Manuscript X-A@1 · X-A.tex · lemma · dc7f5c0a

Lemma: Density expansion

Convention sheet › Metric jet and density coefficients · lines 561–583 · proof included or linked

X-A@1#lem:density, X-A@1#eq:volexp, X-A@1#eq:Aexp

With $- q = - det q$ and $A_{q}^{μν} = - q q^{μν}$ ,

- q A_{q}^{μν} = 1 + κ b_{1} [h] + κ^{2} (b_{1} [k] + b_{2} [h]) (mod κ^{3}), = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} (a_{1}^{μν} [k] + a_{2}^{μν} [h]) (mod κ^{3}) .

Proof. The inverse metric is $q^{μν} = η^{μν} - κ h^{μν} + κ^{2} (h^{μ}_{α} h^{α ν} - k^{μν})$ . Expanding $exp (\frac{1}{2} tr lo g (id + κ η^{- 1} h + κ^{2} η^{- 1} k))$ gives (eq:volexp). Multiplication gives (eq:Aexp).

Displayed formulas in this passage

X-A@1#eq:volexp · line 565
$- q A_{q}^{μν} = 1 + κ b_{1} [h] + κ^{2} (b_{1} [k] + b_{2} [h]) (mod κ^{3}), = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} (a_{1}^{μν} [k] + a_{2}^{μν} [h]) (mod κ^{3}) .$
X-A@1#eq:Aexp · line 565
$- q A_{q}^{μν} = 1 + κ b_{1} [h] + κ^{2} (b_{1} [k] + b_{2} [h]) (mod κ^{3}), = η^{μν} + κ a_{1}^{μν} [h] + κ^{2} (a_{1}^{μν} [k] + a_{2}^{μν} [h]) (mod κ^{3}) .$

Lemma: Density expansion

Displayed formulas in this passage

Nearby in this section