Isolation and the total stress-energy tensor

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 $x^0=ct$. Choose real $s_1<s_2$ and a nondegenerate closed spatial rectangle $\bar{K}={\prod }_{i=1}^3[a_i,b_i]$, $a_i<b_i$. The interval $[s_1,s_2]$ parametrizes $x^0$ in metres; the corresponding coordinate-clock interval is $[s_1\mathrm{/}c,s_2\mathrm{/}c]$ 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 $[s_1,s_2]\times \bar{K}$ (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 $T^{ib}=0$ 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.

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 $f^b={\mathrm{\partial }}_a{T}^{ab}$, which is a separate hypothesis of the integration statements that use this postulate.