Manuscript Oai282-A@1 · Oai282-A.tex · section · 5b6134ec

Why this is not an Einstein equation

Stage 2: conditional improvement and its affine charges › Why this is not an Einstein equation · lines 787–813

This is a counterexample to the algebraic implication, not a counterexample to the quantum framework of the source.

The covariant replacement of the flat formula also requires care. With the curvature convention

[□_{g}, \nabla_{ν}] L = R_{ν ρ} \nabla^{ρ} L,

one has

\nabla^{μ} (\nabla_{μ} \nabla_{ν} - g_{μν} □_{g}) L = R_{ν ρ} \nabla^{ρ} L .

Thus flat conservation cannot simply be promoted to curved conservation without additional terms and hypotheses.

A curved-space matter action, its metric variation, a gravitational action, a state prescription, and renormalization conditions are additional inputs. The stress-tensor and time-ordered-product constraints in curved perturbative QFT illustrate this distinction HollandsWald. Flat local conformal Ward identities do not assert the absence of Weyl anomalies on arbitrary curved backgrounds.

Displayed formulas in this passage

unlabeled · line 794
$[□_{g}, \nabla_{ν}] L = R_{ν ρ} \nabla^{ρ} L,$
unlabeled · line 798
$\nabla^{μ} (\nabla_{μ} \nabla_{ν} - g_{μν} □_{g}) L = R_{ν ρ} \nabla^{ρ} L .$

Nearby in this section

proposition: Proposition: A conserved traceless tensor need not source its declared flat metric