Adopted continuum matter models, evolution equations and observer readings

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 $\mathrm{\Omega }\subseteq A$ 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 $\mathrm{\Omega }$ 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 $\mathrm{\Omega }=\mathrm{\varnothing }$, 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 $\phi :\mathrm{\Omega }\to \mathbb{R}$, a smooth potential $V:\mathbb{R}\to \mathbb{R}$ with values in inverse square metres (${\mathrm{m}}^{-2}$) and a constant $K>0$ in joule per metre ($\mathrm{J}\mathrm{/}\mathrm{m}$) have the field equation $\mathrm{\square }\phi =V^{\mathrm{\prime }}(\phi )$ — 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 $\rho :\mathrm{\Omega }\to (0,\mathrm{\infty })$ (joule per cubic metre) and a $C^1$ future unit timelike field $n:\mathrm{\Omega }\to V$ ($\eta (n,n)=-1$) satisfy ${\mathrm{\partial }}_a(\rho n^a)=0$ and $n^a{\mathrm{\partial }}_a{n}^b=0$; the stress-energy tensor is $T=\rho n\otimes n$.

(iii) Perfect fluid. $C^1$ maps $\epsilon ,P:\mathrm{\Omega }\to \mathbb{R}$ (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 $T=(\epsilon +P)n\otimes n+P{\eta }^{-1}$, where ${\eta }^{-1}$ is the symmetric type-$(2,0)$ inverse metric with components ${\eta }^{ab}$.

(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 $\mathrm{\Omega }$).

(v) Adopted observer readings and units. For a tensor physically assigned to matter in this model, all inertial components $T^{ab}$ have units joule per cubic metre. In length coordinates $x^0=ct$, an ideal observer with future unit direction $n$ and four-velocity $U=cn$ reads the numerical contractions of Observer decomposition of a symmetric stress-energy tensor as energy density ${\epsilon }_n=T^{ab}{n}_a{n}_b$, momentum density ${\pi }_n^a=-c^{-1}{h}^a{}_b{T}^{bc}{n}_c$ and rest stress $S_n^{ab}=h^a{}_c{h}^b{}_d{T}^{cd}$, where $h(v)=v+\eta (n,v)n$. In an adapted inertial frame these are $T^{00}$, $T^{0i}\mathrm{/}c$ and $T^{ij}$ respectively, and the energy-flux components are $c{T}^{i0}=c^2{\pi }_n^i$. 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 $f^b={\mathrm{\partial }}_a{T}^{ab}$ 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 ${\mathrm{\partial }}_t(T^{00})+{\mathrm{\partial }}_i(c{T}^{i0})=c{f}^0$ and ${\mathrm{\partial }}_t(T^{0j}\mathrm{/}c)+{\mathrm{\partial }}_i{T}^{ij}=f^j$. Neither assignment selects a force law or asserts conservation.

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 $ct$ 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).