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Forces and Continuum

1 · Prerequisites

2 · Summary

This page supplies the flat-spacetime force and continuum layer of the framework. It defines the symmetric stress-energy field class, its inertial component and scaling conventions, its four-force density $f^{b} = \partial_{a} T^{a b}$ and the exact conservation terminology, and it proves the pointwise observer decomposition of a symmetric field into energy density, momentum density and rest stress together with its uniqueness. The compact-support machinery (the fundamental lemma, integration by parts and differentiation under the integral sign) supplies the first variation of the adopted scalar-field action, its Euler–Lagrange equation and the Hilbert stress-energy tensor defined by compactly supported metric variations, including the full off-shell divergence identity. Dust and perfect-fluid models are analysed through exact conservation equivalences, and the finite-box balance theorem gives the flux form of conservation by the fundamental theorem of calculus, with explicit side-flux terms and the vanishing-side-flux corollary. The Hodge decomposition of an alternating two-form for an observer, its quadratic invariants and the optional electromagnetic identification are also supplied here for later use. On the physical side, the page adopts the scalar-field, dust and perfect-fluid models with their SI units and observer readings, the total-tensor isolation premise, and then proves conditional local conservation for the three models and integrated conservation of the slice four-momentum of an isolated system on finite boxes. Nothing here asserts gravity, electromagnetic coupling, a force law, an equation of state, energy conditions, an angular-momentum theorem, a four-vector transformation of slice integrals or any empirical fact.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Stress-energy tensor, force density and local balance on flat spacetime

Definition

Let $(A, V, η, C^{+})$ be the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), fix an independent numerical parameter $c > 0$ , and let $Ω \subseteq A$ be a nonempty open set.

(a) A stress-energy tensor on $Ω$ of class $C^{k}$ , $k \geq 0$ (with $C^{0}$ meaning continuous), is a $C^{k}$ symmetric type- $(2, 0)$ tensor field $T$ on $Ω$ in the sense of C^k tensor fields on affine Minkowski space and their inertial components: at every $p \in Ω$ the map $T (p) : V^{*} \times V^{*} \to R$ is bilinear and symmetric, so in every inertial chart its components satisfy $T^{a b} = T^{b a}$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii)).

(b) In each inertial chart $x$ of the model (Inertial frames and inertial charts of the affine Minkowski model) define the component families $ε_{x} : = T^{00}$ , $π_{x}^{i} : = T^{0 i} / c = T^{i 0} / c$ and $S_{x}^{i j} : = T^{i j}$ . These are numerical component and scaling conventions involving the fixed positive number $c$ ; their transformation behaviour is inherited from $T$ and no physical units or measurement interpretation form part of this mathematical definition.

(c) When $k \geq 1$ , the four-force density of $T$ is the $C^{k - 1}$ field whose inertial components are $f^{b} : = \partial_{a} T^{a b}$ , where $\partial_{a}$ are the coordinate partial derivatives of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ and the sum over $a$ is the standard contraction of a contravariant index with a coordinate derivative. The components $\partial_{a} T^{a b}$ transform as a vector, so $f$ is a well-defined vector field on $Ω$ independent of the inertial chart: under an inertial transition $x^{'} = Λ x + a$ the components satisfy $\partial_{a}^{'} T^{' a b} = Λ^{b}_{d} \partial_{c} T^{c d}$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iv)), which is exactly the vector transformation law of clause (i) applied to the components $f^{b}$ . For $k \geq 1$ one says that $T$ is locally conserved, or balanced, on $Ω$ when $f = 0$ on $Ω$ , and that $T$ is balanced against the four-force density $f$ when $\partial_{a} T^{a b} = f^{b}$ .

(d) The definition asserts no conservation of any prescribed symmetric field $T$ : whether a given $T$ is conserved, whether it arises from an action or a matter model, and whether any of its integrals over a slice converges are separate hypotheses stated elsewhere. No choice principle is used. For a merely continuous tensor ( $k = 0$ ) the component conventions and pointwise contractions of (b) are defined, while the divergence, the four-force density and the balance terminology of (c) require the additional $C^{1}$ hypothesis; no derivative of a merely continuous field is asserted.

Remarks

The symmetry condition in (a) is chart-independent by the rank-two component law (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii)): if $T^{a b} = T^{b a}$ in one inertial chart, then $T^{' a b} = Λ^{a}_{c} Λ^{b}_{d} T^{c d} = Λ^{b}_{d} Λ^{a}_{c} T^{d c} = T^{' b a}$ in every other. The constants $ε_{x}, π_{x}, S_{x}$ of (b) are numerical component conventions only; the physical readings of such components — energy density, momentum density, stress, force density — are model premises adopted separately in Adopted continuum matter models, evolution equations and observer readings and used in Local conservation of the adopted matter stress-energy tensors. Nothing in this definition requires the metric contraction $η_{a b} T^{a b}$ to have a sign, requires any energy condition, or asserts that a locally conserved field is the total stress-energy of an isolated system.

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Observer decomposition of a symmetric stress-energy tensor

Statement

Let $(A, V, η, C^{+})$ be the affine Minkowski model, $c > 0$ , $Ω \subseteq A$ open, and let $T$ be a continuous symmetric type- $(2, 0)$ tensor field on $Ω$ (Stress-energy tensor, force density and local balance on flat spacetime). Let $n \in V$ be future unit, $η (n, n) = - 1$ , with rest projection $h (v) = v + η (n, v) n$ and $h^{a}_{b} = δ^{a}_{b} + n^{a} n_{b}$ as in Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c). At a point $p \in Ω$ define $ε : = T^{a b} n_{a} n_{b}$ , $π^{a} : = - c^{- 1} h^{a}_{b} T^{b c} n_{c}$ and $S^{a b} : = h^{a}_{c} h^{b}_{d} T^{c d}$ . Then: (a) $π$ lies in $n^{⊥}$ , $S$ is symmetric and horizontal in both slots ( $S^{a b} n_{b} = n_{a} S^{a b} = 0$ ), and $T^{a b} = ε n^{a} n^{b} + c (n^{a} π^{b} + π^{a} n^{b}) + S^{a b} .$ (b) Contracting with $n$ gives $T^{a b} n_{b} = - ε n^{a} - c π^{a}$ ; hence $ε$ and $π$ are determined by $T$ and $n$ , and the decomposition is unique among decompositions $T^{a b} = ε^{'} n^{a} n^{b} + c (n^{a} π^{' b} + π^{' a} n^{b}) + S^{' a b}$ with $π^{'} \in n^{⊥}$ and $S^{'}$ symmetric horizontal: $ε^{'} = ε$ , $π^{'} = π$ , $S^{'} = S$ . (c) In any adapted inertial orthonormal frame with $n = e_{0}$ , the numerical components of this decomposition are $ε = T^{00}$ , $π^{0} = 0$ , $π^{i} = T^{0 i} / c$ and $S^{0 a} = S^{a 0} = 0$ , $S^{i j} = T^{i j}$ for $i, j = 1, 2, 3$ . Define isotropic rest stress to mean $S^{a b} = p h^{a b}$ for a real scalar $p$ , where $h^{a b} : = η^{a b} + n^{a} n^{b}$ . Thus $S = 0$ means vanishing rest stress, and is the isotropic case $p = 0$ ; isotropy permits any $p$ , including nonzero pressure or tension. These are numerical tensor identities. Physical adoption of the energy/momentum-density reading is a separate model premise; this mathematical lemma asserts no observer measurement. No choice principle is used.

Facts & Assumptions

Given: The affine Minkowski model with inertial chart $x$ , its constant metric components $η_{a b}$ , the inverse components $η^{a b}$ and the index conventions of Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space and Oriented time-oriented affine Minkowski space (the spacetime primitives); a continuous symmetric type- $(2, 0)$ field $T$ on $Ω$ (Stress-energy tensor, force density and local balance on flat spacetime, C^k tensor fields on affine Minkowski space and their inertial components); a future unit vector $n \in V$ , so $n_{a} n^{a} = η (n, n) = - 1$ , with $n^{⊥} = {v : η (n, v) = 0}$ and $h^{a}_{b} = δ^{a}_{b} + n^{a} n_{b}$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)). All components below are taken at one fixed point and the components $T^{a b}$ are those of $T$ in an inertial chart.

[F1]

Rest-space projector. The map $h (v) = v + η (n, v) n$ is linear and idempotent, has image $n^{⊥}$ and kernel $span (n)$ , satisfies $h (n) = 0$ and $η (n, h (v)) = 0$ for all $v$ ; in components $h^{a}_{b} = δ^{a}_{b} + n^{a} n_{b}$ , so $h^{a}_{b} n^{b} = 0 = n_{a} h^{a}_{b}$ and $h^{a}_{c} h^{c}_{b} = h^{a}_{b}$ . (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c))

[F2]

Component conventions and symmetry: $T^{a b} = T^{b a}$ , $δ^{a}_{b} = h^{a}_{b} - n^{a} n_{b}$ , and index lowering commutes with the algebra. (Stress-energy tensor, force density and local balance on flat spacetime)

Proof

technique · direct

1.1F1given

Projector identities. From [F1], $n_{a} n^{a} = - 1$ ; $h^{a}_{b} n^{b} = 0$ , $n_{a} h^{a}_{b} = 0$ and $h^{a}_{c} h^{c}_{b} = h^{a}_{b}$ ; moreover $δ^{a}_{c} = h^{a}_{c} - n^{a} n_{c}$ .

2.1F2step 1.1

Expansion and horizontality. Using $δ^{a}_{c} δ^{b}_{d} = (h^{a}_{c} - n^{a} n_{c}) (h^{b}_{d} - n^{b} n_{d})$ from step 1.1 and the symmetry $T^{c d} = T^{d c}$ of [F2], expand $T^{a b} = (h^{a}_{c} - n^{a} n_{c}) (h^{b}_{d} - n^{b} n_{d}) T^{c d} = S^{a b} - n^{b} h^{a}_{c} T^{c d} n_{d} - n^{a} h^{b}_{d} T^{d c} n_{c} + n^{a} n^{b} n_{c} n_{d} T^{c d};$ with the definitions of $ε$ , $π$ and $S$ this is $T^{a b} = S^{a b} + c n^{b} π^{a} + c n^{a} π^{b} + ε n^{a} n^{b}$ . Moreover $π \in n^{⊥}$ because $n_{a} π^{a} = - c^{- 1} n_{a} h^{a}_{b} T^{b c} n_{c} = 0$ , and $S$ is symmetric because $h^{a}_{c} h^{b}_{d} T^{c d} = h^{b}_{d} h^{a}_{c} T^{d c}$ ; finally $S^{a b} n_{b} = h^{a}_{c} h^{b}_{d} T^{c d} n_{b} = 0$ and $n_{a} S^{a b} = 0$ by step 1.1, so $S$ is horizontal in both slots. This proves (a).

2.2step 1.1given

Frame components (c). In an adapted inertial orthonormal frame with $n = e_{0}$ one has $n^{a} = δ_{0}^{a}$ and $n_{a} = - δ_{a}^{0}$ ; hence $h^{0}_{0} = 0$ , $h^{0}_{j} = h^{i}_{0} = 0$ and $h^{i}_{j} = δ_{j}^{i}$ for $i, j = 1, 2, 3$ . Substituting into the definitions gives $ε = T^{00}$ ; $π^{a} = - c^{- 1} h^{a}_{b} T^{b c} n_{c} = c^{- 1} h^{a}_{b} T^{b 0}$ , so $π^{0} = 0$ and $π^{i} = T^{i 0} / c = T^{0 i} / c$ ; and $S^{a b} = h^{a}_{c} h^{b}_{d} T^{c d}$ gives $S^{0 a} = S^{a 0} = 0$ and $S^{i j} = T^{i j}$ .

3.1step 1.1step 2.1given

Contraction and uniqueness (b). Contracting the expansion of step 2.1 with $n_{b}$ and using step 1.1, $T^{a b} n_{b} = ε n^{a} (n_{b} n^{b}) + c (n^{a} π^{b} n_{b} + π^{a} n_{b} n^{b}) + S^{a b} n_{b} = - ε n^{a} - c π^{a}$ . If $T^{a b} = ε^{'} n^{a} n^{b} + c (n^{a} π^{' b} + π^{' a} n^{b}) + S^{' a b}$ is another decomposition with $π^{'} \in n^{⊥}$ and $S^{'}$ symmetric horizontal, then contracting with $n_{a} n_{b}$ gives $ε^{'} = T^{a b} n_{a} n_{b} = ε$ , contracting with $n_{b}$ and comparing with the identity just proved gives $- c π^{' a} = - ε n^{a} - ε^{'} (- n^{a}) - c π^{a} = - c π^{a}$ , hence $π^{'} = π$ as $c > 0$ ; then $S^{'} = T - ε n \otimes n - c (n \otimes π + π \otimes n) = S$ .

4.1step 2.1step 2.2step 3.1∎

Clauses (a), (b) and (c) are proved in steps 2.1, 3.1 and 2.2. The isotropic rest stress $S^{a b} = p h^{a b}$ is a definition, with $p = 0$ exactly when $S = 0$ ; the identities hold at every point of $Ω$ and use only that $T$ is a continuous symmetric $(2, 0)$ field, while the physical reading of $ε$ , $π$ and $S$ as measured energy density, momentum density and stress is a separate model premise not asserted here. No choice principle is used.

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Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign

Statement

Work in an inertial chart $x$ of the affine Minkowski model, identifying a nonempty open $Ω \subseteq A$ with its image in $R^{4}$ ; all integrals are Riemann (Darboux) integrals. (a) If $g : Ω \to R$ is continuous and $\int_{Ω} g ψ d^{4} x = 0$ for every smooth compactly supported $ψ : Ω \to R$ , then $g = 0$ on $Ω$ . (b) If $f : Ω \to R$ is $C^{1}$ and compactly supported and $g : Ω \to R$ is $C^{1}$ , then for each $a$ , $\int_{Ω} (\partial_{a} f) g d^{4} x = - \int_{Ω} f (\partial_{a} g) d^{4} x$ , provided $g$ is $C^{1}$ on $Ω$ and the products are extended by zero outside $supp f$ . (c) Let $s_{0} > 0$ . If $F : (- s_{0}, s_{0}) \times Ω \to R$ is continuous with continuous $\partial_{s} F$ and there is a compact $K ⋐ Ω$ with $F (s, z) = 0$ for $z \notin K$ and all $s$ , then $s \mapsto \int_{Ω} F (s, z) d^{4} z$ is $C^{1}$ on $(- s_{0}, s_{0})$ with derivative $\int_{Ω} \partial_{s} F (s, z) d^{4} z$ . The lemma is choice-free and is stated for the exact Riemann integration theory on bounded rectangles; no distribution theory, no $L_{l o c}^{1}$ hypothesis and no choice principle is used.

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ with an inertial chart $x$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model), a nonempty open $Ω \subseteq A$ identified with its image $x (Ω) \subseteq R^{4}$ , and coordinates $x^{0}, x^{1}, x^{2}, x^{3}$ whose partial derivatives $\partial_{a}$ are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ . For a continuous $h : Ω \to R$ vanishing outside a compact subset of $Ω$ , the symbol $\int_{Ω} h d^{4} x$ denotes the compactly supported Riemann integral over $R^{4}$ of the zero extension $\tilde{h}$ (The support of a function on $R^{n}$ and its compactly supported Riemann integral); a closed nondegenerate rectangle is called a box.

[F1]

Compactness of the support and a bounding box: a compact set $K \subseteq Ω$ is contained in the interior of some box, and every box in $R^{4}$ with pairwise distinct side endpoints has positive volume. A finite union of boxes is compact and bounded. (The support of a function on $R^{n}$ and its compactly supported Riemann integral, The Riemann integral of a compactly supported function is independent of its bounding rectangle)

[F2]

If $\tilde{h}$ is Riemann integrable on one box whose interior contains $supp \tilde{h}$ , it is integrable on every such box and all the integrals agree. (The Riemann integral of a compactly supported function is independent of its bounding rectangle)

[F3]

The multidimensional integral is linear, monotone and satisfies $∣ \int_{Q} f ∣ \leq \int_{Q} ∣ f ∣$ ; cutting a box along a coordinate hyperplane that meets its interior splits the integral additively. (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ )

[F4]

Every continuous function on a closed nondegenerate rectangle is Riemann integrable. (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable)

[F5]

For a constant $c$ on a nondegenerate box $Q$ , every lower and upper Darboux sum equals $c vol (Q)$ , so $\int_{Q} c = c vol (Q)$ ; for a nondegenerate closed box $Q_{1} \subseteq Q$ , the zero-extended indicator of $Q_{1}$ is integrable on $Q$ with integral equal to the content of $Q_{1}$ , which is its volume. (The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ , A bounded set is Jordan measurable iff its indicator is Riemann integrable, and the integral is its Jordan content)

[F6]

If $A \subseteq R^{p}$ , $B \subseteq R^{q}$ are nondegenerate closed rectangles and $f : A \times B \to R$ is Riemann integrable, then the iterated integral in an order in which every section is integrable equals $\int_{A \times B} f$ . (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections)

[F7]

If $G : [u, v] \to R$ is differentiable at every point of $[u, v]$ as a function on that interval, then $\int_{u}^{v} G^{'} = G (v) - G (u)$ whenever $G^{'}$ is integrable on $[u, v]$ . (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ )

[F8]

The one-variable sum, scalar multiple and product rules hold for coordinate derivatives: $\partial_{a} (f g) = (\partial_{a} f) g + f (\partial_{a} g)$ for $C^{1}$ functions on $Ω$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ )

[F9]

For every compact $K$ inside an open $U \subseteq R^{4}$ there is a smooth $ρ : R^{4} \to [0, 1]$ with $ρ = 1$ on $K$ and $supp ρ \subseteq U$ . (A Euclidean bump for a compact set inside an open set)

[F10]

If $g, h : [u, v] \times [w_{1}, w_{2}] \to R$ are continuous and $x \mapsto g (x, t)$ is differentiable on $(u, v)$ with derivative $h (x, t)$ for every $t$ , then $G (x) = \int_{w_{1}}^{w_{2}} g (x, t) d t$ has $G^{'} (x) = \int_{w_{1}}^{w_{2}} h (x, t) d t$ . (Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral)

[F11]

A continuous function on a compact metric space is uniformly continuous. (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous)

[F12]

Sums, scalar multiples and products of continuous real-valued maps on a topological space are continuous where defined; in particular the product $g ψ$ of a continuous $g : Ω \to R$ and a smooth real-valued $ψ$ on $Ω$ is continuous, and the nonnegativity of a product of a nonnegative and a nonnegative function is pointwise. (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined)

[F13]

Closed boxes in $R^{4}$ are compact, and a subset is compact exactly when it is closed and bounded. (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line,(1),(2))

Proof

technique · direct

1.1givenF1F2F4F13

Zero extensions. If a continuous $u : Ω \to R$ vanishes outside a compact $K \subseteq Ω$ , its zero extension is continuous on $R^{4}$ : it is continuous at points of $Ω$ , and every point outside $Ω$ has a neighbourhood disjoint from the closed set $K$ , on which the extension is zero. Its support lies in $K$ ; [F1], [F2] and [F4] therefore define its finite integral independently of a bounding rectangle. The same argument gives a $C^{1}$ zero extension when $u$ is $C^{1}$ , with derivatives extended by zero; outside $K$ both $u$ and its derivatives vanish. For the parameter family in (c), zero extension of $F$ and $\partial_{s} F$ is jointly continuous on $(- s_{0}, s_{0}) \times R^{4}$ , and the extended parameter derivative is still $\partial_{s} F$ , because the common support $K$ gives the same zero neighbourhoods.

2.1givenF3F5F9F12F13step 1.1

Fundamental lemma. Suppose $g (p) \neq 0$ and set $σ = 1$ if $g (p) > 0$ , $σ = - 1$ otherwise. Continuity and openness give a nondegenerate closed box $Q \subseteq Ω$ and $a > 0$ with $σ g \geq a$ on $Q$ . Choose a smaller nondegenerate closed box $Q_{1} \subseteq int Q$ . By [F9] and [F13] there is a smooth $ρ : R^{4} \to [0, 1]$ , equal to $1$ on $Q_{1}$ , whose support is a compact subset of $int Q$ ; compactness follows because the support is closed and bounded. For $ψ = σ ρ ∣_{Ω}$ , the continuous product $g ψ$ is nonnegative, compactly supported, and at least $a$ on $Q_{1}$ . On a bounding rectangle, [F3] and [F5] give $\int_{Ω} g ψ \geq a vol (Q_{1}) > 0$ , contrary to the test hypothesis. Thus $g = 0$ , proving (a).

2.2givenF1F3F4F6F7F8step 1.1

Integration by parts. For (b), put $K = supp f$ . The function $f g$ is $C^{1}$ on $Ω$ , vanishes outside $K$ , and has a $C^{1}$ zero extension $u$ by step 1.1. Its derivative $\partial_{a} u$ is the zero extension of $(\partial_{a} f) g + f \partial_{a} g$ by [F8]. Choose a nondegenerate bounding rectangle with $K$ in its interior. On each line in the $a$ th coordinate, $u$ vanishes at both endpoints, so [F7] gives integral zero for $\partial_{a} u$ . By [F4] and [F6], integrating over the remaining coordinates gives $0 = \int_{Ω} (\partial_{a} f) g + \int_{Ω} f \partial_{a} g$ . This is (b).

2.3givenF2F4F6F10F11F13step 1.1

Parameter differentiation. For (c), fix a closed interval $[u, v] \subseteq (- s_{0}, s_{0})$ with $u < v$ and a nondegenerate rectangle $Q$ containing $K$ in its interior. Use the joint zero extensions from step 1.1 on $[u, v] \times Q$ . Integrating in one coordinate produces a jointly continuous function of $s$ and the remaining coordinates: its difference at two parameter tuples is bounded by that coordinate's interval length times the uniform oscillation of $F$ on the compact product, which tends to zero by [F11]. Apply [F10] with $s$ as parameter to differentiate this one-coordinate integral. The resulting derivative, the integral of $\partial_{s} F$ , is jointly continuous by the same bound and [F11]. Repeat this argument in each of the four coordinates. [F4] and [F6] identify the repeated integrals with the four-dimensional integrals; hence $G (s) : = \int_{Ω} F (s, z) d^{4} z$ is $C^{1}$ on $[u, v]$ and $G^{'} (s) = \int_{Ω} \partial_{s} F (s, z) d^{4} z$ .

3.1F9F13step 2.1step 2.2step 2.3∎

Every parameter in $(- s_{0}, s_{0})$ lies in the interior of such a closed interval, so step 2.3 proves (c) on the entire open interval. Steps 2.1 and 2.2 prove (a) and (b). All selections are finite, the bump supplier and [F13] are choice-free, and no distributional or local-integrability hypothesis is used.

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First variation of the scalar-field action and the Euler-Lagrange equation

Statement

Let $Ω \subseteq A$ be a nonempty open set with an inertial chart $x$ , let $c, K > 0$ be numerical constants, let $V : R \to R$ be smooth and let $φ : Ω \to R$ be a smooth real scalar field (C^k tensor fields on affine Minkowski space and their inertial components). For every smooth compactly supported $h : Ω \to R$ define the first variation $δ S (φ) [h] : = - \frac{K}{c} \int_{Ω} (η^{a b} \partial_{a} φ \partial_{b} h + V^{'} (φ) h) d^{4} x,$ a finite Riemann integral by compact support of $h$ (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(c)). Then (a) $δ S (φ) [h] = \frac{K}{c} \int_{Ω} (□ φ - V^{'} (φ)) h d^{4} x$ , where $□ φ : = η^{a b} \partial_{a} \partial_{b} φ$ ; (b) $δ S (φ) [h] = 0$ for every such $h$ if and only if $□ φ = V^{'} (φ)$ on $Ω$ . The equation in (b) is called the scalar-field equation of this model. The statement is the first variation of the action on compactly supported variations; it asserts nothing about convergence of $S$ on unbounded domains, no boundary term, no dynamics beyond the equivalence in (b) and no empirical fact. No choice principle is used. Precisely, with $L (ϕ) : = \frac{1}{2} η^{a b} \partial_{a} ϕ \partial_{b} ϕ + V (ϕ)$ , define the relative action $S_{h} (s) : = - (K / c) \int_{Ω} [L (φ + s h) - L (φ)] d^{4} x$ . Its integrand has support in $supp h$ , so it is defined for every real $s$ even when either background action diverges; $S_{h}' (0) = δ S (φ) [h]$ .

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ with a fixed inertial chart $x$ and its coordinate partial derivatives $\partial_{a}$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ), a nonempty open $Ω \subseteq A$ , numerical constants $c, K > 0$ , a smooth $V : R \to R$ , a smooth real scalar field $φ : Ω \to R$ (C^k tensor fields on affine Minkowski space and their inertial components), and smooth compactly supported real $h : Ω \to R$ . In the inertial chart the metric components are the constant matrices $η_{a b} = η^{a b} = diag (- 1, 1, 1, 1)$ , with $η^{a b} = η^{b a}$ (Inertial frames and inertial charts of the affine Minkowski model). We write $□ φ : = η^{a b} \partial_{a} \partial_{b} φ$ and $L (ϕ) : = \frac{1}{2} η^{a b} \partial_{a} ϕ \partial_{b} ϕ + V (ϕ)$ for a smooth scalar function $ϕ$ .

[F1]

Compactly supported differentiation under the integral sign: if $F : (- s_{0}, s_{0}) \times Ω \to R$ is continuous with continuous $\partial_{s} F$ and there is a compact $K ⋐ Ω$ with $F (s, z) = 0$ for $z \notin K$ and all $s$ , then $s \mapsto \int_{Ω} F (s, z) d^{4} z$ is $C^{1}$ on $(- s_{0}, s_{0})$ with derivative $\int_{Ω} \partial_{s} F (s, z) d^{4} z$ . (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(c))

[F2]

Compact-support integration by parts: if $f : Ω \to R$ is $C^{1}$ and compactly supported and $g : Ω \to R$ is $C^{1}$ , then for each $a$ , $\int_{Ω} (\partial_{a} f) g d^{4} x = - \int_{Ω} f (\partial_{a} g) d^{4} x$ . (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(b))

[F3]

Fundamental lemma: if $g : Ω \to R$ is continuous and $\int_{Ω} g ψ d^{4} x = 0$ for every smooth compactly supported $ψ : Ω \to R$ , then $g = 0$ on $Ω$ . (Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign,(a))

[F4]

Chain rule for total derivatives: if $f$ is totally differentiable at $a$ and $g$ at $f (a)$ , then $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ . (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

[F5]

Real one-variable sum, scalar multiple and product rules for derivatives. (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ )

[F6]

If a map is totally differentiable at a point, then all its directional derivatives exist there and equal the total derivative evaluated on the direction; in particular coordinate partial derivatives are the entries of the Jacobian. (A total derivative computes every directional derivative, and its matrix is the Jacobian)

[F7]

Sums, scalar multiples and products of continuous real-valued maps on a topological space are continuous where defined, so every finite polynomial expression in continuous real-valued maps is continuous; a composite of continuous maps is continuous, and pulling a continuous real-valued function back along the coordinate projection $(s, z) \mapsto z$ preserves continuity, both immediately from the definition of continuity through preimages of open sets and from the descriptions of the product and subspace topologies. Consequently the maps $(s, z) \mapsto \partial_{a} φ (z)$ , $(s, z) \mapsto \partial_{a} h (z)$ , $(s, z) \mapsto h (z)$ , $(s, z) \mapsto s$ , $(s, z) \mapsto φ (z) + s h (z)$ and the composites $V (φ (z) + s h (z))$ and $V^{'} (φ (z) + s h (z))$ are continuous on $R \times Ω$ . (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, Continuity of a map of topological spaces at a point and globally, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ )

Proof

technique · direct

1.1givenF4F5F6F7

Fix a smooth compactly supported $h$ . For every real $s$ and every $z \in Ω$ set $Φ_{s} (z) : = L (φ + s h) (z) - L (φ) (z)$ and $F (s, z) : = - (K / c) Φ_{s} (z)$ . Differentiating the real function $s \mapsto L (φ + s h) (z)$ with the sum, scalar-multiple and product rules [F5], the chain rule [F4] and the Jacobian identification [F6] gives $\partial_{s} Φ_{s} (z) = η^{a b} \partial_{a} (φ + s h) (z) \partial_{b} h (z) + V^{'} (φ (z) + s h (z)) h (z),$ and the right-hand side equals $η^{a b} \partial_{a} φ (z) \partial_{b} h (z) + V^{'} (φ (z)) h (z)$ at $s = 0$ . Both $Φ_{s}$ and $\partial_{s} Φ_{s}$ are continuous on $R \times Ω$ : each is a finite sum of products of the continuous maps $(s, z) \mapsto \partial_{a} φ (z)$ , $(s, z) \mapsto \partial_{a} h (z)$ , $(s, z) \mapsto h (z)$ , $(s, z) \mapsto s$ , $(s, z) \mapsto φ (z) + s h (z)$ and of the composites of the last of these with the continuous one-variable functions $V$ and $V^{'}$ , and [F7] gives continuity of all these sums, products and composites; both functions also vanish at every $z \notin supp h$ , because $φ + s h = φ$ there.

2.1F1step 1.1

Apply [F1] to $F$ on $(- 1, 1) \times Ω$ : the relative action $S_{h} (s) = - (K / c) \int_{Ω} Φ_{s} d^{4} x$ is $C^{1}$ on $(- 1, 1)$ with $S_{h}^{'} (0) = - (K / c) \int_{Ω} \partial_{s} Φ_{0} d^{4} x = δ S (φ) [h]$ , using the value of $\partial_{s} Φ_{s}$ at $s = 0$ from step 1.1; the integrand of $S_{h} (s)$ is supported in $supp h$ for every $s$ , so $S_{h} (s)$ is a finite Riemann integral for every real $s$ and $S_{h}^{'} (0) = δ S (φ) [h]$ .

3.1F2step 2.1given

Claim (a): [F2] with $f = h$ and $g = η^{a b} \partial_{b} φ$ (here $η^{a b}$ is constant in the inertial chart) gives $\int_{Ω} η^{a b} \partial_{a} φ \partial_{b} h d^{4} x = - \int_{Ω} h η^{a b} \partial_{a} \partial_{b} φ d^{4} x = - \int_{Ω} (□ φ) h d^{4} x$ . Substituting into the definition of $δ S (φ) [h]$ from step 2.1 yields $δ S (φ) [h] = \frac{K}{c} \int_{Ω} (□ φ - V^{'} (φ)) h d^{4} x$ .

4.1F3step 3.1

Claim (b), forward direction: assume $δ S (φ) [h] = 0$ for every smooth compactly supported $h$ . By step 3.1 the continuous function $g : = \frac{K}{c} (□ φ - V^{'} (φ))$ satisfies $\int_{Ω} g ψ d^{4} x = 0$ for every smooth compactly supported $ψ$ ; [F3] gives $g = 0$ , i.e. $□ φ = V^{'} (φ)$ on $Ω$ .

4.2step 3.1

Claim (b), reverse direction: if $□ φ = V^{'} (φ)$ on $Ω$ , then for every smooth compactly supported $h$ step 3.1 gives $δ S (φ) [h] = \frac{K}{c} \int_{Ω} (□ φ - V^{'} (φ)) h d^{4} x = 0$ , because the integrand is the zero function.

5.1step 2.1step 3.1step 4.1step 4.2∎

Step 3.1 proves (a), steps 4.1 and 4.2 prove the two directions of (b), and step 2.1 identifies the first variation as the derivative of the relative action at $0$ . No background action on $Ω$ need converge, no boundary term and no choice principle is used; the displayed equation is an equivalence on $Ω$ and asserts no dynamics.

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

The Hilbert stress-energy tensor of the scalar-field action on flat spacetime

Statement

Let $Ω \subseteq A$ be any nonempty open set with inertial chart $x$ , let $c, K > 0$ be numerical constants, and let $V : R \to R$ and $φ : Ω \to R$ be smooth. For each smooth symmetric $(0, 2)$ tensor $h$ with compact support, set $g_{a b} (s) = η_{a b} + s h_{a b}$ . Choose $s_{0} > 0$ so that these matrices are nondegenerate with negative determinant for $∣ s ∣ < s_{0}$ ; outside $supp h$ they equal $η$ . Put $D_{h} (s, x) : = [\frac{1}{2} g^{a b} (s, x) \partial_{a} φ (x) \partial_{b} φ (x) + V (φ (x))] \sqrt{- \det g (s, x)}$ and define the relative action $A_{h} (s) : = - \frac{K}{c} \int_{Ω} [D_{h} (s, x) - D_{h} (0, x)] d^{4} x .$ This is the integral of a compactly supported density difference, without assuming either background density has a finite integral. (a) $A_{h}$ is differentiable at $0$ and $A_{h}' (0) = \frac{1}{2 c} \int_{Ω} T^{a b} h_{a b} d^{4} x, T^{a b} : = K [\partial^{a} φ \partial^{b} φ - η^{a b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ))],$ with indices raised by $η$ . (b) $T$ is smooth and symmetric, and is the unique continuous symmetric $(2, 0)$ field for which $\int_{Ω} T^{a b} h_{a b} d^{4} x = 2 c A_{h}' (0)$ for every such $h$ . It is called the Hilbert stress-energy tensor of this scalar-field density. (c) The result holds on unrestricted open $Ω$ , uses only compactly supported metric variations, and imposes no convergence or boundary condition at infinity. If the full background action happens to exist, the relative action is its variation difference and gives the same derivative. No choice principle is used.

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ with a fixed inertial chart $x$ and coordinate partial derivatives $\partial_{a}$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ), a nonempty open $Ω \subseteq A$ , constants $c, K > 0$ , smooth $V : R \to R$ and smooth $φ : Ω \to R$ (C^k tensor fields on affine Minkowski space and their inertial components). In the chart the metric is the constant matrix $η = diag (- 1, 1, 1, 1)$ with inverse $η^{a b} = η^{b a} = diag (- 1, 1, 1, 1)$ and $\det η = - 1$ ; for a smooth symmetric $(0, 2)$ field $h$ with compact support we write $h^{a b} : = η^{a c} η^{b d} h_{c d}$ and raise index pairs of $\partial_{a} φ$ by $η$ , so $\partial^{a} φ = η^{a b} \partial_{b} φ$ and $\partial^{a} φ \partial^{b} φ h_{a b} = h^{a b} \partial_{a} φ \partial_{b} φ$ .

[F1]

[F2]

[F3]

The Leibniz determinant is column-multilinear, alternating and normalized; in particular $\det (η) = - 1$ and, with $A$ invertible, $\det (A^{- 1}) = \det (A)^{- 1}$ . (The Leibniz determinant is column-multilinear, alternating and normalized over every commutative ring, If $A$ is invertible over a commutative ring, then $\det (A^{- 1}) = \det (A)^{- 1}$ )

[F4]

Adjugate identity and Cramer's rule: $A adj (A) = adj (A) A = \det (A) I_{n}$ , and if $\det (A)$ is a unit then $A^{- 1} = \det (A)^{- 1} adj (A)$ ; the adjugate is the transpose of the cofactor matrix and its entries are the cofactors of $A$ . (For every positive-sized square matrix over a commutative ring, $A adj (A) = adj (A) A = \det (A) I$ , If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ , Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring)

[F5]

Real one-variable derivative rules: sum, scalar multiple and product rules; and for $α \in R$ the function $x \mapsto x^{α}$ is continuous and differentiable on $(0, \infty)$ with derivative $α x^{α - 1}$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Rational powers $a^{r}$ of a positive base, Continuity and derivatives of positive-base real powers)

[F6]

Chain rule for total derivatives. (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

[F7]

Sums, scalar multiples, products and, where the denominator is nowhere zero, quotients of continuous real-valued maps on a topological space are continuous; hence every finite polynomial expression in continuous real-valued maps is continuous. A composite of continuous maps is continuous, and pulling a continuous real-valued function back along the coordinate projection $(s, x) \mapsto x$ preserves continuity; both assertions follow immediately from the definition of continuity through preimages of open sets and from the descriptions of the product and subspace topologies. Consequently the maps $(s, x) \mapsto \det g (s, x)$ , $(s, x) \mapsto adj (g (s, x))^{a b}$ and $(s, x) \mapsto g^{a b} (s, x)$ , being finite polynomial expressions in the continuous entries $g_{a b} (s, x) = η_{a b} + s h_{a b} (x)$ with the nowhere-zero denominator $\det g$ , are continuous on $(- s_{0}, s_{0}) \times Ω$ , and with the continuous $\sqrt{\cdot} : (0, \infty) \to R$ the function $D_{h}$ of step 1.1 is continuous there. (Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined, Continuity of a map of topological spaces at a point and globally, For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ )

[F8]

Matrix addition, scalar multiplication, products and transposes obey the usual associative, distributive and identity laws. (Matrix arithmetic over a commutative ring is associative, unital and distributive, and transpose reverses products)

Proof

technique · direct

1.1givenF3F4F7

Fix a smooth symmetric compactly supported $(0, 2)$ field $h$ and set $g_{a b} (s, x) : = η_{a b} + s h_{a b} (x)$ for $s \in R$ , $x \in Ω$ . Since $h_{a b} = 0$ off $supp h$ , one has $g (s, x) = η$ off $supp h$ . The function $(s, x) \mapsto \det g (s, x)$ is, by [F3], a polynomial in $s$ with coefficients continuous in $x$ and equals $- 1$ at $s = 0$ ; for each $x$ continuity in $(s, x^{'})$ at $(0, x)$ yields $ε (x) > 0$ and a neighbourhood $U (x)$ of $x$ with $\det g (s, x^{'}) < 0$ for $∣ s ∣ < ε (x)$ , $x^{'} \in U (x)$ , and compactness of $supp h$ gives finitely many such sets and a common $s_{0} > 0$ with $\det g (s, x) < 0$ for all $∣ s ∣ < s_{0}$ , $x \in Ω$ . Hence for $∣ s ∣ < s_{0}$ the matrix $g (s, x)$ is invertible, its inverse entries are $g^{a b} (s, x) = \det (g (s, x))^{- 1} adj (g (s, x))^{a b}$ by [F4], and $D_{h} (s, x) : = [\frac{1}{2} g^{a b} (s, x) \partial_{a} φ \partial_{b} φ + V (φ)] \sqrt{- \det g (s, x)}$ is defined; the maps $(s, x) \mapsto g^{a b} (s, x)$ and $(s, x) \mapsto \det g (s, x)$ , being polynomials in the continuous entries and quotients with nowhere-vanishing denominator $\det g$ , are continuous on $(- s_{0}, s_{0}) \times Ω$ by [F7], and so are $\sqrt{- \det g (s, x)}$ by [F5] and $D_{h} (s, x)$ by [F7], and $D_{h} (s, x) = D_{h} (0, x)$ whenever $h_{a b} (x) = 0$ , in particular off $supp h$ .

1.2givenF3F4F5F6F8

Derivative identities at $s = 0$ . (i) Differentiating the components of the identity $g (s, x) g (s, x)^{- 1} = I$ with respect to $s$ at $s = 0$ by [F5] and [F8], and using $g_{a b} (0) = η_{a b}$ , $g^{a b} (0) = η^{a b}$ and $η_{a c} η^{c b} = δ_{a}^{b}$ , gives $\partial_{s} g^{a b} (0, x) = - h^{a b} (x)$ , where $h^{a b} : = η^{a c} η^{b d} h_{c d}$ . (ii) Column multilinearity [F3] expands $\det (η + s h) = \det η + s Σ + s^{2} (\dots)$ with $Σ : = \sum_{j} \det (η with its j -th column replaced by the j -th column of h)$ ; expanding each such determinant along the replaced column shows $Σ = \sum_{i, j} h_{i j} C_{i j} (η) = tr (adj (η) h)$ , and by [F4] $adj (η) = \det (η) η^{- 1} = - η^{- 1}$ , so $Σ = - tr (η^{- 1} h) = - η^{a b} h_{a b}$ , because $tr (η^{- 1} h) = η^{a c} h_{c a} = η^{a b} h_{a b}$ by symmetry. (iii) Hence $\frac{d}{d s} \det g (0, x) = - η^{a b} h_{a b} (x)$ , and since $- \det g (0, x) = 1$ the chain rule [F6] with the power rule [F5] gives $\frac{d}{d s} \sqrt{- \det g (s, x)} ∣_{s = 0} = \frac{1}{2} η^{a b} h_{a b} (x)$ .

1.3givenF5F6

The field $T^{a b} : = K [\partial^{a} φ \partial^{b} φ - η^{a b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ))]$ is smooth on $Ω$ by iterating the product and chain rules [F5], [F6], because $φ$ and $V$ are smooth, $V (φ)$ is a smooth composition, and $η^{a b}$ is constant; it is symmetric, because $η^{a b} = η^{b a}$ and $\partial^{a} φ \partial^{b} φ = \partial^{b} φ \partial^{a} φ$ .

2.1F1F5F6step 1.1step 1.2step 1.3

Apply [F1] to $F (s, x) : = - \frac{K}{c} [D_{h} (s, x) - D_{h} (0, x)]$ on $(- s_{0}, s_{0}) \times Ω$ : step 1.1 shows that $F$ is continuous with the stated compact support in $x$ for every $s$ , and $\partial_{s} F$ is continuous by the same arguments applied to the derivative of $D_{h}$ ; hence $A_{h}$ is $C^{1}$ and $A_{h}^{'} (0) = - \frac{K}{c} \int_{Ω} \partial_{s} D_{h} (0, x) d^{4} x$ . By the product and chain rules [F5], [F6] and step 1.2, $\partial_{s} D_{h} (0, x) = - \frac{1}{2} h^{a b} \partial_{a} φ \partial_{b} φ + \frac{1}{2} η^{a b} h_{a b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ)) .$ Therefore $A_{h}^{'} (0) = \frac{K}{2 c} \int_{Ω} [h^{a b} \partial_{a} φ \partial_{b} φ - η^{a b} h_{a b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ))] d^{4} x = \frac{1}{2 c} \int_{Ω} T^{a b} h_{a b} d^{4} x$ , since $h^{a b} \partial_{a} φ \partial_{b} φ = \partial^{a} φ \partial^{b} φ h_{a b}$ .

3.1F2step 1.3step 2.1

Uniqueness in (b). Let $Q$ be a continuous symmetric $(2, 0)$ field with $\int_{Ω} Q^{a b} h_{a b} d^{4} x = 0$ for every smooth symmetric compactly supported $h$ . Fix index pairs and a smooth compactly supported $ψ$ : for $a = b$ take $h_{a a} : = ψ$ ; for $a \neq b$ take $h_{a b} : = h_{b a} : = ψ$ and all other components zero. In the first case $\int_{Ω} Q^{a a} ψ d^{4} x = 0$ , in the second $\int_{Ω} 2 Q^{a b} ψ d^{4} x = 0$ by symmetry of $Q$ ; in both cases the continuous function $Q^{a b}$ pairs to zero against every smooth compactly supported $ψ$ , so $Q^{a b} = 0$ on $Ω$ by [F2]. All sixteen components vanish, hence $Q = 0$ . Applying this to the difference of two continuous symmetric fields with the same pairing against every $h$ shows that the field $T$ of steps 1.3 and 2.1 is the unique such field satisfying $\int_{Ω} T^{a b} h_{a b} d^{4} x = 2 c A_{h}^{'} (0)$ .

4.1step 1.3step 2.1step 3.1∎

Claim (a) is step 2.1, claim (b) is steps 1.3, 2.1 and 3.1, and claim (c) is a statement about the hypotheses used: the argument allows any nonempty open $Ω$ , integrates only the compactly supported density difference $D_{h} (s, \cdot) - D_{h} (0, \cdot)$ (identically zero off $supp h$ ), requires no convergence of $D_{h} (0, \cdot)$ on $Ω$ and invokes no boundary condition; if the background action $\int_{Ω} D_{h} (0, x) d^{4} x$ converges, $A_{h}$ is the difference of the two actions and yields the same derivative. Every choice made is finite; no choice principle is used.

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

Divergence identity for the scalar-field stress-energy tensor

Statement

In the setting of The Hilbert stress-energy tensor of the scalar-field action on flat spacetime, with $φ$ smooth and $T$ the Hilbert stress-energy tensor of the scalar-field action, $\partial_{a} T^{a b} = K (□ φ - V^{'} (φ)) \partial^{b} φ on Ω,$ where $\partial^{b} = η^{b c} \partial_{c}$ and $□ φ = η^{a b} \partial_{a} \partial_{b} φ$ . In particular every solution of the scalar-field equation $□ φ = V' (φ)$ has conserved tensor, $\partial_{a} T^{a b} = 0$ . Conversely conservation implies the scalar-field equation at every point where $d φ \neq 0$ , and hence throughout an open set on which $d φ$ is nowhere zero. In full generality conservation is equivalent to the pointwise product condition $(□ φ - V' (φ)) d φ = 0$ ; it need not imply the scalar equation where $d φ = 0$ . Off shell the balance holds with the four-force density $f^{b} = K (□ φ - V^{'} (φ)) \partial^{b} φ$ . No choice principle is used.

Facts & Assumptions

Given: The setting of The Hilbert stress-energy tensor of the scalar-field action on flat spacetime: the affine Minkowski model with inertial chart $x$ , constant metric components $η_{a b} = η^{a b} = diag (- 1, 1, 1, 1)$ , smooth $V : R \to R$ and smooth $φ : Ω \to R$ on an open $Ω$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, C^k tensor fields on affine Minkowski space and their inertial components), and the field $T^{a b} = K [\partial^{a} φ \partial^{b} φ - η^{a b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ))]$ with $\partial^{a} φ = η^{a b} \partial_{b} φ$ and $□ φ = η^{a b} \partial_{a} \partial_{b} φ$ . The force density of a $C^{1}$ symmetric field is $f^{b} = \partial_{a} T^{a b}$ (Stress-energy tensor, force density and local balance on flat spacetime), and only the coordinate partial derivatives $\partial_{a}$ of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ are used.

[F1]

The Hilbert tensor of the scalar-field density is the field displayed in the Given, and it is smooth and symmetric. (The Hilbert stress-energy tensor of the scalar-field action on flat spacetime)

[F2]

Product, sum and constant-multiple rules for coordinate partial derivatives: $\partial_{a} (f g) = (\partial_{a} f) g + f (\partial_{a} g)$ , and $\partial_{a} (η^{a b} Φ) = η^{a b} \partial_{a} Φ$ for constant $η^{a b}$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ )

[F3]

Young's theorem: if the first partials of a function defined on a disk about a point exist there and are totally differentiable at the point, then the two mixed second partials at the point agree. In particular, for a smooth function all mixed second partials commute at every point. (Young's theorem: total differentiability of the first partials forces equality of mixed partials)

[F4]

The scalar-field equation of this model is $□ φ = V^{'} (φ)$ on $Ω$ . (First variation of the scalar-field action and the Euler-Lagrange equation)

Proof

technique · direct

1.1F1F2

Differentiating the Hilbert tensor. By [F1] and [F2], using that the components $η^{a b}$ are constant, $\partial_{a} T^{a b} = K [\partial_{a} (\partial^{a} φ \partial^{b} φ) - \partial^{b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ))] = K [(□ φ) \partial^{b} φ + \partial^{a} φ \partial_{a} \partial^{b} φ - \partial^{b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ + V (φ))] .$

2.1F2F3step 1.1

Cancellation of the second-order terms. For a smooth $φ$ the two second-order terms of step 1.1 coincide: the first is $X^{b} : = \partial^{a} φ \partial_{a} \partial^{b} φ = η^{a d} η^{b c} \partial_{d} φ \partial_{a} \partial_{c} φ$ , while expanding the derivative of the kinetic term by [F2] gives $\partial^{b} (\frac{1}{2} \partial_{d} φ \partial^{d} φ) = η^{d e} \partial_{d} φ \partial^{b} \partial_{e} φ = : Y^{b}$ and adding $V^{'} (φ) \partial^{b} φ$ for the potential term; after substituting $\partial^{b} \partial_{e} φ = η^{b c} \partial_{c} \partial_{e} φ$ and relabelling the dummy indices, both $X^{b}$ and $Y^{b}$ equal the same contraction $η^{a d} η^{b c} \partial_{a} φ \partial_{c} \partial_{d} φ$ , the identification of the mixed second partials $\partial_{a} \partial_{c} φ = \partial_{c} \partial_{a} φ$ being [F3]. Hence the second-order terms cancel and $\partial_{a} T^{a b} = K (□ φ - V^{'} (φ)) \partial^{b} φ$ on $Ω$ .

3.1F4step 2.1

Consequences. If $□ φ = V^{'} (φ)$ then the right-hand side vanishes, so $T$ is conserved; conversely if $\partial_{a} T^{a b} = 0$ for all $b$ then $(□ φ - V^{'} (φ)) \partial^{b} φ = 0$ for all $b$ , and since $\partial^{b} φ = η^{b c} \partial_{c} φ$ with $η^{b c}$ invertible this is equivalent to $(□ φ - V^{'} (φ)) \partial_{c} φ = 0$ for every $c$ , i.e. to the vanishing of the pointwise product of the scalar $□ φ - V^{'} (φ)$ with the covector $d φ$ ; at a point where some $\partial_{c} φ \neq 0$ this forces $□ φ = V^{'} (φ)$ there, hence on any open set on which $d φ$ is nowhere zero. Off shell the identity is a balance with the four-force density $f^{b} = K (□ φ - V^{'} (φ)) \partial^{b} φ$ .

4.1step 2.1step 3.1∎

The divergence identity and all its consequences are proved in steps 2.1 and 3.1; the identity holds for every smooth $φ$ , the on-shell statement is the specialization of [F4], the converse is stated with its exact $d φ = 0$ caveat, and no choice principle is used.

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Dust: conservation of rho n tensor n is equivalent to continuity and inertial flow

Statement

Let $Ω \subseteq A$ be open, let $ρ : Ω \to R$ be $C^{1}$ with $ρ > 0$ , let $n : Ω \to V$ be a $C^{1}$ field which is future timelike and satisfies $η (n, n) = - 1$ everywhere (a future unit timelike field, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)), and set $T^{a b} : = ρ n^{a} n^{b}$ (a $C^{1}$ symmetric $(2, 0)$ field). Then, on $Ω$ , (a) $\partial_{a} T^{a b} = 0$ if and only if both $\partial_{a} (ρ n^{a}) = 0$ (continuity) and $n^{a} \partial_{a} n^{b} = 0$ (inertial flow); (b) if $ρ \geq 0$ is allowed to vanish, the implication $\partial_{a} T^{a b} = 0 \Rightarrow n^{a} \partial_{a} n^{b} = 0$ is asserted only on the open set ${ρ > 0}$ , and no assertion is made where $ρ = 0$ ; (c) Under the conservation hypothesis $\partial_{a} T^{a b} = 0$ in (a), if $J$ is an interval and $x : J \to Ω$ is a $C^{2}$ curve with $x^{'} (s) = n (x (s))$ for all $s$ (an integral curve of $n$ ), then $x^{''} (s) = 0$ : the flow lines of an inertial unit field are affine straight lines in the affine structure of $A$ . For the nonnegative-density branch in (b), the same conclusion holds for an integral curve whose image lies in ${ρ > 0}$ , under conservation there. The lemma is the exact mathematical content of the dust equations; it asserts no matter model, no interaction and no empirical fact. No choice principle is used.

Facts & Assumptions

Given: The affine Minkowski model $(A, V, η, C^{+})$ with fixed inertial chart $x$ and coordinate partial derivatives $\partial_{a}$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ), an open $Ω \subseteq A$ , $C^{1}$ fields $ρ : Ω \to R$ and $n : Ω \to V$ with $η (n, n) = - 1$ ; $n_{a} : = η_{a b} n^{b}$ and indices are raised and lowered with the constant $η$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c)). For a $C^{1}$ field $n$ the expression $n^{a} \partial_{a} n^{b}$ is the coordinate form of the flat covariant derivative $\nabla_{n} n$ (The flat covariant derivative on affine Minkowski space and fields along curves). For a $C^{1}$ symmetric type- $(2, 0)$ field $T$ one writes $f^{b} = \partial_{a} T^{a b}$ and calls $T$ conserved when $f = 0$ (C^k tensor fields on affine Minkowski space and their inertial components).

[F1]

Product rule for coordinate partial derivatives of products of $C^{1}$ functions: $\partial_{a} (ρ n^{a} n^{b}) = (\partial_{a} ρ) n^{a} n^{b} + ρ (\partial_{a} n^{a}) n^{b} + ρ n^{a} \partial_{a} n^{b}$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ )

[F2]

If $I \subseteq R$ is an interval and $g : I \to R$ is continuous on $I$ and differentiable at every interior point with $g^{'} = 0$ there, then $g$ is constant on $I$ . (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant)

[F3]

Chain rule: a $C^{1}$ curve composed with a $C^{1}$ field is $C^{1}$ and its derivative is obtained by composing the derivative of the field with the curve derivative; concretely $\frac{d}{d s} n^{b} (x (s)) = \partial_{a} n^{b} (x (s)) x^{' a} (s)$ . (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

Proof

technique · direct

1.1F1given

Algebraic identity. Since $ρ$ and $n$ are $C^{1}$ , [F1] gives $\partial_{a} T^{a b} = \partial_{a} (ρ n^{a} n^{b}) = C n^{b} + ρ n^{a} \partial_{a} n^{b}$ with $C : = \partial_{a} (ρ n^{a})$ , the continuity expression.

1.2given

Contraction identity. Because $n_{b} n^{b} = η (n, n) = - 1$ is constant, its derivative vanishes: $0 = \partial_{a} (n_{b} n^{b}) = 2 n_{b} \partial_{a} n^{b}$ , hence multiplying by $n^{a}$ and summing, $n^{a} n_{b} \partial_{a} n^{b} = 0$ ; likewise $n_{b} C n^{b} = - C$ and $n_{b} n^{a} \partial_{a} n^{b} = 0$ .

2.1step 1.1step 1.2

Forward direction of (a) and clause (b). Assume $\partial_{a} T^{a b} = 0$ on $Ω$ . Contracting the identity of step 1.1 with $n_{b}$ and using step 1.2 gives $0 = C n_{b} n^{b} + ρ n_{b} n^{a} \partial_{a} n^{b} = - C$ , hence $C = 0$ . Substituting back gives $ρ n^{a} \partial_{a} n^{b} = 0$ for every $b$ . If $ρ > 0$ everywhere, division by $ρ$ gives $n^{a} \partial_{a} n^{b} = 0$ on $Ω$ ; if only $ρ \geq 0$ is assumed, the same conclusion holds on the open set ${ρ > 0}$ and no claim is made where $ρ = 0$ .

2.2step 1.1

Reverse direction of (a). Conversely, if $C = \partial_{a} (ρ n^{a}) = 0$ and $n^{a} \partial_{a} n^{b} = 0$ on $Ω$ , then step 1.1 gives $\partial_{a} T^{a b} = 0$ on $Ω$ .

3.1F2F3step 2.1

Clause (c). Let $x : J \to Ω$ be a $C^{2}$ curve with $x^{'} (s) = n (x (s))$ . By [F3], $x^{'' b} (s) = \partial_{a} n^{b} (x (s)) x^{' a} (s) = n^{a} \partial_{a} n^{b} (x (s))$ , which vanishes under the conservation hypothesis by step 2.1 (or, in the nonnegative-density branch, at every $s$ with $x (s) \in {ρ > 0}$ by step 2.1 applied there). Hence every component $(x^{b})^{'}$ has zero derivative at every interior point of $J$ , and [F2] gives that $(x^{b})^{'}$ is constant; applying [F2] to the function $s \mapsto x^{b} (s) - x^{b} (s_{0}) - (x^{b})^{'} (s_{0}) (s - s_{0})$ , whose derivative vanishes, gives $x^{b} (s) = x^{b} (s_{0}) + (x^{b})^{'} (s_{0}) (s - s_{0})$ for all $s \in J$ and every $b$ . Thus $x$ is an affine straight line and $x^{''} = 0$ on $J$ .

4.1step 2.1step 2.2step 3.1∎

Steps 2.1 and 2.2 prove (a), step 2.1 proves the restricted implication (b), and step 3.1 proves (c) and its branch on ${ρ > 0}$ . The argument is a pointwise component computation with the product rule and the constancy of $η (n, n)$ ; it asserts no matter model, no interaction, no dynamics of $ρ$ beyond the displayed equations and no empirical fact, and it uses no choice principle.

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Perfect-fluid divergence identity and its equivalence with the energy-balance and Euler equations

Statement

Let $Ω \subseteq A$ be open, let $ε, P : Ω \to R$ be $C^{1}$ , let $n : Ω \to V$ be a $C^{1}$ future unit timelike field ( $η (n, n) = - 1$ ) and set $T^{a b} : = (ε + P) n^{a} n^{b} + P η^{a b}$ . Write $D ε : = n^{a} \partial_{a} ε$ , $ϑ : = \partial_{a} n^{a}$ , $a^{b} : = n^{a} \partial_{a} n^{b}$ and $h^{b c} : = η^{b c} + n^{b} n^{c}$ (these formulas define the first-order expansion scalar, acceleration and raised rest projection for the stipulated $C^{1}$ field; they agree with Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity and The flat covariant derivative on affine Minkowski space and fields along curves when $Ω$ is nonempty and the field is smooth). Then the identity $\partial_{a} T^{a b} = n^{b} (D ε + (ε + P) ϑ) + ((ε + P) a^{b} + h^{b c} \partial_{c} P)$ holds on $Ω$ , and consequently $\partial_{a} T^{a b} = 0$ on $Ω$ if and only if $D ε + (ε + P) ϑ = 0$ and $(ε + P) a^{b} + h^{b c} \partial_{c} P = 0$ on $Ω$ . The two equations are the energy-balance and Euler equations of the model; the lemma asserts no equation of state, no thermodynamic relation and no causality condition. No choice principle is used.

Facts & Assumptions

Given: The affine Minkowski model with inertial chart $x$ , constant $η_{a b} = η^{a b} = diag (- 1, 1, 1, 1)$ and coordinate partials $\partial_{a}$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ); open $Ω$ , $C^{1}$ functions $ε, P$ , a $C^{1}$ future unit timelike field $n$ (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c), C^k tensor fields on affine Minkowski space and their inertial components); $T^{a b} = (ε + P) n^{a} n^{b} + P η^{a b}$ is a $C^{1}$ symmetric tensor field. On nonempty $Ω$ its force density is $f^{b} = \partial_{a} T^{a b}$ and conservation means $f = 0$ (Stress-energy tensor, force density and local balance on flat spacetime). The scalar $ϑ = \partial_{a} n^{a}$ and vector $a^{b} = n^{a} \partial_{a} n^{b}$ are defined by these first-order formulas; on nonempty $Ω$ with smooth $n$ they agree with the expansion and acceleration of Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity and The flat covariant derivative on affine Minkowski space and fields along curves.

[F1]

Product rule and constancy: $\partial_{a} (Φ n^{a}) = (\partial_{a} Φ) n^{a} + Φ \partial_{a} n^{a}$ for $C^{1}$ functions and fields, and $\partial_{a} η^{b c} = 0$ . (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ )

[F2]

Since $n_{b} n^{b} = η (n, n) = - 1$ is constant, $n_{b} n^{a} \partial_{a} n^{b} = \frac{1}{2} n^{a} \partial_{a} (n_{b} n^{b}) = 0$ ; hence $a^{b} = n^{a} \partial_{a} n^{b}$ lies in $n^{⊥}$ , as does $h^{b c} \partial_{c} P$ because $n_{b} h^{b c} = η^{b c} n_{b} + n^{b} n^{c} n_{b} = 0$ . (Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space)

Proof

technique · direct. If $\Omega=\varnothing$, the tensor-field definition supplies the unique empty fields; all displayed component derivatives and products are empty maps, so the identity and both directions of the pointwise equivalence are vacuous. No nonempty-domain stress-energy or congruence definition is invoked in this branch. In steps 1.1–4.1 assume $\Omega\ne\varnothing$

1.1F1given

First expansion. By [F1], using that $η^{a b}$ is constant, $\partial_{a} T^{a b} = \partial_{a} [(ε + P) n^{a} n^{b}] + η^{a b} \partial_{a} P = n^{b} \partial_{a} [(ε + P) n^{a}] + (ε + P) n^{a} \partial_{a} n^{b} + \partial^{b} P,$ where $\partial_{a} [(ε + P) n^{a}] = (\partial_{a} ε + \partial_{a} P) n^{a} + (ε + P) \partial_{a} n^{a} = D ε + D P + (ε + P) ϑ$ with $D P : = n^{a} \partial_{a} P$ , while $n^{a} \partial_{a} n^{b} = a^{b}$ and $\partial_{a} n^{a} = ϑ$ .

2.1F2step 1.1

Reconstruction of the identity. Since $h^{b c} = η^{b c} + n^{b} n^{c}$ , one has $\partial^{b} P = h^{b c} \partial_{c} P - n^{b} D P$ . Substituting into step 1.1 cancels $n^{b} D P$ and gives $\partial_{a} T^{a b} = n^{b} (D ε + (ε + P) ϑ) + (ε + P) a^{b} + h^{b c} \partial_{c} P$ . By [F2] the vector $Y^{b} : = (ε + P) a^{b} + h^{b c} \partial_{c} P$ is in $n^{⊥}$ .

3.1step 2.1

Equivalence with the two equations. Contracting the identity of step 2.1 with $n_{b}$ gives $n_{b} \partial_{a} T^{a b} = n_{b} n^{b} (D ε + (ε + P) ϑ) + n_{b} Y^{b} = - (D ε + (ε + P) ϑ)$ because $n_{b} n^{b} = - 1$ and $Y \in n^{⊥}$ . Hence conservation $\partial_{a} T^{a b} = 0$ implies $D ε + (ε + P) ϑ = 0$ , and substituting this back into the identity of step 2.1 gives $Y^{b} = 0$ ; conversely, if both $D ε + (ε + P) ϑ = 0$ and $Y = 0$ hold, then the identity of step 2.1 gives $\partial_{a} T^{a b} = 0$ . The two equations displayed are exactly the energy-balance equation and the Euler equation.

4.1step 2.1step 3.1∎

The divergence identity and the equivalence with the two conservation equations are proved in steps 2.1 and 3.1. The argument holds for any $C^{1}$ data with $η (n, n) = - 1$ and asserts no equation of state, no thermodynamic relation, no positivity of $ε$ or $P$ , no energy condition and no causality condition; all operations are finite component computations and no choice principle is used.

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Finite-box stress balance and the flux form of conservation

Statement

Let $x$ be an inertial chart of the affine Minkowski model with coordinates $(x^{0}, x^{1}, x^{2}, x^{3})$ , let $B : = \prod_{a = 0}^{3} [c_{a}, d_{a}]$ with $c_{a} < d_{a}$ be a closed axis-parallel box, and let $T$ be a $C^{1}$ symmetric type- $(2, 0)$ tensor field on an open neighbourhood of $B$ ; put $f^{b} : = \partial_{a} T^{a b}$ (continuous on $B$ ) as in Stress-energy tensor, force density and local balance on flat spacetime. (a) (Box identity.) For each $b$ , with $F_{a}^{+} : = {x \in B : x^{a} = d_{a}}$ , $F_{a}^{-} : = {x \in B : x^{a} = c_{a}}$ and each face integral the Riemann integral over the remaining three coordinates, $\int_{B} f^{b} d^{4} x = \sum_{a = 0}^{3} (\int_{F_{a}^{+}} T^{a b} d^{3} x - \int_{F_{a}^{-}} T^{a b} d^{3} x) .$ (b) (Slice balance.) Write $K : = \prod_{i = 1}^{3} [c_{i}, d_{i}]$ and, for $s \in [c_{0}, d_{0}]$ , $Q^{b} (s) : = \int_{K} T^{0 b} (s, x) d^{3} x$ ; this is finite and continuous in $s$ . For $c_{0} \leq s_{1} < s_{2} \leq d_{0}$ , $Q^{b} (s_{2}) - Q^{b} (s_{1}) = \int_{[s_{1}, s_{2}] \times K} f^{b} d^{4} x - \sum_{i = 1}^{3} (\int_{[s_{1}, s_{2}] \times K_{i}^{+}} T^{i b} d^{3} x - \int_{[s_{1}, s_{2}] \times K_{i}^{-}} T^{i b} d^{3} x),$ where $K_{i}^{\pm}$ are the two faces $x^{i} = d_{i}$ , $x^{i} = c_{i}$ of $K$ . (c) (Finite-flux conservation.) If $f = 0$ on $[s_{1}, s_{2}] \times K$ and $T^{i b} = 0$ on the side faces $[s_{1}, s_{2}] \times K_{i}^{\pm}$ for $i = 1, 2, 3$ , then $Q^{b} (s_{1}) = Q^{b} (s_{2})$ for every $b$ . The theorem concerns finite axis-parallel boxes in one inertial chart and is proved by the fundamental theorem of calculus. It does not assert a flux identity for arbitrary regions or hypersurfaces, does not assert that $Q^{b}$ transforms as a four-vector under a change of inertial chart, and asserts no conservation without the stated side-flux hypothesis; no noncompact or limiting statement is made. No choice principle is used.

Facts & Assumptions

Given: The affine Minkowski model with an inertial chart $x$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model), a closed nondegenerate axis-parallel box $B = \prod_{a = 0}^{3} [c_{a}, d_{a}]$ , and a $C^{1}$ symmetric type- $(2, 0)$ field $T$ on an open neighbourhood $U \supseteq B$ (C^k tensor fields on affine Minkowski space and their inertial components); the components $T^{a b}$ are $C^{1}$ functions on $U$ and $f^{b} = \partial_{a} T^{a b}$ with $\partial_{a}$ the coordinate partial derivatives. For a face $F_{a}^{\pm}$ of a box, its face integral is the Riemann integral of the restricted continuous function over the product of the remaining three coordinate intervals.

[F1]

Fundamental theorem of calculus, second part: if $G : [u, v] \to R$ is differentiable at every point of $[u, v]$ as a function on that interval and $G^{'}$ is integrable on $[u, v]$ , then $\int_{u}^{v} G^{'} = G (v) - G (u)$ . (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ )

[F2]

Fubini on product rectangles: if $A$ , $B$ are nondegenerate closed rectangles and $f$ is Riemann integrable on $A \times B$ , then the iterated integral in an order in which every section is integrable equals the multiple integral. (Riemann--Fubini on product rectangles, with lower and upper section integrals and content-zero exceptional sections)

[F3]

Linearity, monotonicity, the absolute-value estimate $∣ \int_{Q} f ∣ \leq \int_{Q} ∣ f ∣$ and additivity under coordinate-slice cuts of the multidimensional Riemann integral. (Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ )

[F4]

Every continuous function on a closed nondegenerate rectangle is Riemann integrable. (Every continuous function on a closed nondegenerate rectangle in $R^{m}$ is Riemann integrable, The lower and upper Darboux integrals over a nondegenerate rectangle in $R^{m}$ )

[F5]

If $n \geq 1$ and $f : Q \to R$ is continuous on a nondegenerate box $Q = \prod_{j < n} [a_{j}, b_{j}]$ , then for every permutation of the coordinates the corresponding repeated Riemann integral exists and equals $\int_{Q} f$ . (A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral)

[F6]

A continuous real function on a nonempty compact metric space is bounded above and below and attains a maximum and a minimum. (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value)

Proof

technique · direct

1.1givenF1F2F3F4F5

Claim (a). Fix $b$ and $a$ . For each fixed tuple $y$ of the remaining three coordinates, the function $t \mapsto T^{a b} (t, y)$ is $C^{1}$ on $[c_{a}, d_{a}]$ as a function on that interval, with derivative $\partial_{a} T^{a b} (t, y)$ , which is continuous and hence integrable by [F4]; [F1] gives $\int_{c_{a}}^{d_{a}} \partial_{a} T^{a b} (t, y) d t = T^{a b} (d_{a}, y) - T^{a b} (c_{a}, y)$ . The function $(t, y) \mapsto \partial_{a} T^{a b} (t, y)$ is continuous on the box $B$ , so its repeated integral in this order exists and equals $\int_{B} \partial_{a} T^{a b} d^{4} x$ by [F5] and [F2]; substituting the pointwise identity above gives $\int_{B} \partial_{a} T^{a b} d^{4} x = \int_{F_{a}^{+}} T^{a b} d^{3} x - \int_{F_{a}^{-}} T^{a b} d^{3} x$ . The integrands are continuous on the box $B$ , hence integrable by [F4]; summing the identities over $a = 0, 1, 2, 3$ with the linearity part of [F3] yields the box identity (a).

1.2givenF1F2F3F4F5F6

Claim (b), finiteness and continuity of $Q^{b}$ . For each $s \in [c_{0}, d_{0}]$ the function $x \mapsto T^{0 b} (s, x)$ is continuous on the compact box $K$ , hence bounded and integrable by [F4] and [F6], so $Q^{b} (s)$ is a finite real number. For $s, r \in [c_{0}, d_{0}]$ and each $x \in K$ , [F1] applied to $t \mapsto T^{0 b} (t, x)$ gives $T^{0 b} (s, x) - T^{0 b} (r, x) = \int_{r}^{s} \partial_{0} T^{0 b} (t, x) d t$ ; integrating over $K$ with [F2] and [F5] gives $Q^{b} (s) - Q^{b} (r) = \int_{K} \int_{r}^{s} \partial_{0} T^{0 b} (t, x) d t d x$ . By [F6] the continuous $∣ \partial_{0} T^{0 b} ∣$ is bounded on the compact box $B$ by some $M_{1} < \infty$ , so the absolute-value estimate [F3] and the constant integral give $∣ Q^{b} (s) - Q^{b} (r) ∣ \leq M_{1} vol (K) ∣ s - r ∣$ , and $Q^{b}$ is (Lipschitz) continuous on $[c_{0}, d_{0}]$ .

2.1F3step 1.1given

Claim (b), slice-balance identity. The sub-box $[s_{1}, s_{2}] \times K$ is a closed nondegenerate axis-parallel box contained in $B$ , and $T$ is $C^{1}$ on a neighbourhood of $B$ , hence on a neighbourhood of the sub-box. Applying the box identity of step 1.1 to this sub-box, the two faces with $a = 0$ give $\int_{K} T^{0 b} (s_{2}, x) d^{3} x - \int_{K} T^{0 b} (s_{1}, x) d^{3} x = Q^{b} (s_{2}) - Q^{b} (s_{1})$ , and for $i = 1, 2, 3$ the faces $F_{i}^{\pm}$ are the side faces $[s_{1}, s_{2}] \times K_{i}^{\pm}$ with the displayed signs; with the sub-box's interior equal to $(s_{1}, s_{2}) \times int K$ , this is exactly the stated identity.

3.1F3step 2.1

Claim (c). If $f = 0$ on $[s_{1}, s_{2}] \times K$ then its integral there vanishes, and if $T^{i b} = 0$ on the side faces $[s_{1}, s_{2}] \times K_{i}^{\pm}$ then the three side-flux differences vanish; step 2.1 then gives $Q^{b} (s_{2}) - Q^{b} (s_{1}) = 0$ , so $Q^{b} (s_{1}) = Q^{b} (s_{2})$ , for every $b$ .

4.1step 1.1step 1.2step 2.1step 3.1∎

Claims (a), (b) and (c) are proved in steps 1.1, 1.2 and 2.1, and 3.1. The argument uses only the fundamental theorem of calculus, Fubini's theorem for continuous functions on boxes and the elementary integral estimates; it concerns one inertial chart and finite axis-parallel boxes, asserts nothing for arbitrary regions, hypersurfaces or noncompact slices, and asserts no conservation without the stated side-flux hypothesis. All choices are finite; no choice principle is used.

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

Hodge decomposition of an alternating two-form for an observer, invariants and the optional field identification

Statement

Let $(V, η)$ be the oriented Lorentz vector space of the affine Minkowski model with orientation and volume form $ε$ of The oriented volume form epsilon and the permutation symbols of an inertial frame, and let $F : V \times V \to R$ be an alternating bilinear form (Alternating $k$ -linear maps, Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms). Define the Hodge dual by $(* F)_{a b} : = \frac{1}{2} ε_{a b}^{c d} F_{c d}$ and set $(α \land β)_{a b} : = α_{a} β_{b} - α_{b} β_{a}$ . Use the standard form normalization $F = \frac{1}{2} \sum_{a, b} F_{a b} θ^{a} \land θ^{b}$ in the dual coframe $θ^{a} (e_{b}) = δ_{b}^{a}$ of an oriented orthonormal frame, and define $(F \land F) (v_{0}, v_{1}, v_{2}, v_{3}) : = 2 [F (v_{0}, v_{1}) F (v_{2}, v_{3}) - F (v_{0}, v_{2}) F (v_{1}, v_{3}) + F (v_{0}, v_{3}) F (v_{1}, v_{2})]$ , the alternating four-form exterior square. Let $n \in V$ be future unit, $η (n, n) = - 1$ , and put $e_{a} : = F_{a b} n^{b}$ , $b_{a} : = (* F)_{a b} n^{b}$ . (a) $e (n) = b (n) = 0$ , and $F = n^{♭} \land e - * (n^{♭} \land b)$ with $n^{♭} : = η (n, \cdot)$ ; also $* (* F) = - F$ , so the Hodge dual squares to $- 1$ on two-forms. (b) In an oriented orthonormal frame with $n = e_{0}$ : $e_{i} = F_{i 0}$ , $b_{i} = - \frac{1}{2} ϵ_{i j k} F_{j k}$ , $F_{0 i} = - e_{i}$ and $F_{i j} = - ϵ_{i j k} b_{k}$ ; hence the six components of $F$ are exactly the six components of the pair $(e, b)$ and the reconstruction in (a) is equivalent to these component formulae. (c) The quadratic invariants are $\frac{1}{2} F_{a b} F^{a b} = ∣ b ∣^{2} - ∣ e ∣^{2}$ and $F_{a b} (* F)^{a b} = - 4 (e \cdot b)$ , where $∣ \cdot ∣$ and $\cdot$ are those of the spatial structure of the frame (The Euclidean spatial structure and cross product of an inertial frame); also $F \land F = 2 (e \cdot b) ε$ . The two scalar expressions $I_{1} : = \frac{1}{2} F_{a b} F^{a b}$ and $I_{2} : = F_{a b} (* F)^{a b}$ form a basis of the homogeneous quadratic polynomials in the six components of $F$ invariant under the proper orthochronous Lorentz group; the square-wedge expression is dependent, $F \land F = - \frac{1}{2} I_{2} ε$ . $I_{2}$ is an oriented pseudoscalar and changes sign if the chosen orientation is reversed with the metric and $F$ held fixed. (d) (Optional SI identification.) Let $c > 0$ . If in an inertial frame one identifies $F^{i 0} = E_{i} / c$ and $F^{i j} = - ϵ_{i j k} B_{k}$ for numerical spatial vectors $E, B \in R^{3}$ , then $e_{i} = - E_{i} / c$ and $b_{i} = B_{i}$ , the invariants of (c) read $\frac{1}{2} F_{a b} F^{a b} = ∣ B ∣^{2} - ∣ E ∣^{2} / c^{2}$ and $F_{a b} (* F)^{a b} = 4 (E \cdot B) / c$ , and $F \land F = - 2 (E \cdot B) ε / c$ . If $κ : = (F \land F) (e_{0}, e_{1}, e_{2}, e_{3}) = (F \land F)_{0123}$ , then $E \cdot B = - c κ / 2$ . Under the passive boost of The passive Lorentz boost with parameter beta along $e_{1}$ with parameter $β \in (- 1, 1)$ one has $E_{1}^{'} = E_{1}$ , $B_{1}^{'} = B_{1}$ , $E_{⊥}^{'} = γ (E + v \times B)_{⊥}$ and $B_{⊥}^{'} = γ (B - v \times E / c^{2})_{⊥}$ where the cross product is the right-handed product (The cross product in $R^{3}$ ), with $γ = (1 - β^{2})^{- 1 / 2}$ and $v = β c$ along $e_{1}$ . In an SI reading, $c$ has units $m / s$ , $E$ has units $V / m$ and $B$ has units tesla, so $E / c$ and all components of $F$ have units tesla; these are conditional unit assignments, not measurements. (e) The theorem is algebra on the oriented Lorentz vector space with the stated index and duality conventions; clause (d) is an explicitly optional identification of the components of $F$ with fields the electromagnetic framework names $E, B$ , and the theorem asserts no field equation, no dynamics, no measurement and no transformation beyond the stated component law. No choice principle is used.

Facts & Assumptions

Given: The oriented Lorentz vector space $(V, η)$ of the affine Minkowski model with its fixed orientation and oriented volume form $ε$ , normalized by $ε (e_{0}, e_{1}, e_{2}, e_{3}) = + 1$ in every positively oriented orthonormal frame (The oriented volume form epsilon and the permutation symbols of an inertial frame, Oriented time-oriented affine Minkowski space (the spacetime primitives), Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)); the raised volume form is $ε^{a b c d} = η^{a a^{'}} η^{b b^{'}} η^{c c^{'}} η^{d d^{'}} ε_{a^{'} b^{'} c^{'} d^{'}}$ , so $ε^{0123} = η^{00} η^{11} η^{22} η^{33} ε_{0123} = - 1$ . An alternating bilinear form $F$ with components $F_{a b} = - F_{b a}$ (Alternating $k$ -linear maps, Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms); a future unit $n \in V$ with $η (n, n) = - 1$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones); and the conventions $(* F)_{a b} : = \frac{1}{2} ε_{a b}^{c d} F_{c d}$ , $(α \land β)_{a b} : = α_{a} β_{b} - α_{b} β_{a}$ , the dual coframe $θ^{a} (e_{b}) = δ_{b}^{a}$ (Linear functionals and the algebraic dual $V^{*} = L (V, F)$ ), and $(F \land F) (v_{0}, \dots, v_{3}) : = 2 [F (v_{0}, v_{1}) F (v_{2}, v_{3}) - F (v_{0}, v_{2}) F (v_{1}, v_{3}) + F (v_{0}, v_{3}) F (v_{1}, v_{2})]$ . The spatial inner product of the frame is the restriction of $η$ to $n^{⊥}$ (The Euclidean spatial structure and cross product of an inertial frame, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ), and $e_{a} : = F_{a b} n^{b}$ , $b_{a} : = (* F)_{a b} n^{b}$ .

[F1]

Observer rest space: for future unit $n$ the subspace $n^{⊥} = {v : η (n, v) = 0}$ is three-dimensional and $η$ restricts to a positive definite form on it; the map $h (v) = v + η (n, v) n$ has image $n^{⊥}$ and satisfies $h^{a}_{b} n^{b} = 0 = n_{a} h^{a}_{b}$ . (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition,(c))

[F2]

Gram–Schmidt: every finite-dimensional inner product space has an orthonormal basis. (Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans)

[F3]

Tensor transformation and invariance of the volume form: for a proper orthochronous Lorentz map $L$ the components of a $(0, 2)$ form satisfy $F_{a b}^{'} = (Λ^{- 1})^{c}_{a} (Λ^{- 1})^{d}_{b} F_{c d}$ for the chart transition matrix $Λ$ , the alternating covariant laws hold on each slot, and $ε_{a b c d}^{'} = ε_{a b c d}$ (the oriented volume form is invariant); the boosts $B (β)$ are proper orthochronous with $B (β)^{- 1} = B (- β)$ and every spatial rotation extension $diag (1, R)$ is a proper orthochronous Lorentz map. (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii),(v), The passive boost is a proper orthochronous Lorentz map with inverse B(-beta), Unique boost-rotation factorization of proper orthochronous Lorentz maps,(iii))

[F4]

Passive boost: for $β \in (- 1, 1)$ and $γ = (1 - β^{2})^{- 1 / 2}$ the boost along the spatial axis $e_{1}$ has the matrix $Λ^{0}_{0} = Λ^{1}_{1} = γ$ , $Λ^{0}_{1} = Λ^{1}_{0} = - γ β$ , and identity on the other two spatial directions; it is proper orthochronous with $B (β)^{- 1} = B (- β)$ . (The passive Lorentz boost with parameter beta, The passive boost is a proper orthochronous Lorentz map with inverse B(-beta))

[F5]

The right-handed cross product of $R^{3}$ : $(u \times v)_{i} = ϵ_{i j k} u_{j} v_{k}$ with $ϵ_{123} = + 1$ . (The cross product in $R^{3}$ )

Proof

technique · direct

1.1givenF1F2

Adapted oriented frame. By [F1] the rest space $n^{⊥}$ is a three-dimensional inner product space; [F2] gives an orthonormal basis $(u_{1}, u_{2}, u_{3})$ of $n^{⊥}$ . The ordered frame $(n, u_{1}, u_{2}, u_{3})$ is orthonormal; if its orientation class is negative, replace $u_{3}$ by $- u_{3}$ , which keeps orthonormality and makes the frame positively oriented. Writing $e_{0} : = n$ , $e_{1}, e_{2}, e_{3}$ for this frame, the components satisfy $η_{a b} = diag (- 1, 1, 1, 1)$ and $ε_{0123} = + 1$ . Only finitely many choices are made.

1.2given

Hodge dual on the six basis forms. Let $θ^{a}$ be the dual coframe and set $a_{i} : = θ^{0} \land θ^{i}$ and $s_{1} : = θ^{2} \land θ^{3}$ , $s_{2} : = θ^{3} \land θ^{1}$ , $s_{3} : = θ^{1} \land θ^{2}$ . Evaluating the definition $(* F)_{a b} = \frac{1}{2} ε_{a b}^{c d} F_{c d}$ on the basis forms, using $ε^{0123} = - 1$ and the antisymmetry of $ε$ , gives $(* a_{i}) = - s_{i}, (* s_{i}) = a_{i} (i = 1, 2, 3) .$ Indeed the only nonzero contributions come from the complementary index pair, and $ε_{0 i}^{j k} = + ϵ_{i j k}$ and $ε_{i j}^{0 k} = - ϵ_{i j k}$ for spatial indices. Consequently $* (* F) = - F$ on every two-form, since $*^{2} = - 1$ on each of the six basis forms. For $F = \frac{1}{2} F_{a b} θ^{a} \land θ^{b}$ the dual is $(* F) = e_{i} s_{i} - b_{i} a_{i}$ whenever the components $e_{i} = F_{i 0}$ and $b_{i} = - \frac{1}{2} ϵ_{i j k} F_{j k}$ are used, by direct substitution into the definition.

2.1step 1.2given

Components of $e$ and $b$ and reconstruction (a),(b). With $n = e_{0}$ one has $n^{a} = δ_{0}^{a}$ , $n_{a} = - δ_{a}^{0}$ , so $e_{0} = F_{00} = 0$ , $e_{i} = F_{i 0}$ and $b_{0} = (* F)_{00} = 0$ , $b_{i} = (* F)_{i 0} = - \frac{1}{2} ϵ_{i j k} F_{j k}$ by step 1.2. Hence $e (n) = e_{0} = 0$ and $b (n) = b_{0} = 0$ . Antisymmetry gives $F_{0 i} = - F_{i 0} = - e_{i}$ , and inverting the relation $b_{i} = - \frac{1}{2} ϵ_{i j k} F_{j k}$ with the identity $ϵ_{i j k} ϵ_{i j l} = 2 δ_{k l}$ gives $F_{i j} = - ϵ_{i j k} b_{k}$ . Comparing components with $n^{♭} \land e$ (whose only nonzero components are $(n^{♭} \land e)_{0 i} = - e_{i}$ ) and with $* (n^{♭} \land b)$ (whose only nonzero components are $* (n^{♭} \land b)_{i j} = ϵ_{i j k} b_{k}$ , computed as in step 1.2) gives $F = n^{♭} \land e - * (n^{♭} \land b)$ , and $* (* F) = - F$ by step 1.2. Identifying the horizontal covectors $e, b$ with their metric dual vectors $e^{♯}, b^{♯} \in n^{⊥}$ , the map $F \mapsto (e^{♯}, b^{♯})$ is thus a linear isomorphism from the six-dimensional space of alternating bilinear forms onto $n^{⊥} \oplus n^{⊥}$ , with inverse given by these component formulae.

3.1step 2.1given

Invariants (c). Raising indices with the diagonal metric, $F^{0 i} = e_{i}$ , $F^{i 0} = - e_{i}$ and $F^{i j} = - ϵ_{i j k} b_{k}$ , while $(* F)^{0 i} = b_{i}$ , $(* F)^{i 0} = - b_{i}$ and $(* F)^{i j} = ϵ_{i j k} e_{k}$ ; counting both orders in each antisymmetric pair, $\frac{1}{2} F_{a b} F^{a b} = \frac{1}{2} (- 2 ∣ e ∣^{2} + 2 ∣ b ∣^{2}) = ∣ b ∣^{2} - ∣ e ∣^{2}, F_{a b} (* F)^{a b} = - 2 (e \cdot b) - 2 (b \cdot e) = - 4 (e \cdot b) .$ Evaluating the normalized alternating four-form on $(e_{0}, \dots, e_{3})$ , only the three terms $2 [F_{01} F_{23} - F_{02} F_{13} + F_{03} F_{12}] = 2 [e_{1} b_{1} + e_{2} b_{2} + e_{3} b_{3}]$ survive, so $F \land F = 2 (e \cdot b) ε$ .

4.1F3F4step 2.1step 3.1

Invariance and observer components. For a proper orthochronous inertial transition with matrix $Λ$ , [F3] gives $F'_{a b} = (Λ^{- 1})^{c}_{a} (Λ^{- 1})^{d}_{b} F_{c d}$ . The contractions $I_{1}, I_{2}$ and the coefficient of $F \land F$ are unchanged, because inverse index factors cancel against the metric factors and the oriented volume form is preserved. For the component lists $(e, b)$ in this paragraph, each frame uses its own time-axis observer, whose components in that frame are $(1, 0, 0, 0)$ ; these lists compare different observers. Under $Λ = diag (1, R)$ , $R \in S O (3)$ , the covariant law and step 2.1 give $(e', b') = (R e, R b)$ . For the passive boost along $e_{1}$ , substitution of the inverse matrix from [F4] gives $e'_{1} = e_{1}$ , $b'_{1} = b_{1}$ and $e'_{2} = γ (e_{2} + β b_{3}), e'_{3} = γ (e_{3} - β b_{2}), b'_{2} = γ (b_{2} - β e_{3}), b'_{3} = γ (b_{3} + β e_{2}) .$ For a fixed physical observer, its primed vector components instead satisfy $n' = Λ n$ , and the associated horizontal covectors transform by the covariant law; that is a distinct comparison.

4.2F3F4F5step 2.1step 3.1

Optional identification (d). Let $c > 0$ and suppose that in the frame one has $F^{i 0} = E_{i} / c$ , $F^{i j} = - ϵ_{i j k} B_{k}$ for spatial data $E, B$ . Then $F_{i 0} = η_{i i} η_{00} F^{i 0} = - E_{i} / c$ , so $e_{i} = - E_{i} / c$ ; and $b_{i} = - \frac{1}{2} ϵ_{i j k} (- ϵ_{j k l} B_{l}) = B_{i}$ by step 2.1. Substituting into step 3.1 gives $\frac{1}{2} F_{a b} F^{a b} = ∣ B ∣^{2} - ∣ E ∣^{2} / c^{2}$ , $F_{a b} (* F)^{a b} = 4 (E \cdot B) / c$ and $F \land F = - 2 (E \cdot B) ε / c$ , so $κ = (F \land F)_{0123} = - 2 E \cdot B / c$ and $E \cdot B = - c κ / 2$ . Under the passive boost along $e_{1}$ the component law $F^{' a b} = Λ^{a}_{c} Λ^{b}_{d} F^{c d}$ of [F3] and [F4], evaluated directly on the six components of $F$ , gives $F^{' 10} = E_{1} / c$ , $F^{' 20} = γ (E_{2} / c - β B_{3})$ , $F^{' 30} = γ (E_{3} / c + β B_{2})$ , $F^{' 23} = - B_{1}$ , $F^{' 13} = γ (B_{2} + β E_{3} / c)$ and $F^{' 12} = γ (- B_{3} + β E_{2} / c)$ ; translating back by the identification and writing $v = β c$ this is exactly $E_{1}^{'} = E_{1}$ , $B_{1}^{'} = B_{1}$ , $E_{⊥}^{'} = γ (E + v \times B)_{⊥}$ , $B_{⊥}^{'} = γ (B - v \times E / c^{2})_{⊥}$ , which are the stated vector formulae under the right-handed cross product [F5]. At $β = 0$ , $γ = 1$ and the identities are immediate.

5.1step 3.1step 4.1

The invariant quadratic polynomials. Let $P$ be a homogeneous quadratic polynomial in the six components of $F$ that is invariant under all proper orthochronous maps; by the isomorphism of step 2.1, $P$ is a quadratic form in $(e, b)$ , say $P = e^{T} A e + b^{T} B b + e^{T} C b$ with $A, B$ symmetric. Invariance under the rotations $R \oplus R$ of step 4.1 means $R^{T} A R = A$ , $R^{T} B R = B$ and $R^{T} C R = C$ for every $R \in S O (3)$ . The half-turns $R = diag (1, - 1, - 1)$ and $R = diag (- 1, - 1, 1)$ are proper rotations, and since $R$ is diagonal the conditions read $R_{i i} M_{i j} R_{j j} = M_{i j}$ entrywise; together they force every off-diagonal entry of $A$ and of $B$ to vanish, while the cyclic permutation matrix (a proper rotation) forces the surviving diagonal entries to be equal, so $A = a I_{3}$ and $B = b I_{3}$ . The same two half-turns force every off-diagonal entry of $C$ to vanish, and the cyclic permutation forces its diagonal entries to agree, so $C = c I_{3}$ . Hence $P = a ∣ e ∣^{2} + b ∣ b ∣^{2} + c (e \cdot b)$ for real $a, b, c$ . Invariance under the boost of step 4.1 applied to $e = (0, 1, 0)$ , $b = 0$ , which is sent to $e^{'} = (0, γ, 0)$ , $b^{'} = (0, 0, γ β)$ , gives $a = a γ^{2} + b γ^{2} β^{2}$ , i.e. $(a + b) γ^{2} β^{2} = 0$ , hence $a + b = 0$ for $β \neq 0$ (and by continuity of the polynomial identity in $β$ also for $β = 0$ ). Thus $P = a (∣ e ∣^{2} - ∣ b ∣^{2}) + c (e \cdot b) = - a I_{1} - \frac{c}{4} I_{2}$ , so $I_{1}$ and $I_{2}$ span the invariant quadratic polynomials. They are linearly independent: at $b = 0$ , $e \neq 0$ one has $I_{1} = - ∣ e ∣^{2} \neq 0$ , $I_{2} = 0$ , and at $e = b \neq 0$ one has $I_{1} = 0$ , $I_{2} = - 4 ∣ e ∣^{2} \neq 0$ . Finally $F \land F = 2 (e \cdot b) ε = - \frac{1}{2} I_{2} ε$ , and $I_{2} = F_{a b} (* F)^{a b}$ changes sign if the orientation is reversed with $η$ and $F$ held fixed, because then $ε$ is replaced by $- ε$ and hence $* F$ by $- * F$ .

6.1step 2.1step 3.1step 5.1step 4.2∎

Clauses (a) and (b) are steps 1.2 and 2.1, clause (c) is steps 3.1 and 5.1, and clause (d) is step 4.2. The proof is finite-dimensional linear algebra on the oriented Lorentz vector space with the stated index conventions; the only choices are the finite Gram–Schmidt and orientation adjustments of step 1.1; clause (d) is an optional identification of components with fields of the electromagnetic framework and the theorem asserts no field equation, no dynamics, no measurement, no energy condition and no transformation beyond the stated component law. No choice principle is used.

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Adopted continuum matter models, evolution equations and observer readings

Physical scope

Flat affine Minkowski spacetime with the SI conventions of the framework, in one inertial chart; matter described only by the adopted scalar-field, dust and perfect-fluid models, their additive total tensor and the adopted observer readings. The postulate contains no gravity, no electromagnetic coupling, no force law, no equation of state, no microscopic derivation, no claim that these models exhaust physical matter, and no empirical premise; it asserts no conservation of the tensors.

Postulate

Adopt, on top of the flat spacetime postulates Adopted inertial covariance of the model equations and Adopted ideal-clock and ideal-vacuum-light interpretation, the SI conventions SI units and coordinate conventions of the special-relativity framework and the particle premises Adopted massive-particle momentum, observer measurement and isolated-collision conservation, the following classical continuum matter models on an open set $Ω \subseteq A$ of the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)).

The adopted matter models, stress-energy terminology, scalar stationarity and Hilbert-tensor identification in clauses (i)–(v) below apply when $Ω$ is nonempty, as required by Stress-energy tensor, force density and local balance on flat spacetime, First variation of the scalar-field action and the Euler-Lagrange equation and The Hilbert stress-energy tensor of the scalar-field action on flat spacetime. If $Ω = \emptyset$ , retain only the degenerate mathematical branch of C^k tensor fields on affine Minkowski space and their inertial components: every field is the unique empty map of its type, tensor addition and the displayed component formulas are empty maps, and all pointwise equations are vacuous. That branch supplies no matter or observer reading, does not invoke the nonempty-domain stress-energy, stationarity or Hilbert-tensor interfaces, and makes no assertion about an action integral.

(i) Scalar field. A real smooth dimensionless field $φ : Ω \to R$ , a smooth potential $V : R \to R$ with values in inverse square metres ( $m^{- 2}$ ) and a constant $K > 0$ in joule per metre ( $J / m$ ) have the field equation $□ φ = V^{'} (φ)$ — the Euler–Lagrange equation of the adopted local action density, with stationarity defined by the finite relative action for compactly supported field variations (First variation of the scalar-field action and the Euler-Lagrange equation); the model's stress-energy tensor is the Hilbert tensor of The Hilbert stress-energy tensor of the scalar-field action on flat spacetime.

(ii) Dust. A $C^{1}$ rest energy density $ρ : Ω \to (0, \infty)$ (joule per cubic metre) and a $C^{1}$ future unit timelike field $n : Ω \to V$ ( $η (n, n) = - 1$ ) satisfy $\partial_{a} (ρ n^{a}) = 0$ and $n^{a} \partial_{a} n^{b} = 0$ ; the stress-energy tensor is $T = ρ n \otimes n$ .

(iii) Perfect fluid. $C^{1}$ maps $ε, P : Ω \to R$ (energy density and pressure), both in joule per cubic metre, and a $C^{1}$ future unit timelike field $n$ satisfy the energy-balance and Euler equations of Perfect-fluid divergence identity and its equivalence with the energy-balance and Euler equations; an equation of state or evolution equation determining $P$ is part of the model and is not supplied by this postulate; the stress-energy tensor is $T = (ε + P) n \otimes n + P η^{- 1}$ , where $η^{- 1}$ is the symmetric type- $(2, 0)$ inverse metric with components $η^{a b}$ .

(iv) Additivity. The total stress-energy tensor of a composite system is the sum of the stress-energy tensors of its constituents (tensor-field addition on $Ω$ ).

(v) Adopted observer readings and units. For a tensor physically assigned to matter in this model, all inertial components $T^{a b}$ have units joule per cubic metre. In length coordinates $x^{0} = c t$ , an ideal observer with future unit direction $n$ and four-velocity $U = c n$ reads the numerical contractions of Observer decomposition of a symmetric stress-energy tensor as energy density $ε_{n} = T^{a b} n_{a} n_{b}$ , momentum density $π_{n}^{a} = - c^{- 1} h^{a}_{b} T^{b c} n_{c}$ and rest stress $S_{n}^{a b} = h^{a}_{c} h^{b}_{d} T^{c d}$ , where $h (v) = v + η (n, v) n$ . In an adapted inertial frame these are $T^{00}$ , $T^{0 i} / c$ and $T^{i j}$ respectively, and the energy-flux components are $c T^{i 0} = c^{2} π_{n}^{i}$ . Momentum density has units kilogram per square metre per second, energy flux has units watt per square metre, and rest stress has units joule per cubic metre. These identifications are adopted model readings, not mathematical facts about an arbitrary symmetric tensor and not reported measurements. If $f^{b} = \partial_{a} T^{a b}$ is assigned to external exchange, the spatial components $f^{i}$ are force per volume and $c f^{0}$ is power per volume; the corresponding numerical balances are $\partial_{t} (T^{00}) + \partial_{i} (c T^{i 0}) = c f^{0}$ and $\partial_{t} (T^{0 j} / c) + \partial_{i} T^{i j} = f^{j}$ . Neither assignment selects a force law or asserts conservation.

Scope and status

This postulate adopts model equations and tensor identifications only. It does not assert conservation of the tensors — that is derived from these equations in Local conservation of the adopted matter stress-energy tensors — does not select any force law, interaction, equation of state, thermodynamic or dissipative behaviour, does not claim that these models exhaust physical matter, and contains no empirical premise, measurement or uncertainty statement. The dimensionless scalar field, the potential with its stated units and the constant $K$ are adopted primitive data of the scalar model; the SI conventions and the identification of the coordinate $x^{0}$ with $c t$ are those of SI units and coordinate conventions of the special-relativity framework, and the observer congruence vocabulary agrees with Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity for smooth fields (C^k tensor fields on affine Minkowski space and their inertial components).

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Isolation and the total stress-energy tensor

Physical scope

Flat affine Minkowski spacetime in one inertial chart; the total stress-energy tensor of a bounded system, including every constituent that carries energy-momentum; the isolation premise concerns a finite closed box and length-coordinate interval with vanishing side flux. The postulate is an idealisation of an isolated bounded system: it makes no apparatus, statistical or empirical claim, adopts no limiting procedure at spatial infinity and asserts nothing about partial tensors or unmodelled matter.

Postulate

Adopt as the boundary and isolation premise of the continuum framework the following.

(i) Total tensor. In every conservation or integration statement of this framework the tensor used is the total stress-energy tensor of the system considered, including every constituent that carries energy-momentum — matter, fields and supporting structures present in the region — assembled by the additivity clause of Adopted continuum matter models, evolution equations and observer readings.

(ii) Localisation. Fix an inertial chart with $x^{0} = c t$ . Choose real $s_{1} < s_{2}$ and a nondegenerate closed spatial rectangle $\overset{ˉ}{K} = \prod_{i = 1}^{3} [a_{i}, b_{i}]$ , $a_{i} < b_{i}$ . The interval $[s_{1}, s_{2}]$ parametrizes $x^{0}$ in metres; the corresponding coordinate-clock interval is $[s_{1} / c, s_{2} / c]$ in seconds. Whenever conservation of a slice four-momentum on this slab is asserted, the model assumes that during that interval the total tensor is $C^{1}$ and symmetric on a neighbourhood of a closed box $[s_{1}, s_{2}] \times \overset{ˉ}{K}$ (Stress-energy tensor, force density and local balance on flat spacetime, Oriented time-oriented affine Minkowski space (the spacetime primitives)), that its components satisfy the zero side-flux condition $T^{i b} = 0$ on the spatial faces of the box for $i = 1, 2, 3$ , and that the slice integrals converge; equivalently, the system is isolated inside the chosen region for the chosen interval.

(iii) Failure of isolation. If the side flux does not vanish, only the exact finite-flux balance of Finite-box stress balance and the flux form of conservation is asserted; no conservation of a slice integral is inferred and no limiting procedure at spatial infinity is adopted.

(iv) Unmodelled matter. For a matter system not governed by one of the models of Adopted continuum matter models, evolution equations and observer readings, conservation of its stress-energy tensor is not inferred; if used it must be adopted or derived separately.

(v) Exclusion caution. Removing a constituent — a support, a field or a container — from the total tensor can make the side-flux condition fail; the premise concerns the total tensor of the system, not an arbitrary partial tensor.

Scope and status

This premise is an idealisation of an isolated bounded system in flat spacetime; it is not a measurement and makes no apparatus, statistical or empirical claim. The localisation clause is a boundary hypothesis on a finite box in a single inertial chart; it neither supplies an equation of motion for the total tensor nor implies the vanishing of the force density $f^{b} = \partial_{a} T^{a b}$ , which is a separate hypothesis of the integration statements that use this postulate.

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Local conservation of the adopted matter stress-energy tensors

Physical scope

Flat affine Minkowski spacetime in every inertial chart, under the adopted scalar-field, dust and perfect-fluid model equations of Adopted continuum matter models, evolution equations and observer readings. The theorem is a conditional deduction from those adopted equations: it asserts no interaction, force law, cross section, measurement or empirical fact, makes no claim for arbitrary prescribed symmetric tensors, and its conclusions inherit the scope and qualifications of the postulate.

Statement

Under the adopted matter premises of Adopted continuum matter models, evolution equations and observer readings on a nonempty open $Ω$ : (a) If $φ$ is a smooth scalar field solving the adopted field equation $□ φ = V^{'} (φ)$ on $Ω$ , then the Hilbert stress-energy tensor of the scalar-field action (The Hilbert stress-energy tensor of the scalar-field action on flat spacetime) satisfies $\partial_{a} T^{a b} = 0$ on $Ω$ , in every inertial chart. (b) If $ρ > 0$ and the future unit field $n$ satisfy the adopted dust equations, then $T = ρ n \otimes n$ satisfies $\partial_{a} T^{a b} = 0$ ; conversely if $\partial_{a} T^{a b} = 0$ and $ρ > 0$ then the dust equations hold. (c) If $ε, P$ and the future unit field $n$ satisfy the adopted fluid equations, then $T = (ε + P) n \otimes n + P η^{- 1}$ satisfies $\partial_{a} T^{a b} = 0$ ; conversely conservation of that $T$ implies the two fluid equations. (d) For a $C^{1}$ symmetric tensor field prescribed on $Ω$ that is not obtained from one of the adopted models together with its field equations, no conservation is asserted; a conservation statement about such a field is an additional explicit premise. (e) The theorem is a conditional deduction from the adopted model equations; it asserts no interaction, force law, cross section, measurement or empirical fact, and its conclusions inherit the scope of the postulate. Under the already stated homogeneous dust reading, the tensor boost computation has the conditional physical interpretation $T^{' 00} = γ^{2} ρ$ . For comoving identical massive particles with fixed rest mass $m$ , rest number density $ν$ and $ρ = ν m c^{2}$ , a rest rectangular parcel has boosted simultaneous volume $1 / γ$ times its rest volume (the parallel length contracts and transverse lengths are unchanged), so its particle density is $γ ν$ ; the adopted massive-particle reading gives each particle energy $γ m c^{2}$ . Thus the two factors in $γ^{2} ρ$ correspond respectively to particle energy and volume/density. This is conditional model interpretation for that homogeneous particle realization, not a derivation of the continuum from microscopic dynamics, a claim about arbitrary tensors, or empirical evidence.

For the postulate's separate mathematical branch $Ω = \emptyset$ , the scalar, dust and fluid component formulas define the unique empty symmetric tensor field and its coordinate derivatives are empty maps. Thus each pointwise conservation equation and each pointwise dust/fluid equivalence in (a)–(c) is vacuously valid; no Hilbert-tensor, stress-energy, action-stationarity or physical-reading supplier is applied to this branch, and no nonempty parcel realization in (e) exists.

Facts & Assumptions

Given: For the adopted-model branch let $Ω$ be nonempty and open; for the separate empty-domain branch use only the empty-field conventions of Adopted continuum matter models, evolution equations and observer readings. The affine Minkowski model with inertial chart $x$ and its index conventions (Oriented time-oriented affine Minkowski space (the spacetime primitives)), the adopted matter model equations and readings of Adopted continuum matter models, evolution equations and observer readings, the particle premises and homogeneous dust reading of Adopted massive-particle momentum, observer measurement and isolated-collision conservation, and the force-density/conservation terminology of Stress-energy tensor, force density and local balance on flat spacetime (conservation of a $C^{1}$ symmetric field $T$ means $\partial_{a} T^{a b} = 0$ , and the force density is chart-independent).

[F1]

Scalar divergence identity: for the smooth scalar field and Hilbert tensor of the adopted model, $\partial_{a} T^{a b} = K (□ φ - V^{'} (φ)) \partial^{b} φ$ ; in particular any solution of $□ φ = V^{'} (φ)$ has conserved Hilbert tensor. (Divergence identity for the scalar-field stress-energy tensor, The Hilbert stress-energy tensor of the scalar-field action on flat spacetime)

[F2]

Dust equivalence: for $C^{1}$ $ρ > 0$ and a $C^{1}$ future unit field $n$ , $\partial_{a} (ρ n^{a} n^{b}) = 0$ if and only if $\partial_{a} (ρ n^{a}) = 0$ and $n^{a} \partial_{a} n^{b} = 0$ . (Dust: conservation of rho n tensor n is equivalent to continuity and inertial flow)

[F3]

Perfect-fluid identity: for the field $T = (ε + P) n \otimes n + P η^{- 1}$ , $\partial_{a} T^{a b} = n^{b} (D ε + (ε + P) ϑ) + ((ε + P) a^{b} + h^{b c} \partial_{c} P)$ , and $\partial_{a} T^{a b} = 0$ if and only if both bracket expressions vanish. (Perfect-fluid divergence identity and its equivalence with the energy-balance and Euler equations)

[F4]

Massive-particle reading: a particle of rest mass $m$ and four-velocity $U = m^{- 1} P$ has energy $E = γ m c^{2}$ in a frame where its speed parameter is $v$ with $γ = (1 - v^{2} / c^{2})^{- 1 / 2}$ ; a rest rectangular parcel's parallel length contracts by the factor $1 / γ$ while transverse lengths are unchanged. (Four-momentum components, energy-velocity relation and passive boost transformation in an inertial frame, Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation, Adopted massive-particle momentum, observer measurement and isolated-collision conservation)

[F5]

Transformation laws: under an inertial transition the components of a $C^{1}$ type- $(2, 0)$ field transform by the rank-two law $T^{' a b} = Λ^{a}_{c} Λ^{b}_{d} T^{c d}$ , and its coordinate divergence transforms as a vector, $\partial_{a}^{'} T^{' a b} = Λ^{b}_{c} \partial_{a} T^{a c}$ , so conservation is a chart-independent statement. (Stress-energy tensor, force density and local balance on flat spacetime, Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii),(iv))

Proof

technique · direct. In steps 1.1–1.5 assume the nonempty adopted-model branch, so the cited nonempty-domain suppliers apply; the separate mathematical empty-domain branch is proved in step 1.6

1.1F1F5given

Clause (a). The adopted scalar model includes the field equation $□ φ = V^{'} (φ)$ on $Ω$ ; [F1] then gives $\partial_{a} T^{a b} = K \cdot 0 \cdot \partial^{b} φ = 0$ on $Ω$ . By [F5] the vanishing is independent of the inertial chart in which the components are computed.

1.2F2given

Clause (b). The adopted dust model is exactly the pair of equations $\partial_{a} (ρ n^{a}) = 0$ and $n^{a} \partial_{a} n^{b} = 0$ with $ρ > 0$ ; the forward direction of [F2] gives $\partial_{a} T^{a b} = 0$ for $T = ρ n \otimes n$ , and the reverse direction gives that conservation with $ρ > 0$ implies both equations.

1.3F3given

Clause (c). The adopted fluid model is the pair of equations obtained by setting both bracket expressions of [F3] to zero; by the equivalence in [F3] this is exactly $\partial_{a} T^{a b} = 0$ for $T = (ε + P) n \otimes n + P η^{- 1}$ , in both directions.

1.4given

Clause (d). The definition Stress-energy tensor, force density and local balance on flat spacetime states explicitly in its clause (d) that it asserts no conservation of any prescribed symmetric field, and the present theorem derives conservation only from the adopted model equations of clauses (a)–(c); accordingly for a $C^{1}$ symmetric field prescribed on $Ω$ that is not obtained from one of the adopted models together with its field equations, no conservation is asserted, and any conservation statement about such a field would be an additional explicit premise.

1.5F4F5given

Clause (e), conditional homogeneous-dust reading. Consider a homogeneous dust realization with comoving identical particles of rest mass $m > 0$ and rest number density $ν > 0$ , so that the adopted reading gives $ρ = ν m c^{2}$ , and consider a rest rectangular parcel of rest volume $V_{0}$ . Take the tensor $T = ρ n \otimes n$ in the rest frame, where $n = e_{0}$ , so that $T^{00} = ρ$ and all other components vanish. The passive boost with parameter $β$ along the parcel's motion direction has $Λ^{0}_{0} = γ$ and is proper orthochronous (The passive Lorentz boost with parameter beta), so the rank-two law of [F5] gives $T^{' 00} = Λ^{0}_{a} Λ^{0}_{b} T^{a b} = (Λ^{0}_{0})^{2} ρ = γ^{2} ρ$ ; thus $T^{' 00} = γ^{2} ν m c^{2} = (γ m c^{2}) (γ ν)$ . In a frame related to the rest frame by that boost the simultaneous volume of the parcel is $V_{0} / γ$ by the length-contraction part of [F4] (parallel length $1 / γ$ , transverse lengths unchanged), so the particle number density in that frame is $ν^{'} = N / (V_{0} / γ) = γ ν$ ; by the massive-particle reading [F4] each particle carries energy $γ m c^{2}$ . Hence the two factors in $γ^{2} ρ$ correspond respectively to particle energy and volume/density in the boosted frame. This is a conditional interpretation of the factorization within the homogeneous particle realization; it is not a derivation of the continuum model from microscopic dynamics.

1.6given

Empty-domain branch. If $Ω = \emptyset$ , Adopted continuum matter models, evolution equations and observer readings supplies only the unique empty fields described by C^k tensor fields on affine Minkowski space and their inertial components. Their component derivatives and products are empty maps; every equality asserted at a point of $Ω$ is therefore vacuous, including the two directions of the dust and fluid equivalences. There is no nonempty parcel in $Ω$ to which clause (e)'s conditional realization can apply. This proves the separate mathematical empty-domain clause without invoking the named nonempty-domain suppliers on that domain.

2.1step 1.1step 1.2step 1.3step 1.4step 1.5step 1.6∎

In the nonempty adopted-model branch, clauses (a), (b), (c) and (d) are steps 1.1–1.4 and clause (e) is step 1.5; the separate mathematical empty-domain clause is step 1.6. Every conservation conclusion is a conditional deduction from the adopted model equations; the theorem asserts no interaction, force law, cross section, measurement or empirical fact, contains no empirical premise and makes no claim about arbitrary prescribed tensors. All choices are finite; no choice principle is used.

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Integrated conservation of the isolated total slice four-momentum

Physical scope

Flat affine Minkowski spacetime in one inertial chart; the total stress-energy tensor of a system subject to the isolation premise on a finite box and length-coordinate interval, with vanishing side flux. The slice integrals are chart-component integrals of a finite box; the theorem asserts no transformation law for them, no noncompact, asymptotic or gravitational-energy statement, and no measurement; the reading of the integrals as four-momentum is the model's interpretation.

Statement

Fix $c > 0$ and an inertial chart $x$ with $x^{0} = c t$ . Let $\overset{ˉ}{K} = \prod_{i = 1}^{3} [a_{i}, b_{i}]$ , $a_{i} < b_{i}$ , be a nondegenerate closed spatial rectangle and let $s_{1} < s_{2}$ be real length-coordinate values, and let $T$ be the total stress-energy tensor of a system in the situation of Isolation and the total stress-energy tensor: $T$ is $C^{1}$ and symmetric on a neighbourhood of $[s_{1}, s_{2}] \times \overset{ˉ}{K}$ , satisfies $\partial_{a} T^{a b} = 0$ there (for the adopted models this is the conclusion of Local conservation of the adopted matter stress-energy tensors; for any other system it is an explicit premise), and satisfies the zero side-flux condition $T^{i b} = 0$ on the spatial faces for $i = 1, 2, 3$ . Put $Q^{b} (s) : = \int_{\overset{ˉ}{K}} T^{0 b} (s, x) d^{3} x$ and $P^{b} : = Q^{b} / c$ . Then: (a) $Q^{b} (s)$ is finite for every $s \in [s_{1}, s_{2}]$ and $Q^{b} (s_{1}) = Q^{b} (s_{2})$ ; (b) consequently $P^{b}$ is independent of $s \in [s_{1}, s_{2}]$ ; if $P$ is future timelike, its invariant mass $\sqrt{- η (P, P)} / c$ is independent of $s$ as well; (c) if the side flux does not vanish, the exact balance is the finite-flux identity of Finite-box stress balance and the flux form of conservation, and no conservation is asserted; (d) no transformation law for $P^{b}$ under a change of inertial chart or slicing is asserted: without an admissible-region flux argument, $P^{b}$ is a chart-component integral and is not claimed to be a four-vector; no noncompact, asymptotic or gravitational-energy statement is made, and the reading of $P^{b}$ as the system's four-momentum is the model's interpretation of the integral, not a measurement.

Facts & Assumptions

Given: The affine Minkowski model with inertial chart $x$ (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model), $c > 0$ , the nondegenerate closed spatial rectangle $\overset{ˉ}{K}$ of the Statement, length-coordinate values $s_{1} < s_{2}$ , and the total tensor $T$ of the isolation premise Isolation and the total stress-energy tensor with its stated hypotheses; $Q^{b} (s) = \int_{\overset{ˉ}{K}} T^{0 b} (s, x) d^{3} x$ and $P^{b} = Q^{b} / c$ ; the conservation of $T$ is either the conclusion of Local conservation of the adopted matter stress-energy tensors for the adopted models or an explicit premise for any other system.

[F1]

Finite-flux theorem: for a $C^{1}$ symmetric field on a neighbourhood of the box, $Q^{b} (s) = \int_{\overset{ˉ}{K}} T^{0 b} (s, x) d^{3} x$ is finite and continuous in $s$ , and for $c_{0} \leq s_{1} < s_{2} \leq d_{0}$ the slice balance $Q^{b} (s_{2}) - Q^{b} (s_{1}) = \int_{[s_{1}, s_{2}] \times \overset{ˉ}{K}} \partial_{a} T^{a b} d^{4} x - \sum_{i = 1}^{3} (\int_{[s_{1}, s_{2}] \times {\overset{ˉ}{K}}_{i}^{+}} T^{i b} d^{3} x - \int_{[s_{1}, s_{2}] \times {\overset{ˉ}{K}}_{i}^{-}} T^{i b} d^{3} x)$ holds; if the force density vanishes on $[s_{1}, s_{2}] \times \overset{ˉ}{K}$ and $T^{i b} = 0$ on the side faces, then $Q^{b} (s_{1}) = Q^{b} (s_{2})$ . (Finite-box stress balance and the flux form of conservation,(b),(c), Stress-energy tensor, force density and local balance on flat spacetime)

[F2]

Total four-momentum and invariant mass: for a vector $P = \sum_{b} P^{b} e_{b}$ built from a chase of chart components $P^{b}$ , if $P$ is future timelike then its invariant mass is $\sqrt{- η (P, P)} / c > 0$ , and if $P$ is future null the convention assigns mass $0$ . (Total four-momentum, invariant mass and center-of-momentum frame of a finite system, Timelike, null, spacelike and causal vectors of a Lorentz vector space)

Proof

technique · direct

1.1F1given

Clause (a). By the isolation premise, $T$ is $C^{1}$ and symmetric on a neighbourhood of $[s_{1}, s_{2}] \times \overset{ˉ}{K}$ ; [F1] applied to this box gives that $Q^{b} (s)$ is finite and continuous for every $s \in [s_{1}, s_{2}]$ , and since $\partial_{a} T^{a b} = 0$ on the box and the zero side-flux condition holds on its spatial faces, the balance identity of [F1] reduces to $Q^{b} (s_{2}) - Q^{b} (s_{1}) = 0$ , i.e. $Q^{b} (s_{1}) = Q^{b} (s_{2})$ .

1.2F1given

Clauses (c) and (d). If the side flux does not vanish, the exact statement available is the full balance identity of [F1] with its side-flux terms; the derivation above never establishes the vanishing of the differences, so no conservation of $Q^{b}$ is asserted in that case. The argument is carried out entirely in one inertial chart: $Q^{b}$ is defined by the chart components $T^{0 b}$ of $T$ and no transformation law for $Q^{b}$ or $P^{b}$ under a change of chart or of slicing is used or claimed; the identification of $P^{b}$ with the system's four-momentum is the model's reading of the integral, not a measurement, and no noncompact or asymptotic limit is taken.

2.1F2step 1.1

Clause (b). Evaluating clause (a) with $(s_{1}, s_{2})$ replaced by any pair $r < s$ in $[s_{1}, s_{2}]$ gives $Q^{b} (s) = Q^{b} (r)$ for all $r, s$ in the interval, so $Q^{b}$ is constant there and $P^{b} = Q^{b} / c$ is constant as well. Form the vector $P : = \sum_{b} P^{b} e_{b}$ in the frame of the chart; since the components are constant in $s$ , $η (P, P)$ is a constant real number, so if $P$ is future timelike its invariant mass $\sqrt{- η (P, P)} / c$ is independent of $s$ , and if it is future null the assigned mass $0$ is likewise independent.

3.1step 1.1step 2.1step 1.2∎

Clauses (a) and (b) are steps 1.1 and 2.1 and clauses (c) and (d) are step 1.2. The theorem is a finite conditional deduction on one box in one inertial chart; it asserts no four-vector transformation law, no noncompact or gravitational-energy statement and no measurement; for the adopted models the conservation hypothesis is supplied by Local conservation of the adopted matter stress-energy tensors, and for any other system it is an explicit premise of this theorem, separate from the isolation postulate. All choices are finite; no choice principle is used.

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The two force carriers and the scope of continuum conservation

Physical scope

Flat affine Minkowski spacetime; scope remarks on the force vocabulary, the conditional character of the conservation statements and the exclusions of the continuum framework. No gravity, electromagnetic coupling, equation of state, energy condition, angular-momentum theorem or empirical premise is asserted anywhere on this page.

Two force carriers

The force vocabulary of this framework has two distinct mathematical carriers. On a massive particle worldline the four-force is $F = \dot{P} = d P / d τ$ (Proper-time derivative of a vector field along a worldline and the four-force notation), and its identities are conditional on the adopted particle dynamics Adopted massive-particle momentum, observer measurement and isolated-collision conservation. In the continuum the force carrier is the four-force density $f^{b} = \partial_{a} T^{a b}$ of a symmetric type- $(2, 0)$ field (Stress-energy tensor, force density and local balance on flat spacetime). Neither object is a force law: the definition of $\dot{P}$ asserts no equation of motion, and the definition of $f$ asserts no conservation and no interaction.

Conservation is conditional

The page's conservation statements are conditional deductions. Local conservation of the scalar, dust and fluid tensors is proved from the adopted model equations of Local conservation of the adopted matter stress-energy tensors; for any other matter the conservation of $T$ must be adopted or derived separately, and no conservation is asserted for an arbitrary prescribed symmetric field. Integrated conservation additionally requires the isolation premise of Isolation and the total stress-energy tensor and is asserted only for finite boxes and slices in one inertial chart Integrated conservation of the isolated total slice four-momentum; it does not supply a four-vector transformation law for the slice integrals and does not extend to noncompact regions.

Exclusions

The framework contains no gravity, no electromagnetic coupling, no derivation of an equation of state, no energy conditions — the printed dominant-energy-condition inequality of the Crowell source p. 191 is defective and is not used — no angular-momentum theorem and no empirical premise; no measurement, apparatus or uncertainty claim is made anywhere on this page. The energy conditions, the electromagnetic field stress tensor and its coupling to charged matter, and angular-momentum flux are developed, if at all, by other frameworks or pairs of the library, not here.

5 · Examples, experiments and counterexamples

None yet.

Forces and Continuum

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Stress-energy tensor, force density and local balance on flat spacetime

Definition

Remarks

Observer decomposition of a symmetric stress-energy tensor

Statement

Facts & Assumptions

Proof

Compact-support variation: the fundamental lemma, integration by parts and differentiation under the integral sign

Statement

Facts & Assumptions

Proof

First variation of the scalar-field action and the Euler-Lagrange equation

Statement

Facts & Assumptions

Proof

The Hilbert stress-energy tensor of the scalar-field action on flat spacetime

Statement

Facts & Assumptions

Proof

Divergence identity for the scalar-field stress-energy tensor

Statement

Facts & Assumptions

Proof

Dust: conservation of rho n tensor n is equivalent to continuity and inertial flow

Statement

Facts & Assumptions

Proof

Perfect-fluid divergence identity and its equivalence with the energy-balance and Euler equations

Statement

Facts & Assumptions

Proof

Finite-box stress balance and the flux form of conservation

Statement

Facts & Assumptions

Proof

Hodge decomposition of an alternating two-form for an observer, invariants and the optional field identification

Statement

Facts & Assumptions

Proof

Adopted continuum matter models, evolution equations and observer readings

Postulate

Scope and status

Isolation and the total stress-energy tensor

Postulate

Scope and status

Local conservation of the adopted matter stress-energy tensors

Statement

Facts & Assumptions

Proof

Integrated conservation of the isolated total slice four-momentum

Statement

Facts & Assumptions

Proof

The two force carriers and the scope of continuum conservation

Two force carriers

Conservation is conditional

Exclusions

5 · Examples, experiments and counterexamples

Sources