Divergence of the measured energy-momentum current and the finite-chain balance identity

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$, let $U\subseteq M$ be a nonempty open subset, let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ with ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $U$, and let $n$ be a smooth observer congruence on $U$. Let $\mu$, $J$, $\mathrm{\Phi }$, $S$ be the fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress and let $a$, $\theta$, $D$ be the kinematic fields of Acceleration, expansion and the along-congruence derivative of an observer congruence.

(a) Divergence identity. On $U$,

${\mathrm{\nabla }}_a{J}^a=-T^{ab}{\mathrm{\nabla }}_a{n}_b\mathrm{.}$

(b) Balance identity. On $U$,

$D\mu +\mu \theta +{\mathrm{\nabla }}_a{\mathrm{\Phi }}^a=-T^{ab}{\mathrm{\nabla }}_a{n}_b\mathrm{.}$

(c) Finite-chain balance. For this clause choose an orientation on $U$ and use its metric volume form ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g{\mathrm{\mid }}_U}$, written ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Let $C$ be a finite smooth singular $4$-chain of $M$ contained in $U$ with induced boundary $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$, where ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2,B$ are smooth singular $3$-chains (The metric volume form, current flux through 3-faces and finite-chain charges clause (b)). Then

${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_J(B)+{\int }_C(-T^{ab}{\mathrm{\nabla }}_a{n}_b){\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

In particular, if $T^{ab}{\mathrm{\nabla }}_a{n}_b=0$ on an open subset containing the image of $C$, then ${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_J(B)$.

(d) Killing contraction. If $\xi$ is a smooth vector field on $U$ with ${\mathrm{\nabla }}_a{\xi }_b+{\mathrm{\nabla }}_b{\xi }_a=0$ (a Killing field, Killing fields of a Lorentzian metric), then $T^{ab}{\mathrm{\nabla }}_a{\xi }_b=0$ on $U$.

• Source of the balance. The right-hand side $-T^{ab}{\mathrm{\nabla }}_a{n}_b$ is a source term produced by the congruence: it vanishes identically for a parallel congruence ($\mathrm{\nabla }n=0$), and for a normalized timelike Killing congruence it reduces to $-{\mathrm{\Phi }}^a{a}_a$ by Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current clause (c). The measured energy current alone therefore need not be conserved; the Killing-weighted current of that lemma is guaranteed to be conserved under its stated hypotheses.

• No positivity or dynamics. The lemma uses only ${\mathrm{\nabla }}^a{T}_{ab}=0$ and the congruence property; no energy condition, no equation of state and no existence of a solution of the field equation is used or asserted.