Noether charge decomposition for algebraic-curvature Lagrangians

Differentiating the coordinate curvature tensor and using that the variation of a connection is a tensor gives $\delta R^a{}_{bcd}=2{\mathrm{\nabla }}_{[c}\delta {\mathrm{\Gamma }}^a{}_{d]b},\delta {\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ae}({\mathrm{\nabla }}_b\delta g_{ec}+{\mathrm{\nabla }}_c\delta g_{eb}-{\mathrm{\nabla }}_e\delta g_{bc})\mathrm{.}$ Substitute these identities into $P^{abcd}\delta R_{abcd}$, use the Riemann symmetries of $P$, and apply the product rule to move derivatives off $\delta g$. The part of the resulting potential that contains a derivative of $\delta g$ is $2({\mathbf{E}}_R^{bcd}){\mathrm{\nabla }}_d\delta g_{bc},({\mathbf{E}}_R^{bcd}{)}_{a_2\dots a_n}:=P^{abcd}({\mathit{ϵ}}_g{)}_{a{a}_2\dots a_n};$ all other curvature terms have no derivative of $\delta g$. For each finite matter-jet term the repeated identity $A^a{\mathrm{\nabla }}_a(\delta T)={\mathrm{\nabla }}_a(A^a\delta T)-({\mathrm{\nabla }}_a{A}^a)\delta T$ moves its finitely many derivatives off the varied tensor slots. Iterating that identity $r$ times writes a term with $r$ derivatives of a variation as a divergence whose coefficients multiply variations of covariant jets of orders $0$ through $r-1$, plus an Euler term with no derivative on the variation. The connection-variation terms are included when those jets are varied as tensors. This supplies a covariant first-variation potential linear in the variations of covariant tensor slots.

By [A2], covariance and the first-variation identity with variation ${\mathcal{L}}_{\zeta }\varphi$ give ${\mathcal{L}}_{\zeta }\mathcal{L}=E_{\varphi }[{\mathcal{L}}_{\zeta }\varphi ]+d\theta (\varphi ;{\mathcal{L}}_{\zeta }\varphi )$. By [F2], ${\mathcal{L}}_{\zeta }\mathcal{L}=d({\iota }_{\zeta }\mathcal{L})+{\iota }_{\zeta }(d\mathcal{L})$; since $\mathcal{L}$ is an $n$-form on an $n$-manifold, $d\mathcal{L}=0$. Subtracting gives $d{J}_{\zeta }=-E_{\varphi }[{\mathcal{L}}_{\zeta }\varphi ]$, so the current is closed on a solution and the assumed canonical local primitive is compatible with the on-shell identity. For a tensor $T$, [F1] gives $({\mathcal{L}}_{\zeta }T{)}^{a_1\dots a_p}{}_{b_1\dots b_q}={\zeta }^e{\mathrm{\nabla }}_e{T}^{a_1\dots a_p}{}_{b_1\dots b_q}-{\sum }_{i=1}^p({\mathrm{\nabla }}_e{\zeta }^{a_i})T^{a_1\dots e\dots a_p}{}_{b_1\dots b_q}+{\sum }_{j=1}^q({\mathrm{\nabla }}_{b_j}{\zeta }^e)T^{a_1\dots a_p}{}_{b_1\dots e\dots b_q}\mathrm{.}$ For each $r\ge 1$ let $D^r\psi :={\mathrm{\nabla }}_{(a_1}\cdots {\mathrm{\nabla }}_{a_r)}\psi$, and set $D^0\psi :=\psi$. Naturality of the Levi-Civita connection gives $D^r\psi [{\mathrm{\Phi }}_t^{\ast }g,{\mathrm{\Phi }}_t^{\ast }\psi ]={\mathrm{\Phi }}_t^{\ast }(D^r\psi [g,\psi ])$ along the local flow ${\mathrm{\Phi }}_t$ of $\zeta$. Differentiating shows that the total variation of this jet, including the variation of the connection, is ${\mathcal{L}}_{\zeta }(D^r\psi )$; applying [F1] to the whole jet tensor shows that it contains only $\zeta$ and $\mathrm{\nabla }\zeta$. Thus every covariant tensor-slot contribution to $\theta (\varphi ;{\mathcal{L}}_{\zeta }\varphi )$ contains only $\zeta$ and $\mathrm{\nabla }\zeta$, even when the finite matter-jet order exceeds one. The metric slot contributes $2{\mathbf{E}}_R^{bcd}{\mathrm{\nabla }}_d({\mathrm{\nabla }}_b{\zeta }_c+{\mathrm{\nabla }}_c{\zeta }_b)$. Since ${\mathbf{E}}_R^{bcd}$ is antisymmetric in $c,d$, ${\mathbf{E}}_R^{bcd}{\mathrm{\nabla }}_d{\mathrm{\nabla }}_c{\zeta }_b=\frac{1}{2}{\mathbf{E}}_R^{bcd}[{\mathrm{\nabla }}_d,{\mathrm{\nabla }}_c]{\zeta }_b$, a curvature-times-$\zeta$ term. Therefore the sole second-derivative term in $J_{\zeta }$ is $2{\mathbf{E}}_R^{bcd}{\mathrm{\nabla }}_d{\mathrm{\nabla }}_b{\zeta }_c$; $-{\iota }_{\zeta }\mathcal{L}$ contains only $\zeta$.

Apply the defining integration-by-parts construction of the canonical local representative to the term in step 2.1. The product rule gives $2{\mathbf{E}}_R^{bcd}{\mathrm{\nabla }}_d{\mathrm{\nabla }}_b{\zeta }_c=2{\mathrm{\nabla }}_d({\mathbf{E}}_R^{bcd}{\mathrm{\nabla }}_b{\zeta }_c)-2({\mathrm{\nabla }}_d{\mathbf{E}}_R^{bcd}){\mathrm{\nabla }}_b{\zeta }_c\mathrm{.}$ In form components, the second-derivative part of $(d(X^{cd}{\mathrm{\nabla }}_{[c}{\zeta }_{d]}){)}_{a_2\dots a_n}$ is $-(n-1)P^{pqcd}({\mathit{ϵ}}_g{)}_{pq[a_3\dots a_n}{\mathrm{\nabla }}_{a_2]}{\mathrm{\nabla }}_{[c}{\zeta }_{d]}\mathrm{.}$ Antisymmetrizing $p,q$, $c,d$, and the form slots, then exchanging the curvature-derivative index pairs of $P$, gives $2P^{abcd}({\mathit{ϵ}}_g{)}_{a{a}_2\dots a_n}{\mathrm{\nabla }}_d{\mathrm{\nabla }}_b{\zeta }_c,$ which is the second-derivative term in step 2.1. Thus the coefficient and sign of $X$ are fixed as stated. For the remaining terms set $Q_{\zeta }^{(1)}:=X^{cd}{\mathrm{\nabla }}_{[c}{\zeta }_{d]}$ and write the residual current uniquely as $J_{\zeta }-d{Q}_{\zeta }^{(1)}={\iota }_{j_{\zeta }}{\mathit{ϵ}}_g$. The second-jet terms just matched cancel, so there are smooth local covariant coefficients, independent of $\zeta$, with $j_{\zeta }^a=A^{ab}{}_c{\mathrm{\nabla }}_b{\zeta }^c+B^a{}_c{\zeta }^c\mathrm{.}$ Fix any solution. Step 2.1 and $d^2=0$ give ${\mathrm{\nabla }}_a{j}_{\zeta }^a=0$ for every smooth vector field $\zeta$. At each point, a coordinate polynomial times a smooth cutoff equal to one near that point realizes arbitrary vector-field jets. Taking $\zeta =0$ and $\mathrm{\nabla }\zeta =0$ there while varying its symmetric second jet gives $A^{(ab)}{}_c=0$. Next take $\zeta =0$ and vary ${\mathrm{\nabla }}_b{\zeta }^c$ arbitrarily. The antisymmetric second-derivative term is $\frac{1}{2}{A}^{ab}{}_c[{\mathrm{\nabla }}_a,{\mathrm{\nabla }}_b]{\zeta }^c$, which vanishes at that point because it is curvature times $\zeta$. The first-jet coefficient therefore gives ${\mathrm{\nabla }}_a{A}^{ab}{}_c+B^b{}_c=0,B^a{}_c={\mathrm{\nabla }}_b{A}^{ab}{}_c,j_{\zeta }^a={\mathrm{\nabla }}_b(A^{ab}{}_c{\zeta }^c)\mathrm{.}$ Define a local covariant $(n-2)$-form on the field neighborhood by $(Q_{\zeta }^{(0)}{)}_{a_3\dots a_n}:=\frac{1}{2}{A}^{[ab]}{}_c{\zeta }^c({\mathit{ϵ}}_g{)}_{ab{a}_3\dots a_n}\mathrm{.}$ Since $\mathrm{\nabla }{\mathit{ϵ}}_g=0$, direct antisymmetrization of its form indices gives $(d{Q}_{\zeta }^{(0)}{)}_{a_2\dots a_n}={\mathrm{\nabla }}_b(A^{[ab]}{}_c{\zeta }^c)({\mathit{ϵ}}_g{)}_{a{a}_2\dots a_n}\mathrm{.}$ On the solution $A^{[ab]}{}_c=A^{ab}{}_c$, so $d{Q}_{\zeta }^{(0)}=J_{\zeta }-d{Q}_{\zeta }^{(1)}$. Thus $Q_{\zeta }^{(0)}+Q_{\zeta }^{(1)}$ is the desired on-shell primitive. The first summand is linear in $\zeta$ with no derivatives; raising the contracted generator index writes it as $W^c{\zeta }_c$. This explicit construction introduces no independent closed-form addition and proves the canonical formula. The stipulated Lagrangian and potential changes contribute respectively ${\iota }_{\zeta }\mu$ and $\mathbf{Y}(\varphi ;{\mathcal{L}}_{\zeta }\varphi )$ to the charge; an exact charge change contributes $dZ$. This gives the full stated class.