The two force carriers and the scope of continuum conservation

The force vocabulary of this framework has two distinct mathematical carriers. On a massive particle worldline the four-force is $F=\dot{P}=dP\mathrm{/}d\tau$ (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 $f^b={\mathrm{\partial }}_a{T}^{ab}$ 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.

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.

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.