The timelike Killing current and the charge of a three-chain

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, choose an orientation on $U$ and its metric volume form (independent of the time orientation), let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold) — in an application this is the stress-energy tensor field of the model's matter sector (Stress-energy tensor field of a model) — and let $\xi$ be a Killing field of $(M,g)$ on $U$ (Killing fields of a Lorentzian metric, Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor clause (a)) 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 $X:=\sqrt{-g(\xi ,\xi )}>0$ on $U$ and $n:=\xi \mathrm{/}X$, an observer congruence on $U$: $g(n,n)=-1$ and $g(n,N)=g(\xi ,N)\mathrm{/}X<0$ (Observer congruences, their observer curves and adapted charts).

(a) Killing current. The Killing current of the pair $(T,\xi )$ is the smooth vector field on $U$

$J_{\xi }{}^a:=T^{ab}{\xi }_b\mathrm{.}$

Since $\xi =Xn$ and $J^a=-T^{ab}{n}_b$, one has $T^{ab}{\xi }_b=-X{J}^a$, where $J=-T(\cdot ,n)$ is the measured energy-momentum current of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress; no conservation of $J_{\xi }$ is asserted by this definition.

(b) Killing charge of a three-chain. For every finite smooth singular $3$-chain $\mathrm{\Sigma }$ of $M$ contained in $U$, the Killing charge of $\mathrm{\Sigma }$ with respect to $\xi$ is the real number

$Q_{\xi }(\mathrm{\Sigma }):=-{\mathrm{\Phi }}_{J_{\xi }}(\mathrm{\Sigma }),$

where ${\mathrm{\Phi }}_{J_{\xi }}$ is the flux of the current $J_{\xi }$ through $3$-chains on the oriented open submanifold $U$ in the sense of The metric volume form, current flux through 3-faces and finite-chain charges clause (b).

When $\xi$ is normalized with a length-valued flow parameter in the SI length-coordinate convention, this charge is read as a Killing energy.

(c) Relation to the measured energy current. Since $J_{\xi }=-XJ$,

$Q_{\xi }(\mathrm{\Sigma })={\mathrm{\Phi }}_{XJ}(\mathrm{\Sigma }),$

the flux of the redshift-weighted measured energy current $XJ$. For a $3$-simplex $\sigma$ that is non-null with velocity basis $(v_1,v_2,v_3)$ and adapted unit normal ${\nu }^{\ast }$ (so that $({\nu }^{\ast },v_1,v_2,v_3)$ is positively oriented and $\epsilon :=g({\nu }^{\ast },{\nu }^{\ast })\in \{-1,1\}$), the pullback of ${\iota }_{XJ}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ is $\epsilon g(XJ,{\nu }^{\ast })\sqrt{\mathrm{\mid }\det (g(v_i,v_j))\mathrm{\mid }}d{t}^1\wedge d{t}^2\wedge d{t}^3$ (Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (c)); when ${\nu }^{\ast }=-n$ and the $v_i$ span $n^{\perp }$ this coefficient equals $-X\mu \sqrt{\mathrm{\mid }\det (g(v_i,v_j))\mathrm{\mid }}$, so up to the orientation sign carried by the adapted normal, $Q_{\xi }(\mathrm{\Sigma })$ is the integral of the redshift-weighted measured energy density $X\mu$ over $\mathrm{\Sigma }$. The factor $X$ carries the normalization and units of $\xi$.

(d) Reading and scope. $Q_{\xi }(\mathrm{\Sigma })$ is the charge that the model assigns to the chain $\mathrm{\Sigma }$ under the symmetry $\xi$, in the framework of the postulates Adopted spacetime model and classical domain of the gravitational framework, Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions with the assigned tensor and symmetry; it does not require the field law for its definition. Under the unit conventions of SI unit and coordinate conventions of the classical gravitational framework, if $T$ is read in $\mathrm{J}{\mathrm{m}}^{-3}$ and the flow parameter of $\xi$ is measured in metres, then $X$ is dimensionless and $Q_{\xi }(\mathrm{\Sigma })$ is read in $\mathrm{J}$. More generally, if that parameter has unit $\mathcal{S}$, then in a length-coordinate chart $X$ has units $\mathrm{m}\mathrm{/}\mathcal{S}$ and $Q_{\xi }$ has units $\mathrm{J}\mathrm{m}\mathrm{/}\mathcal{S}$; converting it to an energy requires an explicitly specified normalization factor. The value depends on the model, on $\xi$ (replacing $\xi$ by a dimensionless constant multiple $\lambda \xi$ with $\lambda >0$ multiplies $X$, $J_{\xi }$ and $Q_{\xi }$ by $\lambda$), and on the chain $\mathrm{\Sigma }$. The definition asserts no conservation, no positivity, no finiteness for unbounded chains and no measured value. If additionally ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $U$, the Killing current lemma gives conservation up to the side flux. The field law (Adopted Einstein field equation in the direct-equation presentation or Adopted classical gravitational action and variational field law) is one way to supply this divergence condition; it is not necessary for the mathematical conservation implication. If no timelike Killing field exists on $U$, $Q_{\xi }$ is not defined by this item.