Killing-multiplier current and its observer density

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ for the tangent-bundle smooth structure of the Lorentzian conventions.

Let $(\mathcal{M},g,N)$ be a smooth time-oriented Lorentzian four-manifold, let $U\subseteq \mathcal{M}$ be open, let $S_{ab}$ be a smooth symmetric covariant $2$-tensor field on $U$, and let $X$ be a smooth Killing field on $U$. For $X_b:=g_{bc}{X}^c$ and $S^{ab}:=g^{ac}{g}^{bd}{S}_{cd}$, define the Killing-multiplier current by

$j_X^a:=-S^{ab}{X}_b\mathrm{.}$

At a point $p\in U$ and for an observer $n\in T_p\mathcal{M}$ with $g_p(n,n)=-1$ and $n$ future-directed, define the associated Killing-multiplier density by

${\rho }_X(p;n):=-g_p(n,j_X)=S_{ab}(p)n^a{X}^b\mathrm{.}$

The current and density are defined for every such $S,X,n$. The definition imposes no conservation equation, sign condition, or coercivity property.

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

Smooth covariant 2-tensor fields on a smooth manifold

The inverse metric, musical isomorphisms and index raising

Observers, rest spaces and the spatial projection at an event

Killing fields of a Lorentzian metric

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The definition makes no conservation, positivity, or coercivity claim for $j_X$ or ${\rho }_X$.

The smooth tangent-bundle convention assumes only countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$; no stronger choice principle is used.