Manuscript XXII-E@1 · XXII-E.tex · section · 51c99d2f

Even a local cone field does not determine Einstein curvature

What the affine cone cannot supply › Even a local cone field does not determine Einstein curvature · lines 386–404

XXII-E@1#eq:curvature-convention

A local geometric extension requires additional structure. The following calculation takes place on a declared auxiliary open set $U \subset R^{m}$ ; it is not an extracted operational spacetime.

Use the curvature convention

R^{ρ}_{σ μν} = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μα} Γ^{α}_{ν σ} - Γ^{ρ}_{ν α} Γ^{α}_{μ σ}, R_{σ ν} = R^{ρ}_{σ ρ ν} .

Displayed formulas in this passage

XXII-E@1#eq:curvature-convention · line 393
$R^{ρ}_{σ μν} = \partial_{μ} Γ^{ρ}_{ν σ} - \partial_{ν} Γ^{ρ}_{μ σ} + Γ^{ρ}_{μα} Γ^{α}_{ν σ} - Γ^{ρ}_{ν α} Γ^{α}_{μ σ}, R_{σ ν} = R^{ρ}_{σ ρ ν} .$

Nearby in this section

proposition: Proposition: Same cone field, different linearized Einstein tensor
section: Even a local cone field does not determine Einstein curvature