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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.

Integrated conservation of the isolated total slice four-momentum

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources