Manuscript X-F-ST@1 · X-F-ST.tex · lemma · e71f03ef

Lemma: Naturality and typed second-order covariance

F1: exact covariance with the Lie term retained · lines 728–767 · proof included or linked

X-F-ST@1#lem:cov2, X-F-ST@1#eq:cov2

For every smooth vector field $ξ$ and smooth symmetric tensor $r$ ,

G_{lin} [K ξ] = 0, 2 G_{2} [K ξ, r] = L_{ξ}^{d} G_{lin} [r] - G_{lin} [L_{ξ}^{c} r] .

No field equation is required.

Proof. The Einstein density is natural: $E [F^{*} q] = F^{*} E [q]$ . This follows either from the natural curvature construction or by differentiating the invariant Einstein–Hilbert action against compact variations.

Let $F_{s}$ be the flow of a compact $ξ$ , and apply naturality to $q = η + λ r$ . The left input is

F_{s}^{*} (η + λ r) = η + sK ξ + λ r + s λ L_{ξ}^{c} r + O (s^{2}) + O (s^{2} λ) .

The coefficient of $s$ is $G_{lin} [K ξ] = 0$ . The coefficient of $s λ$ is

G_{lin} [L_{ξ}^{c} r] + 2 G_{2} [K ξ, r] = L_{ξ}^{d} G_{lin} [r] .

The expressions are local differential expressions, so compact localization of $ξ$ proves the identity for arbitrary smooth germs.

Displayed formulas in this passage

X-F-ST@1#eq:cov2 · line 732
$G_{lin} [K ξ] = 0, 2 G_{2} [K ξ, r] = L_{ξ}^{d} G_{lin} [r] - G_{lin} [L_{ξ}^{c} r] .$
unlabeled · line 752
$F_{s}^{*} (η + λ r) = η + sK ξ + λ r + s λ L_{ξ}^{c} r + O (s^{2}) + O (s^{2} λ) .$
unlabeled · line 760
$G_{lin} [L_{ξ}^{c} r] + 2 G_{2} [K ξ, r] = L_{ξ}^{d} G_{lin} [r] .$

Nearby in this section

section: F1: exact covariance with the Lie term retained