Stress-energy tensor field of a model

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold and let $U\subseteq M$ be a nonempty open subset, the matter domain of the model, carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure.

A stress-energy tensor field on $U$ for the model is a smooth symmetric covariant $2$-tensor field $T$ on $U$ in the sense of Smooth covariant 2-tensor fields on a smooth manifold: at every $p\in U$, $T_p$ is a symmetric bilinear form $T_p:T_{p}M\times T_{p}M\to \mathbb{R}$ varying smoothly with $p$, with component functions $T_{ab}=T({\mathrm{\partial }}_a,{\mathrm{\partial }}_b)$ in every smooth chart of $U$. No nondegeneracy of $T_p$ is required or implied; in particular $T_p$ may have radical, as it does for a dust model whose stress-energy form vanishes on a hyperplane.

Unit and observer reading. At a point choose a $g_p$-orthonormal ordered basis $(e_0,e_1,e_2,e_3)$ with $e_0$ future-directed, meaning $g(e_0,N)<0$, and $g(e_A,e_B)=\mathrm{diag}(-1,1,1,1{)}_{AB}$. Such a basis is obtained from an orthonormal basis by reversing its timelike member if necessary. Write $T_{AB}=T(e_A,e_B)$ and $n=e_0$. Every $T_{AB}$ has the energy-density reading $\mathrm{J}{\mathrm{m}}^{-3}$ of SI unit and coordinate conventions of the classical gravitational framework. The energy density for this observer is ${\rho }_n=T_{00}$; the spatial component ${\varphi }^i=-T_{0i}=T^{0i}$ gives the SI energy-flux component $q_n^i=c{\varphi }^i$ in $\mathrm{W}{\mathrm{m}}^{-2}$ and the SI momentum-density component $(p_n{)}_i={\varphi }^i\mathrm{/}c$ in $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. The stress is $T_{ij}$ in $\mathrm{P}\mathrm{a}=\mathrm{J}{\mathrm{m}}^{-3}$, $i,j=1,2,3$. Thus the off-diagonal tensor entries themselves have energy-density units; their flux and momentum readings require the stated sign and $c$ factors. These names and readings interpret a prescribed tensor and a prescribed future observer, and assert no actual measurement. Different future orthonormal frames generally give different components, related by the corresponding Lorentz change of basis.

Relation to the matter model. This item defines the slot that a matter model fills. The identity of $T$ for a given model — its expression in terms of the model's fields and its equation of motion — comes from that model and is not part of this definition; for a model derived from an action, $T$ is the Hilbert stress-energy tensor of that action, while a model that prescribes $T$ by hand is a separate source model subject to the consistency conditions of the adopted gravitational field law. Nothing here asserts that a prescribed smooth symmetric $T$ is realized by any matter model, that $T$ is conserved, or that $T$ satisfies any positivity condition; conservation and the energy conditions are separate statements, the latter defined separately in this framework.

Scope limitations. The definition is confined to the smooth branch: a matter distribution whose density fails to be smooth across a surface, a distributional point source, or a source supported on a hypersurface is not a stress-energy tensor field of this definition, and such configurations require additional distributional or junction mathematics not supplied here. The matter domain $U$ is an idealization of the region occupied by matter; the model may also include matter sources described on all of $M$ or on a union of several such domains.