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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Local And Global Energy

1 · Prerequisites

2 · Summary

This page treats energy in the classical gravitational framework of gr-einstein-and-matter: what can be said locally about the energy content of matter as measured by an observer congruence, and under which additional hypotheses and a stated normalization a conserved global Killing charge has an energy reading. Six items are mathematics — complete conditional deductions inside a time-oriented smooth Lorentzian four-manifold, with a chosen orientation on the domain for flux statements — and five are physics: one physical reading of the measured fields, one definition of the timelike Killing charge (with its normalized energy reading), one model-conditional theorem and two scope notes. No item reports a measurement, and none asserts that the model or its postulates describe nature.

The observer split. For a smooth symmetric covariant $2$ -tensor field $T$ and a smooth observer congruence $n$ , the measured energy density $μ = T_{a b} n^{a} n^{b}$ , the energy-momentum current $J = - T (\cdot, n)$ , the energy flux $Φ = J - μ n$ and the spatial stress with covariant form $S_{p} (v, w) = T_{p} (h_{p} (v), h_{p} (w))$ for $v, w \in T_{p} M$ , where $h_{p} (v) = v + g_{p} (n_{p}, v) n_{p}$ , are defined in The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress; the orthogonal decomposition $T^{a b} = μ n^{a} n^{b} + n^{a} Φ^{b} + n^{b} Φ^{a} + S^{a b}$ , its uniqueness and the component formulas in a $g$ -orthonormal frame are proved in The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness. The physical reading of these fields as the energy density, energy flux and momentum density, and stress measured by the congruence, with the framework's units, is Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress; it is a reading, not a positivity or conservation statement.

Kinematics and the local balance. The acceleration $a = \nabla_{n} n$ , the expansion $θ = \nabla_{a} n^{a}$ and the along-congruence derivative $D$ of a congruence are defined in Acceleration, expansion and the along-congruence derivative of an observer congruence and characterized in Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative. For on-shell matter, $\nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b}$ , equivalently $D μ + μ θ + \nabla_{a} Φ^{a} = - T^{a b} \nabla_{a} n_{b}$ , and the finite-chain bookkeeping of the current is Divergence of the measured energy-momentum current and the finite-chain balance identity. Fluxes use the metric volume form on the oriented matter domain; the current and its tensor need not extend outside that domain. The balance is a source-term identity: the measured energy current alone need not be conserved.

Stationary models and the Killing energy. If the model admits an everywhere timelike future-directed Killing field $ξ$ , then with $X = \sqrt{- g (ξ, ξ)}$ and $n = ξ / X$ the redshift-weighted current $X J$ is divergence-free and equals minus the Killing current $T (\cdot, ξ)$ (Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current). The Killing charge $Q_{ξ} (Σ) = - Φ_{T (\cdot, ξ)} (Σ)$ of a finite three-chain is defined in The timelike Killing current and the charge of a three-chain; it has energy units in joules when $ξ$ is length-normalized under the framework's length-coordinate convention. The theorem Local energy balance of the measured current and conservation of the Killing charge in stationary models assembles the physical statement: the local balance holds in every smooth model of the adopted field law, and in a stationary model the Killing charge (the Killing energy under that normalization) assigned to homologous slices is conserved up to the side flux. The mathematical conservation implication needs Killing symmetry and $\nabla^{a} T_{a b} = 0$ ; the field law is one source of this divergence condition.

Scope. Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses records what the framework does not provide: an observer-independent local energy density, a local gravitational energy density, or any global energy without a symmetry and control of the flux; asymptotic masses are deferred to a boundary-asymptotic module. Scope and status of the local-and-global-energy page: what is defined, what is conditional, and what is deferred fixes the page's conventions, the smooth and finite-chain restrictions, the choice assumption and the obligations deferred to other pages. The companion page gr-local-and-global-energy-examples works the translating-dust, boost-congruence and perfect-fluid balances, the Killing energy of a static dust, and two counterexamples separating measured energy from observer independence and from on-shell positivity.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress

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 $U \subseteq M$ be a nonempty open subset carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure, let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold), and let $n$ be a smooth observer congruence on $U$ (Observer congruences, their observer curves and adapted charts), so that $g_{p} (n_{p}, n_{p}) = - 1$ and $n_{p}$ is future-directed at every $p \in U$ .

(a) Spatial projection field. Let

$h^{a}_{b} : = δ^{a}_{b} + n^{a} n_{b}$

on $U$ , the component expression of the field of $g$ -orthogonal projections onto the rest spaces $n_{p}^{⊥}$ of Observers, rest spaces and the spatial projection at an event (The observer rest-space decomposition and the spatial projection); at every $p$ one has $h^{a}_{b} n^{b} = 0$ , $n_{a} h^{a}_{b} = 0$ , $h^{a}_{b} h^{b}_{c} = h^{a}_{c}$ and $h^{a}_{a} = 3$ . The field is smooth because $n$ , $g$ and the identity are smooth (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, $C^{r}$ and smooth maps between smooth manifolds).

(b) Measured energy density. The measured energy density of $T$ for the congruence $n$ is the smooth function

$μ : = T_{a b} n^{a} n^{b} : U \to R .$

(c) Energy-momentum current. The energy-momentum current of $T$ for the congruence $n$ is the smooth vector field on $U$ obtained by raising one index of $T$ with the inverse metric and contracting with $n$ ,

$J^{a} : = - T^{a b} n_{b},$

with $T^{a b} = g^{a c} g^{b d} T_{c d}$ (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components); the minus sign is the convention that fixes the energy-component sign convention. A positive $μ$ alone does not imply that $J$ is causal: $J$ may have an arbitrarily large spatial part. No future-causal property of $J$ is imposed by this definition.

(d) Energy flux. The energy flux of $T$ for the congruence $n$ is the smooth vector field

$Φ^{a} : = h^{a}_{b} J^{b} = J^{a} - μ n^{a},$

the spatial projection of $J$ ; it satisfies $n_{a} Φ^{a} = 0$ and $Φ^{a} = - h^{a}_{b} T^{b c} n_{c}$ .

(e) Spatial stress. The spatial stress of $T$ for the congruence $n$ is the smooth field obtained by projecting both tensor slots,

$S^{a b} : = h^{a}_{c} h^{b}_{d} T^{c d},$

which is symmetric in $a, b$ and satisfies $S^{a b} n_{b} = 0$ .

The identities $μ = - J^{a} n_{a}$ , $J = μ n + Φ$ , $S^{a b} n_{b} = 0$ and the orthogonal decomposition

$T^{a b} = μ n^{a} n^{b} + n^{a} Φ^{b} + n^{b} Φ^{a} + S^{a b},$

together with the uniqueness of that decomposition and the component formulas in a $g$ -orthonormal frame, are proved in The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness ↗; this definition asserts none of them. The physical reading and the units of $μ$ , $Φ$ and $S$ are fixed by the framework's reading on this page and are not asserted here.

Remarks

Scope. The objects are defined for every smooth symmetric covariant $2$ -tensor field $T$ on $U$ and every smooth observer congruence $n$ . No symmetry of $T$ beyond its stated symmetry, no conservation equation, no positivity, no dynamics and no measurement is required or implied. In particular $T$ need not be the stress-energy tensor of any matter model.
Choice convention. The smooth structure of the tangent bundle underlying the vector field $n$ assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) (A smooth vector field is a smooth section of the tangent bundle, Observer congruences, their observer curves and adapted charts); this is the only choice principle used in this item.

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

The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $U \subseteq M$ be a nonempty open subset, let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ and let $n$ be a smooth observer congruence on $U$ , and let $μ$ , $J$ , $Φ$ , $S$ and $h$ be as in The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress. Then:

(a) Smoothness and elementary properties. $μ$ is a smooth function, $J$ and $Φ$ are smooth vector fields and $S$ is a smooth symmetric contravariant $2$ -tensor field on $U$ ; moreover

$n_{a} Φ^{a} = 0, S^{a b} n_{b} = 0, μ = - J^{a} n_{a} .$

(b) Orthogonal decomposition. The identity of smooth tensor fields

$T^{a b} = μ n^{a} n^{b} + n^{a} Φ^{b} + n^{b} Φ^{a} + S^{a b}$

holds on $U$ , together with the contraction identity $T^{a b} n_{a} = - μ n^{b} - Φ^{b}$ .

(c) Uniqueness. If $μ^{'}$ is a smooth function, $Ψ$ a smooth vector field with $Ψ^{a} n_{a} = 0$ and $S^{'}$ a smooth symmetric contravariant $2$ -tensor field with $S^{' a b} n_{b} = 0$ on $U$ , and if

$T^{a b} = μ^{'} n^{a} n^{b} + n^{a} Ψ^{b} + n^{b} Ψ^{a} + S^{' a b},$

then $μ^{'} = μ$ , $Ψ = Φ$ and $S^{'} = S$ .

(d) Values on the rest space and frame components. For every $p \in U$ and every $v \in n_{p}^{⊥}$ ,

$T_{p} (n_{p}, n_{p}) = μ (p), T_{p} (n_{p}, v) = - g_{p} (Φ_{p}, v),$

The covariant tensor obtained by lowering both slots of $S_{p}$ is $S_{p}^{♭} (v, w) : = S_{p} (g_{p} (v, \cdot), g_{p} (w, \cdot))$ for $v, w \in T_{p} M$ , with components $S_{a b}^{♭} = g_{a c} g_{b d} S^{c d}$ . The restrictions of $T_{p}$ and $S_{p}^{♭}$ to $n_{p}^{⊥} \times n_{p}^{⊥}$ agree. In every ordered $g_{p}$ -orthonormal basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ with $e_{0} = n_{p}$ ,

$μ (p) = T_{00} (p), Φ^{i} (p) = T^{0 i} (p) = - T_{0 i} (p) (i = 1, 2, 3),$

and $S^{i j} (p) = T^{i j} (p)$ for $i, j \in {1, 2, 3}$ while $S^{0 a} (p) = 0$ . Finally $g_{a b} S^{a b} = g_{a b} T^{a b} + μ$ .

Facts & Assumptions

Given: The time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , the open subset $U$ , the smooth symmetric covariant $2$ -tensor field $T$ on $U$ , the smooth observer congruence $n$ on $U$ , and the fields $μ$ , $J$ , $Φ$ , $S$ , $h$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress.

[L1]

Every $v \in T_{p} M$ has the unique decomposition $v = - g_{p} (v, n_{p}) n_{p} + h_{p} (v)$ with $h_{p} (v) \in n_{p}^{⊥}$ ; $h_{p}$ is linear, $h_{p} \circ h_{p} = h_{p}$ , its kernel is $R n_{p}$ , its image is $n_{p}^{⊥}$ , and $g_{p} (h_{p} (v), h_{p} (w)) = g_{p} (v, w) + g_{p} (v, n_{p}) g_{p} (n_{p}, w)$ (The observer rest-space decomposition and the spatial projection, Observers, rest spaces and the spatial projection at an event).

[L2]

The fields $n$ , $g$ , $g^{- 1}$ and $T$ are smooth, and a tensor field defined from them by the standard index operations and pointwise contractions is smooth: its chart components are obtained from the smooth component functions of $n$ , $g$ , $g^{- 1}$ and $T$ by sums of products, and raising and lowering are the contractions $g^{a c} g^{b d} T_{c d}$ and $g_{a c} g_{b d} X^{c d}$ with the smooth inverse-metric components (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, Smooth covariant 2-tensor fields on a smooth manifold, The inverse metric, musical isomorphisms and index raising, $C^{r}$ and smooth maps between smooth manifolds).

[L3]

$g_{p} (n_{p}, n_{p}) = - 1$ at every $p \in U$ (Observers, rest spaces and the spatial projection at an event, Observer congruences, their observer curves and adapted charts).

[L4]

The inverse-metric components satisfy $g^{a c} g_{c b} = δ^{a}_{b}$ ; raising and lowering an index with $g^{- 1}$ and $g$ are inverse linear operations on each tensor slot (The inverse metric, musical isomorphisms and index raising).

[L5]

At every $p \in U$ there is a $g_{p}$ -orthonormal ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ of $T_{p} M$ ; when $e_{0} = n_{p}$ it satisfies $g_{p} (e_{0}, e_{0}) = - 1$ and $g_{p} (e_{i}, e_{j}) = δ_{i j}$ for $i, j \in {1, 2, 3}$ , and the components of a tensor in that basis are its contractions with the dual basis (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, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

Proof

technique · direct; compute the projections of the tensor in index form, read off the four pieces, and prove uniqueness by contracting the putative decomposition against $n$ and its orthogonal complement

1.1L1L2L3L4algebra

(Smoothness and the elementary identities.) Writing components in a chart, $μ = T_{a b} n^{a} n^{b}$ is a sum of smooth products, and $J^{a} = - g^{a b} T_{b c} n^{c}$ is a sum of smooth products; hence both are smooth, and $Φ = J - μ n$ and $S^{a b} = h^{a}_{c} h^{b}_{d} T^{c d}$ are smooth by [L2]; symmetry of $T^{c d}$ gives symmetry of $S^{a b}$ by interchanging the two contracted indices. The field $h$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress has $h^{a}_{b} n^{b} = n^{a} + n^{a} (n \cdot n) = 0$ and $n_{a} h^{a}_{b} = n_{b} + (n_{a} n^{a}) n_{b} = 0$ by [L3], so $n_{a} Φ^{a} = n_{a} J^{a} - μ (n \cdot n) = n_{a} J^{a} + μ$ and $S^{a b} n_{b} = h^{a}_{c} (n_{b} h^{b}_{d}) T^{c d} = 0$ ; finally $n_{a} J^{a} = - T^{a b} n_{a} n_{b} = - μ$ .

2.1step 1.1L3L4algebra

(The decomposition.) Expanding $h^{a}_{c} = δ^{a}_{c} + n^{a} n_{c}$ and $h^{b}_{d} = δ^{b}_{d} + n^{b} n_{d}$ in $S^{a b} = h^{a}_{c} h^{b}_{d} T^{c d}$ gives $S^{a b} = T^{a b} + n^{a} T^{c b} n_{c} + n^{b} T^{a d} n_{d} + n^{a} n^{b} T^{c d} n_{c} n_{d} = T^{a b} - n^{a} J^{b} - n^{b} J^{a} + μ n^{a} n^{b}$ , using $T^{c b} n_{c} = - J^{b}$ , $T^{a d} n_{d} = - J^{a}$ and $T^{c d} n_{c} n_{d} = μ$ . Substituting $J = μ n + Φ$ and solving for $T^{a b}$ gives $T^{a b} = μ n^{a} n^{b} + n^{a} Φ^{b} + n^{b} Φ^{a} + S^{a b},$ because $n^{a} (μ n^{b} + Φ^{b}) + n^{b} (μ n^{a} + Φ^{a}) - μ n^{a} n^{b} = μ n^{a} n^{b} + n^{a} Φ^{b} + n^{b} Φ^{a}$ . Contracting this identity with $n_{a}$ and using [L3] gives $T^{a b} n_{a} = - μ n^{b} - Φ^{b}$ , since $n_{a} Φ^{a} = 0$ and $n_{a} n^{a} = - 1$ .

3.1step 1.1step 2.1L3L4algebra

(Uniqueness.) Let $μ^{'}, Ψ, S^{'}$ be as in (c). Contracting the assumed decomposition with $n_{a} n_{b}$ kills every term containing $Ψ$ (because $Ψ^{a} n_{a} = 0$ ) or $S^{'}$ (because $S^{' a b} n_{b} = 0$ ) and gives $μ^{'} = T^{a b} n_{a} n_{b} = μ$ by [L3]. Contracting with $n_{a}$ alone and using $n_{a} n^{a} = - 1$ gives $T^{a b} n_{a} = - μ^{'} n^{b} - Ψ^{b}$ , so $Ψ^{b} = - T^{a b} n_{a} - μ^{'} n^{b} = J^{b} - μ n^{b} = Φ^{b}$ , where $- T^{a b} n_{a} = J^{b}$ is the definition of $J$ and $Φ = J - μ n$ . Solving the decomposition for $S^{'}$ now gives $S^{' a b} = T^{a b} - μ n^{a} n^{b} - n^{a} Φ^{b} - n^{b} Φ^{a} = S^{a b}$ by step 2.1.

4.1step 2.1L3L4L5algebra∎

(Rest-space values and frame components.) By definition $μ = T (n, n)$ . For $v \in n_{p}^{⊥}$ , step 2.1 gives $T_{p} (n_{p}, v) = T^{a b} n_{a} v_{b} = (- μ n^{b} - Φ^{b}) v_{b} = - g_{p} (Φ_{p}, v)$ , since $n^{b} v_{b} = 0$ . For $v, w \in n_{p}^{⊥}$ , contract the decomposition of step 2.1 with $v_{a} w_{b}$ , where $v_{a} = g_{a c} v^{c}$ and $w_{b} = g_{b d} w^{d}$ . Every term containing $n^{a}$ or $n^{b}$ vanishes, giving $T_{p} (v, w) = S^{a b} v_{a} w_{b} = S_{p}^{♭} (v, w)$ . In an orthonormal basis with $e_{0} = n_{p}$ , [L5] gives $n^{a} = δ_{0}^{a}$ and $n_{a} = - δ_{a}^{0}$ , so $h^{0}_{b} = δ_{b}^{0} + n^{0} n_{b} = δ_{b}^{0} - δ_{b}^{0} = 0$ and $h^{i}_{b} = δ_{b}^{i}$ ; consequently $S^{0 a} = 0$ , $S^{i j} = T^{i j}$ and $Φ^{i} = J^{i} - μ n^{i} = J^{i} = - T^{i b} n_{b} = - T^{i 0} n_{0} = T^{i 0} = - T_{0 i}$ , while $μ = T_{00}$ . Contracting the decomposition with $g_{a b}$ gives $g_{a b} S^{a b} = g_{a b} T^{a b} - μ (n \cdot n) - 2 g (n, Φ) = g_{a b} T^{a b} + μ$ ; that is, $g_{a b} S^{a b} = g_{a b} T^{a b} + μ$ .

Remarks

Minimal hypotheses. Only symmetry of $T$ and the congruence property $g (n, n) = - 1$ are used; no equation of motion enters the pointwise algebraic decomposition; smoothness of the fields uses the smooth inputs and inverse metric in [L2]. The sign convention is $J = - ♯ (T (n, \cdot))$ and $T (n, v) = - g (Φ, v)$ on the rest space.
Origin of the names. The names measured energy density, energy flux and stress are attached to $μ$ , $Φ$ and $S$ only through the framework's physical reading on this page; this lemma proves the algebraic identities alone.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress

Physical scope

The physical reading of the algebraic observer split of a smooth symmetric two-tensor on the smooth classical branch of the framework: the measured energy density, energy-flux and momentum-density readings and the measured stress of a prescribed tensor for a prescribed observer congruence, with the framework's unit conventions. The item fixes names, component formulas and units only; it asserts no positivity, no conservation, no dynamics, no measurement and no claim that the tensor is realized by matter.

Definition

Adopt the framework's unit and reading conventions (SI unit and coordinate conventions of the classical gravitational framework) and the stress-energy slot (Stress-energy tensor field of a model). Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $U \subseteq M$ be a nonempty open subset, let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ and let $n$ be a smooth observer congruence on $U$ (Observer congruences, their observer curves and adapted charts). Let $μ$ , $J$ , $Φ$ and $S$ be the fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress.

(a) Measured energy density and its unit. The smooth function $μ = T_{a b} n^{a} n^{b}$ is read as the energy density measured by the observers of the congruence $n$ , with numerical values in $J m^{- 3}$ when $T$ is read in the orthonormal-frame convention of SI unit and coordinate conventions of the classical gravitational framework clauses (a) and (d). The same item fixes the energy-density reading of $ρ_{n} = T (n, n)$ for a future unit timelike $n$ .

(b) Energy flux and momentum density. The smooth spatial vector field $Φ = h (J)$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress is the tensor's spatial current in the normalized observer convention. Its components have energy-density units, $J m^{- 3}$ . Define the SI measured energy-flux vector $q : = c Φ$ and SI measured momentum-density covector $p : = Φ^{♭} / c$ , where $Φ^{♭} (a) = g (Φ, a)$ and $c$ is the conversion constant of SI unit and coordinate conventions of the classical gravitational framework. In every ordered orthonormal basis with future $e_{0} = n_{p}$ ,

$Φ^{i} = T^{0 i} = - T (e_{0}, e_{i}), q^{i} = c Φ^{i}, p_{i} = Φ^{i} / c, i = 1, 2, 3,$

by The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness (d). Hence $q^{i}$ has units $W m^{- 2}$ and $p_{i}$ has units $k g m^{- 2} s^{- 1}$ ; equivalently $q^{♭} = c^{2} p$ . When the algebraic balance formulas call $Φ$ an energy flux they use this normalized tensor current, whose SI energy flux is $c Φ$ . The analogous natural-unit momentum notation suppresses the factor $1 / c$ ; it never assigns SI momentum units directly to $Φ^{♭}$ . Symmetry of $T$ gives the two aspects of the same mixed component, with these distinct SI conversions.

(c) Measured stress. Write $S_{a b} = g_{a c} g_{b d} S^{c d}$ for the covariant form of the projected field of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress clause (e). Explicitly, for $p \in U$ and $v, w \in T_{p} M$ , $S_{p} (v, w) : = T_{p} (h_{p} (v), h_{p} (w))$ , where $h_{p} (v) = v + g_{p} (n_{p}, v) n_{p}$ . This agrees with the lowered component formula because $h_{p}$ is self-adjoint for $g_{p}$ . This smooth symmetric covariant field is read as the stress measured by the observers of the congruence: for unit spatial vectors $e, e^{'}$ in the rest space of $n_{p}$ , $S_{p} (e, e^{'})$ is read as the force per unit area in the direction $e$ across a surface whose normal is $e^{'}$ , with components in $J m^{- 3}$ by the conventions.

(d) Current and charge reading. For this clause choose an orientation on $U$ and the volume form of its restricted metric; this choice is independent of the time orientation. The smooth vector field $J$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress clause (c) is read as the energy-momentum current density measured by the congruence, and for a finite smooth singular $3$ -chain $Σ$ of $U$ the flux $Φ_{J} (Σ)$ of The metric volume form, current flux through 3-faces and finite-chain charges clause (b) is read as the energy-momentum assigned to $Σ$ by the congruence, in the sense of the charge reading of that item. For a $3$ -chain carried by a spacelike slice whose normal is (up to sign) $n$ , the index-lowered component $- J_{a} n^{a} = μ$ is the density integrated in that reading.

(e) Scope. The reading is interpretive and applies to every prescribed smooth symmetric $T$ and every observer congruence. No positivity of $μ$ , no conservation of $J$ , no dynamics, no existence of a matter model realizing $T$ and no measurement is asserted; the definition is confined to the smooth branch of the framework, and $T$ need not satisfy any field equation.

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Acceleration, expansion and the along-congruence derivative of an observer congruence

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), let $U \subseteq M$ be a nonempty open subset carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure, and let $n$ be a smooth observer congruence on $U$ (Observer congruences, their observer curves and adapted charts), so that $g_{p} (n_{p}, n_{p}) = - 1$ and $n_{p}$ is future-directed at every $p \in U$ .

(a) Acceleration. The acceleration field of $n$ is the smooth vector field

$a : = \nabla_{n} n,$

the covariant derivative of $n$ along itself; componentwise $a^{b} = n^{a} \nabla_{a} n^{b}$ . The item defines the field; that it is spatial and that it is the acceleration of the observer curves is proved in Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative ↗.

(b) Expansion. The expansion of $n$ is the smooth function

$θ : = \nabla_{a} n^{a} = g^{a b} \nabla_{a} n_{b} .$

Its chart formula and its identification with $h^{a b} \nabla_{a} n_{b}$ are proved in Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative ↗.

(c) Along-congruence derivative. For a smooth function $f : U \to R$ ( $C^{r}$ and smooth maps between smooth manifolds) define

$D f : = n^{a} \nabla_{a} f,$

the derivative of $f$ along $n$ ; $D f$ is smooth and $D$ is a derivation of the smooth functions on $U$ (Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative ↗ clause (d)).

(d) Scope. The acceleration and the expansion are geometric data of the congruence $n$ alone, computed with the connection $\nabla$ . This item asserts neither $a = 0$ (a geodesic congruence) nor $θ = 0$ , and it defines no shear, vorticity, measurement, clock reading or physical interpretation; the reading of observer curves as ideal-clock worldlines belongs to the framework's postulates and is not asserted here.

Remarks

Why only two kinematic fields. On this page the expansion enters the local energy balance through the identity $\nabla_{a} (μ n^{a} + Φ^{a}) = D μ + μ θ + \nabla_{a} Φ^{a}$ of Divergence of the measured energy-momentum current and the finite-chain balance identity. The shear and vorticity tensors are not needed for the statements proved here and are therefore not introduced; no decomposition of $\nabla n$ beyond the trace $θ$ and the acceleration $a$ is asserted.
Choice convention. The smooth structure of the tangent bundle underlying $n$ assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) (A smooth vector field is a smooth section of the tangent bundle); this is the only choice principle used in this item.

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Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ , let $U \subseteq M$ be a nonempty open subset and let $n$ be a smooth observer congruence on $U$ , with acceleration $a = \nabla_{n} n$ , expansion $θ = \nabla_{a} n^{a}$ and along-congruence derivative $D$ as in Acceleration, expansion and the along-congruence derivative of an observer congruence. Then:

(a) The acceleration is spatial. $a$ is smooth and

$g_{p} (a_{p}, n_{p}) = 0 for every p \in U,$

that is, $a_{p} \in n_{p}^{⊥}$ ; equivalently $h (a) = a$ for the projection field $h_{p} : = h_{n_{p}}$ of Observers, rest spaces and the spatial projection at an event clause (c), namely $h_{p} (v) = v + g_{p} (n_{p}, v) n_{p}$ for $v \in T_{p} M$ .

(b) Acceleration of the observer curves. Let $γ : I \to U$ be an observer curve of the congruence, that is, an integral curve of $n$ (Observer congruences, their observer curves and adapted charts clause (b)), and let $D_{s}$ denote the covariant derivative along $γ$ (Covariant derivative along a curve). Then

$D_{s} γ^{'} (s) = a_{γ (s)} for every s \in I .$

Consequently every observer curve is a geodesic in its flow parameter if and only if $a = 0$ on $U$ (Geodesic of an affine connection).

(c) Divergence form of the expansion. In every smooth chart of $U$ , with $g_{a b}$ the metric components and $w : = \sqrt{∣ \det (g_{a b}) ∣} > 0$ ,

$θ = w^{- 1} \partial_{a} (w n^{a}),$

and $θ = g^{a b} \nabla_{a} n_{b} = h^{a b} \nabla_{a} n_{b}$ with $h^{a b} = g^{a b} + n^{a} n^{b}$ . In particular $θ$ is smooth.

(d) Derivation property. For smooth functions $f, h : U \to R$ , $D (f h) = f D h + h D f$ , and $D$ annihilates constants.

Facts & Assumptions

Given: The time-oriented smooth Lorentzian four-manifold $(M, g, N)$ , the open subset $U$ , the smooth observer congruence $n$ on $U$ , and the fields $a$ , $θ$ , $D$ of Acceleration, expansion and the along-congruence derivative of an observer congruence.

[L1]

$\nabla$ is torsion-free and $g$ -compatible: for all smooth vector fields $X, Y, Z$ on $M$ , $X (g (Y, Z)) = g (\nabla_{X} Y, Z) + g (Y, \nabla_{X} Z)$ (Metric compatibility, torsion freedom and the metric-selected connection class clause (a), Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

$\nabla_{X} Y$ is a smooth vector field for smooth $X, Y$ , so $\nabla_{n} n$ is smooth on $U$ ; $\nabla_{X} Y$ is additive and $C^{\infty} (M)$ -linear in $X$ , and additive in $Y$ (Affine connection on a smooth manifold).

[L3]

For an integral curve $γ$ of $n$ , $γ^{'} = n \circ γ$ is the pullback of the supplied smooth field $n$ . The pullback connection defining the curve derivative gives $D_{s} (n \circ γ) = \nabla_{γ^{'}} n = (\nabla_{n} n) \circ γ$ (Covariant derivative along a curve, Geodesic of an affine connection, Observer congruences, their observer curves and adapted charts).

[L4]

$g_{p} (n_{p}, n_{p}) = - 1$ for every $p \in U$ (Observers, rest spaces and the spatial projection at an event, Observer congruences, their observer curves and adapted charts).

[L5]

In every smooth chart of $U$ , for every smooth vector field $J$ on $U$ , $\nabla_{a} J^{a} = w^{- 1} \partial_{a} (w J^{a})$ with $w = \sqrt{∣ \det (g_{a b}) ∣}$ , and the chart formula represents the chart-independent scalar $\nabla_{a} J^{a}$ (The derivative of the metric density and the chart formula for the divergence of a vector field clause (c), Christoffel symbols of an affine connection).

[L6]

Define $h_{p} : = h_{n_{p}}$ by $h_{p} (v) = v + g_{p} (n_{p}, v) n_{p}$ (Observers, rest spaces and the spatial projection at an event clause (c)). This projection field satisfies $h_{p} (a_{p}) = a_{p}$ for $a_{p} \in n_{p}^{⊥}$ and annihilates multiples of $n_{p}$ (The observer rest-space decomposition and the spatial projection, Observers, rest spaces and the spatial projection at an event).

[L7]

For smooth functions $f$ on $U$ , the induced tensor derivative has $\nabla_{a} f = \partial_{a} f$ , so $D f = n (f)$ is the action of the smooth vector field $n$ as a derivation on functions (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), A smooth vector field is a smooth section of the tangent bundle).

Proof

technique · direct; differentiate the normalization $g(n,n)=-1$, identify the covariant derivative along observer curves, and express the divergence of $n$ in a chart

1.1L1L2L4L6algebra

(Spatiality of $a$ .) By [L2], $a = \nabla_{n} n$ is a smooth vector field on $U$ . For any smooth vector field $X$ on $U$ , [L1] gives $X (g (n, n)) = g (\nabla_{X} n, n) + g (n, \nabla_{X} n) = 2 g (\nabla_{X} n, n)$ ; by [L4] the left-hand side is $X (- 1) = 0$ , so $g (\nabla_{X} n, n) = 0$ for every $X$ . Taking $X = n$ gives $g_{p} (a_{p}, n_{p}) = 0$ at every $p \in U$ ; hence $a_{p} \in n_{p}^{⊥}$ and $h_{p} (a_{p}) = a_{p}$ by [L6]. This proves (a).

2.1L3step 1.1algebra

(Acceleration of the observer curves.) Let $γ$ be an integral curve of $n$ ; then $γ^{'} = n \circ γ$ , and [L3] gives $D_{s} γ^{'} (s) = (\nabla_{n} n)_{γ (s)} = a_{γ (s)}$ for every $s \in I$ . If $a = 0$ on $U$ , then $D_{s} γ^{'} = 0$ , so each observer curve is a geodesic in its parameter (Geodesic of an affine connection); conversely, if every observer curve satisfies $D_{s} γ^{'} = 0$ , then $a_{γ (s)} = 0$ for every $s$ and every observer curve, and since every point of $U$ lies on an observer curve (Observer congruences, their observer curves and adapted charts clause (b)), $a = 0$ on $U$ . This proves (b).

2.2L5L1L4step 1.1algebra

(Divergence form.) Applying [L5] to $J : = n$ gives $θ = \nabla_{a} n^{a} = w^{- 1} \partial_{a} (w n^{a})$ ; in particular $θ$ is smooth on $U$ , since the right-hand side is smooth. The trace identities follow from $h^{a b} = g^{a b} + n^{a} n^{b}$ and $n^{b} \nabla_{a} n_{b} = \frac{1}{2} \nabla_{a} (n \cdot n) = 0$ , the last by [L1] and [L4]: $h^{a b} \nabla_{a} n_{b} = g^{a b} \nabla_{a} n_{b} + n^{a} n^{b} \nabla_{a} n_{b} = θ + 0$ . This proves (c).

3.1L7algebra∎

(Derivation property.) For smooth $f, h$ , [L7] gives $D = n$ as a derivation on functions, so $D (f h) = n^{a} \nabla_{a} (f h) = n^{a} (f \nabla_{a} h + h \nabla_{a} f) = f D h + h D f$ ; for a constant function $c$ one has $\nabla_{a} c = 0$ , so $D c = 0$ . This proves (d).

Remarks

Dependence on the congruence only through $n$ and $\nabla$ . The acceleration and expansion are computed from $n$ and the metric-selected connection alone; no adapted chart, tetrad or foliation is required, and no global existence statement for such structures is used.
Choice convention. The smooth structure of the tangent bundle underlying $n$ assumes ${A C}_{ω}$ through Observer congruences, their observer curves and adapted charts; no other choice principle is used here.

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Divergence of the measured energy-momentum current and the finite-chain balance identity

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ , let $U \subseteq M$ be a nonempty open subset, let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ with $\nabla^{a} T_{a b} = 0$ on $U$ , and let $n$ be a smooth observer congruence on $U$ . Let $μ$ , $J$ , $Φ$ , $S$ be the fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress and let $a$ , $θ$ , $D$ be the kinematic fields of Acceleration, expansion and the along-congruence derivative of an observer congruence.

(a) Divergence identity. On $U$ ,

$\nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b} .$

(b) Balance identity. On $U$ ,

$D μ + μ θ + \nabla_{a} Φ^{a} = - T^{a b} \nabla_{a} n_{b} .$

(c) Finite-chain balance. For this clause choose an orientation on $U$ and use its metric volume form ${v o l}_{g ∣_{U}}$ , written ${v o l}_{g}$ . Let $C$ be a finite smooth singular $4$ -chain of $M$ contained in $U$ with induced boundary $\partial C = Σ_{2} - Σ_{1} + B$ , where $Σ_{1}, Σ_{2}, B$ are smooth singular $3$ -chains (The metric volume form, current flux through 3-faces and finite-chain charges clause (b)). Then

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

In particular, if $T^{a b} \nabla_{a} n_{b} = 0$ on an open subset containing the image of $C$ , then $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B)$ .

(d) Killing contraction. If $ξ$ is a smooth vector field on $U$ with $\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0$ (a Killing field, Killing fields of a Lorentzian metric), then $T^{a b} \nabla_{a} ξ_{b} = 0$ on $U$ .

Facts & Assumptions

Given: The time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , the open subset $U$ , the smooth symmetric covariant $2$ -tensor field $T$ with $\nabla^{a} T_{a b} = 0$ on $U$ , the smooth observer congruence $n$ on $U$ , the fields $μ$ , $J$ , $Φ$ , $S$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress, and the fields $a$ , $θ$ , $D$ of Acceleration, expansion and the along-congruence derivative of an observer congruence. In (c) additionally a chosen orientation on $U$ , its metric volume form, and the finite smooth singular $4$ -chain $C \subseteq U$ with boundary $\partial C = Σ_{2} - Σ_{1} + B$ ; in (d) additionally the vector field $ξ$ with $\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0$ .

[L1]

The extension of $\nabla$ to tensor fields satisfies the product rule, and $\nabla g = 0$ , $\nabla g^{- 1} = 0$ ; consequently raising and lowering indices commute with $\nabla$ , and for a smooth covariant $2$ -tensor field $T$ with $\nabla^{a} T_{a b} = 0$ one has $\nabla_{a} T^{a b} = 0$ (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, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, Metric compatibility, torsion freedom and the metric-selected connection class).

[L2]

The fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress satisfy $J = μ n + Φ$ and $μ = - J^{a} n_{a}$ , with $n_{a} Φ^{a} = 0$ (The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness clauses (a) and (b), The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress).

[L3]

On the oriented open submanifold $U$ with restricted metric and connection, for a smooth vector field $K$ on $U$ and a finite smooth singular $4$ -chain $C$ of $U$ , $Φ_{K} (\partial C) = \int_{C} (\nabla_{a} K^{a}) {v o l}_{g}$ ; consequently, if $\partial C = Σ_{2} - Σ_{1} + B$ , then $Φ_{K} (Σ_{2}) - Φ_{K} (Σ_{1}) = - Φ_{K} (B) + \int_{C} (\nabla_{a} K^{a}) {v o l}_{g}$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (b), The metric volume form, current flux through 3-faces and finite-chain charges clause (b)).

[L4]

$D f = n^{a} \nabla_{a} f$ for smooth $f$ , and $θ = \nabla_{a} n^{a}$ (Acceleration, expansion and the along-congruence derivative of an observer congruence clauses (b) and (c), Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative clause (c)).

Proof

technique · direct; differentiate the current, substitute the orthogonal decomposition of $T$, and apply the finite-chain flux balance to the current

1.1L1givenalgebra

(Divergence identity.) By the definition $J^{a} = - T^{a b} n_{b}$ and the product rule [L1], $\nabla_{a} J^{a} = - \nabla_{a} (T^{a b} n_{b}) = - (\nabla_{a} T^{a b}) n_{b} - T^{a b} \nabla_{a} n_{b}$ ; the hypothesis $\nabla^{a} T_{a b} = 0$ gives $\nabla_{a} T^{a b} = 0$ by [L1], so the first term vanishes and $\nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b}$ . This proves (a).

2.1step 1.1L2L4algebra

(Balance identity.) By [L2], $J = μ n + Φ$ ; hence $\nabla_{a} J^{a} = \nabla_{a} (μ n^{a}) + \nabla_{a} Φ^{a} = n^{a} \nabla_{a} μ + μ \nabla_{a} n^{a} + \nabla_{a} Φ^{a}$ , so by [L4] $\nabla_{a} J^{a} = D μ + μ θ + \nabla_{a} Φ^{a}$ . Substituting step 1.1 gives (b).

2.2step 1.1L3algebra

(Finite-chain balance.) A simplex of $M$ with image in $U$ is a simplex of $U$ : restrict its smooth extension to the open inverse image of $U$ , which contains its simplex domain. Apply [L3] on the oriented manifold $U$ , where $J$ is defined, to this chain $C$ . Then $Φ_{J} (\partial C) = \int_{C} (\nabla_{a} J^{a}) {v o l}_{g}$ , and the additivity of the flux over chains gives $Φ_{J} (\partial C) = Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) + Φ_{J} (B)$ . Substituting step 1.1 yields $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B) + \int_{C} (- T^{a b} \nabla_{a} n_{b}) {v o l}_{g}$ ; if the integrand vanishes on an open set containing the image of $C$ , the integral vanishes and the second assertion follows. This proves (c).

3.1givenalgebra∎

(Killing contraction.) Write $T^{a b} \nabla_{a} ξ_{b} = \frac{1}{2} T^{a b} (\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) + \frac{1}{2} T^{a b} (\nabla_{a} ξ_{b} - \nabla_{b} ξ_{a})$ . The first summand vanishes by the Killing equation. In the second, the symmetry $T^{a b} = T^{b a}$ and a relabelling $a \leftrightarrow b$ give $T^{a b} (\nabla_{a} ξ_{b} - \nabla_{b} ξ_{a}) = T^{a b} \nabla_{a} ξ_{b} - T^{b a} \nabla_{a} ξ_{b} = 0$ . Hence $T^{a b} \nabla_{a} ξ_{b} = 0$ . This proves (d).

Remarks

Source of the balance. The right-hand side $- T^{a b} \nabla_{a} n_{b}$ is a source term produced by the congruence: it vanishes identically for a parallel congruence ( $\nabla n = 0$ ), and for a normalized timelike Killing congruence it reduces to $- Φ^{a} a_{a}$ by Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current clause (c). The measured energy current alone therefore need not be conserved; the Killing-weighted current of that lemma is guaranteed to be conserved under its stated hypotheses.
No positivity or dynamics. The lemma uses only $\nabla^{a} T_{a b} = 0$ and the congruence property; no energy condition, no equation of state and no existence of a solution of the field equation is used or asserted.

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Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current

Statement

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\nabla$ , let $U \subseteq M$ be a nonempty open subset, and let $ξ$ be a smooth Killing field on $U$ (Killing fields of a Lorentzian metric) that is everywhere timelike and future-directed,

$g_{p} (ξ_{p}, ξ_{p}) < 0, g_{p} (ξ_{p}, N_{p}) < 0 for every p \in U .$

Put $X : = \sqrt{- g (ξ, ξ)} : U \to (0, \infty)$ (the positive square root) and $n : = ξ / X$ , so that $n$ is a smooth observer congruence on $U$ . Let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ with $\nabla^{a} T_{a b} = 0$ on $U$ , let $μ$ , $J$ , $Φ$ , $S$ be the fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress, and let $a$ , $θ$ , $D$ be the kinematic fields of Acceleration, expansion and the along-congruence derivative of an observer congruence. Then:

(a) Normalization is constant along the congruence, and the acceleration is a spatial gradient.

$n^{a} \nabla_{a} X = 0, a_{a} = \nabla_{a} \ln X = \frac{1}{X} \nabla_{a} X,$

equivalently $\nabla_{a} X = X a_{a}$ and $a = \nabla \ln X$ is the spatial gradient of $\ln X$ .

(b) Vanishing expansion and symmetric part. $θ = 0$ and

$\nabla_{a} n_{b} + \nabla_{b} n_{a} = - (n_{a} a_{b} + n_{b} a_{a}) .$

(c) The divergence source is the acceleration term. $T^{a b} \nabla_{a} n_{b} = J^{a} a_{a} = Φ^{a} a_{a}$ , and consequently

$\nabla_{a} J^{a} = - Φ^{a} a_{a} .$

(d) Conservation of the redshift-weighted current. On $U$ ,

$\nabla_{a} (X J^{a}) = 0, \nabla_{a} (T^{a b} ξ_{b}) = 0.$

For the following flux assertion choose an orientation on $U$ and the volume form of the restricted metric. Then for every finite smooth singular $4$ -chain $C$ of $M$ contained in $U$ with induced boundary $\partial C = Σ_{2} - Σ_{1} + B$ ,

$Φ_{X J} (Σ_{2}) - Φ_{X J} (Σ_{1}) = - Φ_{X J} (B),$

where $Φ_{X J}$ is the flux of the smooth vector field $X J$ (The metric volume form, current flux through 3-faces and finite-chain charges clause (b)).

Facts & Assumptions

Given: The time-oriented smooth Lorentzian four-manifold $(M, g, N)$ with metric-selected connection $\nabla$ , the open subset $U$ , the Killing field $ξ$ with $g (ξ, ξ) < 0$ and $g (ξ, N) < 0$ on $U$ , the functions $X = \sqrt{- g (ξ, ξ)}$ and the congruence $n = ξ / X$ , the smooth symmetric tensor field $T$ with $\nabla^{a} T_{a b} = 0$ , the fields $μ$ , $J$ , $Φ$ , $S$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress and the fields $a$ , $θ$ , $D$ of Acceleration, expansion and the along-congruence derivative of an observer congruence.

[L1]

$ξ$ is a Killing field exactly when $\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a} = 0$ ; equivalently $L_{ξ} g = 0$ (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (a)).

[L2]

$\nabla$ is torsion-free and $g$ -compatible, the extension of $\nabla$ to tensor fields satisfies the product rule, and $\nabla g = 0$ , $\nabla g^{- 1} = 0$ ; in particular indices may be raised and lowered across $\nabla$ , and for smooth $f > 0$ the usual product and quotient rules apply to $f$ -multiples of tensor fields (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (c), Metric compatibility, torsion freedom and the metric-selected connection class, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, The inverse metric, musical isomorphisms and index raising).

[L3]

$- g (ξ, ξ)$ is a positive smooth function on $U$ because $g$ and $ξ$ are smooth and $ξ$ is timelike there; hence $X = \sqrt{- g (ξ, ξ)}$ is smooth: on charts it is the composition of the positive smooth function $- g (ξ, ξ)$ with the square-root function $t \mapsto t^{1 / 2}$ , which is differentiable on $(0, \infty)$ with derivative $\frac{1}{2} t^{- 1 / 2}$ , and iterating the derivative formula exhibits smoothness (Real powers for positive bases, with the zero-base positive-exponent convention, Continuity and derivatives of positive-base real powers, Identity maps and composites of smooth maps are smooth, $C^{r}$ and smooth maps between smooth manifolds). Consequently $n = ξ / X$ is a smooth observer congruence on $U$ : $g (n, n) = g (ξ, ξ) / X^{2} = - 1$ and $g (n, N) = g (ξ, N) / X < 0$ (Observer congruences, their observer curves and adapted charts).

[L4]

The fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress satisfy $J^{a} = - T^{a b} n_{b}$ , $J = μ n + Φ$ and $n_{a} Φ^{a} = 0$ (The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness clauses (a) and (b)).

[L5]

$\nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b}$ and $T^{a b} \nabla_{a} ξ_{b} = 0$ (Divergence of the measured energy-momentum current and the finite-chain balance identity clauses (a) and (d)).

[L6]

$a = \nabla_{n} n$ , $θ = \nabla_{a} n^{a}$ , $D f = n^{a} \nabla_{a} f$ and $θ = h^{a b} \nabla_{a} n_{b}$ (Acceleration, expansion and the along-congruence derivative of an observer congruence, Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative clause (c)).

[L7]

On $U$ with a chosen orientation and its restricted metric and connection, for a smooth vector field $K$ on $U$ with $\nabla_{a} K^{a} = 0$ and a finite smooth singular $4$ -chain $C \subseteq U$ with $\partial C = Σ_{2} - Σ_{1} + B$ , one has $Φ_{K} (Σ_{2}) - Φ_{K} (Σ_{1}) = - Φ_{K} (B)$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (b), The metric volume form, current flux through 3-faces and finite-chain charges).

Proof

technique · direct; contract the Killing equation along $\xi$ to identify the gradient of the normalization and the acceleration, compute the symmetric part of $\nabla n$ and its trace, and integrate the resulting divergence-free current

1.1L1L2L3L6algebra

(Gradient of the normalization and the acceleration.) By [L1] and [L2], $ξ^{a} \nabla_{a} ξ_{b} = - ξ^{a} \nabla_{b} ξ_{a} = - \frac{1}{2} \nabla_{b} (ξ^{a} ξ_{a})$ ; since $ξ^{a} ξ_{a} = - X^{2}$ , this gives $ξ^{a} \nabla_{a} ξ_{b} = \frac{1}{2} \nabla_{b} (X^{2}) = X \nabla_{b} X$ . On one hand $n^{a} \nabla_{a} (X^{2}) = 2 X D X$ by definition of $D$ ; on the other, $n^{a} \nabla_{a} (X^{2}) = - 2 n^{a} ξ^{b} \nabla_{a} ξ_{b} = - \frac{2}{X} (ξ^{a} \nabla_{a} ξ_{b}) ξ^{b} = - 2 X D X$ by [L2] and the identity just proved. Hence $2 X D X = - 2 X D X$ , so $D X = n^{a} \nabla_{a} X = 0$ because $X > 0$ . Next, by the product rule [L2], $\nabla_{a} n_{b} = \frac{1}{X} \nabla_{a} ξ_{b} - \frac{ξ_{b}}{X^{2}} \nabla_{a} X$ ; contracting with $n^{a}$ and using $D X = 0$ gives $a_{b} = \frac{1}{X} n^{a} \nabla_{a} ξ_{b} = \frac{1}{X^{2}} ξ^{a} \nabla_{a} ξ_{b} = \frac{1}{X^{2}} X \nabla_{b} X = \nabla_{b} \ln X$ . This proves (a).

2.1L1step 1.1L2L6algebra

(Vanishing expansion and symmetric part.) Symmetrizing the product formula of step 1.1, $\nabla_{a} n_{b} + \nabla_{b} n_{a} = \frac{1}{X} (\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) - \frac{ξ_{b}}{X^{2}} \nabla_{a} X - \frac{ξ_{a}}{X^{2}} \nabla_{b} X = - (n_{b} \nabla_{a} \ln X + n_{a} \nabla_{b} \ln X) = - (n_{b} a_{a} + n_{a} a_{b})$ by [L1] and step 1.1. For the trace, $θ = \nabla_{a} n^{a} = \frac{1}{X} \nabla_{a} ξ^{a} - \frac{ξ^{a}}{X^{2}} \nabla_{a} X$ , where $\nabla_{a} ξ^{a} = \frac{1}{2} g^{a b} (\nabla_{a} ξ_{b} + \nabla_{b} ξ_{a}) = 0$ and $ξ^{a} \nabla_{a} X = X D X = 0$ ; hence $θ = 0$ . This proves (b).

3.1step 1.1step 2.1L4L5L6algebra

(The divergence source.) Using the product formula and step 1.1, $T^{a b} \nabla_{a} n_{b} = \frac{1}{X} T^{a b} \nabla_{a} ξ_{b} - T^{a b} n_{b} a_{a}$ ; the first term vanishes by [L5], and the second is $- (T^{a b} n_{b}) a_{a} = J^{a} a_{a}$ by [L4]. Since $a$ is spatial and $Φ = J - μ n$ , $J^{a} a_{a} = Φ^{a} a_{a} + μ n^{a} a_{a} = Φ^{a} a_{a}$ by [L6] and clause (a) of Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative. Substituting into [L5] gives $\nabla_{a} J^{a} = - Φ^{a} a_{a}$ . This proves (c).

4.1step 1.1step 3.1L4L2algebra

(Conservation of the weighted current.) By the product rule [L2] and step 1.1, $\nabla_{a} (X J^{a}) = X \nabla_{a} J^{a} + J^{a} \nabla_{a} X = X (- Φ^{a} a_{a}) + J^{a} X a_{a} = 0$ , because $J^{a} a_{a} = Φ^{a} a_{a}$ by step 3.1. Moreover $T^{a b} ξ_{b} = X T^{a b} n_{b} = - X J^{a}$ by [L4], so $\nabla_{a} (T^{a b} ξ_{b}) = - \nabla_{a} (X J^{a}) = 0$ . This proves the first part of (d).

5.1step 4.1L7∎

(Flux balance.) The vector field $X J$ is smooth on $U$ with $\nabla_{a} (X J^{a}) = 0$ by step 4.1, so applying [L7] on the oriented open submanifold $U$ to $K : = X J$ gives $Φ_{X J} (Σ_{2}) - Φ_{X J} (Σ_{1}) = - Φ_{X J} (B)$ for every finite smooth singular $4$ -chain $C \subseteq U$ with the stated boundary. This completes the proof of (d).

Remarks

Interpretation of the weight. $X = \sqrt{- g (ξ, ξ)}$ is the normalization of the Killing field, and $X J$ is the energy-momentum current of the congruence weighted by it; in a static model $X$ is the redshift factor between the congruence observers and the symmetry normalization. The unweighted current $J$ is not conserved in general: its divergence is $- Φ^{a} a_{a}$ , the acceleration source of clause (c).
Hypotheses. Only the Killing equation, the timelike direction of $ξ$ , and $\nabla^{a} T_{a b} = 0$ are used. No extension of $ξ$ or $T$ beyond $U$ , no completeness of the congruence and no asymptotic control of the chains is assumed or asserted.
Choice convention. The smooth structure of the tangent bundle underlying $ξ$ and $n$ assumes ${A C}_{ω}$ through Observer congruences, their observer curves and adapted charts; no other choice principle is used.

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The timelike Killing current and the charge of a three-chain

Physical scope

The charge associated with a timelike Killing symmetry of the smooth classical branch of the gravitational framework: the Killing current built from a smooth stress-energy tensor and its flux through finite smooth singular three-chains on a chosen oriented domain. It has an energy reading in joules when the Killing field is normalized with a length-valued flow parameter under the length-coordinate convention; other normalizations carry the corresponding scale in the charge's units. The item asserts no conservation, no finiteness for unbounded chains, no positivity and no measurement. With divergence-free T and Killing symmetry, the current lemma supplies conservation up to side flux; the field law is one possible source of that divergence condition. The definition applies only where the stated timelike Killing field exists.

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 $U \subseteq M$ be a nonempty open subset carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure, choose an orientation on $U$ and its metric volume form (independent of the time orientation), let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold) — in an application this is the stress-energy tensor field of the model's matter sector (Stress-energy tensor field of a model) — and let $ξ$ be a Killing field of $(M, g)$ on $U$ (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (a)) that is everywhere timelike and future-directed,

$g_{p} (ξ_{p}, ξ_{p}) < 0, g_{p} (ξ_{p}, N_{p}) < 0 for every p \in U .$

Put $X : = \sqrt{- g (ξ, ξ)} > 0$ on $U$ and $n : = ξ / X$ , an observer congruence on $U$ : $g (n, n) = - 1$ and $g (n, N) = g (ξ, N) / X < 0$ (Observer congruences, their observer curves and adapted charts).

(a) Killing current. The Killing current of the pair $(T, ξ)$ is the smooth vector field on $U$

$J_{ξ}^{a} : = T^{a b} ξ_{b} .$

Since $ξ = X n$ and $J^{a} = - T^{a b} n_{b}$ , one has $T^{a b} ξ_{b} = - X J^{a}$ , where $J = - T (\cdot, n)$ is the measured energy-momentum current of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress; no conservation of $J_{ξ}$ is asserted by this definition.

(b) Killing charge of a three-chain. For every finite smooth singular $3$ -chain $Σ$ of $M$ contained in $U$ , the Killing charge of $Σ$ with respect to $ξ$ is the real number

$Q_{ξ} (Σ) : = - Φ_{J_{ξ}} (Σ),$

where $Φ_{J_{ξ}}$ is the flux of the current $J_{ξ}$ through $3$ -chains on the oriented open submanifold $U$ in the sense of The metric volume form, current flux through 3-faces and finite-chain charges clause (b).

When $ξ$ is normalized with a length-valued flow parameter in the SI length-coordinate convention, this charge is read as a Killing energy.

(c) Relation to the measured energy current. Since $J_{ξ} = - X J$ ,

$Q_{ξ} (Σ) = Φ_{X J} (Σ),$

the flux of the redshift-weighted measured energy current $X J$ . For a $3$ -simplex $σ$ that is non-null with velocity basis $(v_{1}, v_{2}, v_{3})$ and adapted unit normal $ν^{*}$ (so that $(ν^{*}, v_{1}, v_{2}, v_{3})$ is positively oriented and $ε : = g (ν^{*}, ν^{*}) \in {- 1, 1}$ ), the pullback of $ι_{X J} {v o l}_{g}$ is $ε g (X J, ν^{*}) \sqrt{∣ \det (g (v_{i}, v_{j})) ∣} d t^{1} \land d t^{2} \land d t^{3}$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (c)); when $ν^{*} = - n$ and the $v_{i}$ span $n^{⊥}$ this coefficient equals $- X μ \sqrt{∣ \det (g (v_{i}, v_{j})) ∣}$ , so up to the orientation sign carried by the adapted normal, $Q_{ξ} (Σ)$ is the integral of the redshift-weighted measured energy density $X μ$ over $Σ$ . The factor $X$ carries the normalization and units of $ξ$ .

(d) Reading and scope. $Q_{ξ} (Σ)$ is the charge that the model assigns to the chain $Σ$ under the symmetry $ξ$ , in the framework of the postulates Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions with the assigned tensor and symmetry; it does not require the field law for its definition. Under the unit conventions of SI unit and coordinate conventions of the classical gravitational framework, if $T$ is read in $J m^{- 3}$ and the flow parameter of $ξ$ is measured in metres, then $X$ is dimensionless and $Q_{ξ} (Σ)$ is read in $J$ . More generally, if that parameter has unit $S$ , then in a length-coordinate chart $X$ has units $m / S$ and $Q_{ξ}$ has units $J m / S$ ; converting it to an energy requires an explicitly specified normalization factor. The value depends on the model, on $ξ$ (replacing $ξ$ by a dimensionless constant multiple $λ ξ$ with $λ > 0$ multiplies $X$ , $J_{ξ}$ and $Q_{ξ}$ by $λ$ ), and on the chain $Σ$ . The definition asserts no conservation, no positivity, no finiteness for unbounded chains and no measured value. If additionally $\nabla^{a} T_{a b} = 0$ on $U$ , the Killing current lemma gives conservation up to the side flux. The field law (Adopted Einstein field equation in the direct-equation presentation or Adopted classical gravitational action and variational field law) is one way to supply this divergence condition; it is not necessary for the mathematical conservation implication. If no timelike Killing field exists on $U$ , $Q_{ξ}$ is not defined by this item.

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Local energy balance of the measured current and conservation of the Killing charge in stationary models

Physical scope

The smooth classical branch of the adopted gravitational framework with the field law in one of its two presentations and a smooth matter stress-energy tensor on the model's matter domain: the local balance identity for the energy density, energy flux and energy-momentum current measured by a prescribed ideal-clock observer congruence, the finite-chain bookkeeping of that current, and, when the model admits an everywhere timelike future-directed Killing field, conservation of its associated Killing charge up to side flux. The charge is an energy in joules when its Killing field is length-normalized in the framework's length-coordinate convention; other normalizations carry their units through the charge. The theorem is a conditional consequence of the adopted field law and the stated smooth hypotheses; it asserts no existence of Killing fields, no convergence or finiteness for unbounded chains, no asymptotic or quasilocal mass, no positivity and no measured value, and it carries no empirical premise.

Statement

Adopt one of the two presentations of the framework's field law, Adopted Einstein field equation in the direct-equation presentation or Adopted classical gravitational action and variational field law, 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; the two presentations define one field law by Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling. Let $(M, g, N)$ be the model's smooth time-oriented Lorentzian four-manifold with an orientation, let $U \subseteq M$ be the matter domain with the induced orientation, let $T$ be the model's smooth symmetric stress-energy tensor field on $U$ (Stress-energy tensor field of a model), and let $n$ be a smooth observer congruence on $U$ , read as the field of directions of ideal-clock observers of the model (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework). Let $μ$ , $J$ , $Φ$ , $S$ be the measured fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress, with the physical reading and units of Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress: $Φ$ is the normalized spatial tensor current and its SI measured energy-flux vector is $q = c Φ$ , and let $a$ , $θ$ , $D$ be the kinematic fields of Acceleration, expansion and the along-congruence derivative of an observer congruence.

(a) On-shell conservation. The adopted field law implies

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

(b) Local energy balance. On $U$ ,

$D μ + μ θ + \nabla_{a} Φ^{a} = \nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b} .$

Thus the energy density measured by the congruence, its normalized spatial current $Φ$ (SI energy flux $q = c Φ$ ) and its expansion obey a local balance whose source is the congruence term $- T^{a b} \nabla_{a} n_{b}$ ; the measured energy current is conserved on an open set exactly where that source vanishes there.

(c) Finite-chain bookkeeping. For every finite smooth singular $4$ -chain $C$ of $M$ contained in $U$ with induced boundary $\partial C = Σ_{2} - Σ_{1} + B$ ,

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

In particular, if the source term vanishes on an open set containing the image of $C$ and the side flux $Φ_{J} (B)$ vanishes, then $Φ_{J} (Σ_{2}) = Φ_{J} (Σ_{1})$ : equal measured energy needs a vanishing side flux, not merely on-shell conservation.

(d) Stationary models and the Killing charge. Suppose in addition that $U$ carries a Killing field $ξ$ of $(M, g)$ that is everywhere timelike and future-directed, with $X : = \sqrt{- g (ξ, ξ)} > 0$ and $n = ξ / X$ , and let $Q_{ξ}$ be the Killing charge of The timelike Killing current and the charge of a three-chain. Then:

$\nabla_{a} (X J^{a}) = 0$ and $\nabla_{a} (T^{a b} ξ_{b}) = 0$ on $U$ .
For every finite smooth singular $4$ -chain $C \subseteq U$ with $\partial C = Σ_{2} - Σ_{1} + B$ ,

$Q_{ξ} (Σ_{2}) - Q_{ξ} (Σ_{1}) = - Q_{ξ} (B);$

in particular, if $Q_{ξ} (B) = 0$ then $Q_{ξ} (Σ_{2}) = Q_{ξ} (Σ_{1})$ : the model's Killing charge assigned to homologous slices is conserved up to the side flux, and is conserved when that flux vanishes. With a length-normalized $ξ$ , this charge is the model's energy in joules. 3. In this stationary case the measured energy current itself satisfies $\nabla_{a} J^{a} = - Φ^{a} a_{a}$ with $a = \nabla \ln X$ , so $\nabla_{a} J^{a} = 0$ on $U$ if and only if $Φ^{a} a_{a} = 0$ there.

(e) Qualification. The theorem asserts no existence of a timelike Killing field, no finiteness or convergence for chains outside the finite smooth singular class, no asymptotic or quasilocal mass, no positivity of $μ$ , and no measured value. Its conclusions are conditional on the adopted field law, the stated smoothness and the stated chains; the theorem uses no empirical premise, and any later use of a numerical energy must carry its own measurement, calibration and uncertainty qualifications.

Facts & Assumptions

Given: The adopted field law in one of its two presentations; a model with smooth $(M, g, N)$ , matter domain $U$ , smooth symmetric stress-energy tensor $T$ ; a smooth observer congruence $n$ on $U$ ; the measured fields $μ$ , $J$ , $Φ$ , $S$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress and the kinematic fields $a$ , $θ$ , $D$ of Acceleration, expansion and the along-congruence derivative of an observer congruence. In (d) additionally the timelike future-directed Killing field $ξ$ on $U$ with $X = \sqrt{- g (ξ, ξ)}$ and $n = ξ / X$ , and the Killing energy $Q_{ξ}$ of The timelike Killing current and the charge of a three-chain.

[L1]

The field equation holds on the matter domain $U$ in the direct presentation, and in the action presentation its equation form is that of Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling (for prescribed tensors the equation is adopted directly). Restrict the metric, time orientation and connection to the open manifold $U$ and apply Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid clause (a) with ambient manifold $U$ : it yields $\nabla^{a} T_{a b} = 0$ on $U$ , without an extension of $T$ beyond $U$ .

[L2]

On the stated hypotheses, $\nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b}$ , $D μ + μ θ + \nabla_{a} Φ^{a} = \nabla_{a} J^{a}$ , and for every finite smooth singular $4$ -chain $C \subseteq U$ with $\partial C = Σ_{2} - Σ_{1} + B$ , $Φ_{J} (Σ_{2}) - Φ_{J} (Σ_{1}) = - Φ_{J} (B) + \int_{C} (- T^{a b} \nabla_{a} n_{b}) {v o l}_{g}$ (Divergence of the measured energy-momentum current and the finite-chain balance identity clauses (a)-(c), Acceleration, expansion and the along-congruence derivative of an observer congruence).

[L3]

If $ξ$ is Killing and timelike on $U$ with $X = \sqrt{- g (ξ, ξ)}$ and $n = ξ / X$ , then $\nabla_{a} (X J^{a}) = 0$ , $\nabla_{a} (T^{a b} ξ_{b}) = 0$ , $T^{a b} \nabla_{a} n_{b} = Φ^{a} a_{a}$ and $\nabla_{a} J^{a} = - Φ^{a} a_{a}$ with $a = \nabla \ln X$ ; moreover $T^{a b} ξ_{b} = - X J^{a}$ and $Q_{ξ} (Σ) = - (Φ_{J_{ξ}}) (Σ) = Φ_{X J} (Σ)$ (Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current clauses (a)-(d), The timelike Killing current and the charge of a three-chain clauses (a)-(c)).

[L4]

On the oriented open submanifold $U$ with its restricted metric and connection, for a smooth vector field $K$ on $U$ with $\nabla_{a} K^{a} = 0$ and a finite smooth singular $4$ -chain $C \subseteq U$ with $\partial C = Σ_{2} - Σ_{1} + B$ , one has $Φ_{K} (Σ_{2}) - Φ_{K} (Σ_{1}) = - Φ_{K} (B)$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (b), The metric volume form, current flux through 3-faces and finite-chain charges).

Proof

technique · direct; instantiate the adopted field law's necessary conservation, apply the divergence and balance identities of the measured current, and use the conservation of the redshift-weighted Killing current for the stationary statement

1.1L1given

(On-shell conservation.) For the action sector with a Hilbert tensor, stationarity implies the field equation locally on $U$ : chart-supported test variations inside $U$ give the coefficient-vanishing argument of Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling. Such variations extend by zero away from their compact chart support, so they are among the adopted variations. The direct presentation, including prescribed-tensor sectors, adopts the same equation on $U$ . Apply the consistency theorem [L1] on the restricted spacetime $U$ to obtain $\nabla^{a} T_{a b} = 0$ on $U$ . This proves (a).

2.1step 1.1L2algebra

(Local balance and bookkeeping.) With $\nabla^{a} T_{a b} = 0$ now available, [L2] gives $\nabla_{a} J^{a} = - T^{a b} \nabla_{a} n_{b}$ and $D μ + μ θ + \nabla_{a} Φ^{a} = \nabla_{a} J^{a}$ on $U$ , which is the displayed identity of (b); moreover [L2] gives the finite-chain identity of (c), and the special case follows when the source integrand vanishes on an open set containing the image of $C$ and $Φ_{J} (B) = 0$ . This proves (b) and (c).

3.1step 2.1L3L4algebra

(Stationary conservation.) Assume now the additional hypotheses of (d). By [L3], $\nabla_{a} (X J^{a}) = 0$ and $\nabla_{a} (T^{a b} ξ_{b}) = 0$ on $U$ , which is (d)1. Applying (c) with the divergence-free field $X J$ in place of $J$ — the finite-chain flux balance [L4] on the oriented open manifold $U$ , where $X J$ is defined, with $K = X J$ — gives $Φ_{X J} (Σ_{2}) - Φ_{X J} (Σ_{1}) = - Φ_{X J} (B)$ ; since $Q_{ξ} = Φ_{X J}$ by [L3], this is $Q_{ξ} (Σ_{2}) - Q_{ξ} (Σ_{1}) = - Q_{ξ} (B)$ . If $Q_{ξ} (B) = 0$ , the charges are equal, and the equality $Q_{ξ} (B) = 0$ holds in particular when the flux of $X J$ through every simplex of the side chain vanishes. This proves (d)1 and (d)2.

4.1step 3.1L3algebra∎

(The measured current in the stationary case.) By [L3], $T^{a b} \nabla_{a} n_{b} = Φ^{a} a_{a}$ and $\nabla_{a} J^{a} = - Φ^{a} a_{a}$ with $a = \nabla \ln X$ ; hence $\nabla_{a} J^{a} = 0$ on $U$ if and only if $Φ^{a} a_{a} = 0$ there. This proves (d)3 and completes the proof.

Remarks

What is and is not empirical. No experiment, measurement, calibration or sampling enters the argument; the theorem is a conditional deduction from the adopted field law and the stated smooth geometry. The quantities $μ$ , $Φ$ , $S$ and $Q_{ξ}$ are read through Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress and The timelike Killing current and the charge of a three-chain; a later quantitative use of any of them must supply its own empirical premises and uncertainties.
Scope of the global statement. Clause (d)2 is a finite-chain statement. It does not assert that a total energy of an unbounded slice exists, nor that the limit of the side flux at infinity vanishes; those are additional hypotheses, recorded in Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses.
Choice convention. The smooth structure of the tangent bundle underlying $n$ and $ξ$ assumes ${A C}_{ω}$ through Observer congruences, their observer curves and adapted charts; no other choice principle is used.

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Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses

Physical scope

Scope and status note for energy questions of the classical gravitational framework: the observer dependence of measured energy density, the absence of any local gravitational energy density in the framework as built, the additional hypotheses a global energy requires, and the deferral of asymptotic masses. The note asserts no theorem about nature, no no-go theorem and no measured value.

Scope and status

This note fixes what the framework does and does not provide under the name "energy". It proves nothing by itself and asserts no fact about nature.

Observer dependence of measured energy. The measured energy density $μ = T_{a b} n^{a} n^{b}$ and the SI measured energy flux $q = c Φ$ of Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress depend on the observer congruence $n$ ; only the tensor $T$ itself is observer independent. For a prescribed model different congruences generally assign different energies to the same region, and the framework contains no congruence-independent local energy density of matter.
No local gravitational energy in the framework as built. The framework defines a stress-energy tensor only for the matter sector (Stress-energy tensor field of a model, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions): $T$ is the matter content of the field law, and the geometric side $G_{a b} + Λ g_{a b}$ has identically vanishing divergence (The second Bianchi identity and the vanishing divergence of the Einstein tensor). The local balance of Local energy balance of the measured current and conservation of the Killing charge in stationary models is a balance of the matter current; no item of the framework defines a local gravitational energy density or flux that would complete it into a conservation law for a total energy. The framework's normal-coordinate material (Point-normal and exponential coordinate charts of the metric-selected connection, Scope of normal coordinates and of the local-inertial-frame reading) records that the Christoffel symbols can be made to vanish at a point in a point-normal chart; it asserts nothing about second metric derivatives or curvature. Separately, the curvature operator $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors clause (a) is a tensor of type $(1, 3)$ by Tensoriality of the curvature, its coordinate formula and its point-normal metric formula clause (a). Thus if $R_{p} \neq 0$ , no chart can make all its components zero: zero components in a coordinate basis would imply, by multilinearity, that $R_{p} (u, v) w = 0$ for all $u, v, w \in T_{p} M$ . This tensorial observation does not establish a no-go theorem for gravitational energy. The normal-coordinate identities alone define no local gravitational energy density; the cited discussions of gravitational-energy localization provide context, not a no-go theorem proved or used here.
A global energy needs a symmetry and control of the flux. The Killing charge $Q_{ξ}$ of The timelike Killing current and the charge of a three-chain exists only for a model that admits an everywhere timelike future-directed Killing field $ξ$ ; the theorem is conditional on this field and supplies no existence result for any specified metric (Killing fields of a Lorentzian metric). Even with such a $ξ$ , the finite-chain statement of Local energy balance of the measured current and conservation of the Killing charge in stationary models gives $Q_{ξ} (Σ_{2}) - Q_{ξ} (Σ_{1}) = - Q_{ξ} (B)$ : conservation requires the side flux to vanish. It is read as an energy in joules when $ξ$ is length-normalized under the framework's length-coordinate convention; for another flow-parameter normalization, its units carry that scale. For a slice of unbounded spatial extent one further needs the limiting flux and the integrals to converge; the flux theorem Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current is stated for finite smooth singular chains and supplies no such limit. Smooth noncompact manifolds and open matter domains are allowed; it is integration over unbounded slices and passage to asymptotic limits that need additional convergence hypotheses. The scope note Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations excludes distributional sources and low-regularity fields, not noncompact smooth domains.
A Killing field need not be timelike everywhere. A Killing field that is timelike on one region may be spacelike or past-directed elsewhere, in which case the charge is not an energy there; the de Sitter example recorded in the cited Tong reference has no globally defined positive conserved energy of this kind. Nothing in the framework selects a preferred Killing field or normalization: the normalization is part of the model data, and changing $ξ$ changes $Q_{ξ}$ and can change its unit scale as recorded in The timelike Killing current and the charge of a three-chain clause (d).
Asymptotic and quasilocal masses are deferred. The ADM energy and related Bondi-type quantities are defined by surface integrals at infinity for asymptotically flat spacetimes, and Komar-type integrals by fluxes at a chosen boundary; both require an asymptotic or boundary structure (fall-off conditions, a sphere at infinity, limit arguments) or a chosen quasilocal surface. None of this structure is defined on this page or on the pages it depends on. Such masses are deferred to a future boundary-asymptotic module; no asymptotic mass, quasilocal mass or radiative energy loss is defined, computed or used in this framework here.
Empirical status. The note contains no experiment, no measured input and no empirical premise. It records the scope of the framework's energy statements and the obligations deferred to other pages; it does not assert that the framework's energy notions are the only possible ones, nor that the model describes nature.

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Scope and status of the local-and-global-energy page: what is defined, what is conditional, and what is deferred

Physical scope

Scope, convention and status note for the energy material of the classical gravitational framework: the objects defined on this page, the status of the balance and conservation claims, the smooth and finite-chain restrictions, the choice assumptions, and the obligations deferred to other pages. The note asserts no theorem, no measured value and no claim about nature.

Scope and status

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

What is defined. For a smooth symmetric covariant $2$ -tensor field $T$ and a smooth observer congruence $n$ , the page defines the measured energy density, energy-momentum current, energy flux and spatial stress (The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress), their physical reading and units (Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress), the acceleration and expansion of the congruence (Acceleration, expansion and the along-congruence derivative of an observer congruence), and, for a model with an everywhere timelike future-directed Killing field, the Killing current and the Killing charge of a finite three-chain (read as an energy in joules when the field is length-normalized under the framework's length-coordinate convention) (The timelike Killing current and the charge of a three-chain). The algebraic decomposition behind the measured fields is proved in The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness, and the divergence and finite-chain identities in Divergence of the measured energy-momentum current and the finite-chain balance identity and Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current.
What is conditional. The local energy balance and conservation of the Killing charge (whose energy reading requires the stated normalization) are established by theorem Local energy balance of the measured current and conservation of the Killing charge in stationary models: they are conditional on the adopted field law, on the smooth hypotheses 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 conservation statement, on the existence of the timelike Killing field and on the vanishing of the side flux. Nothing on the page establishes that a Killing field exists for a given metric (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor), and the finite-chain flux statements are those of The metric volume form, current flux through 3-faces and finite-chain charges and Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current.
Smooth and finite branch. The local constructions and identities use smooth fields on nonempty open domains, which may be noncompact; the flux statements integrate over finite smooth singular chains. This restricts the integration objects, not the ambient manifold or matter domain. Distributional sources, interfaces with nonsmooth fields and rough regions need the additional mathematics deferred in Scope and conventions of the Einstein-and-matter page: presentations, matter sectors and open obligations. Integration over unbounded slices and passage to asymptotic limits require additional convergence and boundary hypotheses not supplied by the finite-chain statements. The observer congruence is an idealization of a family of ideal-clock worldlines; the physical reading of measured quantities is the framework's reading of SI unit and coordinate conventions of the classical gravitational framework, and no numerical value is asserted by any item of the page.
Deferred obligations. The absence of a universal global energy, the observer dependence of measured energy and the deferral of ADM-, Bondi- and Komar-type masses to a future boundary-asymptotic module are recorded in Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses. In particular, no asymptotic energy, quasilocal mass, radiative loss or initial-data energy is defined or used here. Later pages of this framework may consume the redshift-weighted Killing current and the measured-energy identities for stationary and spherical models and for linearized waves; each such use must verify the hypotheses of the items it consumes and carry the item's scope qualifications with it.
Choice convention. The smooth-bundle structure used throughout the page assumes the axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) through Observer congruences, their observer curves and adapted charts (and The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress); no other choice principle is used, and no foundational item of this page introduces a choice principle of its own.
Empirical status. The page contains no experiment, no measured input and no empirical premise. Its physics claims are conditional deductions from the adopted postulates and stated hypotheses; any later comparison with reported observations tests the combined model and apparatus assumptions and is evidence rather than proof.

5 · Examples, experiments and counterexamples

None yet.

Local And Global Energy

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress

Definition

Remarks

The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness

Statement

Facts & Assumptions

Proof

Remarks

Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress

Definition

Acceleration, expansion and the along-congruence derivative of an observer congruence

Definition

Remarks

Spatiality and curve identity of the acceleration, the divergence form of the expansion, and the along-congruence derivative

Statement

Facts & Assumptions

Proof

Remarks

Divergence of the measured energy-momentum current and the finite-chain balance identity

Statement

Facts & Assumptions

Proof

Remarks

Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current

Statement

Facts & Assumptions

Proof

Remarks

The timelike Killing current and the charge of a three-chain

Definition

Local energy balance of the measured current and conservation of the Killing charge in stationary models

Statement

Facts & Assumptions

Proof

Remarks

Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses

Scope and status

Scope and status of the local-and-global-energy page: what is defined, what is conditional, and what is deferred

Scope and status

5 · Examples, experiments and counterexamples

Sources