Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

Let $(M,g,N)$ be an oriented time-oriented smooth Lorentzian four-manifold with metric-selected connection $\mathrm{\nabla }$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, Oriented smooth manifolds and oriented charts).

(a) Characterization of Killing fields. For every smooth vector field $\xi$ (A smooth vector field is a smooth section of the tangent bundle), with ${\xi }_b=g_{bc}{\xi }^c$ the lowered field,

${\mathcal{L}}_{\xi }g=0⟺{\mathrm{\nabla }}_a{\xi }_b+{\mathrm{\nabla }}_b{\xi }_a=0.$

(b) Contraction along geodesics. If $\xi$ is a Killing field and $\gamma :I\to M$ is a geodesic with $D_t{\gamma }^{\mathrm{\prime }}=0$ in its affine parameter (Geodesic of an affine connection, Covariant derivative along a curve), then $t\mapsto g_{\gamma (t)}({\xi }_{\gamma (t)},{\gamma }^{\mathrm{\prime }}(t))$ is constant on $I$.

(c) Conserved current. Let $U\subseteq M$ be open, let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold) with ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $U$, and let $\xi$ be a Killing field on $U$. Then $J^a:=T^{ab}{\xi }_b$ is a smooth vector field on $U$ with ${\mathrm{\nabla }}_a{J}^a=0$; consequently the flux balance of Divergence, Stokes flux balance and the non-null face formula for a Lorentzian current clause (b) applies to $J$ on every smooth singular $4$-chain contained in $U$.

(d) On-shell conservation of the Hilbert tensor. Let $c>0$, let $S:\mathcal{D}\to \mathbb{R}$ be a model action with Hilbert stress-energy tensor $T$ at $(g,\phi )$ with respect to $c$ (The Hilbert stress-energy tensor of a metric-dependent action). For the flow diffeomorphisms $F$ below, suppose the configuration set $\mathcal{X}$ carries specified maps $P_F:\mathcal{X}\to \mathcal{X}$ with $P_{\mathrm{i}\mathrm{d}}=\mathrm{i}\mathrm{d}$; write $F^{\ast }\phi :=P_F(\phi )$. For geometric matter fields these are the model's pullbacks; no pullback operation on an arbitrary set is implicit. For every smooth compactly supported vector field $\xi$, its flow ${\mathrm{\Phi }}_v$ is global (Complete vector fields, Local and global flows generated by a vector field); this holds for every compactly supported smooth field, using the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) through Compactly supported smooth vector fields are complete. Write $G(u,v):=S({\mathrm{\Phi }}_v^{\ast }g,{\mathrm{\Phi }}_u^{\ast }\phi )$ for the simultaneous pullback by the flow (this presupposes that the pairs lie in $\mathcal{D}$, which is part of the hypotheses below). Suppose (H1)–(H4) hold for every such $\xi$ (with its own $G$ and parameter neighborhood):

• (H1) the map $G$ is defined on a neighborhood of $(0,0)$ in ${\mathbb{R}}^2$ and is differentiable at $(0,0)$;
• (H2) ${\mathrm{\partial }}_{u}G(0,0)=0$ (the matter configuration $\phi$ satisfies its model equations in this direction);
• (H3) ${\mathrm{\partial }}_{v}G(0,0)=-\frac{1}{2c}{\int }_M{T}_{ab}{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, where $q^{ab}:={\mathrm{\partial }}_v({\mathrm{\Phi }}_v^{\ast }g{)}^{ab}{\mathrm{\mid }}_{v=0}$ (the metric variation is represented by the Hilbert tensor);
• (H4) $G(s,s)=S(g,\phi )$ whenever $(s,s)$ lies in that neighborhood (invariance of the action under simultaneous pullback by the flow).

Then ${\mathrm{\nabla }}^a{T}_{ab}=0$ on $M$.

• Choice convention. Clause (d) uses the completeness of the flow of a compactly supported smooth vector field, which is proved from the axiom ${\mathrm{A}\mathrm{C}}_{\omega }$ of The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) in Compactly supported smooth vector fields are complete; this is the only choice principle used in the item, and it is inherited by consumers of clause (d). Clauses (a)-(c) use no choice principle.

• On-shell means matter equations plus Hilbert representation. The hypothesis (H2) is the model's matter equation at the configuration $\phi$ in the flow direction; (H3) is the Hilbert representation of the metric variation. Neither is implied by the other, and no conservation is asserted off shell or for a tensor prescribed without a functional.

• Global versus local Killing fields. The current in (c) is defined on the open set where both $T$ obeys the divergence condition and $\xi$ is Killing; the flux balance of (c) concerns chains contained in that set and carries the side-flux hypotheses of the flux theorem.