Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress

Adopt the framework's unit and reading conventions (SI unit and coordinate conventions of the classical gravitational framework) and the stress-energy slot (Stress-energy tensor field of a model). Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold, let $U\subseteq M$ be a nonempty open subset, let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ and let $n$ be a smooth observer congruence on $U$ (Observer congruences, their observer curves and adapted charts). Let $\mu$, $J$, $\mathrm{\Phi }$ and $S$ be the fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress.

(a) Measured energy density and its unit. The smooth function $\mu =T_{ab}{n}^a{n}^b$ is read as the energy density measured by the observers of the congruence $n$, with numerical values in $\mathrm{J}{\mathrm{m}}^{-3}$ when $T$ is read in the orthonormal-frame convention of SI unit and coordinate conventions of the classical gravitational framework clauses (a) and (d). The same item fixes the energy-density reading of ${\rho }_n=T(n,n)$ for a future unit timelike $n$.

(b) Energy flux and momentum density. The smooth spatial vector field $\mathrm{\Phi }=h(J)$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress is the tensor's spatial current in the normalized observer convention. Its components have energy-density units, $\mathrm{J}{\mathrm{m}}^{-3}$. Define the SI measured energy-flux vector $q:=c\mathrm{\Phi }$ and SI measured momentum-density covector $p:={\mathrm{\Phi }}^{\mathrm{♭}}\mathrm{/}c$, where ${\mathrm{\Phi }}^{\mathrm{♭}}(a)=g(\mathrm{\Phi },a)$ and $c$ is the conversion constant of SI unit and coordinate conventions of the classical gravitational framework. In every ordered orthonormal basis with future $e_0=n_p$,

${\mathrm{\Phi }}^i=T^{0i}=-T(e_0,e_i),q^i=c{\mathrm{\Phi }}^i,p_i={\mathrm{\Phi }}^i\mathrm{/}c,i=1,2,3,$

by The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness (d). Hence $q^i$ has units $\mathrm{W}{\mathrm{m}}^{-2}$ and $p_i$ has units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$; equivalently $q^{\mathrm{♭}}=c^{2}p$. When the algebraic balance formulas call $\mathrm{\Phi }$ an energy flux they use this normalized tensor current, whose SI energy flux is $c\mathrm{\Phi }$. The analogous natural-unit momentum notation suppresses the factor $1\mathrm{/}c$; it never assigns SI momentum units directly to ${\mathrm{\Phi }}^{\mathrm{♭}}$. Symmetry of $T$ gives the two aspects of the same mixed component, with these distinct SI conversions.

(c) Measured stress. Write $S_{ab}=g_{ac}{g}_{bd}{S}^{cd}$ for the covariant form of the projected field of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress clause (e). Explicitly, for $p\in U$ and $v,w\in T_{p}M$, $S_p(v,w):=T_p(h_p(v),h_p(w))$, where $h_p(v)=v+g_p(n_p,v)n_p$. This agrees with the lowered component formula because $h_p$ is self-adjoint for $g_p$. This smooth symmetric covariant field is read as the stress measured by the observers of the congruence: for unit spatial vectors $e,e^{\mathrm{\prime }}$ in the rest space of $n_p$, $S_p(e,e^{\mathrm{\prime }})$ is read as the force per unit area in the direction $e$ across a surface whose normal is $e^{\mathrm{\prime }}$, with components in $\mathrm{J}{\mathrm{m}}^{-3}$ by the conventions.

(d) Current and charge reading. For this clause choose an orientation on $U$ and the volume form of its restricted metric; this choice is independent of the time orientation. The smooth vector field $J$ of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress clause (c) is read as the energy-momentum current density measured by the congruence, and for a finite smooth singular $3$-chain $\mathrm{\Sigma }$ of $U$ the flux ${\mathrm{\Phi }}_J(\mathrm{\Sigma })$ of The metric volume form, current flux through 3-faces and finite-chain charges clause (b) is read as the energy-momentum assigned to $\mathrm{\Sigma }$ by the congruence, in the sense of the charge reading of that item. For a $3$-chain carried by a spacelike slice whose normal is (up to sign) $n$, the index-lowered component $-J_a{n}^a=\mu$ is the density integrated in that reading.

(e) Scope. The reading is interpretive and applies to every prescribed smooth symmetric $T$ and every observer congruence. No positivity of $\mu$, no conservation of $J$, no dynamics, no existence of a matter model realizing $T$ and no measurement is asserted; the definition is confined to the smooth branch of the framework, and $T$ need not satisfy any field equation.