $Φ_{J}$ as the integral over a chain this is $Φ_{J} (\partial C) = \int_{C} (\nabla_{a} J^{a}) {v o l}_{g}$ .

3.1step 2.1L4algebra

(Charge balance.) If $\partial C = Σ_{2} - Σ_{1} + B$ , additivity of the chain integral gives $Φ_{J} (\partial C) = Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) + Φ_{J} (B)$ , so step 2.1 yields the displayed balance. If the integrand $\nabla_{a} J^{a}$ vanishes on $C$ , the integral is zero and $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B)$ ; if in addition $Φ_{J} (B) = 0$ , then $Φ_{J} (Σ_{2}) = Φ_{J} (Σ_{1})$ .

4.1step 1.2L5L6algebra∎

(Flux through the face.) By [L5] and the definition of the interior product, $σ^{*} (ι_{J} {v o l}_{g}) (\partial_{1}, \partial_{2}, \partial_{3}) ∣_{t} = {v o l}_{g, p} (J, v_{1}, v_{2}, v_{3})$ . Splitting $J = α ν^{*} + w^{'}$ and using multilinearity, ${v o l}_{g, p} (J, v_{1}, v_{2}, v_{3}) = α {v o l}_{g, p} (ν^{*}, v_{1}, v_{2}, v_{3}) + {v o l}_{g, p} (w^{'}, v_{1}, v_{2}, v_{3})$ . The second term vanishes because $w^{'} \in H$ is a linear combination of $v_{1}, v_{2}, v_{3}$ and the $4$ -form is alternating (Interior product on alternating covectors). By [L6] the first term is $α \sqrt{∣ \det (g_{p} (v_{i}, v_{j})) ∣} = ε g_{p} (J, ν^{*}) \sqrt{∣ \det (g_{p} (v_{i}, v_{j})) ∣}$ , which is the displayed formula of (c) and completes the proof.

Remarks

The side flux is a genuine hypothesis. The balance in (b) is an identity, not an inequality: a divergence-free current can have nonzero flux through a bounded region's side wall, and the charges of the two slices differ by exactly that side flux. The compact-support case (B absent or of zero flux) is the setting used for conserved charges.
Regularity. All chains are finite smooth singular chains; no arbitrary rough or infinite region is integrated, no boundary at infinity is treated and no convergence of an exhaustion is asserted. Finiteness of the chain makes every sum involved finite.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Lorentzian metrics are stable under small compactly supported symmetric perturbations

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), so that at every $p \in M$ the form $g_{p}$ has inertia $(3, 1, 0)$ , and let $h$ be a smooth symmetric covariant $2$ -tensor field on $M$ (Smooth covariant 2-tensor fields on a smooth manifold) whose support is compact.

(a) Pointwise openness. Let $V$ be a real vector space of dimension $4$ and let $G$ be a symmetric bilinear form on $V$ of inertia $(3, 1, 0)$ (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space). If in some basis of $V$ the matrix of $G$ has the form $diag (- 1, 1, 1, 1) + E$ , then $G$ has inertia $(3, 1, 0)$ whenever $∣ E_{i j} ∣ < 1 / 8$ for all $i, j$ . Consequently, for every basis the set of matrices of symmetric forms of inertia $(3, 1, 0)$ is open in the space $M_{4}^{s y m} (R)$ with respect to the maximum-entry norm, and equivalently $g_{p}$ has a neighborhood in the space of symmetric forms at $p$ consisting entirely of forms of inertia $(3, 1, 0)$ .

(b) Global stability. There is $λ_{0} > 0$ such that for every real $λ$ with $∣ λ ∣ < λ_{0}$ the smooth symmetric field $g + λ h$ is a smooth Lorentzian metric on $M$ of signature $(-, +, +, +)$ , with time orientation inherited from $N$ (the field $N$ remains timelike for $g + λ h$ for $∣ λ ∣$ small).

Facts & Assumptions

Given: A time-oriented smooth Lorentzian four-manifold $(M, g, N)$ and a smooth symmetric compactly supported $2$ -tensor field $h$ .

[L1]

At every $p \in M$ the form $g_{p}$ has inertia $(3, 1, 0)$ : in some basis its matrix is $diag (- 1, 1, 1, 1)$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

[L2]

Congruent symmetric forms have the same inertia; the inertia of a form that is positive definite on a subspace $W$ and negative on a vector $u$ is read off by completing the square (Sylvester's law of inertia: every real symmetric form is congruent to $diag (I_{p}, - I_{q}, 0_{r})$ , and $(p, q, r)$ is unique).

[L3]

A positive definite form on a subspace has an orthonormal basis, by Gram–Schmidt applied in that subspace (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans).

[L4]

$h$ is smooth hence continuous with compact support $K$ ; in charts, the component functions of $g$ and $h$ are continuous on compact sets, and finitely many charts cover $K$ since $K$ is compact (Smooth manifolds and their smooth charts, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, Smooth covariant 2-tensor fields on a smooth manifold).

Proof

technique · direct; bound the perturbation in a normal form, read the inertia from a subspace and a vector, then make the pointwise bound uniform over the compact support in finitely many charts

1.1algebra

(Positive and negative directions.) Let $B$ be the symmetric form with matrix $D + E$ , $D = diag (- 1, 1, 1, 1)$ , in a basis $e_{0}, e_{1}, e_{2}, e_{3}$ and suppose $∣ E_{i j} ∣ < 1 / 8$ for all $i, j$ . For $w = \sum_{i = 1}^{3} w_{i} e_{i} \neq 0$ , writing $∣ w ∣^{2} = \sum w_{i}^{2}$ and using $(∣ w_{1} ∣ + ∣ w_{2} ∣ + ∣ w_{3} ∣)^{2} \leq 3 ∣ w ∣^{2}$ , $B (w, w) = ∣ w ∣^{2} + E (w, w) \geq ∣ w ∣^{2} - \max_{i, j} ∣ E_{i j} ∣ (\sum_{i = 1}^{3} ∣ w_{i} ∣)^{2} \geq ∣ w ∣^{2} - 3 \cdot \frac{1}{8} ∣ w ∣^{2} > 0.$ Also $B (e_{0}, e_{0}) = - 1 + E_{00} < 0$ .

2.1step 1.1L2L3algebra

(Inertia from the two directions.) Let $(w_{1}, w_{2}, w_{3})$ be a $B$ -orthonormal basis of $W = span (e_{1}, e_{2}, e_{3})$ , which exists by [L3] because $B$ is positive definite there by step 1.1; write $u : = e_{0}$ and $β_{i} : = B (u, w_{i})$ . In the basis $(u, w_{1}, w_{2}, w_{3})$ the matrix of $B$ is $(\begin{array}{cc} B (u, u) & β^{T} \\ β & I_{3} \end{array})$ , and the congruence with the invertible lower triangular matrix $(\begin{array}{cc} 1 & 0 \\ - β & I_{3} \end{array})$ transforms it into $diag (B (u, u) - β^{T} β, I_{3})$ : expanding the product gives $B (u, w_{j}) - β_{j} = 0$ in the off-diagonal block and $I_{3}$ in the lower block. Since $β^{T} β \geq 0$ and $B (u, u) < 0$ , the first diagonal entry is strictly negative; by Sylvester's law [L2] the inertia of $B$ is $(3, 1, 0)$ .

3.1step 2.1L1L2algebra

(Pointwise openness.) Every symmetric form whose matrix in the basis $e$ is within maximum-entry distance $1 / 8$ of $D$ has inertia $(3, 1, 0)$ by steps 1.1 and 2.1. Given any symmetric form $G$ of inertia $(3, 1, 0)$ on $V$ , [L1] supplies a basis in which its matrix is $D$ ; applying the previous sentence in that basis shows the set of such matrices is open around the matrix of $G$ in the maximum-entry norm of that basis. Changing bases multiplies matrices by congruences $P^{T} (\cdot) P$ with $P$ invertible, a linear isomorphism of the space of symmetric matrices onto itself; hence openness holds in every basis and, at any $p \in M$ , around $g_{p}$ in the space of symmetric forms at $p$ .

4.1step 3.1L4algebra

(Uniform openness on the support.) If $K = \emptyset$ , this uniform statement is vacuous; assume $K \neq \emptyset$ . For each point of the compact set $K = supp h$ , choose a coordinate ball whose closure is compact and lies inside its chart domain; such balls exist by taking a smaller Euclidean ball with closure inside the chart image. Compactness gives finitely many such balls $V_{1}, \dots, V_{m}$ covering $K$ . Write $U_{j}$ for the chart domain containing $V_{j}$ and $K_{j} : = \overline{V_{j}} \subseteq U_{j}$ for its compact closure. Let $G_{j} (q)$ be the matrix of $g_{q}$ in that chart, continuous on $K_{j}$ . I claim there is $δ_{j} > 0$ such that every symmetric matrix $B$ with $\max_{i, j} ∣ B_{i j} - G_{j} (q)_{i j} ∣ < δ_{j}$ for some $q \in K_{j}$ has inertia $(3, 1, 0)$ . Otherwise there are $q_{n} \in K_{j}$ and matrices $B_{n}$ with $\max ∣ B_{n} - G_{j} (q_{n}) ∣ < 1 / n$ and $B_{n}$ of inertia different from $(3, 1, 0)$ ; passing to a subsequence with $q_{n} \to q \in K_{j}$ (compactness, [L4]), continuity gives $B_{n} \to G_{j} (q)$ , which is a matrix of inertia $(3, 1, 0)$ ; but step 3.1 makes a whole maximum-entry neighborhood of $G_{j} (q)$ consist of such matrices, so $B_{n}$ has inertia $(3, 1, 0)$ for large $n$ , a contradiction.

5.1step 4.1L4algebra∎

(Conclusion.) If $h = 0$ , equivalently $K = supp h = \emptyset$ , then $g + λ h = g$ is a smooth Lorentzian metric with the same time orientation for every $λ$ , so the conclusion holds with any $λ_{0} > 0$ ; assume henceforth $K \neq \emptyset$ . Let $δ : = \min_{j} δ_{j} > 0$ , let $M_{h} : = \max_{j} \max_{q \in K_{j}} \max_{a, b} ∣ h_{a b}^{(j)} (q) ∣$ where $h^{(j)}$ denotes components in chart $j$ , a finite number because each $K_{j}$ is compact and $h$ is smooth (Smooth covariant 2-tensor fields on a smooth manifold). For $∣ λ ∣ < λ_{0} : = δ / (1 + M_{h})$ and $q \in K$ , the matrix of $(g + λ h)_{q}$ differs from that of $g_{q}$ by $∣ λ ∣ ∣ h_{i j} (q) ∣ < δ$ , so it has inertia $(3, 1, 0)$ by step 4.1; off $K$ the field equals $g$ . Hence $g + λ h$ is a smooth symmetric nondegenerate field of signature $(-, +, +, +)$ on $M$ for every such $λ$ . Since $N$ is nowhere zero and $g (N, N) < 0$ , the same smallness argument applied to the continuous function $q \mapsto g_{q} (N_{q}, N_{q})$ on $M$ (which is negative and, on the support of $h$ , bounded away from $0$ ; off the support it is unchanged) shows that $N$ remains timelike for $g + λ h$ after shrinking $λ_{0}$ if necessary.

Remarks

Sharpness. The lemma is a stability statement only: it does not assert that an arbitrary symmetric perturbation of a Lorentzian metric is Lorentzian, and it gives no uniform bound valid for perturbations with non-compact support or for all amplitudes.
Use. The statement is used to realize prescribed compactly supported symmetric variations of the metric by genuine families of Lorentzian metrics: it supplies the existence step of the Hilbert-tensor uniqueness lemma and of the compact metric-variation theorem on this page.

LemmaStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes

Statement

Let $M$ be a smooth four-manifold, let $g$ be a smooth symmetric nondegenerate covariant $2$ -tensor field on $M$ (Smooth covariant 2-tensor fields on a smooth manifold), and let $(U, x)$ be any smooth chart, with indices $a, b, c, d = 0, 1, 2, 3$ . Write $g_{a b} = g (\partial_{a}, \partial_{b})$ , let $(g^{a b})$ be the inverse matrix of $(g_{a b})$ , and define the strictly positive chart weight $w : = \sqrt{∣ \det (g_{a b}) ∣}$ on $U$ . These are local matrix constructions; no signature, time orientation or orientation is assumed. A continuous covariant tensor field means a family of bilinear forms whose chart components are continuous; smooth means that they are $C^{\infty}$ . For a covector or vector field the same regularity convention applies to its components in the coordinate dual basis or coordinate basis, respectively. Let $F$ be a continuous symmetric covariant $2$ -tensor field on $U$ . Suppose that for every smooth symmetric covariant $2$ -tensor field $q$ on $M$ whose support is a compact subset of $U$ ,

$\int_{U} F_{a b} (x) q^{a b} (x) w (x) d^{4} x = 0,$

where $q^{a b} = g^{a c} g^{b d} q_{c d}$ are the components of $q$ raised with the inverse metric and the integral is the ordinary Riemann integral of the continuous compactly supported function $F_{a b} q^{a b} w$ in the chart coordinates (The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ ). Then $F = 0$ on $U$ . Moreover, if $W$ is a continuous covector field on $U$ with $\int_{U} W_{a} (x) q^{a} (x) w (x) d^{4} x = 0$ for every smooth vector field $q$ on $M$ whose support is a compact subset of $U$ , then $W = 0$ on $U$ .

Facts & Assumptions

Given: A smooth four-manifold $M$ , a smooth symmetric nondegenerate covariant $2$ -tensor field $g$ , a chart $(U, x)$ with metric density $w > 0$ , and a continuous symmetric field $F$ on $U$ with the stated orthogonality property.

[L1]

In each chart $g^{a b}$ is smooth: the adjugate formula writes each entry as a polynomial in the smooth $g_{c d}$ divided by the nowhere-zero determinant (If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ ). Matrix multiplication gives $g^{a c} g_{c b} = δ^{a}_{b}$ . Thus raising $q^{a b} = g^{a c} g^{b d} q_{c d}$ and lowering $q_{a b} = g_{a c} g_{b d} q^{c d}$ are inverse operations preserving symmetry, smoothness and support. Also $w$ is smooth and positive, since the determinant is smooth, nowhere zero and has constant sign near each point.

[L2]

Components prescribed smoothly with compact support inside one chart extend by zero to a smooth field on $M$ . For a covariant $2$ -tensor this follows from the smooth Jacobian transformation law of Chart independence and the transformation law of a smooth covariant 2-tensor field on overlaps; outside its closed support it is zero on a neighborhood. For vectors and covectors the same assertion follows from the coordinate-basis change $\partial / \partial y^{a} = (\partial x^{c} / \partial y^{a}) \partial / \partial x^{c}$ and its dual: all transition coefficients are smooth (Smooth manifolds and their smooth charts). Compact support in the chart ensures smoothness at points outside the chart as well.

[L3]

For $0 < r < R$ there is a smooth function $ρ : R^{4} \to [0, 1]$ with $ρ = 1$ on the closed ball of radius $r$ about $0$ and support in the open ball of radius $R$ ; after translating and rescaling the chart coordinates this gives, about every point $p \in U$ , nonnegative smooth bumps supported in any prescribed coordinate ball contained in $U$ and equal to $1$ near $p$ (A smooth bump between concentric Euclidean balls).

[L4]

A continuous compactly supported function on a chart image is extended by zero to $R^{4}$ and integrated on a nondegenerate closed rectangle containing its support in its interior. The extension is continuous, hence integrable by Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable. The integral does not depend on this rectangle: extending a grid to a larger rectangle adds only zero cells, and a common containing rectangle compares any two choices. If this function is nonnegative and positive at a point, continuity gives a smaller closed rectangle $Q_{0}$ and $m > 0$ on which it is at least $m$ . A grid containing the endpoints of $Q_{0}$ has lower sum at least $m vol (Q_{0}) > 0$ ; the integral is at least this sum by The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , Lower and upper Darboux sums over a grid partition in $R^{m}$ , and Grid partitions of a rectangle in $R^{m}$ , their cells, refinements and mesh. A negative value is treated by changing sign.

Proof

technique · direct; suppose the coefficient is nonzero at a point, contract it with a fixed symmetric matrix, multiply by a bump and contradict the hypothesis

1.1L1algebra

(Nonvanishing coefficient at a point.) Suppose for contradiction that $F_{a b} (p_{0}) \neq 0$ for some $p_{0} \in U$ . Since the symmetric bilinear form $F (p_{0})$ is nonzero, there are $v, w \in T_{p_{0}} M$ with $F (p_{0}) (v, w) \neq 0$ . Write $v = \sum_{a} v^{a} \partial_{a}$ , $w = \sum_{b} w^{b} \partial_{b}$ at $p_{0}$ and put $s^{a b} : = \frac{1}{2} (v^{a} w^{b} + w^{a} v^{b})$ , a symmetric matrix with $F_{a b} (p_{0}) s^{a b} = F (p_{0}) (v, w) \neq 0$ . Replacing $s$ by $- s$ if necessary, the continuous function $f : = F_{a b} s^{a b}$ on $U$ satisfies $f (p_{0}) > 0$ ; fix such an $s$ .

2.1step 1.1L1L2L3construct

(A bump test field.) By continuity, $f > 0$ on a coordinate ball $B (p_{0}, r_{0}) \subseteq U$ . Choose $0 < r < R < r_{0}$ with the closed coordinate ball of radius $R$ about $x (p_{0})$ contained in the positive region $B (p_{0}, r_{0})$ and let $ρ$ be the bump of [L3], equal to $1$ on the closed ball of radius $r$ and supported in the open ball of radius $R$ . Define the chart components $q^{a b} : = ρ s^{a b}$ and lower them, $q_{a b} : = g_{a c} g_{b d} q^{c d}$ ; these are smooth, symmetric and supported in the compact coordinate ball of radius $R$ , so by [L2] they define a smooth symmetric tensor field $q$ on $M$ with support compact in $U$ .

3.1step 2.1L4algebra

(Contradiction.) For this $q$ , $F_{a b} q^{a b} w = ρ f w$ is continuous and nonnegative with a strictly positive value at $p_{0}$ (there $ρ = 1$ , $w > 0$ and $f > 0$ ), so its integral over $U$ is strictly positive by [L4], contradicting the hypothesis that the integral vanishes for every such $q$ . Hence no point with $F (p_{0}) \neq 0$ exists and $F = 0$ on $U$ , as claimed.

4.1step 1.1step 3.1L2L3L4algebra∎

(Covector version.) Let $W$ be a continuous covector field with the stated property and suppose $W_{a} (p_{0}) \neq 0$ . Choose $s \in T_{p_{0}} M$ with $W_{a} (p_{0}) s^{a} > 0$ (replacing $s$ by $- s$ if necessary); by continuity $f : = W_{a} s^{a}$ (with $s$ read in the chart components) is strictly positive on a coordinate ball about $p_{0}$ . Taking the bump $ρ$ of [L3] equal to $1$ near $p_{0}$ and supported in a smaller ball, the chart components $q^{a} : = ρ s^{a}$ , extended by zero, define a smooth compactly supported vector field by [L2] applied slotwise, and $\int_{U} W_{a} q^{a} w = \int_{U} ρ f w > 0$ by [L4], contradicting the hypothesis. Hence $W = 0$ on $U$ ; together with step 3.1 this proves both assertions.

Remarks

Local form. The lemma is stated in one chart because that is the form in which the compact variation argument is applied; a global version follows by applying it on each chart of a cover, and no choice beyond finitely many charts per point is used.
Why compact support is needed. The bump localizes the test variation about the point where the coefficient is nonzero; without compact support inside $U$ the boundary of the chart would contribute and the conclusion could fail.

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The Hilbert stress-energy tensor of a metric-dependent action

Definition

$S : D ⟶ R$

be a real-valued functional defined on a subset $D$ of the set of pairs $(\tilde{g}, φ)$ with $\tilde{g}$ a smooth Lorentzian metric on $M$ of signature $(-, +, +, +)$ and $φ \in X$ . Fix $(\tilde{g}, φ) \in D$ ; below we always take $\tilde{g} = g$ . The elements $φ$ are the model's matter configurations; nothing about their nature is assumed here. Assume that $D$ admits every compactly supported metric direction at $(g, φ)$ : for each smooth symmetric covariant $2$ -tensor field $h$ with compact support, there is $ε_{h} > 0$ such that $(g + s h, φ) \in D$ for every $∣ s ∣ < ε_{h}$ . Since $D$ contains only pairs whose metric is Lorentzian, these admitted linear families are Lorentzian throughout that interval. This condition makes every compactly supported inverse-metric variation available as well, since $h \mapsto - g^{a c} g^{b d} h_{c d}$ is a bijection on compactly supported smooth symmetric tensor fields.

(a) Metric families. A smooth metric family at $g$ with compactly supported variation is a map $s \mapsto g_{s}$ , $∣ s ∣ < ε$ for some $ε > 0$ , such that $(g_{s}, φ) \in D$ for all $s$ , such that $g_{s}$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ for every $s$ , such that in every chart $(U, x)$ the component functions $(s, x (p)) \mapsto (g_{s})_{a b} (p)$ are jointly $C^{\infty}$ on $(- ε, ε) \times x (U)$ , such that $g_{0} = g$ , and such that there is a compact subset $K \subseteq M$ with $g_{s} = g$ outside $K$ for every $∣ s ∣ < ε$ . For such a family define

$q^{a b} : = \frac{d}{d s} g_{s}^{a b} ∣_{s = 0},$

the derivative at $s = 0$ of the components of the inverse metric of $g_{s}$ in any smooth chart; the family of functions $q^{a b}$ is a smooth symmetric contravariant $2$ -tensor field of compact support, well defined because the components $g_{s}^{a b}$ depend smoothly on $s$ (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

(b) Hilbert tensor. A smooth symmetric covariant $2$ -tensor field $T$ on $M$ is a Hilbert stress-energy tensor of $S$ at $(g, φ)$ with respect to $c$ when, for every smooth metric family $s \mapsto g_{s}$ at $g$ with compactly supported variation,

$\frac{d}{d s} S (g_{s}, φ) ∣_{s = 0} = - \frac{1}{2 c} \int_{M} T_{a b} q^{a b} {v o l}_{g},$

where ${v o l}_{g}$ is the metric volume form of The metric volume form, current flux through 3-faces and finite-chain charges and $\int_{M} ω$ is the integral of the compactly supported smooth top form $ω : = T_{a b} q^{a b} {v o l}_{g}$ over the oriented manifold $M$ (Finite chart localization gives choice-free integration and compact Stokes). The right-hand side is defined because $q^{a b}$ has compact support, so $ω$ is compactly supported. If such a $T$ exists it is unique, by Uniqueness of the Hilbert stress-energy tensor ↗; one then calls it the Hilbert stress-energy tensor of $S$ at $(g, φ)$ .

(c) Conventions and scope. The sign and the factor $1 / (2 c)$ are fixed conventions of this library: they are the ones that turn the framework's adopted Einstein-Hilbert normalization into the field equation displayed in the framework's physical postulate. The condition is a property of the functional $S$ at the point $(g, φ)$ ; it is not automatic, and this definition does not assert that any functional has a Hilbert tensor. The definition uses the orientation of $M$ ; a general functional $S$ has no orientation-reversal law, so no transformation of $S$ or $T$ is asserted. When $S$ is given by integrating $L (g, φ) {v o l}_{g}$ with an orientation-independent scalar coefficient $L$ and the integral exists (or compact action differences are used), define the reversed-orientation action by the same construction. Reversing orientation replaces ${v o l}_{g}$ by $- {v o l}_{g}$ and reverses the sign of integration of a fixed top form. These two signs cancel, so both the action variation and the integral in the defining identity are unchanged, as is its tensor $T$ .

Remarks

Relation to the invariant-coordinate form. In a chart, writing $δ$ generically for the derivative at $s = 0$ , the defining identity reads $δ S = - \frac{1}{2 c} \int T_{a b} δ g^{a b} w d^{4} x$ with $w$ the metric density of The metric volume form, current flux through 3-faces and finite-chain charges. The normalization must be compared at the same functional $S$ . When its smooth symmetric coordinate variational derivative $D_{a b} : = δ S / δ g^{a b}$ exists, meaning $δ S = \int D_{a b} δ g^{a b} d^{4} x$ for compactly supported variations, our convention gives $T_{a b} = - 2 c w^{- 1} D_{a b}$ . Carroll's (4.70) instead gives $T_{a b}^{C} = - w^{- 1} D_{a b}$ , so $T_{a b} = 2 c T_{a b}^{C}$ for the same $S$ . Tong's (4.46) gives $T_{a b}^{T} = - 2 w^{- 1} D_{a b}$ ; Reall's (9.28)-(9.29), after $δ g_{a b} = - g_{a c} g_{b d} δ g^{c d}$ , gives that same coefficient. Thus our tensor is $c$ times their tensors for the same $S$ . Rescaling $S$ also rescales its tensor; matching physical normalizations therefore requires specifying the action normalization as well as the factor $c$ .
The matter configuration is held fixed. The definition concerns the partial derivative of $S$ in the metric direction at fixed $φ$ ; the matter equations of motion are a separate condition on the same functional and are not part of this definition.

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Uniqueness of the Hilbert stress-energy tensor

Statement

Let $(M, g, N)$ be an oriented time-oriented smooth Lorentzian four-manifold, let $c > 0$ , let $S : D \to R$ and $φ$ be as in The Hilbert stress-energy tensor of a metric-dependent action; in particular, $D$ admits every compactly supported metric direction at $(g, φ)$ . Suppose that $T$ and $T^{'}$ are both Hilbert stress-energy tensors of $S$ at $(g, φ)$ with respect to $c$ relative to the metric volume form of The metric volume form, current flux through 3-faces and finite-chain charges. Then $T = T^{'}$ on $M$ ; hence the definite article in "the Hilbert stress-energy tensor" is justified.

Facts & Assumptions

Given: The setting of The Hilbert stress-energy tensor of a metric-dependent action and two smooth symmetric covariant $2$ -tensor fields $T, T^{'}$ satisfying its defining property.

[L1]

$T$ satisfies the defining identity of The Hilbert stress-energy tensor of a metric-dependent action clause (b) for every smooth metric family at $g$ with compactly supported variation, and $T^{'}$ satisfies the same identity. (given)

[L2]

For every smooth symmetric covariant $2$ -tensor field $h$ of compact support there is $λ_{0} > 0$ such that $g + μ h$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ for every $∣ μ ∣ < λ_{0}$ (Lorentzian metrics are stable under small compactly supported symmetric perturbations).

[L3]

If $F$ is a continuous symmetric covariant $2$ -tensor field on a chart domain $U$ and $\int_{U} F_{a b} q^{a b} w d^{4} x = 0$ for every smooth symmetric compactly supported $q$ , then $F = 0$ on $U$ (A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes, The metric volume form, current flux through 3-faces and finite-chain charges).

[L4]

The inverse-metric components $g^{a b}$ depend smoothly on the metric components and satisfy $g^{a c} g_{c b} = δ^{a}_{b}$ ; the raising operation $h \mapsto h^{a b} = g^{a c} g^{b d} h_{c d}$ is a linear bijection of the space of smooth symmetric compactly supported $(0, 2)$ -fields onto the space of smooth symmetric compactly supported $(2, 0)$ -fields (The inverse metric, musical isomorphisms and index raising, Smooth covariant 2-tensor fields on a smooth manifold, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L5]

The integral of a compactly supported smooth top form over the oriented $M$ is additive and local: if the support of the form lies in a chart domain $U$ , the integral equals the ordinary chart integral $\int_{U} f w d^{4} x$ (Finite chart localization gives choice-free integration and compact Stokes, Smooth manifolds and their smooth charts).

Proof

technique · direct; compare the two defining identities along localized metric perturbations and apply the coefficient-vanishing lemma on charts

1.1L1given

(The difference is orthogonal to all variations.) Put $D : = T - T^{'}$ , a smooth symmetric covariant $2$ -tensor field. For any smooth metric family $s \mapsto g_{s}$ at $g$ with compactly supported variation, [L1] gives the same derivative of $S$ computed with $T$ and with $T^{'}$ , so $\int_{M} D_{a b} q^{a b} {v o l}_{g} = 0$ , where $q^{a b} = \frac{d}{d s} g_{s}^{a b} ∣_{s = 0}$ .

2.1step 1.1L2L4algebra

(Realizing perturbations.) Fix a smooth symmetric compactly supported field $h$ . By [L2], choose $λ_{0} > 0$ with $g + μ h$ Lorentzian for $∣ μ ∣ < λ_{0}$ ; shrink $λ_{0}$ also below the domain-admission radius $ε_{h}$ of the defining hypotheses. For $0 < λ < λ_{0}$ the map $s \mapsto g + s λ h$ (with $∣ s ∣ < 1$ ) is a smooth metric family at $g$ with compactly supported variation. Differentiating $g_{s}^{a c} g_{s c b} = δ^{a}_{b}$ in $s$ by [L4], $q^{a c} g_{c b} + g^{a c} λ h_{c b} = 0$ , so $q^{a b} = - λ g^{a c} g^{b d} h_{c d}$ . Substituting into step 1.1 and dividing by $- λ \neq 0$ , $\int_{M} D_{a b} h^{a b} {v o l}_{g} = 0$ for every smooth symmetric compactly supported $h$ , where $h^{a b} = g^{a c} g^{b d} h_{c d}$ .

3.1step 2.1L3L4L5

(Coefficient vanishing on charts.) As $h$ ranges over all smooth symmetric compactly supported $(0, 2)$ -fields, $h^{a b}$ ranges over all smooth symmetric compactly supported $(2, 0)$ -fields by [L4]; equivalently, step 2.1 states $\int_{M} D_{a b} q^{a b} {v o l}_{g} = 0$ for every smooth symmetric compactly supported $q$ written in covariant form. Fix a chart $(U, x)$ and restrict to $q$ supported in $U$ ; by [L5] the integral is the chart integral $\int_{U} D_{a b} q^{a b} w d^{4} x$ . Since $D$ is smooth, hence continuous, [L3] gives $D = 0$ on $U$ .

4.1step 3.1step 2.1given∎

(Conclusion.) Every point of $M$ lies in a chart, so step 3.1 gives $D = 0$ on all of $M$ , that is, $T = T^{'}$ .

Remarks

What is established. The lemma proves uniqueness of a Hilbert tensor when one exists; existence remains a property of the particular functional $S$ and is checked model by model where the framework's matter actions are treated.
Choice. The construction uses the finite chart integrals of [L5] and the bump of the coefficient-vanishing lemma, both choice-free; no selection of frames or atlases beyond finitely many charts is made.

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First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ , curvature tensors $R^{a}_{b c d}$ , ${Ric}_{a b}$ , scalar curvature $S$ and Einstein tensor $G_{a b}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). Let $(U, x)$ be a smooth chart, let $K \subseteq U$ be compact, let $I \subseteq R$ be an open interval with $0 \in I$ , and let $s \mapsto g (s)$ , $s \in I$ , be a family of smooth Lorentzian metrics of signature $(-, +, +, +)$ on $M$ such that

$g (0) = g$ and $g (s) = g$ on $M ∖ K$ for every $s \in I$ ; and
the components $g (s)_{a b}$ in the chart are smooth functions of $(s, x)$ on $I \times x (U)$ (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

Write $h_{a b} : = \partial_{s} g (s)_{a b} ∣_{s = 0}$ , a smooth symmetric covariant $2$ -tensor field with support in $K$ , and $q^{a b} : = \partial_{s} g (s)^{a b} ∣_{s = 0}$ (The inverse metric, musical isomorphisms and index raising).

(a) Variation of the inverse metric; realizability. $q^{a b} = - g^{a c} g^{b d} h_{c d}$ ; conversely, for every smooth symmetric covariant $2$ -tensor field $h_{0}$ with compact support in $U$ there is $λ_{0} > 0$ such that for every $∣ λ ∣ < λ_{0}$ the family $s \mapsto g + s λ h_{0}$ satisfies the hypotheses above on the parameter interval $(- 1, 1)$ (with $K = supp h_{0}$ ). In particular any prescribed smooth compactly supported inverse-metric direction $q$ is realized by $g + s h$ on a sufficiently small interval, with $h_{a b} = - g_{a c} g_{b d} q^{c d}$ .

(b) Variation of the Christoffel symbols. The field $δ Γ^{a}_{b c} : = \partial_{s} Γ (s)^{a}_{b c} ∣_{s = 0}$ is a smooth tensor field on $U$ and

$δ Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\nabla_{b} h_{c d} + \nabla_{c} h_{b d} - \nabla_{d} h_{b c}) .$

(c) Variation of the Ricci tensor (Palatini identity). $δ {Ric}_{b d} : = \partial_{s} Ric (s)_{b d} ∣_{s = 0}$ satisfies

$δ {Ric}_{b d} = \nabla_{a} δ Γ^{a}_{d b} - \nabla_{d} δ Γ^{a}_{a b} .$

(d) Variation of the curvature density. With $V^{a} : = g^{b d} δ Γ^{a}_{d b} - g^{a b} δ Γ^{d}_{d b}$ and $w (s) : = \sqrt{∣ \det (g (s)_{a b}) ∣}$ the metric density of The metric volume form, current flux through 3-faces and finite-chain charges,

$\partial_{s} (w (s) S (s)) ∣_{s = 0} = w G_{a b} q^{a b} + \partial_{a} (w V^{a}) .$

(e) Variation of the chart action integral. For every constant $Λ \in R$ , the function $F (s) : = \int_{U} [(S (s) - 2 Λ) w (s) - (S (0) - 2 Λ) w (0)] d^{4} x$ is the compactly supported chart action difference, differentiable at $0$ , and

$F^{'} (0) = \int_{U} (G_{a b} + Λ g_{a b}) q^{a b} w d^{4} x,$

where the integrand is compactly supported in $K$ and the integral is an ordinary Riemann integral in the chart coordinates (The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ ).

(f) Variation of the timelike length and particle action. Let $γ : [a, b] \to U$ and $H$ be as in First variation of the length of a timelike curve: stationarity is the geodesic equation, so that $γ$ is $C^{2}$ , future timelike, parametrized with $g (γ^{'}, γ^{'}) = - c^{2}$ , the variation is $C^{2}$ with fixed endpoints and future timelike fibres, and $L (s)$ is the length functional of that lemma. Then the first-variation formulas (1)-(3) of that lemma hold verbatim. Consequently for every $m > 0$ the particle action $S_{p} (γ) : = - m c \int_{a}^{b} \sqrt{- g (γ^{'} (t), γ^{'} (t))} d t$ satisfies $\partial_{s} S_{p} (H (s, \cdot)) ∣_{s = 0} = - m c L^{'} (0)$ , so within this class of curves and variations $γ$ is stationary for $S_{p}$ if and only if $D_{t} γ^{'} = 0$ , i.e. if and only if $γ$ is a geodesic in its proper-time parameter (Geodesic of an affine connection, Covariant derivative along a curve).

Facts & Assumptions

Given: The chart $(U, x)$ , the compact set $K \subseteq U$ , the interval $I$ about $0$ , the smooth family $s \mapsto g (s)$ of Lorentzian metrics equal to $g$ off $K$ , and the curve and variation of clause (f).

[L1]

The metric-selected connection $\nabla$ of $g$ exists, is unique and is torsion-free and $g$ -compatible; in a chart its Christoffel symbols are $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ , and its extension to tensor fields commutes with contractions, annihilates $g, g^{- 1}$ and has coordinate expression $T^{\dots}_{; c} = \partial_{c} T^{\dots}$ plus one $Γ$ -term per slot (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Torsion freedom and metric compatibility in chart components).

[L2]

$\nabla$ satisfies the point-normal chart property: about every $p \in U$ there is a chart with $g_{a b} (p) = diag (- 1, 1, 1, 1)$ and $\partial_{a} g_{b c} (p) = 0$ , hence $Γ^{a}_{b c} (p) = 0$ (Point-normal and exponential coordinate charts of the metric-selected connection).

[L3]

The curvature components have the chart expression $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , and ${Ric}_{b d} = R^{a}_{b a d}$ , $S = g^{b d} {Ric}_{b d}$ , $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Tensoriality of the curvature, its coordinate formula and its point-normal metric formula).

[L4]

For a smooth metric family with compactly supported variation: if $w (s)$ is the density of $g (s)$ , then $\partial_{s} w (s) ∣_{0} = - \frac{1}{2} w g_{a b} q^{a b}$ ; equivalently $\partial_{s} w (s) ∣_{0} = \frac{1}{2} w g^{a b} h_{a b}$ (The derivative of the metric density and the chart formula for the divergence of a vector field clause (a), Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

[L5]

For a differentiable matrix curve $A$ with invertible values, $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ , and the inverse-metric components are the matrix inverse of $g_{a b}$ ; hence differentiating $g (s)^{a c} g (s)_{c b} = δ^{a}_{b}$ gives $q^{a c} g_{c b} + g^{a c} h_{c b} = 0$ (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ ).

[L6]

If $g$ is Lorentzian and $h_{0}$ is smooth symmetric with compact support, then $g + s λ h_{0}$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ for every $∣ s ∣ < 1$ and $∣ λ ∣$ smaller than some $λ_{0}$ (Lorentzian metrics are stable under small compactly supported symmetric perturbations).

[L7]

On the chart domain, $\nabla_{a} J^{a} = w^{- 1} \partial_{a} (w J^{a})$ for every smooth vector field $J$ (The derivative of the metric density and the chart formula for the divergence of a vector field clause (c)).

[L8]

A continuous chart coefficient supported in a fixed compact subset of $x (U)$ is extended by zero to $R^{4}$ and integrated on a containing closed nondegenerate rectangle. It is integrable by Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable, with the integral of The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ . For a $C^{1}$ compactly supported coefficient $f$ , zero extension is $C^{1}$ ; Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections and The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ give $\int \partial_{a} f = 0$ because the endpoint values in that coordinate vanish. For a jointly smooth parameter coefficient $A (s, x)$ with fixed compact support, choose a compact parameter interval about $0$ . Uniform continuity of $\partial_{s} A$ on its product with the containing rectangle (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous) and The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ imply $(A (s, x) - A (0, x)) / s \to \partial_{s} A (0, x)$ uniformly in $x$ . The Darboux sums give $∣ \int f - \int h ∣ \leq vol (Q) \sup_{Q} ∣ f - h ∣$ for continuous functions on $Q$ (Lower and upper Darboux sums over a grid partition in $R^{m}$ ); hence the derivative passes through this integral.

[L9]

The proper-time length functional of a $C^{2}$ future timelike curve in a chart, with $g (γ^{'}, γ^{'}) = - c^{2}$ and $C^{2}$ endpoint-fixing variations with timelike fibres, has the first-variation formulas $L^{'} (0) = - \int_{a}^{b} \frac{g (D_{t} J, γ^{'})}{\sqrt{- g (γ^{'}, γ^{'})}} d t = \frac{1}{c} \int_{a}^{b} g (J, D_{t} γ^{'}) d t$ and is stationary for every such variation exactly when $D_{t} γ^{'} = 0$ (First variation of the length of a timelike curve: stationarity is the geodesic equation, Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, Covariant derivative along a curve).

Proof

technique · direct; differentiate the defining algebraic identities and the Christoffel and curvature formulas in point-normal charts, integrate the total-divergence term, then cite the standalone curve-variation lemma for the particle functional

1.1L5algebra

(Variation of the inverse metric.) Differentiate the identity $g (s)^{a c} g (s)_{c b} = δ^{a}_{b}$ at $s = 0$ using [L5]: $q^{a c} g_{c b} + g^{a c} h_{c b} = 0$ . Multiplying by $g^{- 1}$ twice gives $q^{a b} = - g^{a c} g^{b d} h_{c d}$ , the first assertion of (a).

1.2L1L2algebra

(Variation of the Christoffel symbols.) Fix $p$ , use the point-normal chart of [L2] at $p$ , and let $g (s)$ be the given family; at $p$ , the Christoffel formula of [L1] applied to $g (s)$ gives $Γ (s)^{a}_{b c} (p) = \frac{1}{2} g (s)^{a d} (p) (\partial_{b} g (s)_{d c} + \partial_{c} g (s)_{b d} - \partial_{d} g (s)_{b c}) (p)$ . Differentiating at $s = 0$ and using $g (0) = g$ , the derivatives of the metric at $p$ may be replaced by covariant derivatives because $Γ^{a}_{b c} (p) = 0$ ; this gives $δ Γ^{a}_{b c} (p) = \frac{1}{2} g^{a d} (\nabla_{b} h_{c d} + \nabla_{c} h_{b d} - \nabla_{d} h_{b c}) (p)$ , the displayed formula. Both sides are tensor fields: the left side is a difference of Christoffel-symbol families, whose transformation law differs only by a metric-independent term that cancels, and the right side is built from tensors. Since the identity holds at every point in some point-normal chart, it holds on $U$ ; this proves (b).

2.1L6algebra

(Realizability.) Let $h_{0}$ be smooth symmetric with compact support in $U$ . By [L6] applied to the Lorentzian metric $g$ and the field $h_{0}$ , there is $λ_{0} > 0$ such that $g + μ h_{0}$ is a smooth Lorentzian metric of signature $(-, +, +, +)$ for every $∣ μ ∣ < λ_{0}$ . For $∣ λ ∣ < λ_{0}$ and $∣ s ∣ < 1$ put $g (s) : = g + s λ h_{0}$ , so that $μ = s λ$ ranges over $(- λ_{0}, λ_{0})$ ; then $g (s)$ is Lorentzian for every $s \in (- 1, 1)$ , equals $g$ off $K = supp h_{0}$ , and its components are smooth in $(s, x)$ . For an arbitrary prescribed $q$ , take $h_{a b} = - g_{a c} g_{b d} q^{c d}$ and restrict $g + s h$ to the stability interval; step 1.1 gives exactly that inverse-metric derivative. This proves the remaining assertions of (a).

2.2L1L2L3step 1.2algebra

(Palatini identity.) In the point-normal chart at $p$ , the chart expression of [L3] reads $R (s)^{a}_{b c d} = \partial_{c} Γ (s)^{a}_{d b} - \partial_{d} Γ (s)^{a}_{c b} + \sum_{e} (Γ (s)^{a}_{c e} Γ (s)^{e}_{d b} - Γ (s)^{a}_{d e} Γ (s)^{e}_{c b})$ . At $p$ the quadratic terms have a factor $Γ (0)^{e}_{x y} (p) = 0$ , so differentiating at $s = 0$ gives $δ R^{a}_{b c d} (p) = \partial_{c} δ Γ^{a}_{d b} (p) - \partial_{d} δ Γ^{a}_{c b} (p) = \nabla_{c} δ Γ^{a}_{d b} (p) - \nabla_{d} δ Γ^{a}_{c b} (p)$ , where the last equality again uses $Γ (p) = 0$ . Both sides are tensors, so the identity holds on $U$ . Contracting $a = c$ with [L3], $δ {Ric}_{b d} = δ R^{a}_{b a d} = \nabla_{a} δ Γ^{a}_{d b} - \nabla_{d} δ Γ^{a}_{a b}$ , which is (c).

3.1step 1.1step 2.2L1L3L4L7algebra

(Variation of the density times scalar curvature.) From [L4], $\partial_{s} w (s) ∣_{0} = - \frac{1}{2} w g_{a b} q^{a b}$ , and from step 1.1 and (c), $\partial_{s} S (s) ∣_{0} = \partial_{s} (g (s)^{b d} Ric (s)_{b d}) ∣_{0} = q^{b d} {Ric}_{b d} + g^{b d} δ {Ric}_{b d}$ . Therefore $\partial_{s} (w (s) S (s)) ∣_{0} = w [{Ric}_{b d} q^{b d} - \frac{1}{2} S g_{b d} q^{b d} + g^{b d} (\nabla_{a} δ Γ^{a}_{d b} - \nabla_{d} δ Γ^{a}_{a b})] .$ The first two terms are $w G_{b d} q^{b d}$ by [L3]. For the last two terms, metric compatibility ( $\nabla g^{- 1} = 0$ , [L1]) gives $g^{b d} \nabla_{a} δ Γ^{a}_{d b} = \nabla_{a} (g^{b d} δ Γ^{a}_{d b})$ and $g^{b d} \nabla_{d} δ Γ^{a}_{a b} = \nabla_{d} (g^{b d} δ Γ^{a}_{a b}) = \nabla_{a} (g^{a b} δ Γ^{d}_{d b})$ after relabelling the second index; their difference is $\nabla_{a} V^{a}$ with $V^{a} = g^{b d} δ Γ^{a}_{d b} - g^{a b} δ Γ^{d}_{d b}$ , so $\partial_{s} (w S) ∣_{0} = w G_{a b} q^{a b} + w \nabla_{a} V^{a}$ . Since $\partial_{a} (w V^{a}) = w \nabla_{a} V^{a}$ by the density-divergence formula [L7], this is (d).

4.1step 3.1L4L7L8algebra

(Chart integral.) The action-difference integrand $[(S (s) - 2 Λ) w (s) - (S (0) - 2 Λ) w (0)]$ is smooth in $(s, x)$ on $I \times x (U)$ and supported in the fixed compact set $K$ ; all its $s$ -derivatives are supported in $K$ . By [L8], $F$ is differentiable at $0$ with $F^{'} (0) = \int_{U} \partial_{s} ((S (s) - 2 Λ) w (s)) ∣_{0} d^{4} x$ . By [L4] and (d), $\partial_{s} ((S (s) - 2 Λ) w (s)) ∣_{0} = w G_{a b} q^{a b} + \partial_{a} (w V^{a}) + Λ g_{a b} q^{a b} w = w (G_{a b} + Λ g_{a b}) q^{a b} + \partial_{a} (w V^{a})$ , the last step by [L7]. The function $\partial_{a} (w V^{a})$ has compact support in $K \subseteq U$ , so its integral vanishes by [L8]; hence $F^{'} (0) = \int_{U} (G_{a b} + Λ g_{a b}) q^{a b} w d^{4} x$ , which is (e).

5.1L9algebra∎

(Particle functional.) Clause (f) is exactly [L9]: the lemma's hypotheses are the stated ones, and its conclusions are the formulas (1)-(3) quoted in (f). Multiplying the length functional by the positive constant $m c$ gives $S_{p} (H (s, \cdot)) = - m c L (s)$ , so $\partial_{s} S_{p} (H (s, \cdot)) ∣_{0} = - m c L^{'} (0)$ ; since $m c > 0$ , the vanishing of $L^{'} (0)$ for every admissible variation is equivalent to the vanishing of $\partial_{s} S_{p} ∣_{0}$ , and [L9] identifies this with $D_{t} γ^{'} = 0$ .

Remarks

Hypotheses used. The compact support of the variation is used twice: to make the parameter integral differentiable under the integral sign and to kill the total-divergence term. Nonzero boundary variations of the metric are not covered and require boundary terms that this theorem does not supply. The chart $U$ is only a localization of the integral; the formulas (a)-(d) are chart-independent tensor identities.
No physical premise. The theorem is mathematics: it identifies the first variations of the stated functionals. The physical adoption of an action principle and its reading as the field law of the model occur in the framework's postulates, not here.
Relation to the particle functional's class. The curve class in (f) is the restricted one of First variation of the length of a timelike curve: stationarity is the geodesic equation: a single chart, fixed endpoints, timelike fibres, parametrization with $g (γ^{'}, γ^{'}) = - c^{2}$ . Null curves and maximality statements are outside it.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Killing fields of a Lorentzian metric

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) and let $ξ$ be a smooth vector field on $M$ (A smooth vector field is a smooth section of the tangent bundle). The Lie derivative of the metric along $ξ$ , written $L_{ξ} g$ , is the smooth symmetric covariant $2$ -tensor field obtained by the flow definition of The Lie derivative of a tensor field applied to $g$ : for the local flow $Φ_{t}$ of $ξ$ ,

$L_{ξ} g : = \frac{d}{d t} ∣_{t = 0} Φ_{t}^{*} g .$

The field $ξ$ is a Killing field of $(M, g)$ (an isometry generator, or a Killing vector) when

$L_{ξ} g = 0.$

The equivalent local characterization

$L_{ξ} g = 0 ⟺ \nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0,$

where $\nabla$ is the metric-selected connection of $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and $ξ_{b} : = g_{b c} ξ^{c}$ is the lowered field, is proved in Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor ↗ clause (a); the definition itself asserts neither direction. Whether a given metric has Killing fields is a property of that metric: neither existence nor uniqueness of a Killing field is asserted here, and a general metric has none beyond the trivial field $ξ = 0$ .

Remarks

What is defined. Only the object "Killing field of the pair $(g, ξ)$ " is defined, through the Lie derivative of the metric. No conserved quantity, charge or physical interpretation is attached to a Killing field by this item; conservation along geodesics and the divergence-free current constructed from a conserved stress-energy tensor are separate results of Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor ↗.
No global isometry is asserted. A Killing field generates a local flow of local isometries where it is defined; a global one-parameter group of isometries requires the flow of $ξ$ to be global, which is a separate hypothesis (for instance compact support of $ξ$ ).

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Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

Statement

Let $(M, g, N)$ be an oriented time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Oriented smooth manifolds and oriented charts).

(a) Characterization of Killing fields. For every smooth vector field $ξ$ (A smooth vector field is a smooth section of the tangent bundle), with $ξ_{b} = g_{b c} ξ^{c}$ the lowered field,

$L_{ξ} g = 0 ⟺ \nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0.$

(b) Contraction along geodesics. If $ξ$ is a Killing field and $γ : I \to M$ is a geodesic with $D_{t} γ^{'} = 0$ in its affine parameter (Geodesic of an affine connection, Covariant derivative along a curve), then $t \mapsto g_{γ (t)} (ξ_{γ (t)}, γ^{'} (t))$ is constant on $I$ .

(c) Conserved current. Let $U \subseteq M$ be open, let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold) with $\nabla^{a} T_{a b} = 0$ on $U$ , and let $ξ$ be a Killing field on $U$ . Then $J^{a} : = T^{a b} ξ_{b}$ is a smooth vector field on $U$ with $\nabla_{a} J^{a} = 0$ ; consequently the flux balance of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (b) applies to $J$ on every smooth singular $4$ -chain contained in $U$ .

(d) On-shell conservation of the Hilbert tensor. Let $c > 0$ , let $S : D \to R$ be a model action with Hilbert stress-energy tensor $T$ at $(g, φ)$ with respect to $c$ (The Hilbert stress-energy tensor of a metric-dependent action). For the flow diffeomorphisms $F$ below, suppose the configuration set $X$ carries specified maps $P_{F} : X \to X$ with $P_{i d} = i d$ ; write $F^{*} φ : = P_{F} (φ)$ . For geometric matter fields these are the model's pullbacks; no pullback operation on an arbitrary set is implicit. For every smooth compactly supported vector field $ξ$ , its flow $Φ_{v}$ is global (Complete vector fields, Local and global flows generated by a vector field); this holds for every compactly supported smooth field, using the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) through Compactly supported smooth vector fields are complete. Write $G (u, v) : = S (Φ_{v}^{*} g, Φ_{u}^{*} φ)$ for the simultaneous pullback by the flow (this presupposes that the pairs lie in $D$ , which is part of the hypotheses below). Suppose (H1)–(H4) hold for every such $ξ$ (with its own $G$ and parameter neighborhood):

(H1) the map $G$ is defined on a neighborhood of $(0, 0)$ in $R^{2}$ and is differentiable at $(0, 0)$ ;
(H2) $\partial_{u} G (0, 0) = 0$ (the matter configuration $φ$ satisfies its model equations in this direction);
(H3) $\partial_{v} G (0, 0) = - \frac{1}{2 c} \int_{M} T_{a b} q^{a b} {v o l}_{g}$ , where $q^{a b} : = \partial_{v} (Φ_{v}^{*} g)^{a b} ∣_{v = 0}$ (the metric variation is represented by the Hilbert tensor);
(H4) $G (s, s) = S (g, φ)$ whenever $(s, s)$ lies in that neighborhood (invariance of the action under simultaneous pullback by the flow).

Then $\nabla^{a} T_{a b} = 0$ on $M$ .

Facts & Assumptions

Given: The oriented time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ ; in (c) an open $U$ , a smooth symmetric $T$ with $\nabla^{a} T_{a b} = 0$ and a Killing field $ξ$ on $U$ ; in (d) the action $S$ , the Hilbert tensor $T$ , and the hypotheses (H1)-(H4) for every compactly supported smooth vector field $ξ$ , with its global flow $Φ_{v}$ and the supplied configuration pullbacks.

[L1]

The coordinate formula $(L_{ξ} g)_{a b} = ξ^{c} \partial_{c} g_{a b} + g_{c b} \partial_{a} ξ^{c} + g_{a c} \partial_{b} ξ^{c}$ holds for the Lie derivative of a covariant tensor (The coordinate formula for the Lie derivative of a covariant tensor, The Lie derivative of a tensor field).

[L2]

$\nabla$ is torsion-free and $g$ -compatible, with $g_{a b; c} = 0$ and Christoffel symbols $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ ; the compatibility identity $\partial_{d} g_{a b} = g_{b c} Γ^{c}_{a d} + g_{a c} Γ^{c}_{b d}$ holds in every chart; the extension of $\nabla$ to tensor fields satisfies the product rule and $\nabla g^{- 1} = 0$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Torsion freedom and metric compatibility in chart components, Metric compatibility, torsion freedom and the metric-selected connection class).

[L3]

For smooth fields $V, W$ along a smooth curve, $\frac{d}{d t} g (V, W) = g (D_{t} V, W) + g (V, D_{t} W)$ ; the covariant derivative along the curve of the restriction of an ambient vector field is the ambient derivative restricted to the curve (Covariant differentiation along curves: metric pairing, parallel transport and geodesic speed, Covariant derivative along a curve).

[L4]

The integral of the exterior derivative of a compactly supported smooth $3$ -form on the oriented four-manifold $M$ vanishes; in particular $\int_{M} \nabla_{a} J^{a} {v o l}_{g} = 0$ for every smooth compactly supported vector field $J$ , since $\nabla_{a} J^{a} {v o l}_{g} = d (ι_{J} {v o l}_{g})$ and $ι_{J} {v o l}_{g}$ has compact support (Finite chart localization gives choice-free integration and compact Stokes, Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (a), The metric volume form, current flux through 3-faces and finite-chain charges).

[L5]

If a continuous covector field $W$ on a chart domain satisfies $\int_{U} W_{a} q^{a} w d^{4} x = 0$ for every smooth compactly supported vector field $q$ supported in the chart, then $W = 0$ ; and the chart integral of a compactly supported divergence is zero (A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes, The metric volume form, current flux through 3-faces and finite-chain charges).

[L7]

For a global flow $Φ_{v}$ of $ξ$ , the derivative $\partial_{v} (Φ_{v}^{*} K) ∣_{v = 0}$ is the Lie derivative $L_{ξ} K$ of the tensor field $K$ ; moreover $(Φ_{v}^{*} g)^{- 1} = Φ_{v}^{*} (g^{- 1})$ for every $v$ (The Lie derivative of a tensor field, Local and global flows generated by a vector field).

[L8]

If a map $F : R^{2} \to R$ is differentiable at $(0, 0)$ , then $s \mapsto F (s, s)$ is differentiable at $0$ with derivative $\partial_{1} F (0, 0) + \partial_{2} F (0, 0)$ (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ , Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Proof

technique · direct; convert the Lie derivative of the metric into the symmetrized covariant derivative, differentiate the pairing along geodesics, then differentiate the invariance identity of the pullback family and integrate by parts

1.1L1L2algebra

(Characterization.) Fix smooth vector fields $X, Y$ . Evaluating the coordinate formula of [L1] on $X, Y$ , $(L_{ξ} g) (X, Y) = ξ^{c} \partial_{c} g_{a b} X^{a} Y^{b} + g_{c b} \partial_{a} ξ^{c} X^{a} Y^{b} + g_{a c} \partial_{b} ξ^{c} X^{a} Y^{b}$ . On the other hand, by [L2], $\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = g_{b c} (\partial_{a} ξ^{c} + Γ^{c}_{a d} ξ^{d}) + g_{a c} (\partial_{b} ξ^{c} + Γ^{c}_{b d} ξ^{d}) = g_{b c} \partial_{a} ξ^{c} + g_{a c} \partial_{b} ξ^{c} + ξ^{d} (g_{b c} Γ^{c}_{a d} + g_{a c} Γ^{c}_{b d})$ ; metric compatibility in the form $\partial_{d} g_{a b} = g_{b c} Γ^{c}_{a d} + g_{a c} Γ^{c}_{b d}$ (which is the Christoffel formula of [L2]) makes the last term $ξ^{d} \partial_{d} g_{a b}$ . Hence $(\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) X^{a} Y^{b} = (L_{ξ} g) (X, Y)$ for all $X, Y$ , and since $X, Y$ are arbitrary, $L_{ξ} g$ vanishes exactly when the symmetrized derivative vanishes. This proves (a).

1.2L8givenalgebra

(The invariance identity.) Fix any compactly supported smooth $ξ$ with global flow $Φ_{v}$ and let $G$ be as in (d). By [L8], $s \mapsto G (s, s)$ is differentiable at $0$ with derivative $\partial_{u} G (0, 0) + \partial_{v} G (0, 0)$ ; by (H4), $G (s, s) = S (g, φ)$ is constant in $s$ , so this derivative vanishes. Substituting (H2) and (H3), $0 = 0 - \frac{1}{2 c} \int_{M} T_{a b} q^{a b} {v o l}_{g}, q^{a b} = \partial_{v} (Φ_{v}^{*} g)^{a b} ∣_{v = 0} .$

2.1step 1.1L3algebra

(Contraction along geodesics.) Let $γ$ be a geodesic with $D_{t} γ^{'} = 0$ and $ξ$ Killing. By [L3], $\frac{d}{d t} g (ξ, γ^{'}) = g (D_{t} ξ, γ^{'}) + g (ξ, D_{t} γ^{'}) = g (D_{t} ξ, γ^{'})$ , and $D_{t} ξ$ is the restriction to $γ$ of $\nabla_{γ^{'}} ξ$ . By (a), $g (\nabla_{γ^{'}} ξ, γ^{'}) = \frac{1}{2} (\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) γ^{' a} γ^{' b} = 0$ . Hence the derivative vanishes identically and the function is constant on $I$ ; this proves (b).

2.2step 1.1L2algebra

(Conserved current.) On $U$ put $J^{a} = T^{a b} ξ_{b}$ . By [L2] and $\nabla g^{- 1} = 0$ , $\nabla_{a} J^{a} = (\nabla_{a} T^{a b}) ξ_{b} + T^{a b} \nabla_{a} ξ_{b}$ . The first term vanishes by hypothesis, and the second is $\frac{1}{2} T^{a b} (\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) = 0$ because $T$ is symmetric and $ξ$ Killing, using (a). Hence $\nabla_{a} J^{a} = 0$ ; the flux-balance assertion is clause (b) of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current applied to this $J$ .

2.3step 1.1L2L7algebra

(The inverse-metric variation.) By [L7], $(Φ_{v}^{*} g)^{- 1} = Φ_{v}^{*} (g^{- 1})$ , so $q^{a b} = \partial_{v} (Φ_{v}^{*} (g^{- 1}))^{a b} ∣_{v = 0} = (L_{ξ} g^{- 1})^{a b}$ . Differentiating $g^{a c} (v) g_{c b} (v) = δ^{a}_{b}$ for the family $g (v) : = Φ_{v}^{*} g$ with $\partial_{v} g_{c b} ∣_{v = 0} = (L_{ξ} g)_{c b} = \nabla_{c} ξ_{b} + \nabla_{b} ξ_{c}$ (step 1.1) gives $(L_{ξ} g^{- 1})^{a b} = - g^{a c} g^{b d} (L_{ξ} g)_{c d} = - (\nabla^{a} ξ^{b} + \nabla^{b} ξ^{a})$ , where indices are raised with $g^{- 1}$ by [L2].

3.1step 1.2step 2.3L2L4algebra

(Conclusion of (d).) Substituting step 2.3 into step 1.2, $0 = \frac{1}{2 c} \int_{M} T_{a b} (\nabla^{a} ξ^{b} + \nabla^{b} ξ^{a}) {v o l}_{g} = \frac{1}{c} \int_{M} T_{a b} \nabla^{a} ξ^{b} {v o l}_{g}$ , by symmetry of $T$ . By the product rule [L2], $T_{a b} \nabla^{a} ξ^{b} = \nabla^{a} (T_{a b} ξ^{b}) - (\nabla^{a} T_{a b}) ξ^{b}$ ; the vector field $J^{a} : = T^{a b} ξ_{b}$ is smooth and compactly supported (since $ξ$ is), so by [L4] $\int_{M} \nabla_{a} J^{a} {v o l}_{g} = 0$ and therefore $0 = \int_{M} T_{a b} \nabla^{a} ξ^{b} {v o l}_{g} = - \int_{M} (\nabla^{a} T_{a b}) ξ^{b} {v o l}_{g}$ .

4.1step 3.1L5algebra∎

(Vanishing of the divergence.) Fix a chart $(U, x)$ ; the hypotheses hold for every compactly supported smooth $ξ$ supported in $U$ . Writing the last integral in the chart and putting $W_{b} : = \nabla^{a} T_{a b}$ , a smooth covector field on $U$ , step 3.1 gives $\int_{U} W_{b} ξ^{b} w d^{4} x = 0$ for every such $ξ$ , so $W = 0$ on $U$ by [L5]; the integral-of-a-divergence identity in [L5] was already used through [L4]. Since every point lies in a chart, $\nabla^{a} T_{a b} = 0$ on $M$ , which proves (d).

Remarks

Choice convention. Clause (d) uses the completeness of the flow of a compactly supported smooth vector field, which is proved from the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) in Compactly supported smooth vector fields are complete; this is the only choice principle used in the item, and it is inherited by consumers of clause (d). Clauses (a)-(c) use no choice principle.
On-shell means matter equations plus Hilbert representation. The hypothesis (H2) is the model's matter equation at the configuration $φ$ in the flow direction; (H3) is the Hilbert representation of the metric variation. Neither is implied by the other, and no conservation is asserted off shell or for a tensor prescribed without a functional.
Global versus local Killing fields. The current in (c) is defined on the open set where both $T$ obeys the divergence condition and $ξ$ is Killing; the flux balance of (c) concerns chains contained in that set and carries the side-flux hypotheses of the flux theorem.

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The scalar-field matter model of the classical gravitational framework

Definition

Let $(M, g, N)$ be an oriented time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Oriented smooth manifolds and oriented charts), let $c > 0$ be a fixed real constant, and let $\nabla$ be the metric-selected connection of $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

(a) Data. A scalar-field matter model on $M$ is a triple $(φ, V, K)$ consisting of

a smooth function $φ : M \to R$ ( $C^{r}$ and smooth maps between smooth manifolds, with target the standard one-dimensional smooth manifold $R$ );
a smooth function $V : R \to R$ , the potential; and
a real constant $K > 0$ , the coupling.

The physical reading of $K$ and of the units of $φ$ and $V$ is fixed by the framework's matter postulate, not by this definition.

(b) Scalar-action density and action differences. For every smooth function $φ^{'}$ and every smooth Lorentzian metric $g^{'}$ of signature $(-, +, +, +)$ on $M$ define the scalar-action density

$ℓ (φ^{'}, g^{'}) : = - \frac{K}{c} [\frac{1}{2} g^{' a b} \partial_{a} φ^{'} \partial_{b} φ^{'} + V (φ^{'})] {v o l}_{g^{'}} \in Ω^{4} (M),$

where $g^{' a b}$ are the inverse-metric components (The inverse metric, musical isomorphisms and index raising) and ${v o l}_{g^{'}}$ the metric volume form (The metric volume form, current flux through 3-faces and finite-chain charges). For two pairs $(φ, g)$ and $(φ^{'}, g^{'})$ with $φ^{'} - φ$ and all components of $g^{'} - g$ supported in a compact set, the difference $ℓ (φ^{'}, g^{'}) - ℓ (φ, g)$ is a compactly supported smooth $4$ -form and the action difference

$Δ S : = \int_{M} [ℓ (φ^{'}, g^{'}) - ℓ (φ, g)]$

is a real number (Finite chart localization gives choice-free integration and compact Stokes), the integral being independent of how the pair $(φ, g)$ is extended outside the support. Only such differences are used; no integrality of the total action on a noncompact $M$ is asserted.

(c) Scalar equation and candidate stress-energy tensor. The model's scalar equation of motion is

$□_{g} φ - V^{'} (φ) = 0, □_{g} φ : = \nabla^{a} \partial_{a} φ,$

and the model's candidate stress-energy tensor is the smooth symmetric covariant $2$ -tensor field

$T_{a b} : = K [\partial_{a} φ \partial_{b} φ - g_{a b} (\frac{1}{2} \partial^{c} φ \partial_{c} φ + V (φ))],$

with $\partial^{c} φ \partial_{c} φ = g^{c d} \partial_{c} φ \partial_{d} φ$ (The inverse metric, musical isomorphisms and index raising, Smooth covariant 2-tensor fields on a smooth manifold). That this candidate is indeed the Hilbert stress-energy tensor of the model, and that the scalar equation implies its conservation, are proved in First variation, Hilbert tensor and conservation of the scalar-field model ↗; the definition itself asserts neither proved property, and this item does not assert that any solution of the scalar equation exists.

Remarks

Sign conventions. The overall sign of $ℓ$ is chosen so that, under the Hilbert convention of The Hilbert stress-energy tensor of a metric-dependent action, the metric variation of the action difference reproduces the tensor displayed in clause (c). The kinetic term has the sign convention $+ \frac{1}{2} g^{a b} \partial_{a} φ \partial_{b} φ$ ; the sign of the potential term is $+ V (φ)$ inside the bracket of $ℓ$ and $- V (φ)$ inside the metric bracket of $T_{a b}$ .
Scope. The item defines a model on the given smooth $(M, g, N)$ ; it makes no claim that the model is realized by any physical system, that solutions exist for given initial data, or that the scalar field is fundamental.

PropositionStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

First variation, Hilbert tensor and conservation of the scalar-field model

Statement

Let $(M, g, N)$ be an oriented time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ and let $(φ, V, K)$ be a scalar-field matter model on $M$ with action differences $Δ S$ , scalar equation $□_{g} φ - V^{'} (φ) = 0$ and candidate stress-energy tensor $T$ (The scalar-field matter model of the classical gravitational framework, Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

(a) Matter first variation. Let $u \mapsto φ_{u}$ , $∣ u ∣ < u_{0}$ , be a family of smooth functions with $φ_{u = 0} = φ$ such that $(φ_{u} - φ)$ is supported in a fixed compact set for all $u$ and the functions $(u, p) \mapsto φ_{u} (p)$ are smooth. Then $u \mapsto Δ S (φ_{u}, g)$ is differentiable at $0$ and

$\frac{d}{d u} ∣_{u = 0} Δ S (φ_{u}, g) = \frac{K}{c} \int_{M} (□_{g} φ - V^{'} (φ)) δ φ {v o l}_{g}, δ φ : = \partial_{u} φ_{u} ∣_{u = 0} .$

In particular the family is stationary in this direction, for every such family, exactly when $□_{g} φ = V^{'} (φ)$ .

(b) Metric variation and Hilbert tensor. Let $s \mapsto g_{s}$ , $∣ s ∣ < s_{0}$ , be a smooth metric family at $g$ with compactly supported variation, and let $q^{a b} = \partial_{s} g_{s}^{a b} ∣_{s = 0}$ . Then

$\frac{d}{d s} ∣_{s = 0} Δ S (φ, g_{s}) = - \frac{1}{2 c} \int_{M} T_{a b} q^{a b} {v o l}_{g} .$

Consequently $T$ is the Hilbert stress-energy tensor of the scalar-field model at $(g, φ)$ with respect to $c$ (The Hilbert stress-energy tensor of a metric-dependent action), and by Uniqueness of the Hilbert stress-energy tensor it is the only one.

(c) Conservation identity. On $M$ ,

$\nabla^{a} T_{a b} = K (□_{g} φ - V^{'} (φ)) \partial_{b} φ .$

Consequently the scalar equation implies $\nabla^{a} T_{a b} = 0$ . Conversely conservation implies the scalar equation at each point where $d φ \neq 0$ , and therefore everywhere if that set is dense, by continuity. No converse follows on an open constant-field region.

Facts & Assumptions

Given: The oriented time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , and a scalar-field matter model $(φ, V, K)$ .

[L1]

$\nabla$ is torsion-free and $g$ -compatible, $\nabla g = 0$ , $\nabla g^{- 1} = 0$ , and for a smooth scalar $f$ the field $\nabla_{a} f = \partial_{a} f$ is a covector; $\nabla_{a} \nabla_{b} f = \nabla_{b} \nabla_{a} f$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Metric compatibility, torsion freedom and the metric-selected connection class).

[L2]

The metric volume form satisfies $δ {v o l}_{g} = - \frac{1}{2} g_{a b} q^{a b} {v o l}_{g}$ along a metric family with inverse-metric variation $q$ , and $\nabla_{a} J^{a} {v o l}_{g} = d (ι_{J} {v o l}_{g})$ ; the integral of a compactly supported divergence vanishes (The derivative of the metric density and the chart formula for the divergence of a vector field, The metric volume form, current flux through 3-faces and finite-chain charges, Finite chart localization gives choice-free integration and compact Stokes, Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (a) — the flux identity is used only through the vanishing of the compactly supported divergence integral).

[L3]

Along the metric family, $q^{a b} = \partial_{s} g_{s}^{a b} ∣_{0}$ and the inverse-metric variation identity of First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional clause (a) holds (First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional).

[L4]

The action difference of The scalar-field matter model of the classical gravitational framework clause (b) is an integral of a compactly supported smooth top-form difference, not of either possibly nonintegrable total density. For fixed compact support choose finitely many fixed smooth chart cutoffs with sum $1$ near that support, as in Finite chart localization gives choice-free integration and compact Stokes. In each chart extend the cutoff coefficient by zero to a containing rectangle; it is continuous and integrable by Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable and The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ . Its parameter derivative is uniformly continuous on a compact parameter interval times this rectangle by Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous. The mean value theorem (The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ ) gives uniform convergence of parameter difference quotients; the estimate $∣ \int f - \int h ∣ \leq vol (Q) \sup ∣ f - h ∣$ , obtained from Lower and upper Darboux sums over a grid partition in $R^{m}$ , passes the derivative through the chart integral. Summing the finitely many chart derivatives proves differentiation under this global compact-difference integral.

[L5]

On each chart, a continuous symmetric covariant coefficient whose metric-density integral against every smooth compactly supported symmetric test tensor vanishes is zero (A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes). The coefficient may be a scalar multiple of the nondegenerate metric.

Proof

technique · direct; vary the Lagrangian density in the field and in the metric, integrate by parts using the vanishing of compactly supported divergence integrals, and differentiate the stress-energy formula

1.1L4algebra

(Field variation of the density.) Write $Φ (u) : = \frac{1}{2} g^{a b} \partial_{a} φ_{u} \partial_{b} φ_{u} + V (φ_{u})$ , so that $Δ S (φ_{u}, g) = - \frac{K}{c} \int_{M} [Φ (u) - Φ (0)] {v o l}_{g}$ . Differentiating the integrand at $u = 0$ , $\partial_{u} Φ = (g^{a b} \partial_{a} φ) \partial_{b} δ φ + V^{'} (φ) δ φ$ , and by [L4] the derivative of the integral is the integral of the derivative.

2.1step 1.1L1L2L5algebra

(Integration by parts.) Consider the smooth compactly supported vector field $J^{b} : = δ φ g^{a b} \partial_{a} φ$ ; by [L1], $\nabla_{b} J^{b} = (\partial_{b} δ φ) g^{a b} \partial_{a} φ + δ φ □_{g} φ$ , and $\int_{M} \nabla_{b} J^{b} {v o l}_{g} = 0$ by [L2]. Hence $\int_{M} (g^{a b} \partial_{a} φ) \partial_{b} δ φ {v o l}_{g} = - \int_{M} δ φ □_{g} φ {v o l}_{g}$ , and step 1.1 gives $\frac{d}{d u} Δ S = \frac{K}{c} \int_{M} (□_{g} φ - V^{'} (φ)) δ φ {v o l}_{g}$ , which is the variation formula in (a). Every compactly supported smooth scalar $f$ is an admissible first variation, by $φ_{u} = φ + u f$ . Let $E = □_{g} φ - V' (φ)$ . If all such variations are stationary, positivity of $K / c$ gives $\int E f {v o l}_{g} = 0$ . On any oriented coordinate chart and for every compactly supported symmetric tensor $q$ , take $f = g_{a b} q^{a b}$ , extended by zero. Then $\int (E g_{a b}) q^{a b} w d^{4} x = 0$ , so [L5] gives $E g = 0$ . Nondegeneracy of $g$ forces $E = 0$ at every point. Conversely $E = 0$ makes every variation derivative vanish by the displayed formula. This proves all of (a).

2.2step 1.1L2L3L4algebra

(Metric variation.) With $Ψ (s) : = \frac{1}{2} g_{s}^{a b} \partial_{a} φ \partial_{b} φ + V (φ)$ , the difference is $Δ S (s) = - \frac{K}{c} \int_{M} [Ψ (s) {v o l}_{g_{s}} - Ψ (0) {v o l}_{g}]$ along the metric family, differentiate at $s = 0$ : $\partial_{s} Ψ = \frac{1}{2} \partial_{a} φ \partial_{b} φ q^{a b}$ , and by [L2] $\partial_{s} {v o l}_{g} = - \frac{1}{2} g_{a b} q^{a b} {v o l}_{g}$ . Therefore $\partial_{s} (Ψ {v o l}_{g}) ∣_{0} = \frac{1}{2} [\partial_{a} φ \partial_{b} φ - g_{a b} Ψ] q^{a b} {v o l}_{g} = \frac{1}{2 K} T_{a b} q^{a b} {v o l}_{g}$ by the definition of $T$ in The scalar-field matter model of the classical gravitational framework clause (c). Differentiating under the integral by [L4] and multiplying by $- K / c$ gives (b). More precisely fix the base pair $(g, φ)$ and define the real-valued functional on all smooth Lorentzian/scalar pairs differing from it in a compact set by the action difference of the Definition. Its domain admits every compact metric direction by the stability lemma, and this calculation applies to every jointly smooth compact metric family at the base pair, so it satisfies the exact Hilbert definition; uniqueness then follows from Uniqueness of the Hilbert stress-energy tensor.

3.1step 2.2L1algebra∎

(Conservation identity.) By [L1], $\nabla^{a} T_{a b} = K [\nabla^{a} (\partial_{a} φ \partial_{b} φ) - \nabla_{b} (\frac{1}{2} \partial^{c} φ \partial_{c} φ + V (φ))]$ . The first term is $K [(\nabla^{a} \partial_{a} φ) \partial_{b} φ + \partial_{a} φ \nabla^{a} \partial_{b} φ] = K [□_{g} φ \partial_{b} φ + \partial_{a} φ \nabla_{b} \partial^{a} φ]$ by torsion freedom, and the second is $K [\partial_{a} φ \nabla_{b} \partial^{a} φ + V^{'} (φ) \partial_{b} φ]$ . Subtracting gives $\nabla^{a} T_{a b} = K (□_{g} φ - V^{'} (φ)) \partial_{b} φ$ , which is the identity in (c). The forward conservation consequence follows by substitution. At a point with $d φ \neq 0$ choose a tangent vector $X$ with $d φ (X) \neq 0$ ; if conservation holds, evaluating the identity on $X$ and using $K > 0$ gives $□_{g} φ - V' (φ) = 0$ . If such points are dense, smoothness gives the same equation everywhere. For the necessity of this caveat take $φ = a$ constant and a smooth potential with $V' (a) \neq 0$ : then $T = - K V (a) g$ is conserved by metric compatibility, while the scalar equation fails. This proves all of (c).

Remarks

Existence is not claimed. The proposition identifies the equation satisfied by stationary configurations of the action difference; it asserts neither that $□_{g} φ = V^{'} (φ)$ has solutions on a given $(M, g)$ nor any well-posedness property.
Dependence on the sign convention. Changing the overall sign of $ℓ$ reverses the sign of $Δ S$ and of the Hilbert tensor, and correspondingly the field equation; the proposition is stated for the convention fixed in The scalar-field matter model of the classical gravitational framework.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The dust matter model: rest density, four-velocity, stress-energy and dust system

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and let $U \subseteq M$ be a nonempty open subset, the matter domain, carrying its restricted smooth structure (An open subset of a smooth manifold has a canonical restricted smooth structure).

(a) Data. A dust model on $U$ is a pair $(ρ, u)$ consisting of

a smooth function $ρ : U \to R$ with $ρ (p) > 0$ for every $p \in U$ , the rest density; and
a smooth future-directed unit timelike vector field $u$ on $U$ ( $C^{r}$ and smooth maps between smooth manifolds with target $T M$ described through the tangent bundle; concretely, $u$ is a smooth vector field on $U$ with $g_{p} (u_{p}, u_{p}) = - 1$ and $g_{p} (u_{p}, N_{p}) < 0$ at every $p$ , Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

The parameter $ρ$ is the coefficient in the unit-normalized tensor of clause (b); this mathematical definition assigns it no physical unit. In the framework's SI stress-energy reading it is the rest energy density $ε : = ρ$ , in $J m^{- 3}$ . The rest-mass density is instead $ρ_{m} : = ε / c^{2}$ , in $k g m^{- 3}$ , and the physical four-velocity is $U : = c u$ , with $g (U, U) = - c^{2}$ . Thus the same tensor is $T^{a b} = ε u^{a} u^{b} = ρ_{m} U^{a} U^{b}$ . A purely mathematical or $c = 1$ convention may rescale the density coefficient; interpreting $ρ$ as an SI mass density while keeping the unit tangent $u$ would require the explicit tensor $c^{2} ρ u \otimes u$ . Physical realizations and readings are adopted by the matter postulate, not deduced here.

(b) Stress-energy tensor. The model's stress-energy tensor has the smooth symmetric contravariant components and their covariant lowering

$T^{a b} : = ρ u^{a} u^{b} equivalently T_{a b} = ρ u_{a} u_{b},$

with $u_{a} = g_{a c} u^{c}$ (The inverse metric, musical isomorphisms and index raising, Smooth covariant 2-tensor fields on a smooth manifold); the two expressions correspond under raising both indices, by $g^{a c} g^{b d} T_{c d} = T^{a b}$ .

(c) Dust system. The dust system on $U$ is the pair of equations

$\nabla_{a} (ρ u^{a}) = 0, u^{a} \nabla_{a} u^{b} = 0.$

The first is a continuity equation for the density current $ρ u$ ; the second says that the integral curves of $u$ are geodesics in the parameter that makes $u$ their unit tangent, i.e. that the dust particles are in free fall (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

Remarks

What is and is not asserted. The definition introduces the data, the tensor and the system; it asserts neither that the dust system holds, nor that $T$ is divergence-free, nor that $T$ is the Hilbert tensor of any action. The equivalence of the dust system with $\nabla^{a} T_{a b} = 0$ (where $ρ > 0$ ) is proved in Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations.
Smooth-branch restriction. Caustics or shocks, where the congruence $u$ fails to be a smooth unit field or $ρ$ ceases to be smooth, lie outside this model; a pressureless fluid with vanishing density in some region does not determine $u$ there, which is why $ρ > 0$ on the whole matter domain is part of the data.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and let $U \subseteq M$ be a nonempty open matter domain (An open subset of a smooth manifold has a canonical restricted smooth structure).

(a) Data. A perfect-fluid model on $U$ is a triple $(ε, p, n)$ consisting of two smooth functions $ε, p : U \to R$ , the rest energy density and the pressure, and a smooth future-directed unit timelike vector field $n$ on $U$ with $g_{p} (n_{p}, n_{p}) = - 1$ and $g_{p} (n_{p}, N_{p}) < 0$ at every $p$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, $C^{r}$ and smooth maps between smooth manifolds). No positivity of $ε$ or of $p$ is required or implied. A model may carry the additional data of an equation of state: a function $P : D_{P} \to R$ on a subset $D_{P} \subseteq R$ with $ε (U) \subseteq D_{P}$ , together with the relation $p = P (ε)$ ; the relation is a further assumption and is not part of the basic triple.

(b) Stress-energy tensor and derived fields. The model's stress-energy tensor is

$T^{a b} : = (ε + p) n^{a} n^{b} + p g^{a b},$

a smooth symmetric contravariant $2$ -tensor field (Smooth covariant 2-tensor fields on a smooth manifold, The inverse metric, musical isomorphisms and index raising). The comoving derivative, expansion and acceleration of $n$ are

$D f : = n^{a} \partial_{a} f (f \in C^{\infty} (U)), ϑ : = \nabla_{a} n^{a}, a^{b} : = n^{a} \nabla_{a} n^{b},$

and the rest-space projection is the smooth tensor field

$h^{a b} : = g^{a b} + n^{a} n^{b} .$

The physical reading of $ε$ and $p$ as a rest-frame energy density and pressure, and their units, are fixed by the framework's matter postulate, not by this definition.

(c) Fluid equations. The perfect-fluid equations on $U$ are

$D ε + (ε + p) ϑ = 0, (ε + p) a^{b} + h^{b a} \partial_{a} p = 0.$

That the stress-energy tensor is divergence-free if and only if these two equations hold is proved in Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations; the definition asserts neither divergence-freedom nor the equations.

Remarks

Projection convention. Since $n$ has unit length, $h^{a b}$ satisfies $h^{a b} n_{b} = 0$ and $h^{a b} g_{b c} h^{c d} = h^{a d}$ ; it is the $g$ -orthogonal projection onto the rest space $n^{⊥}$ raised with $g^{- 1}$ . The vector $a^{b}$ is automatically orthogonal to $n$ : $n_{b} a^{b} = \frac{1}{2} n^{a} \nabla_{a} (n_{b} n^{b}) = 0$ .
Scope. A model with $ε + p = 0$ has a stress-energy tensor proportional to $g^{a b}$ on $U$ ; the equations above then degenerate and the acceleration is not determined by the pressure gradient. Shock waves, dissipation and constitutive transport lie outside this model, as do equations of state with phase transitions, unless the smooth model is restricted to a phase.

PropositionStatement: AI-adaptedProof: AI-generatedprecheck passOpen item page →

Divergence-free dust is exactly the dust system, and divergence-free perfect fluid is exactly the fluid equations

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ and let $U \subseteq M$ be a nonempty open matter domain (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, An open subset of a smooth manifold has a canonical restricted smooth structure).

(a) Dust. Let $(ρ, u)$ be a dust model on $U$ , and let $T_{a b} : = ρ u_{a} u_{b}$ be its derived tensor (The dust matter model: rest density, four-velocity, stress-energy and dust system). Then

$\nabla^{a} T_{a b} = 0 on U ⟺ \nabla_{a} (ρ u^{a}) = 0 and u^{a} \nabla_{a} u^{b} = 0 on U .$

(b) Perfect fluid. Let $(ε, p, n)$ be a perfect-fluid model on $U$ , with derived tensor $T_{a b} : = (ε + p) n_{a} n_{b} + p g_{a b}$ , comoving derivative $D$ , expansion $ϑ$ , acceleration $a$ and projection $h$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy). Then

$\nabla^{a} T_{a b} = 0 on U ⟺ D ε + (ε + p) ϑ = 0 and (ε + p) a^{b} + h^{b a} \partial_{a} p = 0 on U .$

Facts & Assumptions

Given: The time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , the open matter domain $U$ , and the dust and perfect-fluid models of The dust matter model: rest density, four-velocity, stress-energy and dust system and The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy.

[L1]

$\nabla$ is torsion-free and $g$ -compatible; the extension of $\nabla$ to tensor fields is a derivation commuting with contractions and $\nabla g = 0$ , $\nabla g^{- 1} = 0$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Metric compatibility, torsion freedom and the metric-selected connection class).

[L2]

For a smooth vector field $V$ on $U$ , $\nabla_{a} V^{a} = g^{a b} \nabla_{a} V_{b}$ ; for smooth functions $f$ , $\nabla_{a} f = \partial_{a} f$ and $\nabla_{a} \nabla_{b} f = \nabla_{b} \nabla_{a} f$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, The inverse metric, musical isomorphisms and index raising).

[L4]

$u$ and $n$ are unit fields: $g (u, u) = - 1$ and $g (n, n) = - 1$ (The dust matter model: rest density, four-velocity, stress-energy and dust system, The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy).

Proof

technique · direct; expand the divergence of each stress-energy tensor, contract and project along the unit timelike field, and use nondegeneracy to conclude the converse

1.1L1L4algebra

(Dust divergence.) By [L1], $\nabla^{a} T_{a b} = \nabla^{a} (ρ u_{a} u_{b}) = u_{b} \nabla_{a} (ρ u^{a}) + ρ u^{a} \nabla_{a} u_{b}$ . Contracting with $u^{b}$ and using [L4], $u^{b} \nabla^{a} T_{a b} = - \nabla_{a} (ρ u^{a}) + ρ u^{a} u_{b} \nabla_{a} u^{b}$ , and $u_{b} \nabla_{a} u^{b} = \frac{1}{2} \nabla_{a} (u_{b} u^{b}) = 0$ because $g (u, u) = - 1$ is constant; so $u^{b} \nabla^{a} T_{a b} = - \nabla_{a} (ρ u^{a})$ .

1.2L1L2algebra

(Fluid divergence.) By [L1], $\nabla^{a} T_{a b} = \nabla^{a} ((ε + p) n_{a} n_{b}) + \nabla_{b} p$ , and $\nabla^{a} (n_{a} n_{b}) = n_{b} \nabla_{a} n^{a} + n^{a} \nabla_{a} n_{b} = ϑ n_{b} + a_{b}$ . Hence $\nabla^{a} T_{a b} = n_{b} (D ε + D p + (ε + p) ϑ) + (ε + p) a_{b} + \partial_{b} p,$ where $D f = n^{a} \partial_{a} f = n^{a} \nabla_{a} f$ and $a_{b} = g_{b c} a^{c}$ by [L2].

2.1step 1.1L4algebra

(Dust equivalence.) If $\nabla^{a} T_{a b} = 0$ , then step 1.1 gives $\nabla_{a} (ρ u^{a}) = 0$ ; substituting back into step 1.1 gives $ρ u^{a} \nabla_{a} u_{b} = 0$ , and $ρ > 0$ (The dust matter model: rest density, four-velocity, stress-energy and dust system) gives $u^{a} \nabla_{a} u^{b} = 0$ . Conversely, if both dust-system equations hold, step 1.1 gives $\nabla^{a} T_{a b} = u_{b} \cdot 0 + ρ \cdot 0 = 0$ . This proves (a).

2.2step 1.2L2L4algebra

(Two projections.) Contracting step 1.2 with $n^{b}$ and using [L4], $n^{b} n_{b} = - 1$ and $n^{b} a_{b} = 0$ : $n^{b} \nabla^{a} T_{a b} = - (D ε + D p + (ε + p) ϑ) + n^{b} \partial_{b} p = - (D ε + (ε + p) ϑ),$ since $n^{b} \partial_{b} p = D p$ and $D p = n^{b} \nabla_{b} p$ by [L2]. Applying the projection $h^{c b} = g^{c b} + n^{c} n^{b}$ to step 1.2 kills the $n_{b}$ -term and gives $h^{c b} \nabla^{a} T_{a b} = (ε + p) h^{c b} a_{b} + h^{c b} \partial_{b} p = (ε + p) a^{c} + h^{c b} \partial_{b} p$ , where $h^{c b} a_{b} = a^{c}$ by the remark of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy ( $n^{b} a_{b} = 0$ ).

3.1step 1.2step 2.2L2algebra∎

(Fluid equivalence.) If $\nabla^{a} T_{a b} = 0$ , then contracting step 1.2 with $n^{b}$ gives, by step 2.2, $D ε + (ε + p) ϑ = 0$ , and projecting step 1.2 with $h^{c b}$ gives, again by step 2.2, $(ε + p) a^{c} + h^{c b} \partial_{b} p = 0$ . Conversely, suppose both fluid equations hold. By the projection identity $h_{b}^{c} \partial_{c} p = \partial_{b} p + n_{b} D p$ , which follows from $h_{b}^{c} = δ_{b}^{c} + n_{b} n^{c}$ , the second equation reads $(ε + p) a_{b} = - \partial_{b} p - n_{b} D p$ . Substituting this and the first equation into step 1.2, $\nabla^{a} T_{a b} = n_{b} (D ε + (ε + p) ϑ) + n_{b} D p - (\partial_{b} p + n_{b} D p) + \partial_{b} p = 0.$ This proves (b).

Remarks

Role of $ρ > 0$ and of the projections. In (a) the strict positivity of the rest density is used in deriving the dust system from conservation to divide by $ρ$ ; on a region where the model has $ρ = 0$ the equation $ρ u^{a} \nabla_{a} u_{b} = 0$ carries no information about $u$ . In (b) the two displayed equations are the contraction and the rest-space projection of the divergence; together they are equivalent to the full covector equation because $W_{b} = - (W_{c} n^{c}) n_{b} + (W_{b} + (W_{c} n^{c}) n_{b})$ for every covector $W$ .
No equation of state used. Neither equivalence uses an equation of state: the equations are identities for the tensor defined from $ε, p, n$ alone. A model that adds $p = P (ε)$ restricts the admissible tuples $(ε, p, n)$ , it does not change the equivalence.

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Adopted classical gravitational action and variational field law

Physical scope

Adoption of the action formulation of the classical gravitational field law: the Einstein-Hilbert action of the smooth Lorentzian metric with cosmological parameter Lambda, the supplied scalar-field matter action (dust and perfect-fluid sectors use the separate direct-equation presentation), the Hilbert convention for the matter stress-energy tensor, and the variational principle as the field law of the model. Excludes quantum gravity and quantum fields, non-smooth and distributional sources, boundaries and junction conditions, gravity-mediated matter self-force corrections, and any claim that an action principle or a particular matter model describes nature. The adopted fields, constants and equations are model input; deductions from them are conditional on this adoption.

Postulate

Adopt as physical assumptions of the classical gravitational framework, for models whose geometric data satisfy Adopted spacetime model and classical domain of the gravitational framework and whose matter sector satisfies Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, the following statements. All symbols $c, G_{N}, Λ, κ$ are those of SI unit and coordinate conventions of the classical gravitational framework.

(i) Oriented models. The spacetime of a model in this formulation carries, in addition to $(M, g, N)$ , an orientation (Oriented smooth manifolds and oriented charts); the metric volume form ${v o l}_{g}$ of The metric volume form, current flux through 3-faces and finite-chain charges and its integrals are understood with respect to it. A model without a chosen orientation, or with a nonorientable $M$ , requires the density formulation of the action, which is not developed here and is outside the scope of this postulate.

(ii) Einstein-Hilbert action density. The Einstein-Hilbert action density of a smooth Lorentzian metric $g^{'}$ of signature $(-, +, +, +)$ on $M$ is the smooth $4$ -form

$ℓ_{E H} (g^{'}) : = \frac{c^{3}}{16 π G_{N}} (S_{g^{'}} - 2 Λ) {v o l}_{g^{'}},$

where $S_{g^{'}}$ is the scalar curvature of $g^{'}$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors) and $Λ \in R$ is a constant parameter of dimension $m^{- 2}$ . For two metrics whose difference is supported in a compact set, the action difference $Δ S_{E H} : = \int_{M} [ℓ_{E H} (g^{'}) - ℓ_{E H} (g)]$ is a real number. Only such differences are used; no value of the total action of a noncompact spacetime is asserted. The coupling

$κ : = \frac{8 π G_{N}}{c^{4}}$

has dimension $m^{- 1} {k g}^{- 1} s^{2}$ .

(iii) Matter sectors and available presentations. A model adopting this action presentation uses the supplied scalar-field action in sector (1). Sectors (2) and (3) are described here only to distinguish the alternative direct-equation presentation; they do not adopt this variational postulate. The three sector descriptions use the units of SI unit and coordinate conventions of the classical gravitational framework:

a scalar-field sector, specified by the data $(φ, V, K)$ of The scalar-field matter model of the classical gravitational framework with the physical readings: $φ$ dimensionless, $V$ a geometric potential with values in $m^{- 2}$ , and $K > 0$ in $J m^{- 1}$ . Length coordinates give $\partial_{a} φ$ dimension $m^{- 1}$ , so the kinetic term and $V$ have matching dimensions; the physical potential energy density is $K V$ in $J m^{- 3}$ . The sector's action differences are those defined there, and its stress-energy tensor is the candidate tensor of that definition;
a dust sector, specified by $(ρ, u)$ of The dust matter model: rest density, four-velocity, stress-energy and dust system, whose unit-tangent coefficient is the rest energy density $ε : = ρ$ in $J m^{- 3}$ ; the rest-mass density is $ρ_{m} = ε / c^{2}$ in $k g m^{- 3}$ and the physical four-velocity is $U = c u$ with $g (U, U) = - c^{2}$ . Its stress-energy tensor $T^{a b} = ρ u^{a} u^{b} = ρ_{m} U^{a} U^{b}$ is prescribed by the model and its matter equations are the dust system of that definition;
a perfect-fluid sector, specified by $(ε, p, n)$ (with optional equation of state) of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy with $ε, p$ in $J m^{- 3}$ , whose stress-energy tensor is prescribed by the model and whose matter equations are the fluid equations of that definition.

In sectors (2) and (3) no matter action is supplied: the model prescribes $T$ and the matter equations directly and adopts the direct-equation presentation, with its consistency conditions. Clause (iv) does not apply to these sectors. In sector (1) the matter action is the scalar action of The scalar-field matter model of the classical gravitational framework; its metric variation is required to be represented by its stress-energy tensor in the Hilbert convention

$δ S_{m} = - \frac{1}{2 c} \int_{M} T_{a b} δ g^{a b} {v o l}_{g} off shell in matter variables,$

which is the content of The Hilbert stress-energy tensor of a metric-dependent action clause (b) for that action; for the scalar action this representation is a proved property of the model, recorded in First variation, Hilbert tensor and conservation of the scalar-field model clause (b), and is adopted here as the reading of $T$ in the framework's units.

(iv) Variational field law. For the supplied scalar-field action, the classical field law adopted for the model is: the metric $g$ is stationary for the total action difference $Δ S_{t o t} : = Δ S_{E H} + Δ S_{m}$ under every jointly smooth metric family with compactly supported variation as defined in The Hilbert stress-energy tensor of a metric-dependent action clause (a), and the model's matter configuration satisfies its matter equations of motion. This variational principle is the adopted field law of the action presentation; the displayed field equation it is equivalent to is derived, not adopted a second time, in Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling.

(v) Alternative presentation. The direct-equation presentation of the same field law is adopted separately in Adopted Einstein field equation in the direct-equation presentation. A model adopts exactly one of the two presentations; they are alternative presentations of one classical field law and are never combined as two independent premises.

(vi) Status and scope. Clauses (i)-(v) are adopted assumptions. No geometry, symmetry principle or observation derives them; the action formulation is not claimed to be selected by nature, and no numerical value of $c$ , $G_{N}$ or $Λ$ is asserted. The smooth-branch exclusions of Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions apply verbatim: lower-dimensional or distributional sources, boundaries with junction conditions, and non-smooth fields require mathematics and physical assumptions not supplied here, and the comparison of any model prediction with data tests the combined model, including this postulate, rather than proving it.

Status, scope and limitations

Deductions are conditional. Every quantity derived from this postulate — the field equations, conservation identities and consistency conditions — is a logical consequence of these adopted assumptions and the mathematics of the framework, not a statement about nature.
Choices inside the formulation. The cosmological parameter may be fixed to zero for a restricted branch of a model; the matter sector is a model input; an equation of state in the fluid sector is a further assumption of that model. Different choices give different models of the same framework.
No boundary terms are adopted. Nonzero variations of the metric on a boundary are not covered by clause (iv); their treatment requires additional boundary terms that this framework does not supply.

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Adopted Einstein field equation in the direct-equation presentation

Physical scope

Adoption of the direct-equation presentation of the classical gravitational field law: the Einstein tensor of the metric-selected connection plus the cosmological term equals the coupling times the model's stress-energy tensor on the matter domain. Excludes quantum gravity and quantum fields, non-smooth and distributional sources, boundaries and junction conditions, and any claim that the equation or a particular matter model describes nature. The equation, the cosmological parameter and the matter content are model input; all deductions from this postulate are conditional on it.

Postulate

Adopt as the field law of the direct-equation presentation of the classical gravitational framework, for models satisfying Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions, the following statement.

(i) The equation. Let $(M, g, N)$ be the model's smooth time-oriented Lorentzian four-manifold with metric-selected connection $\nabla$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) and let $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ be the Einstein tensor of that connection (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Let $T$ be the model's stress-energy tensor field on its matter domain $U \subseteq M$ (Stress-energy tensor field of a model) and let $Λ \in R$ be a constant parameter with dimension $m^{- 2}$ and $G_{N} > 0$ , $c > 0$ the constants of SI unit and coordinate conventions of the classical gravitational framework. Adopt

$G_{a b} + Λ g_{a b} = \frac{8 π G_{N}}{c^{4}} T_{a b} on U$

as the field law of the model. Writing $κ : = 8 π G_{N} / c^{4}$ and $T_{a b} = g_{a c} g_{b d} T^{c d}$ for the lowered tensor, the equation reads $G_{a b} + Λ g_{a b} = κ T_{a b}$

(ii) Matter content. The equation constrains the metric and the matter of the model jointly. A model specifies its matter sector as in Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions: a scalar-field model with its action and stress-energy tensor (The scalar-field matter model of the classical gravitational framework), or a dust or perfect-fluid model with prescribed stress-energy tensor and matter equations. The consistency condition that every solution must satisfy, $\nabla^{a} T_{a b} = 0$ , is not adopted separately; it is a consequence of the equation and of the contracted second Bianchi identity and is derived in Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid.

(iii) Alternative presentation. The action formulation of the same field law is adopted separately in Adopted classical gravitational action and variational field law. A model adopts exactly one of the two presentations. They are alternative presentations of one classical field law: the action-to-equation theorem of this page identifies the equation obtained from stationarity of the total action with the equation displayed in clause (i), and the direct presentation is not an additional premise of an action-formulated model.

(iv) Status and scope. The equation is an adopted law of the model. It is not derived from mathematical symmetry, from the equivalence principle, or from observation; covariance and correspondence arguments motivate but do not prove it. No numerical value of $c$ , $G_{N}$ or $Λ$ is asserted, and no claim is made that any particular metric or matter configuration solves the equation or describes nature. The smooth-branch exclusions of Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions apply: distributional or lower-dimensional sources, boundaries with junction conditions, and non-smooth fields are outside this formulation. Comparisons of a model's predictions with data test the combined model — geometry, matter content and this law — and agreement is evidence about that combination, not a proof of the postulate.

Status, scope and limitations

Restricted branches. A model may fix $Λ = 0$ or choose a particular matter sector; each such restriction is part of the model, not of the framework.
No well-posedness claim. The postulate asserts the equation on the matter domain; it asserts neither existence nor uniqueness of solutions for any data, which are separate mathematical problems with their own hypotheses.

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Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling

Physical scope

Smooth classical domain of the action formulation of the framework: an oriented time-oriented smooth Lorentzian four-manifold, a matter sector whose stress-energy tensor is represented by its action in the Hilbert convention (the supplied scalar-field sector), and compactly supported metric variations. Excludes quantum gravity, boundaries and junction terms, non-smooth or distributional sources, and constraint or propagation claims; no empirical statement and no existence claim for solutions. Every conclusion is a conditional consequence of the adopted action postulate and the mathematics of the framework.

Statement

Adopt the action formulation of the framework, that is, the action density and matter sectors of Adopted classical gravitational action and variational field law, for a model whose geometric data and matter sector satisfy Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions; let $(M, g, N)$ be oriented with metric volume form ${v o l}_{g}$ and metric-selected connection $\nabla$ (Oriented smooth manifolds and oriented charts, The metric volume form, current flux through 3-faces and finite-chain charges, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), and suppose the model's matter sector supplies a stress-energy tensor field $T$ on $M$ (Stress-energy tensor field of a model, Smooth covariant 2-tensor fields on a smooth manifold) which is the Hilbert stress-energy tensor of the matter action at $(g, φ)$ with respect to $c$ (The Hilbert stress-energy tensor of a metric-dependent action); in particular this holds in the scalar-field sector, where the model is $(φ, V, K)$ of The scalar-field matter model of the classical gravitational framework and the claim is supplied by First variation, Hilbert tensor and conservation of the scalar-field model clause (b).

(a) Equivalence. With $κ : = 8 π G_{N} / c^{4}$ and $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ the Einstein tensor (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), the following are equivalent:

the metric $g$ is stationary for the total action difference $Δ S_{t o t} = Δ S_{E H} + Δ S_{m}$ under every smooth family of smooth Lorentzian metrics $s \mapsto g_{s}$ with $g_{0} = g$ and compactly supported variation;
$G_{a b} + Λ g_{a b} = κ T_{a b}$ at every point of $M$ .

(b) The two presentations agree. Consequently, in the action presentation the classical field law of clause (iv) of Adopted classical gravitational action and variational field law has the same equation form $G_{a b} + Λ g_{a b} = κ T_{a b}$ as the field law adopted in the direct-equation presentation Adopted Einstein field equation in the direct-equation presentation; the two presentations are equivalent descriptions of one classical field law, and a model adopts exactly one of them. In the prescribed- $T$ sectors (dust and perfect fluid) the equation is adopted directly, as recorded in clause (iii) of Adopted classical gravitational action and variational field law, and its consistency condition is derived separately in pthm-gr-einstein-consistency-and-energy-conditions on the owning page gr-einstein-and-matter.

Facts & Assumptions

Given: The adopted action formulation of Adopted classical gravitational action and variational field law for a model with an oriented $(M, g, N)$ , matter sector with Hilbert tensor $T$ at $(g, φ)$ , and the constants of SI unit and coordinate conventions of the classical gravitational framework.

[L1]

For every chart-supported smooth family $s \mapsto g_{s}$ of smooth Lorentzian metrics at $g$ with compactly supported variation and every constant $Λ$ , the compactly supported chart action difference $F (s) = \int_{U} [(S (s) - 2 Λ) w (s) - (S (0) - 2 Λ) w (0)] d^{4} x$ satisfies $F^{'} (0) = \int_{U} (G_{a b} + Λ g_{a b}) q^{a b} w d^{4} x,$ with $q^{a b} = \partial_{s} g_{s}^{a b} ∣_{0}$ , and every smooth symmetric compactly supported $q$ arises from such a family (First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional clauses (a),(e)).

[L2]

The matter action's metric variation is represented by $T$ : $\frac{d}{d s} S_{m} (g_{s}, φ) ∣_{0} = - \frac{1}{2 c} \int_{M} T_{a b} q^{a b} {v o l}_{g}$ (The Hilbert stress-energy tensor of a metric-dependent action clause (b)). In the scalar sector this is proved in First variation, Hilbert tensor and conservation of the scalar-field model clause (b).

[L3]

The action differences of clauses (ii)-(iii) of Adopted classical gravitational action and variational field law are integrals of compactly supported smooth top forms. Finite smooth cutoffs with compact chart supports and sum $1$ near any compact set exist, and they define the global integral by a finite sum independently of the cutoffs. For a chart-supported top form this global integral agrees with its chart integral; this applies both to the localized gravitational variations and to the Hilbert convention (Finite chart localization gives choice-free integration and compact Stokes, The metric volume form, current flux through 3-faces and finite-chain charges).

[L4]

If a continuous symmetric covariant $2$ -tensor field $F$ on a chart domain satisfies $\int_{U} F_{a b} q^{a b} w d^{4} x = 0$ for every smooth symmetric compactly supported $q$ , then $F = 0$ (A continuous symmetric coefficient orthogonal to every compactly supported variation vanishes).

[L5]

$Δ S_{t o t}$ is the sum of the action differences of Adopted classical gravitational action and variational field law clauses (ii)-(iii), and the constants satisfy $\frac{c^{3}}{16 π G_{N}} \cdot \frac{8 π G_{N}}{c^{4}} = \frac{1}{2 c}$ (SI unit and coordinate conventions of the classical gravitational framework).

Proof

technique · direct; differentiate the total action difference along an arbitrary compactly supported metric variation and apply the coefficient-vanishing lemma

1.1L1L3algebra

(Einstein-Hilbert part for arbitrary compact support.) Let $g_{s}$ be an arbitrary family in the Statement, equal to $g$ off a compact set $K$ , and put $h = \partial_{s} g_{s} ∣_{0}$ , $q^{a b} = - g^{a c} g^{b d} h_{c d}$ . By [L3] choose finitely many smooth cutoffs $χ_{i}$ with compact chart supports and $\sum_{i} χ_{i} = 1$ near $K$ . Put $h_{i} = χ_{i} h$ , so $h = \sum_{i} h_{i}$ and $q = \sum_{i} q_{i}$ with $q_{i} = χ_{i} q$ . The derivative at $g$ of the top form $(S - 2 Λ) {v o l}_{g}$ is linear in $h$ and its first and second derivatives: this follows directly from the inverse-metric, connection and Ricci variation formulas of First variation of the Einstein-Hilbert density under compactly supported metric variations, and of the timelike length functional clauses (a)-(d), applied locally: near any point one may multiply $g_{s} - g$ by a compactly chart-supported cutoff equal to $1$ on a neighborhood of that point. For small parameter values the resulting metric remains Lorentzian by compactness and agrees with the original family on that neighborhood, so the same local variation jets and formulas apply. Thus this derivative for $h$ is the sum of its derivatives for the $h_{i}$ , including all derivatives of the cutoffs; none is discarded. Each $h_{i}$ is realizable by a Lorentzian family after restricting its parameter interval about $0$ , by [L1]. For each such chart-supported family [L1] gives the integrated derivative $\int_{M} (G_{a b} + Λ g_{a b}) q_{i}^{a b} {v o l}_{g}$ , where chart and global integrals agree by [L3]. Differentiation of the original global action difference is legitimate by the same finite localization: in each compact chart support its smooth coefficient and parameter derivative are uniformly continuous on a compact parameter interval times that support, so difference quotients converge uniformly to the derivative and hence their integrals converge. Finite linearity and $\sum_{i} q_{i} = q$ therefore give $\frac{d}{d s} Δ S_{E H} ∣_{0} = \frac{c^{3}}{16 π G_{N}} \int_{M} (G_{a b} + Λ g_{a b}) q^{a b} {v o l}_{g} .$ No single chart containing $K$ and no partition on the whole noncompact manifold are required.

1.2L2

(Matter part.) By [L2], $\frac{d}{d s} Δ S_{m} ∣_{s = 0} = - \frac{1}{2 c} \int_{M} T_{a b} q^{a b} {v o l}_{g}$ .

2.1step 1.1step 1.2L5algebra

(Total variation.) Adding steps 1.1 and 1.2 and using $c^{3} / (16 π G_{N}) = (1 / (2 c)) \cdot c^{4} / (8 π G_{N}) = (1 / (2 c)) κ^{- 1}$ by [L5], $\frac{d}{d s} Δ S_{t o t} ∣_{0} = \frac{c^{3}}{16 π G_{N}} \int_{M} (G_{a b} + Λ g_{a b} - κ T_{a b}) q^{a b} {v o l}_{g} .$

3.1step 2.1L1L3L4algebra

(Coefficient vanishes.) Suppose $g$ is stationary in the sense of clause (a)(1): the derivative in step 2.1 vanishes for every smooth family with compactly supported variation. Every smooth symmetric test field $q$ compactly supported in a chosen chart is realizable by [L1], so $\int_{U} F_{a b} q^{a b} w d^{4} x = 0$ for the continuous symmetric field $F_{a b} : = G_{a b} + Λ g_{a b} - κ T_{a b}$ and every smooth symmetric compactly supported $q$ ; [L4] gives $F = 0$ on every chart, that is, $G_{a b} + Λ g_{a b} = κ T_{a b}$ . Conversely, if the equation holds, the integrand of step 2.1 vanishes identically for every family, so the derivative vanishes and $g$ is stationary. This proves the equivalence (a).

4.1step 3.1given∎

(Identification of the presentations.) Clause (b) is now immediate: clause (iv) of Adopted classical gravitational action and variational field law states the variational principle whose equation form is, by step 3.1, exactly $G_{a b} + Λ g_{a b} = κ T_{a b}$ , the equation adopted in Adopted Einstein field equation in the direct-equation presentation clause (i); the prescribed- $T$ sectors adopt that same equation directly by clause (iii) of the action postulate. Since each model adopts exactly one presentation, no model uses both as independent premises.

Remarks

Conditional content. The theorem is a deduction from the adopted action postulate and the mathematics of the variational calculus; it does not assert that nature realizes the action principle, that the matter sector is realized physically, or that any solution of the equation exists.
No empirical premises. No experiment, measurement or empirical estimate enters this item, and no empirical qualification is inherited by its conclusions.
Boundary terms excluded. The equivalence is stated for compactly supported metric variations; the Einstein-Hilbert action of a region with boundary would require additional boundary terms and is not covered here.

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Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid

Physical scope

Smooth classical domain of the framework in either presentation of the adopted field law: an oriented time-oriented smooth Lorentzian four-manifold, a constant cosmological parameter, a smooth stress-energy tensor on the model's matter domain, and the field equation G_{ab}+Lambda g_{ab}=kappa T_{ab}. The conclusions are conditional consequences of the adopted field law; they assert no existence, no propagation and no empirical fact, and none of the four energy conditions is universally forced by the field equation; particular matter models may imply some, and otherwise they are additional assumptions.

Statement

Adopt one of the two presentations of the framework's field law, that is, Adopted classical gravitational action and variational field law or Adopted Einstein field equation in the direct-equation presentation, for a model with geometric data and matter sector satisfying Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions. Let $(M, g, N)$ be the model's smooth time-oriented Lorentzian four-manifold with metric-selected connection $\nabla$ and curvature tensors $R^{a}_{b c d}$ , ${Ric}_{a b}$ , $S$ , $G_{a b}$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), let $T$ be the model's smooth symmetric stress-energy tensor field on the matter domain (Stress-energy tensor field of a model), let $Λ \in R$ be the constant parameter and $κ = 8 π G_{N} / c^{4}$ as in SI unit and coordinate conventions of the classical gravitational framework. Let $U \subseteq M$ be the model's open matter domain and suppose the field law holds on $U$ ,

$G_{a b} + Λ g_{a b} = κ T_{a b} on U .$

(a) Necessary conservation. $\nabla^{a} T_{a b} = 0$ on $U$ . Consequently a prescribed smooth symmetric $T$ with $\nabla^{a} T_{a b} \neq 0$ at some point cannot satisfy the field law there with that proposed metric; the divergence is computed using its metric-selected connection, and may change if the metric is changed.

(b) Trace. On $U$ , $g^{a b} T_{a b}$ satisfies $- S + 4 Λ = κ g^{a b} T_{a b}$ .

(c) Parameter variation. If instead $Λ : U \to R$ is a smooth function and the field law $G_{a b} + Λ g_{a b} = κ T_{a b}$ holds on $U$ with that function in place of a constant, then on $U$

$κ \nabla^{a} T_{a b} = \partial_{b} Λ .$

Consequently, at every point where $d Λ \neq 0$ , the equation requires $\nabla^{a} T_{a b} \neq 0$ ; where $d Λ = 0$ , this identity imposes conservation. A smooth $Λ$ can be locally constant but nonconstant across disconnected components of $U$ , in which case this identity does not force nonconservation.

(d) A vacuum solution. In the global vacuum example take $U = M = R^{4}$ with the identity chart and the flat metric $g_{a b} = diag (- 1, 1, 1, 1)$ (a smooth Lorentzian metric with time orientation $N = \partial_{0}$ ), the field law with $T = 0$ and $Λ = 0$ holds: $G = 0$ and $S = 0$ for the flat metric. With $T = 0$ and $Λ \neq 0$ it fails, since $G + Λ g = Λ g \neq 0$ .

(e) Energy conditions are additional assumptions. Neither adopted presentation of the field law contains a clause imposing any of the weak, null, dominant or strong energy conditions of The pointwise energy conditions WEC, NEC, DEC and SEC; none of the four is a universal consequence of that equation, as the direct-equation perfect-fluid model in the proof shows. Particular matter models may imply energy conditions; when not derived, they are additional assumptions in the sense of clause (iv) of Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions.

(f) Perfect-fluid rest-frame reading. Let $(ε, p, n)$ be a perfect-fluid model on its matter domain, with derived tensor $T = (ε + p) n^{♭} \otimes n^{♭} + p g$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy), and at any $p \in U$ let $(e_{0}, e_{1}, e_{2}, e_{3})$ be a $g_{p}$ -orthonormal basis with $e_{0} = n_{p}$ . Then $T_{A B} = diag (ε, p, p, p)$ in that basis, and at $p$

$T satisfies WEC ⟺ ε \geq 0 and ε + p \geq 0,$

$T satisfies NEC ⟺ ε + p \geq 0, T satisfies DEC ⟺ ε \geq ∣ p ∣,$

$T satisfies SEC ⟺ ε + 3 p \geq 0 and ε + p \geq 0.$

In particular a model with $ε > 0$ and $p = - ε$ satisfies the NEC, WEC and DEC and violates the SEC; the inequality signs here are those of the strict-timelike convention of The pointwise energy conditions WEC, NEC, DEC and SEC, under which continuity extensions to the causal cone are not assumed.

Facts & Assumptions

Given: The adopted field law in one of its two presentations on the open matter domain $U$ ; a model with smooth $(M, g, N)$ , smooth symmetric $T$ on $U$ , constant $Λ$ and coupling $κ = 8 π G_{N} / c^{4}$ ; in (f) additionally a perfect-fluid model on $U$ .

[L1]

The Einstein tensor of the metric-selected connection satisfies $\nabla^{a} G_{a b} = 0$ , and $g^{a b} G_{a b} = - S$ with $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ (The second Bianchi identity and the vanishing divergence of the Einstein tensor, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors).

[L2]

$\nabla g = 0$ and $\nabla g^{- 1} = 0$ ; the extension of $\nabla$ to tensor fields obeys the product rule, and for a smooth function $f$ , $\nabla^{a} (f g_{a b}) = \partial_{b} f$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Metric compatibility, torsion freedom and the metric-selected connection class).

[L3]

The flat metric on the identity chart of $R^{4}$ has vanishing Christoffel symbols, curvature and scalar curvature; the identity chart metric is smooth, symmetric and Lorentzian of signature $(-, +, +, +)$ , and $N = \partial_{0}$ is a time orientation (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L4]

An observer $n_{p}$ with $g_{p} (n_{p}, n_{p}) = - 1$ has a positive definite three-dimensional rest space $n_{p}^{⊥}$ ; choosing a $g_{p}$ -orthonormal basis of $n_{p}^{⊥}$ by Gram–Schmidt and adjoining $e_{0} : = n_{p}$ gives a $g_{p}$ -orthonormal basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ (Observers, rest spaces and the spatial projection at an event, The observer rest-space decomposition and the spatial projection, Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans).

[L5]

The four energy conditions are the pointwise inequalities of The pointwise energy conditions WEC, NEC, DEC and SEC: WEC $T (v, v) \geq 0$ for timelike $v$ ; NEC $T (v, v) \geq 0$ for null $v$ ; DEC $- ♯ (T (v, \cdot)) \in cl (C_{p}^{+})$ for future timelike $v$ ; SEC $(T - \frac{1}{2} ({tr}_{g} T) g) (v, v) \geq 0$ for timelike $v$ .

[L6]

The coordinate curvature formula of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (b) applies to every smooth Lorentzian metric with its metric-selected connection; contraction gives ${Ric}_{b d} = R^{a}_{b a d}$ , $S = g^{b d} {Ric}_{b d}$ and $G = Ric - \frac{1}{2} S g$ .

Proof

technique · direct; differentiate the field equation using the contracted Bianchi identity, contract traces, test the flat metric, and evaluate the quadratic inequalities of a rest-frame diagonal tensor

1.1L1L2algebra

(Conservation and trace.) Applying $\nabla^{a}$ to the field equation and using [L1] and [L2]: $κ \nabla^{a} T_{a b} = \nabla^{a} G_{a b} + Λ \nabla^{a} g_{a b} = 0$ , so $\nabla^{a} T_{a b} = 0$ since $κ > 0$ ; this is (a). Contracting the field equation with $g^{a b}$ and using [L1]: $g^{a b} (G_{a b} + Λ g_{a b}) = - S + 4 Λ = κ g^{a b} T_{a b}$ , which is (b).

1.2L1L2algebra

(Function parameter.) If $Λ$ is a smooth function and the equation holds, then by [L1] and [L2], $κ \nabla^{a} T_{a b} = \nabla^{a} (G_{a b} + Λ g_{a b}) = \partial_{b} Λ$ , which is (c).

1.3L3algebra

(Flat solution.) By [L3], the flat metric has $Γ = 0$ , hence $R = 0$ , $Ric = 0$ , $S = 0$ and $G = 0$ ; with $T = 0$ and $Λ = 0$ the field equation reads $0 = 0$ , while for $Λ \neq 0$ it would require $Λ g_{a b} = 0$ , which fails because $g$ is nondegenerate; this is (d).

1.4L4algebra

(Rest-frame components.) By [L4] choose the orthonormal basis with $e_{0} = n_{p}$ and write $T_{A B} = T (e_{A}, e_{B})$ and let $e^{A}$ be the dual covectors, $e^{A} (e_{B}) = δ^{A}_{B}$ . From $T_{a b} = (ε + p) n_{a} n_{b} + p g_{a b}$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy) and $n = n_{p} = e_{0}$ : $T_{00} = (ε + p) (- 1)^{2} + p (- 1) = ε$ , $T_{0 i} = 0$ , $T_{i j} = p δ_{i j}$ ; so $T_{A B} = diag (ε, p, p, p)$ . For a vector $v = v^{0} e_{0} + v$ with $v = \sum_{i \geq 1} v^{i} e_{i}$ , $T (v, v) = ε (v^{0})^{2} + p ∣ v ∣^{2}$ .

2.1step 1.4L5algebra

(WEC and NEC.) A vector $v \neq 0$ is timelike exactly when $(v^{0})^{2} > ∣ v ∣^{2}$ and null exactly when $(v^{0})^{2} = ∣ v ∣^{2}$ with $v \neq 0$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model). Writing $s : = ∣ v ∣ / ∣ v^{0} ∣ \in [0, 1)$ for timelike $v$ , $T (v, v) = (v^{0})^{2} [ε + p s^{2}]$ . This is nonnegative for all such $v$ if and only if $ε \geq 0$ (take $v = 0$ ) and $ε + p \geq 0$ (let $s \to 1$ , i.e. approach the null cone); conversely these inequalities make every value nonnegative. For null $v$ , $T (v, v) = (v^{0})^{2} (ε + p)$ , so the NEC is exactly $ε + p \geq 0$ .

2.2step 1.4L5algebra

(DEC.) By [L5] and [L2], the DEC requires $w : = - ♯ (T (v, \cdot))$ to lie in the closed future cone, including zero, for every future timelike $v$ . In the orthonormal basis, $T (v, \cdot) = ε v^{0} e^{0} + p \sum_{i} v^{i} e^{i}$ and $♯$ raises with $diag (- 1, 1, 1, 1)$ , so $w = ε v^{0} e_{0} - p \sum_{i} v^{i} e_{i}$ . Since $v$ is future and $e_{0} = n_{p}$ is the future unit normal of the rest frame, $v^{0} > 0$ . By The pointwise energy conditions WEC, NEC, DEC and SEC and Classification of future and past causal vectors by a timelike field, $w$ lies in $cl (C_{p}^{+})$ exactly when $w = 0$ or $w$ is future-directed and causal; for $w = ε v^{0} e_{0} - p \sum_{i} v^{i} e_{i}$ this is exactly the pair of conditions $ε v^{0} \geq 0$ and $(ε v^{0})^{2} \geq p^{2} ∣ v ∣^{2}$ . Requiring them for all future timelike $v$ forces $ε \geq 0$ (take $v = 0$ , so that $w = ε e_{0}$ must be $0$ or future causal) and $ε^{2} \geq p^{2}$ , that is $ε \geq ∣ p ∣$ (let $∣ v ∣ / v^{0} \to 1$ ); conversely $ε \geq ∣ p ∣$ gives both conditions. This is the DEC equivalence.

3.1step 1.4step 2.1L5algebra

(SEC.) By [L5], with ${tr}_{g} T = g^{a b} T_{a b} = - ε + 3 p$ (from step 1.4), the tensor $T - \frac{1}{2} ({tr}_{g} T) g$ has rest-frame components $\frac{ε + 3 p}{2}$ in the $(0, 0)$ slot and $\frac{ε - p}{2} δ_{i j}$ in the spatial slots, so its value on $v = v^{0} e_{0} + v$ is $\frac{(v^{0})^{2}}{2} [(ε + 3 p) + (ε - p) s^{2}]$ with $s = ∣ v ∣ / ∣ v^{0} ∣ \in [0, 1)$ . This is nonnegative for every $s \in [0, 1)$ exactly when $ε + 3 p \geq 0$ (at $s = 0$ ) and $2 (ε + p) \geq 0$ (letting $s \to 1$ ), which is the SEC equivalence; for $ε > 0$ , $p = - ε$ the conditions NEC, WEC and DEC hold with the equalities $ε + p = 0$ and $ε = ∣ p ∣$ , while SEC fails since $ε + 3 p = - 2 ε < 0$ .

4.1step 2.1step 2.2step 3.1L1L2L6givenconstructalgebra∎

(Energy conditions are not universal consequences.) The absence of inequalities from the adopted law is a formulation fact; nonimplication also needs a model. On $M = R^{4}$ with coordinates $(t, x^{1}, x^{2}, x^{3})$ , choose $L > 0$ , put $a (t) = \exp ((t / L)^{2})$ and $H = a^{'} / a = 2 t / L^{2}$ , and set $g = - d t^{2} + a^{2} \sum_{i} (d x^{i})^{2}$ , $N = n = \partial_{t}$ , $Λ = 0$ . This metric is smooth Lorentzian and $n$ is future unit timelike. Its only nonzero Christoffel symbols are $Γ^{0}_{i j} = a a^{'} δ_{i j}$ and $Γ^{i}_{0 j} = Γ^{i}_{j 0} = H δ^{i}_{j}$ . Substitution into the coordinate curvature formula of Tensoriality of the curvature, its coordinate formula and its point-normal metric formula and contraction give ${Ric}_{00} = - 3 (H^{'} + H^{2})$ , ${Ric}_{0 i} = 0$ , ${Ric}_{i j} = a^{2} (H^{'} + 3 H^{2}) δ_{i j}$ , hence $S = 6 H^{'} + 12 H^{2}$ , $G_{00} = 3 H^{2}$ , $G_{i j} = - a^{2} (2 H^{'} + 3 H^{2}) δ_{i j}$ . Choose the prescribed perfect fluid $ε = 3 H^{2} / κ$ , $p = - (2 H^{'} + 3 H^{2}) / κ$ and $T = (ε + p) n^{♭} \otimes n^{♭} + p g$ . Then $G = κ T$ , and its fluid equations hold: acceleration is zero, $ϑ = 3 H$ , spatial pressure gradient is zero, and $D ε + (ε + p) ϑ = 6 H H^{'} / κ - 6 H H^{'} / κ = 0$ . Thus it is an admissible direct-equation model, with no energy condition additionally adopted. At $t = 0$ , $ε = 0$ and $p = - 4 / (κ L^{2}) < 0$ ; steps 2.1–3.1 show that WEC, NEC, DEC and SEC all fail there. This proves (e) without inferring nonimplication merely from the wording of a postulate. In length coordinates $L$ has dimension $m$ , $H$ dimension $m^{- 1}$ , and $ε, p$ have the required energy-density units; the construction reports no observation.

Remarks

What is not proved. The theorem does not assert that any matter model satisfies any energy condition, that solutions of the field equation exist for prescribed data, or that a given solution propagates or is unique; those are separate mathematical problems.
Empirical status. The energy conditions and the field law are adopted assumptions; experimental tests compare predictions of a fully specified model with reported observations and do not prove or refute the postulate deductively.
No empirical premises. No measured input enters this item, so no empirical uncertainty is inherited by its conclusions.

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Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations

Physical scope

Scope, convention and status note for the field-law and matter-content material of the classical gravitational framework in its smooth oriented branch: the two presentations of the field law, the three matter sectors, the coupling and parameter conventions, the separation between adopted assumptions and conditional deductions, and the obligations deferred to other pages. The note asserts no law, no solution and no empirical fact.

Scope, status and conventions

This note fixes the scope of the page's field-law and matter content and the status of its claims; it proves nothing by itself.

Two presentations, one field law. The framework adopts the classical field law in exactly one of two presentations for any given model: the action formulation Adopted classical gravitational action and variational field law or the direct-equation formulation Adopted Einstein field equation in the direct-equation presentation. The theorem Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling identifies the equation form of the two presentations, so they define one classical field law rather than two independent premises; a model never adopts both.
Adopted versus derived. The field law, the matter sectors, the cosmological parameter and the Hilbert convention are adopted model input. The equation form of the action presentation, the conservation and trace identities, the parameter-variation identity and the rest-frame readings of the energy conditions are conditional deductions recorded in Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling and Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid. None of the adopted statements is derived from mathematics, symmetry or observation, and none of the deductions asserts that the model describes nature.
Matter sectors. The matter content is one of the sectors of clause (iii) of Adopted classical gravitational action and variational field law: the scalar-field model with a supplied action and Hilbert tensor in the action presentation, or the dust and perfect-fluid models with prescribed stress-energy tensor and matter equations in the direct-equation presentation. The stress-energy tensor occupies the slot defined in Stress-energy tensor field of a model; its component reading in a $g$ -orthonormal frame is in $J m^{- 3}$ , and the coupling satisfies $κ T \sim m^{- 2}$ with $κ = 8 π G_{N} / c^{4}$ in the conventions of SI unit and coordinate conventions of the classical gravitational framework. The equation of state of a fluid sector is a further assumption of that model; the energy conditions of The pointwise energy conditions WEC, NEC, DEC and SEC are optional additional assumptions unless a particular matter model implies them; the field equation alone does not universally force them.
Smooth oriented branch. The material assumes the smooth-branch restriction of Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions and, for the action and flux constructions, an orientation of the model's spacetime. Distributional or lower-dimensional sources, boundaries with junction conditions, low-regularity fields and quantum matter lie outside this page; they require further mathematics and, where physical, further adopted assumptions.
Empirical status. The page contains no experiment, no measured input and no empirical premise. Predictions computed from a fully specified model — geometry, matter sector and field law — may later be compared with reported observations; such a comparison tests the combined model and its auxiliary assumptions, and agreement is evidence rather than a proof of any postulate.
Deferred obligations. The broader observer-dependent energy-current discussion and the limits on global gravitational energy are treated on the local-and-global energy page; this page already proves the divergence and finite-chain flux balance for smooth currents, including Killing currents. The flat and Newtonian limits, the spherical vacuum, the homogeneous cosmologies and the linearized wave models are treated on their own pages. This page states no existence, uniqueness or propagation theorem for solutions of the field equation, and no initial-value or global causal claim.

5 · Examples, experiments and counterexamples

None yet.