Manuscript X-A-ST@1 · X-A-ST.tex · section · d9ccbde1

Naturality and plateau covariance

Relational expansion and covariance: Route I › Naturality and plateau covariance · lines 1034–1061

X-A-ST@1#eq:naturality

For an orientation-preserving diffeomorphism, a contravariant tensor density of weight one transforms as

(F^{*} A)^{μν} (x) = det D F (x) (D F^{- 1})^{μ}_{α} (D F^{- 1})^{ν}_{β} A^{α β} (F (x)) .

Consequently,

\partial_{μ} ((F^{*} A)^{μν} \partial_{ν} F^{*} ψ) = F_{dens}^{*} (\partial_{μ} (A^{μν} \partial_{ν} ψ)) .

This follows from the chain rule and change of variables, or from integration against compact tests. Metric density and volume satisfy

A_{F^{*} q} = F^{*} A_{q}, v_{F^{*} q} = F_{dens}^{*} v_{q} .

For formal jets these identities follow coefficientwise by applying the genuine identities on each compact neighbourhood for sufficiently small numerical $κ$ , then taking Taylor coefficients. No globally uniform numerical nondegeneracy is needed.

Displayed formulas in this passage

X-A-ST@1#eq:naturality · line 1044
$\partial_{μ} ((F^{*} A)^{μν} \partial_{ν} F^{*} ψ) = F_{dens}^{*} (\partial_{μ} (A^{μν} \partial_{ν} ψ)) .$
unlabeled · line 1038
$(F^{*} A)^{μν} (x) = det D F (x) (D F^{- 1})^{μ}_{α} (D F^{- 1})^{ν}_{β} A^{α β} (F (x)) .$
unlabeled · line 1053
$A_{F^{*} q} = F^{*} A_{q}, v_{F^{*} q} = F_{dens}^{*} v_{q} .$

Naturality and plateau covariance

Displayed formulas in this passage

Nearby in this section