Stress-energy tensor, force density and local balance on flat spacetime

Let $(A,V,\eta ,C^{+})$ be the affine Minkowski model (Oriented time-oriented affine Minkowski space (the spacetime primitives)), fix an independent numerical parameter $c>0$, and let $\mathrm{\Omega }\subseteq A$ be a nonempty open set.

(a) A stress-energy tensor on $\mathrm{\Omega }$ of class $C^k$, $k\ge 0$ (with $C^0$ meaning continuous), is a $C^k$ symmetric type-$(2,0)$ tensor field $T$ on $\mathrm{\Omega }$ in the sense of C^k tensor fields on affine Minkowski space and their inertial components: at every $p\in \mathrm{\Omega }$ the map $T(p):V^{\ast }\times V^{\ast }\to \mathbb{R}$ is bilinear and symmetric, so in every inertial chart its components satisfy $T^{ab}=T^{ba}$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii)).

(b) In each inertial chart $x$ of the model (Inertial frames and inertial charts of the affine Minkowski model) define the component families ${\epsilon }_x:=T^{00}$, ${\pi }_x^i:=T^{0i}\mathrm{/}c=T^{i0}\mathrm{/}c$ and $S_x^{ij}:=T^{ij}$. These are numerical component and scaling conventions involving the fixed positive number $c$; their transformation behaviour is inherited from $T$ and no physical units or measurement interpretation form part of this mathematical definition.

(c) When $k\ge 1$, the four-force density of $T$ is the $C^{k-1}$ field whose inertial components are $f^b:={\mathrm{\partial }}_a{T}^{ab}$, where ${\mathrm{\partial }}_a$ are the coordinate partial derivatives of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ and the sum over $a$ is the standard contraction of a contravariant index with a coordinate derivative. The components ${\mathrm{\partial }}_a{T}^{ab}$ transform as a vector, so $f$ is a well-defined vector field on $\mathrm{\Omega }$ independent of the inertial chart: under an inertial transition $x^{\mathrm{\prime }}=\mathrm{\Lambda }x+a$ the components satisfy ${\mathrm{\partial }}_a^{\mathrm{\prime }}{T}^{\mathrm{\prime }ab}={\mathrm{\Lambda }}^b{}_d{\mathrm{\partial }}_c{T}^{cd}$ (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iv)), which is exactly the vector transformation law of clause (i) applied to the components $f^b$. For $k\ge 1$ one says that $T$ is locally conserved, or balanced, on $\mathrm{\Omega }$ when $f=0$ on $\mathrm{\Omega }$, and that $T$ is balanced against the four-force density $f$ when ${\mathrm{\partial }}_a{T}^{ab}=f^b$.

(d) The definition asserts no conservation of any prescribed symmetric field $T$: whether a given $T$ is conserved, whether it arises from an action or a matter model, and whether any of its integrals over a slice converges are separate hypotheses stated elsewhere. No choice principle is used. For a merely continuous tensor ($k=0$) the component conventions and pointwise contractions of (b) are defined, while the divergence, the four-force density and the balance terminology of (c) require the additional $C^1$ hypothesis; no derivative of a merely continuous field is asserted.

The symmetry condition in (a) is chart-independent by the rank-two component law (Transformation of tensor, covector, derivative and current components under affine Lorentz maps,(iii)): if $T^{ab}=T^{ba}$ in one inertial chart, then $T^{\mathrm{\prime }ab}={\mathrm{\Lambda }}^a{}_c{\mathrm{\Lambda }}^b{}_d{T}^{cd}={\mathrm{\Lambda }}^b{}_d{\mathrm{\Lambda }}^a{}_c{T}^{dc}=T^{\mathrm{\prime }ba}$ in every other. The constants ${\epsilon }_x,{\pi }_x,S_x$ of (b) are numerical component conventions only; the physical readings of such components — energy density, momentum density, stress, force density — are model premises adopted separately in Adopted continuum matter models, evolution equations and observer readings and used in Local conservation of the adopted matter stress-energy tensors. Nothing in this definition requires the metric contraction ${\eta }_{ab}{T}^{ab}$ to have a sign, requires any energy condition, or asserts that a locally conserved field is the total stress-energy of an isolated system.