Local energy balance of the measured current and conservation of the Killing charge in stationary models

Adopt one of the two presentations of the framework's field law, Adopted Einstein field equation in the direct-equation presentation or Adopted classical gravitational action and variational field law, for a model with geometric data and matter sector satisfying Adopted spacetime model and classical domain of the gravitational framework and Adopted matter-content assumptions: fields, equations, stress-energy and energy conditions; the two presentations define one field law by Stationarity of the adopted total action is equivalent to the Einstein equation with the framework's coupling. Let $(M,g,N)$ be the model's smooth time-oriented Lorentzian four-manifold with an orientation, let $U\subseteq M$ be the matter domain with the induced orientation, let $T$ be the model's smooth symmetric stress-energy tensor field on $U$ (Stress-energy tensor field of a model), and let $n$ be a smooth observer congruence on $U$, read as the field of directions of ideal-clock observers of the model (Adopted clock, free-fall and light-ray assumptions of the classical gravitational framework). Let $\mu$, $J$, $\mathrm{\Phi }$, $S$ be the measured fields of The observer split of a symmetric two-tensor: energy density, energy-momentum current, flux and stress, with the physical reading and units of Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress: $\mathrm{\Phi }$ is the normalized spatial tensor current and its SI measured energy-flux vector is $q=c\mathrm{\Phi }$, and let $a$, $\theta$, $D$ be the kinematic fields of Acceleration, expansion and the along-congruence derivative of an observer congruence.

(a) On-shell conservation. The adopted field law implies

${\mathrm{\nabla }}^a{T}_{ab}=0\text{on }U\mathrm{.}$

(b) Local energy balance. On $U$,

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

Thus the energy density measured by the congruence, its normalized spatial current $\mathrm{\Phi }$ (SI energy flux $q=c\mathrm{\Phi }$) and its expansion obey a local balance whose source is the congruence term $-T^{ab}{\mathrm{\nabla }}_a{n}_b$; the measured energy current is conserved on an open set exactly where that source vanishes there.

(c) Finite-chain bookkeeping. For every finite smooth singular $4$-chain $C$ of $M$ contained in $U$ with induced boundary $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$,

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

In particular, if the source term vanishes on an open set containing the image of $C$ and the side flux ${\mathrm{\Phi }}_J(B)$ vanishes, then ${\mathrm{\Phi }}_J({\mathrm{\Sigma }}_2)={\mathrm{\Phi }}_J({\mathrm{\Sigma }}_1)$: equal measured energy needs a vanishing side flux, not merely on-shell conservation.

(d) Stationary models and the Killing charge. Suppose in addition that $U$ carries a Killing field $\xi$ of $(M,g)$ that is everywhere timelike and future-directed, with $X:=\sqrt{-g(\xi ,\xi )}>0$ and $n=\xi \mathrm{/}X$, and let $Q_{\xi }$ be the Killing charge of The timelike Killing current and the charge of a three-chain. Then:

1. ${\mathrm{\nabla }}_a(X{J}^a)=0$ and ${\mathrm{\nabla }}_a(T^{ab}{\xi }_b)=0$ on $U$.
2. For every finite smooth singular $4$-chain $C\subseteq U$ with $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$,

$Q_{\xi }({\mathrm{\Sigma }}_2)-Q_{\xi }({\mathrm{\Sigma }}_1)=-Q_{\xi }(B);$

in particular, if $Q_{\xi }(B)=0$ then $Q_{\xi }({\mathrm{\Sigma }}_2)=Q_{\xi }({\mathrm{\Sigma }}_1)$: the model's Killing charge assigned to homologous slices is conserved up to the side flux, and is conserved when that flux vanishes. With a length-normalized $\xi$, this charge is the model's energy in joules. 3. In this stationary case the measured energy current itself satisfies ${\mathrm{\nabla }}_a{J}^a=-{\mathrm{\Phi }}^a{a}_a$ with $a=\mathrm{\nabla }\ln X$, so ${\mathrm{\nabla }}_a{J}^a=0$ on $U$ if and only if ${\mathrm{\Phi }}^a{a}_a=0$ there.

(e) Qualification. The theorem asserts no existence of a timelike Killing field, no finiteness or convergence for chains outside the finite smooth singular class, no asymptotic or quasilocal mass, no positivity of $\mu$, and no measured value. Its conclusions are conditional on the adopted field law, the stated smoothness and the stated chains; the theorem uses no empirical premise, and any later use of a numerical energy must carry its own measurement, calibration and uncertainty qualifications.

• What is and is not empirical. No experiment, measurement, calibration or sampling enters the argument; the theorem is a conditional deduction from the adopted field law and the stated smooth geometry. The quantities $\mu$, $\mathrm{\Phi }$, $S$ and $Q_{\xi }$ are read through Physical reading of the observer split: measured energy density, energy flux and momentum density, and measured stress and The timelike Killing current and the charge of a three-chain; a later quantitative use of any of them must supply its own empirical premises and uncertainties.

• Scope of the global statement. Clause (d)2 is a finite-chain statement. It does not assert that a total energy of an unbounded slice exists, nor that the limit of the side flux at infinity vanishes; those are additional hypotheses, recorded in Absence of a universal global energy: observer dependence, no local gravitational energy, and deferred asymptotic masses.

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