Finite Killing energy current and reference normalization

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-+++)$. Here $N$ is the supplied smooth timelike vector field selecting the future cone, as in clause (c) of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Let $U\subseteq M$ be an open chart domain and let $\mathrm{\nabla }$ be the unique torsion-free, $g$-compatible connection supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Let $x=(x^0,x^1,x^2,x^3)$ be a positively oriented smooth chart on $U$ whose image contains a neighborhood of the compact closure $\bar{K}=[x_0^{-},x_0^{+}]\times \bar{\mathrm{\Omega }}$ of the slab $K=[x_0^{-},x_0^{+}]\times \mathrm{\Omega }$, where $x_0^{-}<x_0^{+}$, $\mathrm{\Omega }={\prod }_{i=1}^3(a_i,b_i)$ and $\bar{\mathrm{\Omega }}={\prod }_{i=1}^3[a_i,b_i]$ with $a_i<b_i$. The chart orients $U$, which inherits the time orientation from $N$. Let $T^{ab}$ be a smooth symmetric contravariant rank-two tensor on $U$ satisfying ${\mathrm{\nabla }}_a{T}^{ab}=0$, and let $\xi$ be a smooth Killing field on $U$. Use the induced Levi-Civita derivative on all tensor fields. With the inverse metric and index conventions of The inverse metric, musical isomorphisms and index raising and Tensor fields of type (r,s) on a Lorentzian four-manifold and their components, set ${\xi }_b:=g_{bc}{\xi }^c$ and $T_{ab}:=g_{ac}{g}_{bd}{T}^{cd}$. Define the Killing current by $j_{\xi }^a:=-T^{ab}{\xi }_b\mathrm{.}$ On an open neighborhood of $\bar{K}$, write $h_{ij}:=g_{ij}$, assume $h$ is positive definite and $g^{00}<0$, and choose the time orientation so that $n^a=-\alpha g^{a0}$ is future-directed, where $\alpha :=(-g^{00}{)}^{-1\mathrm{/}2}$; the future unit normal is $n_a=-\alpha {\delta }_a^0$. Put $w:=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}$, and let $d{V}_h:=\sqrt{\det h}{d}^{3}x$ be the induced positive spatial volume element. For $x^0=s\in [x_0^{-},x_0^{+}]$, define the finite slice charge $H_{\xi }(s):={\int }_{\mathrm{\Omega }}[-g(j_{\xi },n)]d{V}_h={\int }_{\mathrm{\Omega }}{T}_{ab}{n}^a{\xi }^{b}d{V}_h\mathrm{.}$ Let $B$ be the union of the six side faces of $K$, and let ${\mathrm{\Phi }}_J(B)$ be the net outward side flux defined in Finite spacelike slab and normal flux charge; explicitly it is the sum of the three upper-minus-lower face integrals of $w{J}^i$ over the remaining coordinate variables. Conditional on the exact finite-slab balance stated in Complete finite-slab production and outward-flux balance, $H_{\xi }(x_0^{+})-H_{\xi }(x_0^{-})+{\mathrm{\Phi }}_{j_{\xi }}(B)=0.$ Thus the endpoint charges are equal exactly when the net outward side flux is zero; divergence-free current alone does not imply that equality. The balance uses precisely the open spatial rectangle and compact-closure convention of that theorem.

In the static subcase on $U$, assume $\xi ={\mathrm{\partial }}_{x^0}$ is future timelike and orthogonal to these coordinate slices, so $g_{0i}=0$ and the future unit normal is $u^a=n^a={\xi }^a\mathrm{/}{\mathcal{N}}_{\xi }$, where ${\mathcal{N}}_{\xi }:=\sqrt{-g(\xi ,\xi )}>0$. Define $ϵ:=T_{ab}{u}^a{u}^b$. Then the same finite charge is the redshifted energy functional $H_{\xi }(s)={\int }_{\mathrm{\Omega }}{\mathcal{N}}_{\xi }ϵ\sqrt{\det h}{d}^{3}x\mathrm{.}$ This is a matter-current charge on the specified finite slice, not a general gravitational energy.

For every constant $a>0$, replacing $\xi$ by $\tilde{\xi }=a\xi$ preserves the Killing equation and gives $\tilde{j}=aj$ and $\tilde{H}=aH$. In the static subcase it also gives ${\tilde{\mathcal{N}}}_{\xi }=a{\mathcal{N}}_{\xi }$ and the same normalized field $u=\xi \mathrm{/}{\mathcal{N}}_{\xi }$. In the SI convention below, $a$ is dimensionless. If a function $\vartheta :U\to (0,\mathrm{\infty })$ is supplied and interpreted as the local temperature assigned to the observer field $u$, with SI unit kelvins, define ${\mathcal{T}}_{\xi }(q):={\mathcal{N}}_{\xi }(q)\vartheta (q)$ for $q\in U$. No regularity of $\vartheta$ is needed for this pointwise definition. Then ${\mathcal{T}}_{a\xi }=a{\mathcal{T}}_{\xi }$; no equilibrium or constancy of ${\mathcal{T}}_{\xi }$ is asserted.

For the SI reading, take length coordinates $x^0=ct$ and spatial coordinates $x^i$ in metres, where $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$; the metric components are dimensionless, and $\xi$ is normalized so that its flow parameter has units of length, hence its components are dimensionless. At any $q\in U$, let $p,k,\pi \in T_q^{\ast }U$ be covectors whose components have units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, ${\mathrm{m}}^{-1}$, and $\mathrm{J}$, respectively, interpreted as mechanical momentum, wave, and energy-valued covectors. These labels assign physical interpretations and units only; no motion law or identification between $p,k,\pi$ is assumed. Let $\mathrm{\hslash }$ be a supplied constant with SI unit $\mathrm{J}\mathrm{s}$. These are pointwise pairings, so no regularity beyond the stated covectors is needed. If $T_{ab}$ has components in $\mathrm{J}{\mathrm{m}}^{-3}$, then $H_{\xi }$ is in joules. At each $q\in U$, the mechanical pairing $E_{\xi }(p;q):=-c{p}_a{\xi }^a(q)$ has units of joules. The wave pairing $E_{\xi }(k;q):=-\mathrm{\hslash }c{k}_a{\xi }^a(q)$ also has units of joules, while the energy-valued covector pairs as $E_{\xi }(\pi ;q):=-{\pi }_a{\xi }^a(q)$ with no extra factor of $c$. The rescaled covector $p_a^{\pi }:={\pi }_a\mathrm{/}c$ has mechanical momentum units. Each pairing scales linearly with $\xi$.

For an additional horizon clause, let $\mathcal{H}\subset U$ be a smooth embedded codimension-one submanifold whose induced metric is degenerate at every point (the null-hypersurface condition). Assume $\xi$ is nonzero and tangent to $\mathcal{H}$, and suppose a function ${\kappa }_{\xi }:\mathcal{H}\to \mathbb{R}$ is defined there by ${\mathrm{\nabla }}_{\xi }\xi ={\kappa }_{\xi }\xi$ along $\mathcal{H}$. Then the rescaled generator satisfies ${\kappa }_{a\xi }=a{\kappa }_{\xi }$. In geometric units where the flow parameter has length, ${\kappa }_{\xi }$ has units ${\mathrm{m}}^{-1}$; the associated acceleration normalization is $c^2{\kappa }_{\xi }$ in $\mathrm{m}{\mathrm{s}}^{-2}$. No existence or constancy of a horizon surface gravity, or temperature law, is asserted.

The statement concerns the matter current built from $T$ and $\xi$ on a finite coordinate slab. It does not establish positivity, a global gravitational energy, an unbounded-slice charge, an exhaustion limit, an equilibrium theorem, or a relation between temperature and surface gravity. The finite-slab balance uses the exact MI01 hypotheses and suppliers.