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

Finite-dimensional metric jets

A well-defined linearized comparison › Finite-dimensional metric jets · lines 718–763

XXII-E@1#eq:residual, XXII-E@1#eq:ricci-matching

Equations (eq:ricci-jet)–(eq:einstein-jet) are the first variation of the curvature convention (eq:curvature-convention). They follow by substituting

Γ^{(1) ρ}_{μν} = \frac{1}{2} η^{ρ σ} (\partial_{μ} h_{σ ν} + \partial_{ν} h_{σ μ} - \partial_{σ} h_{μν})

and retaining the terms linear in $h$ . The comparator agrees with the standard flat-background linearized Einstein calculation Carroll (Section 6) Tong (Section 5).

A proposed rule must now supply

j_{D} = M (D), τ_{D} = S (D), R (D) = E_{η} (j_{D}) - κ_{cmp} τ_{D} .

Here $κ_{cmp} \neq = 0$ is a predeclared comparison coupling. Its normalization, physical dimensions, and possible relationship to Newton's constant are not determined by the cone.

In dimension $m \geq 3$ , matching the Einstein tensor is equivalent to the trace-adjusted Ricci relation

R^{(1)}_{μν} = κ_{cmp} (τ_{μν} - \frac{η _{μν}}{m - 2} η^{α β} τ_{α β}) .

It is not an equality between the full four-index Riemann tensor and a two-index source.

Displayed formulas in this passage

unlabeled · line 724
$Γ^{(1) ρ}_{μν} = \frac{1}{2} η^{ρ σ} (\partial_{μ} h_{σ ν} + \partial_{ν} h_{σ μ} - \partial_{σ} h_{μν})$
XXII-E@1#eq:residual · line 739
$j_{D} = M (D), τ_{D} = S (D), R (D) = E_{η} (j_{D}) - κ_{cmp} τ_{D} .$
XXII-E@1#eq:ricci-matching · line 751
$R^{(1)}_{μν} = κ_{cmp} (τ_{μν} - \frac{η _{μν}}{m - 2} η^{α β} τ_{α β}) .$

Finite-dimensional metric jets

Displayed formulas in this passage

Nearby in this section