The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model), let $U\subseteq M$ be a nonempty open subset carrying the restricted smooth structure of An open subset of a smooth manifold has a canonical restricted smooth structure, let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold), and let $n$ be a smooth observer congruence on $U$ (Observer congruences, their observer curves and adapted charts), so that $g_p(n_p,n_p)=-1$ and $n_p$ is future-directed at every $p\in U$.

(a) Spatial projection field. Let

$h^a{}_b:={\delta }^a{}_b+n^a{n}_b$

on $U$, the component expression of the field of $g$-orthogonal projections onto the rest spaces $n_p^{\perp }$ of Observers, rest spaces and the spatial projection at an event (The observer rest-space decomposition and the spatial projection); at every $p$ one has $h^a{}_b{n}^b=0$, $n_a{h}^a{}_b=0$, $h^a{}_b{h}^b{}_c=h^a{}_c$ and $h^a{}_a=3$. The field is smooth because $n$, $g$ and the identity are smooth (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, $C^r$ and smooth maps between smooth manifolds).

(b) Measured energy density. The measured energy density of $T$ for the congruence $n$ is the smooth function

$\mu :=T_{ab}{n}^a{n}^b:U\to \mathbb{R}\mathrm{.}$

(c) Energy-momentum current. The energy-momentum current of $T$ for the congruence $n$ is the smooth vector field on $U$ obtained by raising one index of $T$ with the inverse metric and contracting with $n$,

$J^a:=-T^{ab}{n}_b,$

with $T^{ab}=g^{ac}{g}^{bd}{T}_{cd}$ (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components); the minus sign is the convention that fixes the energy-component sign convention. A positive $\mu$ alone does not imply that $J$ is causal: $J$ may have an arbitrarily large spatial part. No future-causal property of $J$ is imposed by this definition.

(d) Energy flux. The energy flux of $T$ for the congruence $n$ is the smooth vector field

${\mathrm{\Phi }}^a:=h^a{}_b{J}^b=J^a-\mu n^a,$

the spatial projection of $J$; it satisfies $n_a{\mathrm{\Phi }}^a=0$ and ${\mathrm{\Phi }}^a=-h^a{}_b{T}^{bc}{n}_c$.

(e) Spatial stress. The spatial stress of $T$ for the congruence $n$ is the smooth field obtained by projecting both tensor slots,

$S^{ab}:=h^a{}_c{h}^b{}_d{T}^{cd},$

which is symmetric in $a,b$ and satisfies $S^{ab}{n}_b=0$.

The identities $\mu =-J^a{n}_a$, $J=\mu n+\mathrm{\Phi }$, $S^{ab}{n}_b=0$ and the orthogonal decomposition

$T^{ab}=\mu n^a{n}^b+n^a{\mathrm{\Phi }}^b+n^b{\mathrm{\Phi }}^a+S^{ab},$

together with the uniqueness of that decomposition and the component formulas in a $g$-orthonormal frame, are proved in The orthogonal observer decomposition of a symmetric two-tensor and its uniqueness ↗; this definition asserts none of them. The physical reading and the units of $\mu$, $\mathrm{\Phi }$ and $S$ are fixed by the framework's reading on this page and are not asserted here.

• Scope. The objects are defined for every smooth symmetric covariant $2$-tensor field $T$ on $U$ and every smooth observer congruence $n$. No symmetry of $T$ beyond its stated symmetry, no conservation equation, no positivity, no dynamics and no measurement is required or implied. In particular $T$ need not be the stress-energy tensor of any matter model.

• Choice convention. The smooth structure of the tangent bundle underlying the vector field $n$ assumes the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) (A smooth vector field is a smooth section of the tangent bundle, Observer congruences, their observer curves and adapted charts); this is the only choice principle used in this item.