Kinematics of a timelike Killing congruence and conservation of the redshift-weighted energy current

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, and let $\xi$ be a smooth Killing field on $U$ (Killing fields of a Lorentzian metric) that is everywhere timelike and future-directed,

$g_p({\xi }_p,{\xi }_p)<0,g_p({\xi }_p,N_p)<0\text{for every }p\in U\mathrm{.}$

Put $X:=\sqrt{-g(\xi ,\xi )}:U\to (0,\mathrm{\infty })$ (the positive square root) and $n:=\xi \mathrm{/}X$, so that $n$ is a smooth observer congruence on $U$. Let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ with ${\mathrm{\nabla }}^a{T}_{ab}=0$ 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. Then:

(a) Normalization is constant along the congruence, and the acceleration is a spatial gradient.

$n^a{\mathrm{\nabla }}_{a}X=0,a_a={\mathrm{\nabla }}_a\ln X=\frac{1}{X}{\mathrm{\nabla }}_{a}X,$

equivalently ${\mathrm{\nabla }}_{a}X=X{a}_a$ and $a=\mathrm{\nabla }\ln X$ is the spatial gradient of $\ln X$.

(b) Vanishing expansion and symmetric part. $\theta =0$ and

${\mathrm{\nabla }}_a{n}_b+{\mathrm{\nabla }}_b{n}_a=-(n_a{a}_b+n_b{a}_a)\mathrm{.}$

(c) The divergence source is the acceleration term. $T^{ab}{\mathrm{\nabla }}_a{n}_b=J^a{a}_a={\mathrm{\Phi }}^a{a}_a$, and consequently

${\mathrm{\nabla }}_a{J}^a=-{\mathrm{\Phi }}^a{a}_a\mathrm{.}$

(d) Conservation of the redshift-weighted current. On $U$,

${\mathrm{\nabla }}_a(X{J}^a)=0,{\mathrm{\nabla }}_a\left(T^{ab}{\xi }_b\right)=0.$

For the following flux assertion choose an orientation on $U$ and the volume form of the restricted metric. Then for every finite smooth singular $4$-chain $C$ of $M$ contained in $U$ with induced boundary $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$,

${\mathrm{\Phi }}_{XJ}({\mathrm{\Sigma }}_2)-{\mathrm{\Phi }}_{XJ}({\mathrm{\Sigma }}_1)=-{\mathrm{\Phi }}_{XJ}(B),$

where ${\mathrm{\Phi }}_{XJ}$ is the flux of the smooth vector field $XJ$ (The metric volume form, current flux through 3-faces and finite-chain charges clause (b)).

• Interpretation of the weight. $X=\sqrt{-g(\xi ,\xi )}$ is the normalization of the Killing field, and $XJ$ is the energy-momentum current of the congruence weighted by it; in a static model $X$ is the redshift factor between the congruence observers and the symmetry normalization. The unweighted current $J$ is not conserved in general: its divergence is $-{\mathrm{\Phi }}^a{a}_a$, the acceleration source of clause (c).

• Hypotheses. Only the Killing equation, the timelike direction of $\xi$, and ${\mathrm{\nabla }}^a{T}_{ab}=0$ are used. No extension of $\xi$ or $T$ beyond $U$, no completeness of the congruence and no asymptotic control of the chains is assumed or asserted.

• Choice convention. The smooth structure of the tangent bundle underlying $\xi$ and $n$ assumes ${\mathrm{A}\mathrm{C}}_{\omega }$ through Observer congruences, their observer curves and adapted charts; no other choice principle is used.